Step by step 2D example#
A simple 2D case with symbolic work to illustrate the two-scale library implementation and approach.
Initialization#
The TS library will to be used in parallel in this test. Thus we start by launching an MPI cluster of 2 MPI engines with ipyparallel.
import ipyparallel as ipp
cluster = ipp.Cluster(engines="mpi", n=2)
rc = cluster.start_and_connect_sync()
view=rc[:]
/dolfinx-env/lib/python3.12/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html
from .autonotebook import tqdm as notebook_tqdm
Use Twoscale dolfinx implementation
In this context (%%px), we load the components of the TS library and dolfinx
%%px
from dolfinx import mesh
from dolfinx import fem
from dolfinx.fem import petsc
from dolfinx import la
import ufl
import basix
from twoscale import core
from twoscale import linear
from twoscale import util
This test case is very basic and uses, at the coarse scale, a rectangular plate of dimension 2LxL.
%%px
domain= mesh.create_rectangle(MPI.COMM_WORLD,[[0.0, 0.], [2*L, L]],[2,1],ghost_mode=mesh.GhostMode.none)
The numerical application used throughout this document is:
\[\displaystyle c_x=0.0, c_y=0.0, i_x=0.1, f=500.0, L=1, E=5000.0, \nu=0.333333333333333, \mu=1875.0, \lambda=3750.0\]
where
\(i_x\) is the imposed displacement along x component to stretch the plate by pulling on right-hand side of the plate.
\(c_x\),\(c_y\) are the imposed displacement along x,y component to block rigid-body modes and clamp left hand side of the plate. Usually set to zero, these arbitrary values are made visible by considering that they can be non-null. Retaining them as explicit variables gives a way to see more clearly their influence in the computation.
\(f\) is the value of the constant imposed volumic load along -x axis
\(E\) is the Young’s modulus
\(\nu\) is the Posson’s ratio
\(L\) is one dimension of the plate
\(\lambda\) and \(\mu\) are the Lamè coefficients
First of all the jump between scale has to be defined. For that the set of enriched nodes and refinement criteria around these nodes needs to be created. In this simple test case all coarse nodes are enriched and all coarse elements are split once. Thus refinement localisation criterion can be anything as TS library always refine at least once all element in the support which is what we are looking for. The following simple function, which returns true for all mesh coordinates, will do the job:
%%px
def crit(coords):
return coords[0]>-L
And the exact same function can be used for enriched nodes selection (all).
To create a scaleJump object sj that stores all the information required by TS method the only function proposed for now is topDown.
%px sj=core.topDown(domain,crit,crit,1)
This object sj provides the mesh at both scales (the new coarse mesh is in general a re distributed version of the original coarse mesh according to the set of enriched nodes).
Warning
The mesh argument given to topDown (here domain) should not be used after the call, as it has been transformed and moved inside this function.
Thus, the two meshes at both scales are:
%%px
fdomain=sj.getFineMesh
cdomain=sj.getCoarseMesh
The following shows the discretisation of the 2D problem for the symbolic approach and with TS library
2026-09-23 15:50:42.262 ( 1.725s) [ 7F825B351140]vtkXOpenGLRenderWindow.:1450 WARN| bad X server connection. DISPLAY=
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
Cell In[16], line 22
18 plt.subplot(0, 0)
19 plt.add_axes()
20 plt.add_mesh(grid,show_edges=True,show_scalar_bar=False,scalars=np.array(range(0,4)),edge_color='w')
21 plt.add_point_labels(xx, labels, point_size=2, font_size=14,shape_color='white')
---> 22 plt.camera_position =cam_pos
23 plt.add_title("symbolic coarse scale",font_size=12)
24 idxmgf=domainf.geometry.index_map()
25 nbdfb=idxmgf.size_local
File /dolfinx-env/lib/python3.12/site-packages/pyvista/core/utilities/misc.py:572, in _NoNewAttrMixin.__setattr__(self, key, value)
570 if object.__getattribute__(self, '__dict__').get('__frozen_by_class') is type(self):
571 _NoNewAttrMixin._check_new_attribute(self, key)
--> 572 object.__setattr__(self, key, value)
File /dolfinx-env/lib/python3.12/site-packages/pyvista/plotting/plotter.py:2299, in BasePlotter.camera_position(self, camera_location)
2297 @camera_position.setter
2298 def camera_position(self, camera_location: CameraPositionOptions) -> None:
-> 2299 self.renderer.camera_position = camera_location
File /dolfinx-env/lib/python3.12/site-packages/pyvista/core/utilities/misc.py:572, in _NoNewAttrMixin.__setattr__(self, key, value)
570 if object.__getattribute__(self, '__dict__').get('__frozen_by_class') is type(self):
571 _NoNewAttrMixin._check_new_attribute(self, key)
--> 572 object.__setattr__(self, key, value)
File /dolfinx-env/lib/python3.12/site-packages/pyvista/plotting/renderer.py:584, in Renderer.camera_position(self, camera_location)
582 self.camera.position = scale_point(self.camera, camera_location[0], invert=False)
583 self.camera.focal_point = scale_point(self.camera, camera_location[1], invert=False)
--> 584 self.camera.up = camera_location[2]
586 # reset clipping range
587 self.reset_camera_clipping_range()
File /dolfinx-env/lib/python3.12/site-packages/pyvista/core/utilities/misc.py:572, in _NoNewAttrMixin.__setattr__(self, key, value)
570 if object.__getattribute__(self, '__dict__').get('__frozen_by_class') is type(self):
571 _NoNewAttrMixin._check_new_attribute(self, key)
--> 572 object.__setattr__(self, key, value)
File /dolfinx-env/lib/python3.12/site-packages/pyvista/plotting/camera.py:510, in Camera.up(self, vector)
508 if np.allclose(vector, 0.0):
509 msg = 'Camera up vector cannot be zero.'
--> 510 raise ValueError(msg)
511 self.SetViewUp(vector)
512 self.is_set = True
ValueError: Camera up vector cannot be zero.
[output:0]
[output:1]
[output:0]
Fine scale level#
The fine scale system#
Fine-scale triangles have edge sizes of \(\frac{L}{2}\) and \(\frac{L.\sqrt{2}}{2}\) and the elementary matrix associated with one fine triangle is given by \(ke = t_e.area_e.C^t.M.C\) with:
\(t_e\) the thickness taken as 1.
\(area_e\) the area, which is \(\frac{1}{2}.\frac{L}{2}.\frac{L}{2}=\frac{L^2}{8}\)
Using the lamè coefficients \(\lambda\), \(\mu\) the isotropic plan strain stiffness matrix \(M\) is:
\[\begin{split}\displaystyle M = \left[\begin{matrix}\lambda + 2 \mu & \lambda & 0\\\lambda & \lambda + 2 \mu & 0\\0 & 0 & \mu\end{matrix}\right]\end{split}\]
And C for a triangle of category 0 is
\[\begin{split}\displaystyle C_0 = \left[\begin{matrix}0 & 0 & \frac{2}{L} & 0 & - \frac{2}{L} & 0\\0 & - \frac{2}{L} & 0 & 0 & 0 & \frac{2}{L}\\- \frac{2}{L} & 0 & 0 & \frac{2}{L} & \frac{2}{L} & - \frac{2}{L}\end{matrix}\right]\end{split}\]
and for a triangle of category 1 C is
\[\begin{split}\displaystyle C_1 = \left[\begin{matrix}- \frac{2}{L} & 0 & \frac{2}{L} & 0 & 0 & 0\\0 & 0 & 0 & - \frac{2}{L} & 0 & \frac{2}{L}\\0 & - \frac{2}{L} & - \frac{2}{L} & \frac{2}{L} & \frac{2}{L} & 0\end{matrix}\right]\end{split}\]
and for a triangle of category 2 C is
\[\begin{split}\displaystyle C_2 = \left[\begin{matrix}- \frac{2}{L} & 0 & \frac{2}{L} & 0 & 0 & 0\\0 & - \frac{2}{L} & 0 & 0 & 0 & \frac{2}{L}\\- \frac{2}{L} & - \frac{2}{L} & 0 & \frac{2}{L} & \frac{2}{L} & 0\end{matrix}\right]\end{split}\]
and for triangle of category 3 C is
\[\begin{split}\displaystyle C_3 = \left[\begin{matrix}0 & 0 & \frac{2}{L} & 0 & - \frac{2}{L} & 0\\0 & - \frac{2}{L} & 0 & \frac{2}{L} & 0 & 0\\- \frac{2}{L} & 0 & \frac{2}{L} & \frac{2}{L} & 0 & - \frac{2}{L}\end{matrix}\right]\end{split}\]
Thus, the elementary matrices for triangles of categories 0, 1, 2 and 3 are:
\[\begin{split}\displaystyle Ke_0 = \left[\begin{matrix}\frac{\mu}{2} & 0 & 0 & - \frac{\mu}{2} & - \frac{\mu}{2} & \frac{\mu}{2}\\0 & \frac{\lambda}{2} + \mu & - \frac{\lambda}{2} & 0 & \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu\\0 & - \frac{\lambda}{2} & \frac{\lambda}{2} + \mu & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2}\\- \frac{\mu}{2} & 0 & 0 & \frac{\mu}{2} & \frac{\mu}{2} & - \frac{\mu}{2}\\- \frac{\mu}{2} & \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & \frac{\mu}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2} & - \frac{\lambda}{2} - \frac{\mu}{2}\\\frac{\mu}{2} & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & - \frac{\mu}{2} & - \frac{\lambda}{2} - \frac{\mu}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2}\end{matrix}\right]\end{split}\]
\[\begin{split}\displaystyle Ke_1 = \left[\begin{matrix}\frac{\lambda}{2} + \mu & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & 0 & - \frac{\lambda}{2}\\0 & \frac{\mu}{2} & \frac{\mu}{2} & - \frac{\mu}{2} & - \frac{\mu}{2} & 0\\- \frac{\lambda}{2} - \mu & \frac{\mu}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2} & - \frac{\lambda}{2} - \frac{\mu}{2} & - \frac{\mu}{2} & \frac{\lambda}{2}\\\frac{\lambda}{2} & - \frac{\mu}{2} & - \frac{\lambda}{2} - \frac{\mu}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2} & \frac{\mu}{2} & - \frac{\lambda}{2} - \mu\\0 & - \frac{\mu}{2} & - \frac{\mu}{2} & \frac{\mu}{2} & \frac{\mu}{2} & 0\\- \frac{\lambda}{2} & 0 & \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & 0 & \frac{\lambda}{2} + \mu\end{matrix}\right]\end{split}\]
\[\begin{split}\displaystyle Ke_2 = \left[\begin{matrix}\frac{\lambda}{2} + \frac{3 \mu}{2} & \frac{\lambda}{2} + \frac{\mu}{2} & - \frac{\lambda}{2} - \mu & - \frac{\mu}{2} & - \frac{\mu}{2} & - \frac{\lambda}{2}\\\frac{\lambda}{2} + \frac{\mu}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2} & - \frac{\lambda}{2} & - \frac{\mu}{2} & - \frac{\mu}{2} & - \frac{\lambda}{2} - \mu\\- \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} & \frac{\lambda}{2} + \mu & 0 & 0 & \frac{\lambda}{2}\\- \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\mu}{2} & \frac{\mu}{2} & 0\\- \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\mu}{2} & \frac{\mu}{2} & 0\\- \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & 0 & 0 & \frac{\lambda}{2} + \mu\end{matrix}\right]\end{split}\]
\[\begin{split}\displaystyle Ke_3 = \left[\begin{matrix}\frac{\mu}{2} & 0 & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\mu}{2}\\0 & \frac{\lambda}{2} + \mu & - \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & 0\\- \frac{\mu}{2} & - \frac{\lambda}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2} & \frac{\lambda}{2} + \frac{\mu}{2} & - \frac{\lambda}{2} - \mu & - \frac{\mu}{2}\\- \frac{\mu}{2} & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} + \frac{\mu}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2} & - \frac{\lambda}{2} & - \frac{\mu}{2}\\0 & \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} & \frac{\lambda}{2} + \mu & 0\\\frac{\mu}{2} & 0 & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\mu}{2}\end{matrix}\right]\end{split}\]
With TS library/FEniCSx elementary matrices are defined through the formulation:
%%px
#space
space = fem.functionspace(fdomain, fdomain.ufl_domain().ufl_coordinate_element())
#lamè
mu=fem.Constant(fdomain,E / (2.0 * (1.0 + nu)))
lmbda = fem.Constant(fdomain,E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu)))
#trial/test
u = ufl.TrialFunction(space)
v = ufl.TestFunction(space)
# strain
def eps(v):
return 0.5*(ufl.grad(v) + ufl.grad(v).T)
# stress
def sigma(strain):
return 2.0*mu*strain + lmbda*ufl.tr(strain)*ufl.Identity(2)
# bilinear formulation
a=ufl.inner(sigma(eps(u)), eps(v)) * ufl.dx
In symolic treatment standard system matrix A at fine scale is than given by assembling \(Ke_0\), \(Ke_1\), \(Ke_2\) and \(Ke_3\),for each element of each category:
\[\begin{split}\displaystyle A =\left[\begin{array}{cccccccccccccccccccccccccccccc}\frac{\lambda}{2} + \frac{3 \mu}{2} & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & \frac{\mu}{2} & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{\lambda}{2} + \frac{3 \mu}{2} & \frac{\mu}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{\lambda}{2} - \mu & \frac{\mu}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\lambda}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & - \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & - \mu & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & \frac{\mu}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & \frac{\mu}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & - \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\lambda}{2} + \frac{3 \mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & - \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{\mu}{2} & \frac{\lambda}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda + 3 \mu & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & - \frac{\lambda}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\mu}{2} & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda + 3 \mu & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{\lambda}{2} - \frac{\mu}{2} & - \mu & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 2 \lambda + 6 \mu & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & - \mu & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0\\- \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 2 \lambda + 6 \mu & 0 & - \mu & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 2 \lambda + 6 \mu & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 2 \lambda + 6 \mu & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & - \mu & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 2 \lambda + 6 \mu & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & - \mu & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2}\\0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 2 \lambda + 6 \mu & 0 & - \mu & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & - \frac{\lambda}{2} - \frac{\mu}{2} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & - \frac{\lambda}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & \lambda + 3 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & \frac{\lambda}{2}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & \lambda + 3 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\mu}{2} & - \frac{\lambda}{2} - \mu\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{3 \mu}{2} & 0 & - \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{3 \mu}{2} & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & - \frac{\mu}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & \frac{\mu}{2} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & - \mu & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & - \frac{\mu}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & \frac{\mu}{2}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & \frac{\lambda}{2} & - \frac{\mu}{2}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & - \frac{\mu}{2} & \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & \frac{\lambda}{2} + \frac{3 \mu}{2} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & \frac{\lambda}{2} & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\lambda}{2} + \frac{3 \mu}{2}\end{array}\right]\end{split}\]
All symbolic numerical conterpart are identified with a trealing \(N\) in their name and are optionaly visible using drop down widget.
The elementary load vector associated to one fine triangle is given by \(Fe = t_e.\int N^t.fn.dV\) with \(fn=[-f,0]\) considering only constant volumic load along -x component. Integration gives for all element category:
\[\begin{split}\displaystyle Fe = \frac{L^{2}}{24}.\left[\begin{matrix}- f\\0\\- f\\0\\- f\\0\end{matrix}\right]\end{split}\]
Standard system rhs B is than by assemble \(Fe\) for each element :
\[\begin{split}\displaystyle B = \frac{L^{2}}{24}.\left[\begin{matrix}- 2 f\\0\\- 2 f\\0\\- 4 f\\0\\- 2 f\\0\\- 2 f\\0\\- 2 f\\0\\- 8 f\\0\\- 4 f\\0\\- 8 f\\0\\- 2 f\\0\\- 2 f\\0\\- 2 f\\0\\- 4 f\\0\\- 2 f\\0\\- 2 f\\0\end{matrix}\right]\end{split}\]
Equivalently, with the TS library/FEniCSx, the volumic load is set by imposing \([-f,0]\) as a constant field
%%px
#constant
fv=fem.Constant(fdomain,[-f,0.])
#linear formulation
b=ufl.dot(fv,v)*ufl.dx
Dirichlet operator used in this work follows PETSc/FEniCSx logic:
Dirichlet dofs rows and columns of the system matrix are nullified by application of a \(D\) operator (identity matrix with 0 on Dirichlet dofs diagonal position)
The Dirichlet dofs diagonal position are set to 1 by adding a \(U=\mathbb{I}-D\) matrix to filtered \(A\) matrix (i.e. \(D^t.AD\))
The Dirichlet dofs values are set in a vector \(X_D\)
The rhs \(B\) is filtered by \(D\)
\(X_D\) is added to rhs
The coupling terms related to imposed Dirichlet values (\(D^t.A.X_D\)) is substracted to rhs
For this test case the following Dirichlet boundary conditions values are set for symbolic problem:
dofs (0,1,20) are fixed repectively with \(c_x\), \(c_y\) and \(c_x\) arbitrary values.
dofs (8,18,28) are fixed with \(i_x\) (traction)
Dirichlet boundary condition with TS library/FEniCSx resume to identify dofs and associate values to them:
%%px
bc_nodes_1=mesh.locate_entities(fdomain, 0, lambda x: np.isclose(x[0], 0.))
bc_dofs_1=fem.locate_dofs_topological(space.sub(0),0,bc_nodes_1)
bc1=fem.dirichletbc(np.array(cx),bc_dofs_1,space.sub(0))
bc_nodes_2=mesh.locate_entities(fdomain, 0, lambda x: np.isclose(x[0], 2*L))
bc_dofs_2=fem.locate_dofs_topological(space.sub(0),0,bc_nodes_2)
bc2=fem.dirichletbc(np.array(ix),bc_dofs_2,space.sub(0))
bc_nodes_3=mesh.locate_entities(fdomain, 0, lambda x: np.logical_and(np.isclose(x[0], 0.),np.isclose(x[1], 0.)))
bc_dofs_3=fem.locate_dofs_topological(space.sub(1),0,bc_nodes_3)
bc3=fem.dirichletbc(np.array(cy),bc_dofs_3,space.sub(1))
bcs=[bc1,bc2,bc3]
In symbolic approach, has mentionned above, applying Dirichlet BC resume to:
\(AD=D^t.A.D+U\)
\(BD=-D^t.A.X_D+X_D+D^t.B\)
Has \(D^t=D\) it simplify to:
\(AD=D.A.D+U\)
\(BD=-D.A.X_D+X_D+D.B\)
Sytem matrix after Dirichlet BC is thus:
\[\begin{split}\displaystyle AD = \left[\begin{array}{cccccccccccccccccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & \lambda + 3 \mu & - \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & - \mu & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & \frac{\mu}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & \frac{\mu}{2} & \lambda + 3 \mu & 0 & 0 & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & 0 & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\lambda}{2} + \frac{3 \mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda + 3 \mu & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - \mu & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 2 \lambda + 6 \mu & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & - \mu & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0\\0 & 0 & 0 & - \lambda - 2 \mu & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 2 \lambda + 6 \mu & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 2 \lambda + 6 \mu & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 2 \lambda + 6 \mu & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & - \mu & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 2 \lambda + 6 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & - \mu & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2}\\0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 2 \lambda + 6 \mu & 0 & - \mu & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & \lambda + 3 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\lambda}{2} + \frac{3 \mu}{2} & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & \frac{\mu}{2} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & - \mu & 0 & 0 & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & \frac{\lambda}{2} & \lambda + 3 \mu & 0 & - \frac{\lambda}{2} - \mu & - \frac{\lambda}{2} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & - \lambda - 2 \mu & \frac{\lambda}{2} + \frac{\mu}{2} & 0 & 0 & 0 & 0 & 0 & \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & - \frac{\mu}{2} & - \frac{\mu}{2} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \mu & - \frac{\mu}{2} & \lambda + 3 \mu & 0 & 0 & \frac{\mu}{2}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \lambda - 2 \mu & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} & - \frac{\mu}{2} & 0 & \lambda + 3 \mu & 0 & - \frac{\mu}{2}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\lambda}{2} - \frac{\mu}{2} & 0 & 0 & - \frac{\lambda}{2} - \mu & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\mu}{2} & - \frac{\mu}{2} & 0 & \frac{\lambda}{2} + \frac{3 \mu}{2}\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle BD = \left[\begin{matrix}c_{x}\\c_{y}\\- \frac{L^{2} f}{12} + c_{x} \left(\frac{\lambda}{2} + \mu\right) - \frac{c_{y} \mu}{2}\\- \frac{c_{x} \lambda}{2} + \frac{c_{y} \mu}{2}\\- \frac{L^{2} f}{6}\\0\\- \frac{L^{2} f}{12} + i_{x} \left(\frac{\lambda}{2} + \mu\right)\\\frac{i_{x} \lambda}{2}\\i_{x}\\\frac{i_{x} \lambda}{2}\\c_{x}\\c_{y} \left(\frac{\lambda}{2} + \mu\right)\\- \frac{L^{2} f}{3} + c_{x} \left(\lambda + 2 \mu\right) + c_{y} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right)\\c_{x} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) + c_{x} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right)\\- \frac{L^{2} f}{6}\\0\\- \frac{L^{2} f}{3} + i_{x} \left(\lambda + 2 \mu\right)\\i_{x} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) + i_{x} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right)\\i_{x}\\0\\c_{x}\\\frac{c_{x} \lambda}{2}\\- \frac{L^{2} f}{12} + c_{x} \left(\frac{\lambda}{2} + \mu\right)\\\frac{c_{x} \lambda}{2}\\- \frac{L^{2} f}{6}\\0\\- \frac{L^{2} f}{12} + i_{x} \left(\frac{\lambda}{2} + \mu\right)\\- \frac{i_{x} \lambda}{2}\\i_{x}\\- \frac{i_{x} \lambda}{2}\end{matrix}\right]\end{split}\]
For TS library/FEniCSx the linear, bilinear ufl formulation and the Dirichlet BC can now be used to create \(A\),\(B\),\(AD\),\(BD\) matrices.
To simplifify implementation the use of createFineScaleSytems will hide all the creation and assembly aspect of those matrices.
%%px
[A,AD,B,BD]=util.createFineScaleSytems(a,b,bcs=bcs)
Solution at fine scale#
\(S_F=AD^{-1}.BD\)
\[\begin{split}\displaystyle S_F = \left[\begin{matrix}c_{x}\\c_{y}\\\frac{- 28 L^{2} f \lambda^{4} - 341 L^{2} f \lambda^{3} \mu - 1493 L^{2} f \lambda^{2} \mu^{2} - 2766 L^{2} f \lambda \mu^{3} - 1824 L^{2} f \mu^{4} + 360 c_{x} \lambda^{4} \mu + 3456 c_{x} \lambda^{3} \mu^{2} + 11646 c_{x} \lambda^{2} \mu^{3} + 16128 c_{x} \lambda \mu^{4} + 7704 c_{x} \mu^{5} + 120 i_{x} \lambda^{4} \mu + 1152 i_{x} \lambda^{3} \mu^{2} + 3882 i_{x} \lambda^{2} \mu^{3} + 5376 i_{x} \lambda \mu^{4} + 2568 i_{x} \mu^{5}}{24 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{24 L^{2} f \lambda^{4} + 197 L^{2} f \lambda^{3} \mu + 519 L^{2} f \lambda^{2} \mu^{2} + 460 L^{2} f \lambda \mu^{3} + 60 L^{2} f \mu^{4} + 480 c_{y} \lambda^{4} \mu + 4608 c_{y} \lambda^{3} \mu^{2} + 15528 c_{y} \lambda^{2} \mu^{3} + 21504 c_{y} \lambda \mu^{4} + 10272 c_{y} \mu^{5}}{24 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{- 28 L^{2} f \lambda^{4} - 303 L^{2} f \lambda^{3} \mu - 1201 L^{2} f \lambda^{2} \mu^{2} - 2070 L^{2} f \lambda \mu^{3} - 1312 L^{2} f \mu^{4} + 120 c_{x} \lambda^{4} \mu + 1152 c_{x} \lambda^{3} \mu^{2} + 3882 c_{x} \lambda^{2} \mu^{3} + 5376 c_{x} \lambda \mu^{4} + 2568 c_{x} \mu^{5} + 120 i_{x} \lambda^{4} \mu + 1152 i_{x} \lambda^{3} \mu^{2} + 3882 i_{x} \lambda^{2} \mu^{3} + 5376 i_{x} \lambda \mu^{4} + 2568 i_{x} \mu^{5}}{12 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{26 L^{2} f \lambda^{4} + 221 L^{2} f \lambda^{3} \mu + 603 L^{2} f \lambda^{2} \mu^{2} + 536 L^{2} f \lambda \mu^{3} + 36 L^{2} f \mu^{4} + 240 c_{y} \lambda^{4} \mu + 2304 c_{y} \lambda^{3} \mu^{2} + 7764 c_{y} \lambda^{2} \mu^{3} + 10752 c_{y} \lambda \mu^{4} + 5136 c_{y} \mu^{5}}{12 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{- 28 L^{2} f \lambda^{4} - 341 L^{2} f \lambda^{3} \mu - 1493 L^{2} f \lambda^{2} \mu^{2} - 2766 L^{2} f \lambda \mu^{3} - 1824 L^{2} f \mu^{4} + 120 c_{x} \lambda^{4} \mu + 1152 c_{x} \lambda^{3} \mu^{2} + 3882 c_{x} \lambda^{2} \mu^{3} + 5376 c_{x} \lambda \mu^{4} + 2568 c_{x} \mu^{5} + 360 i_{x} \lambda^{4} \mu + 3456 i_{x} \lambda^{3} \mu^{2} + 11646 i_{x} \lambda^{2} \mu^{3} + 16128 i_{x} \lambda \mu^{4} + 7704 i_{x} \mu^{5}}{24 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{80 L^{2} f \lambda^{4} + 687 L^{2} f \lambda^{3} \mu + 1893 L^{2} f \lambda^{2} \mu^{2} + 1684 L^{2} f \lambda \mu^{3} + 84 L^{2} f \mu^{4} + 480 c_{y} \lambda^{4} \mu + 4608 c_{y} \lambda^{3} \mu^{2} + 15528 c_{y} \lambda^{2} \mu^{3} + 21504 c_{y} \lambda \mu^{4} + 10272 c_{y} \mu^{5}}{24 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\i_{x}\\\frac{26 L^{2} f \lambda^{4} + 221 L^{2} f \lambda^{3} \mu + 603 L^{2} f \lambda^{2} \mu^{2} + 536 L^{2} f \lambda \mu^{3} + 36 L^{2} f \mu^{4} + 120 c_{y} \lambda^{4} \mu + 1152 c_{y} \lambda^{3} \mu^{2} + 3882 c_{y} \lambda^{2} \mu^{3} + 5376 c_{y} \lambda \mu^{4} + 2568 c_{y} \mu^{5}}{6 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\c_{x}\\\frac{26 L^{2} f \lambda^{5} + 273 L^{2} f \lambda^{4} \mu + 1045 L^{2} f \lambda^{3} \mu^{2} + 1742 L^{2} f \lambda^{2} \mu^{3} + 1108 L^{2} f \lambda \mu^{4} + 72 L^{2} f \mu^{5} + 60 c_{x} \lambda^{5} \mu + 576 c_{x} \lambda^{4} \mu^{2} + 1941 c_{x} \lambda^{3} \mu^{3} + 2688 c_{x} \lambda^{2} \mu^{4} + 1284 c_{x} \lambda \mu^{5} + 240 c_{y} \lambda^{5} \mu + 2784 c_{y} \lambda^{4} \mu^{2} + 12372 c_{y} \lambda^{3} \mu^{3} + 26280 c_{y} \lambda^{2} \mu^{4} + 26640 c_{y} \lambda \mu^{5} + 10272 c_{y} \mu^{6} - 60 i_{x} \lambda^{5} \mu - 576 i_{x} \lambda^{4} \mu^{2} - 1941 i_{x} \lambda^{3} \mu^{3} - 2688 i_{x} \lambda^{2} \mu^{4} - 1284 i_{x} \lambda \mu^{5}}{12 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\\frac{- 26 L^{2} f \lambda^{4} - 273 L^{2} f \lambda^{3} \mu - 1041 L^{2} f \lambda^{2} \mu^{2} - 1707 L^{2} f \lambda \mu^{3} - 1014 L^{2} f \mu^{4} + 180 c_{x} \lambda^{4} \mu + 1728 c_{x} \lambda^{3} \mu^{2} + 5823 c_{x} \lambda^{2} \mu^{3} + 8064 c_{x} \lambda \mu^{4} + 3852 c_{x} \mu^{5} + 60 i_{x} \lambda^{4} \mu + 576 i_{x} \lambda^{3} \mu^{2} + 1941 i_{x} \lambda^{2} \mu^{3} + 2688 i_{x} \lambda \mu^{4} + 1284 i_{x} \mu^{5}}{12 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{26 L^{2} f \lambda^{5} + 273 L^{2} f \lambda^{4} \mu + 1045 L^{2} f \lambda^{3} \mu^{2} + 1742 L^{2} f \lambda^{2} \mu^{3} + 1108 L^{2} f \lambda \mu^{4} + 72 L^{2} f \mu^{5} + 60 c_{x} \lambda^{5} \mu + 576 c_{x} \lambda^{4} \mu^{2} + 1941 c_{x} \lambda^{3} \mu^{3} + 2688 c_{x} \lambda^{2} \mu^{4} + 1284 c_{x} \lambda \mu^{5} + 240 c_{y} \lambda^{5} \mu + 2784 c_{y} \lambda^{4} \mu^{2} + 12372 c_{y} \lambda^{3} \mu^{3} + 26280 c_{y} \lambda^{2} \mu^{4} + 26640 c_{y} \lambda \mu^{5} + 10272 c_{y} \mu^{6} - 60 i_{x} \lambda^{5} \mu - 576 i_{x} \lambda^{4} \mu^{2} - 1941 i_{x} \lambda^{3} \mu^{3} - 2688 i_{x} \lambda^{2} \mu^{4} - 1284 i_{x} \lambda \mu^{5}}{12 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\\frac{- 26 L^{2} f \lambda^{4} - 295 L^{2} f \lambda^{3} \mu - 1197 L^{2} f \lambda^{2} \mu^{2} - 2046 L^{2} f \lambda \mu^{3} - 1256 L^{2} f \mu^{4} + 120 c_{x} \lambda^{4} \mu + 1152 c_{x} \lambda^{3} \mu^{2} + 3882 c_{x} \lambda^{2} \mu^{3} + 5376 c_{x} \lambda \mu^{4} + 2568 c_{x} \mu^{5} + 120 i_{x} \lambda^{4} \mu + 1152 i_{x} \lambda^{3} \mu^{2} + 3882 i_{x} \lambda^{2} \mu^{3} + 5376 i_{x} \lambda \mu^{4} + 2568 i_{x} \mu^{5}}{12 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{26 L^{2} f \lambda^{5} + 273 L^{2} f \lambda^{4} \mu + 1045 L^{2} f \lambda^{3} \mu^{2} + 1742 L^{2} f \lambda^{2} \mu^{3} + 1108 L^{2} f \lambda \mu^{4} + 72 L^{2} f \mu^{5} + 60 c_{x} \lambda^{5} \mu + 576 c_{x} \lambda^{4} \mu^{2} + 1941 c_{x} \lambda^{3} \mu^{3} + 2688 c_{x} \lambda^{2} \mu^{4} + 1284 c_{x} \lambda \mu^{5} + 240 c_{y} \lambda^{5} \mu + 2784 c_{y} \lambda^{4} \mu^{2} + 12372 c_{y} \lambda^{3} \mu^{3} + 26280 c_{y} \lambda^{2} \mu^{4} + 26640 c_{y} \lambda \mu^{5} + 10272 c_{y} \mu^{6} - 60 i_{x} \lambda^{5} \mu - 576 i_{x} \lambda^{4} \mu^{2} - 1941 i_{x} \lambda^{3} \mu^{3} - 2688 i_{x} \lambda^{2} \mu^{4} - 1284 i_{x} \lambda \mu^{5}}{12 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\\frac{- 26 L^{2} f \lambda^{4} - 273 L^{2} f \lambda^{3} \mu - 1041 L^{2} f \lambda^{2} \mu^{2} - 1707 L^{2} f \lambda \mu^{3} - 1014 L^{2} f \mu^{4} + 60 c_{x} \lambda^{4} \mu + 576 c_{x} \lambda^{3} \mu^{2} + 1941 c_{x} \lambda^{2} \mu^{3} + 2688 c_{x} \lambda \mu^{4} + 1284 c_{x} \mu^{5} + 180 i_{x} \lambda^{4} \mu + 1728 i_{x} \lambda^{3} \mu^{2} + 5823 i_{x} \lambda^{2} \mu^{3} + 8064 i_{x} \lambda \mu^{4} + 3852 i_{x} \mu^{5}}{12 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{26 L^{2} f \lambda^{5} + 273 L^{2} f \lambda^{4} \mu + 1045 L^{2} f \lambda^{3} \mu^{2} + 1742 L^{2} f \lambda^{2} \mu^{3} + 1108 L^{2} f \lambda \mu^{4} + 72 L^{2} f \mu^{5} + 60 c_{x} \lambda^{5} \mu + 576 c_{x} \lambda^{4} \mu^{2} + 1941 c_{x} \lambda^{3} \mu^{3} + 2688 c_{x} \lambda^{2} \mu^{4} + 1284 c_{x} \lambda \mu^{5} + 240 c_{y} \lambda^{5} \mu + 2784 c_{y} \lambda^{4} \mu^{2} + 12372 c_{y} \lambda^{3} \mu^{3} + 26280 c_{y} \lambda^{2} \mu^{4} + 26640 c_{y} \lambda \mu^{5} + 10272 c_{y} \mu^{6} - 60 i_{x} \lambda^{5} \mu - 576 i_{x} \lambda^{4} \mu^{2} - 1941 i_{x} \lambda^{3} \mu^{3} - 2688 i_{x} \lambda^{2} \mu^{4} - 1284 i_{x} \lambda \mu^{5}}{12 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\i_{x}\\\frac{26 L^{2} f \lambda^{5} + 273 L^{2} f \lambda^{4} \mu + 1045 L^{2} f \lambda^{3} \mu^{2} + 1742 L^{2} f \lambda^{2} \mu^{3} + 1108 L^{2} f \lambda \mu^{4} + 72 L^{2} f \mu^{5} + 60 c_{x} \lambda^{5} \mu + 576 c_{x} \lambda^{4} \mu^{2} + 1941 c_{x} \lambda^{3} \mu^{3} + 2688 c_{x} \lambda^{2} \mu^{4} + 1284 c_{x} \lambda \mu^{5} + 240 c_{y} \lambda^{5} \mu + 2784 c_{y} \lambda^{4} \mu^{2} + 12372 c_{y} \lambda^{3} \mu^{3} + 26280 c_{y} \lambda^{2} \mu^{4} + 26640 c_{y} \lambda \mu^{5} + 10272 c_{y} \mu^{6} - 60 i_{x} \lambda^{5} \mu - 576 i_{x} \lambda^{4} \mu^{2} - 1941 i_{x} \lambda^{3} \mu^{3} - 2688 i_{x} \lambda^{2} \mu^{4} - 1284 i_{x} \lambda \mu^{5}}{12 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\c_{x}\\\frac{26 L^{2} f \lambda^{5} + 273 L^{2} f \lambda^{4} \mu + 1045 L^{2} f \lambda^{3} \mu^{2} + 1742 L^{2} f \lambda^{2} \mu^{3} + 1108 L^{2} f \lambda \mu^{4} + 72 L^{2} f \mu^{5} + 60 c_{x} \lambda^{5} \mu + 576 c_{x} \lambda^{4} \mu^{2} + 1941 c_{x} \lambda^{3} \mu^{3} + 2688 c_{x} \lambda^{2} \mu^{4} + 1284 c_{x} \lambda \mu^{5} + 120 c_{y} \lambda^{5} \mu + 1392 c_{y} \lambda^{4} \mu^{2} + 6186 c_{y} \lambda^{3} \mu^{3} + 13140 c_{y} \lambda^{2} \mu^{4} + 13320 c_{y} \lambda \mu^{5} + 5136 c_{y} \mu^{6} - 60 i_{x} \lambda^{5} \mu - 576 i_{x} \lambda^{4} \mu^{2} - 1941 i_{x} \lambda^{3} \mu^{3} - 2688 i_{x} \lambda^{2} \mu^{4} - 1284 i_{x} \lambda \mu^{5}}{6 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\\frac{- 28 L^{2} f \lambda^{4} - 341 L^{2} f \lambda^{3} \mu - 1493 L^{2} f \lambda^{2} \mu^{2} - 2766 L^{2} f \lambda \mu^{3} - 1824 L^{2} f \mu^{4} + 360 c_{x} \lambda^{4} \mu + 3456 c_{x} \lambda^{3} \mu^{2} + 11646 c_{x} \lambda^{2} \mu^{3} + 16128 c_{x} \lambda \mu^{4} + 7704 c_{x} \mu^{5} + 120 i_{x} \lambda^{4} \mu + 1152 i_{x} \lambda^{3} \mu^{2} + 3882 i_{x} \lambda^{2} \mu^{3} + 5376 i_{x} \lambda \mu^{4} + 2568 i_{x} \mu^{5}}{24 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{80 L^{2} f \lambda^{5} + 847 L^{2} f \lambda^{4} \mu + 3267 L^{2} f \lambda^{3} \mu^{2} + 5470 L^{2} f \lambda^{2} \mu^{3} + 3452 L^{2} f \lambda \mu^{4} + 168 L^{2} f \mu^{5} + 240 c_{x} \lambda^{5} \mu + 2304 c_{x} \lambda^{4} \mu^{2} + 7764 c_{x} \lambda^{3} \mu^{3} + 10752 c_{x} \lambda^{2} \mu^{4} + 5136 c_{x} \lambda \mu^{5} + 480 c_{y} \lambda^{5} \mu + 5568 c_{y} \lambda^{4} \mu^{2} + 24744 c_{y} \lambda^{3} \mu^{3} + 52560 c_{y} \lambda^{2} \mu^{4} + 53280 c_{y} \lambda \mu^{5} + 20544 c_{y} \mu^{6} - 240 i_{x} \lambda^{5} \mu - 2304 i_{x} \lambda^{4} \mu^{2} - 7764 i_{x} \lambda^{3} \mu^{3} - 10752 i_{x} \lambda^{2} \mu^{4} - 5136 i_{x} \lambda \mu^{5}}{24 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\\frac{- 28 L^{2} f \lambda^{4} - 303 L^{2} f \lambda^{3} \mu - 1201 L^{2} f \lambda^{2} \mu^{2} - 2070 L^{2} f \lambda \mu^{3} - 1312 L^{2} f \mu^{4} + 120 c_{x} \lambda^{4} \mu + 1152 c_{x} \lambda^{3} \mu^{2} + 3882 c_{x} \lambda^{2} \mu^{3} + 5376 c_{x} \lambda \mu^{4} + 2568 c_{x} \mu^{5} + 120 i_{x} \lambda^{4} \mu + 1152 i_{x} \lambda^{3} \mu^{2} + 3882 i_{x} \lambda^{2} \mu^{3} + 5376 i_{x} \lambda \mu^{4} + 2568 i_{x} \mu^{5}}{12 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{26 L^{2} f \lambda^{5} + 273 L^{2} f \lambda^{4} \mu + 1045 L^{2} f \lambda^{3} \mu^{2} + 1742 L^{2} f \lambda^{2} \mu^{3} + 1108 L^{2} f \lambda \mu^{4} + 72 L^{2} f \mu^{5} + 120 c_{x} \lambda^{5} \mu + 1152 c_{x} \lambda^{4} \mu^{2} + 3882 c_{x} \lambda^{3} \mu^{3} + 5376 c_{x} \lambda^{2} \mu^{4} + 2568 c_{x} \lambda \mu^{5} + 240 c_{y} \lambda^{5} \mu + 2784 c_{y} \lambda^{4} \mu^{2} + 12372 c_{y} \lambda^{3} \mu^{3} + 26280 c_{y} \lambda^{2} \mu^{4} + 26640 c_{y} \lambda \mu^{5} + 10272 c_{y} \mu^{6} - 120 i_{x} \lambda^{5} \mu - 1152 i_{x} \lambda^{4} \mu^{2} - 3882 i_{x} \lambda^{3} \mu^{3} - 5376 i_{x} \lambda^{2} \mu^{4} - 2568 i_{x} \lambda \mu^{5}}{12 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\\frac{- 28 L^{2} f \lambda^{4} - 341 L^{2} f \lambda^{3} \mu - 1493 L^{2} f \lambda^{2} \mu^{2} - 2766 L^{2} f \lambda \mu^{3} - 1824 L^{2} f \mu^{4} + 120 c_{x} \lambda^{4} \mu + 1152 c_{x} \lambda^{3} \mu^{2} + 3882 c_{x} \lambda^{2} \mu^{3} + 5376 c_{x} \lambda \mu^{4} + 2568 c_{x} \mu^{5} + 360 i_{x} \lambda^{4} \mu + 3456 i_{x} \lambda^{3} \mu^{2} + 11646 i_{x} \lambda^{2} \mu^{3} + 16128 i_{x} \lambda \mu^{4} + 7704 i_{x} \mu^{5}}{24 \mu \left(20 \lambda^{4} + 192 \lambda^{3} \mu + 647 \lambda^{2} \mu^{2} + 896 \lambda \mu^{3} + 428 \mu^{4}\right)}\\\frac{24 L^{2} f \lambda^{5} + 245 L^{2} f \lambda^{4} \mu + 913 L^{2} f \lambda^{3} \mu^{2} + 1498 L^{2} f \lambda^{2} \mu^{3} + 980 L^{2} f \lambda \mu^{4} + 120 L^{2} f \mu^{5} + 240 c_{x} \lambda^{5} \mu + 2304 c_{x} \lambda^{4} \mu^{2} + 7764 c_{x} \lambda^{3} \mu^{3} + 10752 c_{x} \lambda^{2} \mu^{4} + 5136 c_{x} \lambda \mu^{5} + 480 c_{y} \lambda^{5} \mu + 5568 c_{y} \lambda^{4} \mu^{2} + 24744 c_{y} \lambda^{3} \mu^{3} + 52560 c_{y} \lambda^{2} \mu^{4} + 53280 c_{y} \lambda \mu^{5} + 20544 c_{y} \mu^{6} - 240 i_{x} \lambda^{5} \mu - 2304 i_{x} \lambda^{4} \mu^{2} - 7764 i_{x} \lambda^{3} \mu^{3} - 10752 i_{x} \lambda^{2} \mu^{4} - 5136 i_{x} \lambda \mu^{5}}{24 \mu \left(20 \lambda^{5} + 232 \lambda^{4} \mu + 1031 \lambda^{3} \mu^{2} + 2190 \lambda^{2} \mu^{3} + 2220 \lambda \mu^{4} + 856 \mu^{5}\right)}\\i_{x}\\\frac{c_{x} \lambda + 2 c_{y} \lambda + 4 c_{y} \mu - i_{x} \lambda}{2 \left(\lambda + 2 \mu\right)}\end{matrix}\right]\end{split}\]
Semi numerical application
\[\displaystyle \mu=1875.0~ then ~\mu=1875.0,c_x=0.0, c_y=0.0~ then ~\mu=1875.0, c_x=0.0, c_y=0.0,f=0\]
\[\begin{split}\displaystyle S_f = \left[\begin{matrix}c_{x}\\c_{y}\\- 5.50353474723223 \cdot 10^{-5} L^{2} f + 0.75 c_{x} + 0.25 i_{x}\\1.67266906762705 \cdot 10^{-5} L^{2} f + 1.0 c_{y}\\- 8.75550220088035 \cdot 10^{-5} L^{2} f + 0.5 c_{x} + 0.5 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f + 1.0 c_{y}\\- 5.50353474723223 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 0.75 i_{x}\\5.93570761637988 \cdot 10^{-5} L^{2} f + 1.0 c_{y}\\i_{x}\\7.60837668400693 \cdot 10^{-5} L^{2} f + 1.0 c_{y}\\c_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f + 0.125 c_{x} + 1.0 c_{y} - 0.125 i_{x}\\- 7.46431906095772 \cdot 10^{-5} L^{2} f + 0.75 c_{x} + 0.25 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f + 0.125 c_{x} + 1.0 c_{y} - 0.125 i_{x}\\- 8.61144457783113 \cdot 10^{-5} L^{2} f + 0.5 c_{x} + 0.5 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f + 0.125 c_{x} + 1.0 c_{y} - 0.125 i_{x}\\- 7.46431906095772 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 0.75 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f + 0.125 c_{x} + 1.0 c_{y} - 0.125 i_{x}\\i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f + 0.125 c_{x} + 1.0 c_{y} - 0.125 i_{x}\\c_{x}\\7.60837668400694 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 1.0 c_{y} - 0.25 i_{x}\\- 5.50353474723223 \cdot 10^{-5} L^{2} f + 0.75 c_{x} + 0.25 i_{x}\\5.93570761637989 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 1.0 c_{y} - 0.25 i_{x}\\- 8.75550220088035 \cdot 10^{-5} L^{2} f + 0.5 c_{x} + 0.5 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 1.0 c_{y} - 0.25 i_{x}\\- 5.50353474723223 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 0.75 i_{x}\\1.67266906762705 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 1.0 c_{y} - 0.25 i_{x}\\i_{x}\\0.25 c_{x} + 1.0 c_{y} - 0.25 i_{x}\end{matrix}\right]=\left[\begin{matrix}0.0\\0.0\\- 5.50353474723223 \cdot 10^{-5} L^{2} f + 0.25 i_{x}\\1.67266906762705 \cdot 10^{-5} L^{2} f\\- 8.75550220088035 \cdot 10^{-5} L^{2} f + 0.5 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f\\- 5.50353474723223 \cdot 10^{-5} L^{2} f + 0.75 i_{x}\\5.93570761637988 \cdot 10^{-5} L^{2} f\\i_{x}\\7.60837668400693 \cdot 10^{-5} L^{2} f\\0.0\\3.80418834200347 \cdot 10^{-5} L^{2} f - 0.125 i_{x}\\- 7.46431906095772 \cdot 10^{-5} L^{2} f + 0.25 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f - 0.125 i_{x}\\- 8.61144457783113 \cdot 10^{-5} L^{2} f + 0.5 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f - 0.125 i_{x}\\- 7.46431906095772 \cdot 10^{-5} L^{2} f + 0.75 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f - 0.125 i_{x}\\i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f - 0.125 i_{x}\\0.0\\7.60837668400694 \cdot 10^{-5} L^{2} f - 0.25 i_{x}\\- 5.50353474723223 \cdot 10^{-5} L^{2} f + 0.25 i_{x}\\5.93570761637989 \cdot 10^{-5} L^{2} f - 0.25 i_{x}\\- 8.75550220088035 \cdot 10^{-5} L^{2} f + 0.5 i_{x}\\3.80418834200347 \cdot 10^{-5} L^{2} f - 0.25 i_{x}\\- 5.50353474723223 \cdot 10^{-5} L^{2} f + 0.75 i_{x}\\1.67266906762705 \cdot 10^{-5} L^{2} f - 0.25 i_{x}\\i_{x}\\- 0.25 i_{x}\end{matrix}\right]=\left[\begin{matrix}0.0\\0.0\\0.25 i_{x}\\0\\0.5 i_{x}\\0\\0.75 i_{x}\\0\\i_{x}\\0\\0.0\\- 0.125 i_{x}\\0.25 i_{x}\\- 0.125 i_{x}\\0.5 i_{x}\\- 0.125 i_{x}\\0.75 i_{x}\\- 0.125 i_{x}\\i_{x}\\- 0.125 i_{x}\\0.0\\- 0.25 i_{x}\\0.25 i_{x}\\- 0.25 i_{x}\\0.5 i_{x}\\- 0.25 i_{x}\\0.75 i_{x}\\- 0.25 i_{x}\\i_{x}\\- 0.25 i_{x}\end{matrix}\right]\end{split}\]
\[\begin{split}\displaystyle S_f =\left[\begin{matrix}0.0\\0.0\\-0.00251767373616113\\0.00836334533813525\\0.00622248899559824\\0.0190209417100173\\0.0474823262638389\\0.0296785380818994\\0.1\\0.0380418834200347\\0.0\\0.00652094171001734\\-0.0123215953047886\\0.00652094171001734\\0.00694277711084434\\0.00652094171001734\\0.0376784046952114\\0.00652094171001734\\0.1\\0.00652094171001734\\0.0\\0.0130418834200347\\-0.00251767373616113\\0.00467853808189942\\0.00622248899559824\\-0.00597905828998266\\0.0474823262638389\\-0.0166366546618647\\0.1\\-0.025\end{matrix}\right]\end{split}\]
With TS library/FEniCSx \(AD\), \(BD\) system is solve with PETSc:
%%px
petsc_options={"ksp_type":"preonly","pc_type":"lu","pc_factor_mat_solver_type":"mumps"}
solverf = petsc4py.PETSc.KSP().create(fdomain.comm)
solverf.setOperators(AD)
opts = petsc4py.PETSc.Options()
for k, v in petsc_options.items():
opts[k] = v
solverf.setFromOptions()
sf=fem.Function(space)
x_sf = sf.x.petsc_vec
solverf.solve(BD, x_sf)
sf.x.scatter_forward()
A sequential FEniCSx implementation (in drop down bellow) provides a way to do direct comparison with symbolic computation:
Numerical field difference between symbolic and sequential FEniCSx resolution
x y
[[ 1.90819582e-17 -1.21430643e-17]
[ 0.00000000e+00 -2.25514052e-17]
[ 0.00000000e+00 -1.12757026e-17]
[ 8.67361738e-18 -1.73472348e-17]
[ 0.00000000e+00 0.00000000e+00]
[ 1.38777878e-17 -7.80625564e-18]
[ 8.23993651e-18 -1.04083409e-17]
[ 1.56125113e-17 -9.54097912e-18]
[ 1.38777878e-17 -1.38777878e-17]
[ 1.38777878e-17 -9.54097912e-18]
[ 1.38777878e-17 0.00000000e+00]
[ 6.93889390e-18 -1.38777878e-17]
[ 0.00000000e+00 -6.93889390e-18]
[ 0.00000000e+00 -3.46944695e-17]
[ 0.00000000e+00 -1.99493200e-17]]
visualisation with scale factor
[output:0]
Coarse scale level#
Coarse non enriched system#
Compare to fine scale we have only two element categories.
The elementary stifness matrix are the same as fine scale ones.
The elementary load are the same as fine scale ones muliplied by 4
This gives the folowing matrices:
Solution at coarse scale without enrichment#
\[\begin{split}\displaystyle Scs = \left[\begin{matrix}c_{x}\\c_{y}\\\frac{- 2 L^{2} f \lambda - 5 L^{2} f \mu + c_{x} \lambda^{2} + 9 c_{x} \lambda \mu + 10 c_{x} \mu^{2} + i_{x} \lambda^{2} + 9 i_{x} \lambda \mu + 10 i_{x} \mu^{2}}{2 \left(\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}\right)}\\\frac{\frac{L^{2} f \lambda}{2} + c_{y} \lambda^{2} + 9 c_{y} \lambda \mu + 10 c_{y} \mu^{2}}{\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}}\\i_{x}\\\frac{L^{2} f \lambda + c_{y} \lambda^{2} + 9 c_{y} \lambda \mu + 10 c_{y} \mu^{2}}{\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}}\\c_{x}\\\frac{2 L^{2} f \lambda^{2} + 4 L^{2} f \lambda \mu + c_{x} \lambda^{3} + 9 c_{x} \lambda^{2} \mu + 10 c_{x} \lambda \mu^{2} + 2 c_{y} \lambda^{3} + 22 c_{y} \lambda^{2} \mu + 56 c_{y} \lambda \mu^{2} + 40 c_{y} \mu^{3} - i_{x} \lambda^{3} - 9 i_{x} \lambda^{2} \mu - 10 i_{x} \lambda \mu^{2}}{2 \left(\lambda^{3} + 11 \lambda^{2} \mu + 28 \lambda \mu^{2} + 20 \mu^{3}\right)}\\\frac{- 2 L^{2} f \lambda - 5 L^{2} f \mu + c_{x} \lambda^{2} + 9 c_{x} \lambda \mu + 10 c_{x} \mu^{2} + i_{x} \lambda^{2} + 9 i_{x} \lambda \mu + 10 i_{x} \mu^{2}}{2 \left(\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}\right)}\\\frac{L^{2} f \lambda^{2} + 2 L^{2} f \lambda \mu + c_{x} \lambda^{3} + 9 c_{x} \lambda^{2} \mu + 10 c_{x} \lambda \mu^{2} + 2 c_{y} \lambda^{3} + 22 c_{y} \lambda^{2} \mu + 56 c_{y} \lambda \mu^{2} + 40 c_{y} \mu^{3} - i_{x} \lambda^{3} - 9 i_{x} \lambda^{2} \mu - 10 i_{x} \lambda \mu^{2}}{2 \left(\lambda^{3} + 11 \lambda^{2} \mu + 28 \lambda \mu^{2} + 20 \mu^{3}\right)}\\i_{x}\\\frac{c_{x} \lambda + 2 c_{y} \lambda + 4 c_{y} \mu - i_{x} \lambda}{2 \left(\lambda + 2 \mu\right)}\end{matrix}\right]\end{split}\]
Semi numerical application
\[\displaystyle \mu=1875.0~ then ~\mu=1875.0,c_x=0.0, c_y=0.0~ then ~\mu=1875.0, c_x=0.0, c_y=0.0,f=0\]
\[\begin{split}\displaystyle Scs = \left[\begin{matrix}c_{x}\\c_{y}\\- 7.5 \cdot 10^{-5} L^{2} f + 0.5 c_{x} + 0.5 i_{x}\\1.66666666666667 \cdot 10^{-5} L^{2} f + 1.0 c_{y}\\i_{x}\\3.33333333333333 \cdot 10^{-5} L^{2} f + 1.0 c_{y}\\c_{x}\\3.33333333333333 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 1.0 c_{y} - 0.25 i_{x}\\- 7.5 \cdot 10^{-5} L^{2} f + 0.5 c_{x} + 0.5 i_{x}\\1.66666666666667 \cdot 10^{-5} L^{2} f + 0.25 c_{x} + 1.0 c_{y} - 0.25 i_{x}\\i_{x}\\0.25 c_{x} + 1.0 c_{y} - 0.25 i_{x}\end{matrix}\right]=\left[\begin{matrix}0.0\\0.0\\- 7.5 \cdot 10^{-5} L^{2} f + 0.5 i_{x}\\1.66666666666667 \cdot 10^{-5} L^{2} f\\i_{x}\\3.33333333333333 \cdot 10^{-5} L^{2} f\\0.0\\3.33333333333333 \cdot 10^{-5} L^{2} f - 0.25 i_{x}\\- 7.5 \cdot 10^{-5} L^{2} f + 0.5 i_{x}\\1.66666666666667 \cdot 10^{-5} L^{2} f - 0.25 i_{x}\\i_{x}\\- 0.25 i_{x}\end{matrix}\right]=\left[\begin{matrix}0.0\\0.0\\0.5 i_{x}\\0\\i_{x}\\0\\0.0\\- 0.25 i_{x}\\0.5 i_{x}\\- 0.25 i_{x}\\i_{x}\\- 0.25 i_{x}\end{matrix}\right]\end{split}\]
Numerical application
\[\begin{split}\displaystyle Scs =\left[\begin{matrix}0.0\\0.0\\0.0125\\0.00833333333333333\\0.1\\0.0166666666666667\\0.0\\-0.00833333333333334\\0.0125\\-0.0166666666666667\\0.1\\-0.025\end{matrix}\right]\end{split}\]
Equivalent solution with TS library/FEniCSx#
In the following hidden cells, a FEniCSx definition of the coarse non enriched problem is given:
It’s resolution is given by:
%%px
problem_cs = petsc.LinearProblem(a_cs, b_cs,bcs=bc_cs,petsc_options=petsc_options,petsc_options_prefix="cs_")
disp_cs = problem_cs.solve()
[stdout:0] [ 0. 0. 0.0125 -0.01666667 0. -0.00833333
0.0125 0.00833333]
[stdout:1] [ 0.0125 0.00833333 0.1 0.01666667 0.1 -0.025
0.0125 -0.01666667]
Two scales level#
Coarse enriched field#
At coarse scale the field must be enriched to be able to promote fine scale results at coarse scale. In this test case, as all coarse dofs are enriched, the enriched space is equivalent to the standard coarse dof space.
For the symbolic application, we will consider that standard DOFs comme first (0,…11) and enriched dof after (12,…23) with the same ordering (see figure in sections).
Library has been compiled for 'nested' strategy
Thus all mention of 'monolithic' can be ignored in remaining part of this document
For TS library/FEniCSx we will consider a mixed space of two simple Lagrange order 1 spaces grabed from mesh definition for monolithic strategy:
%%px
el_mixed = basix.ufl.mixed_element([cdomain.ufl_domain().ufl_coordinate_element(), cdomain.ufl_domain().ufl_coordinate_element()])
coarse_enriched_space = fem.functionspace(cdomain, el_mixed)
[output:1]
[output:0]
And the enriched field is thus:
%%px
Sce=fem.Function(coarse_enriched_space,name='field_coarse_enriched')
if not nested_strategy:
coarse_field=Sce
coarse_space=coarse_enriched_space
For TS library/FEniCSx using nested strategy we will consider two separate simple Lagrange order-1 spaces grabbed from the mesh definition. But only one will be given to the library: the standard coarse space. The other one (the enriched space) is constructed internally by the library and may span only on a submesh (If not all nodes are enriched which is not the case in this test case). A simple way to implement this is to use directly the field obtained in the coarse computation:
%%px
if nested_strategy:
coarse_field=disp_cs
coarse_space=space_cs
Boundary conditions at both scale#
In all approaches the Dirichlet boundary conditions are imposed in the same way at both scale. At fine scale it has already been taken into account with the creation of \(AD\),\(BD\) from \(A\),\(B\) with createFineScaleSytems function. At coarse scale the TS library expects with the monolithic approach a unique DirichletBC object to impose BC. The detailled procedure is given in “exemple of dirichlet boundary condition setting for the coarse enriched problem “ and adapted to the current test case where only standard dofs are imposed. Thus to selectively impose boundary condition sub space are needed with monolithic approach:
%%px
std_=coarse_enriched_space.sub(0)
std, std_to_mix=std_.collapse()
enr_=coarse_enriched_space.sub(1)
enr, enr_to_mix=enr_.collapse()
Like at fine scale, nodes with imposed BS need to be identified:
%%px
bc_cnodes_1=mesh.locate_entities(cdomain, 0, lambda x: np.isclose(x[0], 0.))
bc_cnodes_2=mesh.locate_entities(cdomain, 0, lambda x: np.isclose(x[0], 2*L))
bc_cnodes_3=mesh.locate_entities(cdomain, 0, lambda x: np.logical_and(np.isclose(x[0], 0.),np.isclose(x[1], 0.)))
Dofs are then selectively identified from these nodes and subspace. Unique boundary condition object is then created (again see here for more detail) :
%%px
bc_cdofs_1=fem.locate_dofs_topological(std.sub(0),0,bc_cnodes_1)
bc_cdofs_2=fem.locate_dofs_topological(std.sub(0),0,bc_cnodes_2)
bc_cdofs_3=fem.locate_dofs_topological(std.sub(1),0,bc_cnodes_3)
#imposed values will be stored in a field
all_BC_values=fem.Function(coarse_enriched_space)
#by default this fiel is with null values
#only dof related to bc_cnodes_2 need to be set to ix
#For that indirection is created
std_to_mix_np=np.asarray(std_to_mix[0])
bc_cdofsce_2=std_to_mix_np[bc_cdofs_2]
all_BC_values.x.array[bc_cdofsce_2]=ix
# the unique bc object is then
all_BC_dof=np.concat((std_to_mix_np[bc_cdofs_1],bc_cdofsce_2,std_to_mix_np[bc_cdofs_3]))
all_BC_dof.sort()
if not nested_strategy:
bcc=[fem.dirichletbc(all_BC_values,all_BC_dof)]
With nested strategy bcc is simply the boundary condition created at coarse scale level
%%px
if nested_strategy:
bcc=bc_cs
Fine to coarse operator: standard part#
Operator \(PS\) (S for standard part)
rows (fine): dofs (0,..29)
columns (coarse): std dof (0,…11) first enriched (12,…23) second
only standard part expressed thus column from 12 to 23 are null. By hand for symbolic computation it is:
\[\begin{split}\displaystyle PS = \left[\begin{array}{cccccccccccccccccccccccc}1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
Apply this operator to A we should normaly obtain Acs:
\[\begin{split}\displaystyle PS^t.A.PS = \left[\begin{array}{cccccccccccccccccccccccc}0.5 \lambda + 1.5 \mu & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0 & 0 & - 0.5 \mu & 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.5 \lambda + 1.5 \mu & 0.5 \mu & - 0.5 \mu & 0 & 0 & 0.5 \lambda & - 0.5 \lambda - 1.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- 0.5 \lambda - 1.0 \mu & 0.5 \mu & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 \lambda & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0.5 \mu & - 0.5 \mu & 0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & 0.5 \lambda + 1.5 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0 & - 0.5 \mu & 0.5 \lambda & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 \lambda & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- 0.5 \mu & 0.5 \lambda & 0 & 0 & 0 & 0 & 0.5 \lambda + 1.5 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0 & 0 & - 0.5 \lambda - 0.5 \mu & 0.5 \lambda + 1.5 \mu & 0.5 \lambda & - 0.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - 0.5 \lambda - 0.5 \mu & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0.5 \lambda & - 0.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & - 0.5 \lambda - 0.5 \mu & - 0.5 \mu & 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle (PS^t.A.PS).block(S,S) - Acs = \left[\begin{array}{cccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
Which is the case.
With the TS library \(PS\) is created when coarseManager object is created with the use of the function generateCoarseManager:
%%px
cm =core.generateCoarseManager(sj,space,coarse_space,None,bcc)
TwoScale loop initialization#
To initialize TwoScale loop in general we use the solution Scs and project it at fine scale with PS to get a starting approximation \(Sts_0=PS.[Scs, 0]^t\):
\[\begin{split}\displaystyle Sts_0 = \left[\begin{matrix}1.0 c_{x}\\1.0 c_{y}\\\frac{- 0.5 L^{2} f \lambda - 1.25 L^{2} f \mu + 0.75 c_{x} \lambda^{2} + 6.75 c_{x} \lambda \mu + 7.5 c_{x} \mu^{2} + 0.25 i_{x} \lambda^{2} + 2.25 i_{x} \lambda \mu + 2.5 i_{x} \mu^{2}}{1.0 \lambda^{2} + 9.0 \lambda \mu + 10.0 \mu^{2}}\\\frac{0.25 L^{2} f \lambda + 1.0 c_{y} \lambda^{2} + 9.0 c_{y} \lambda \mu + 10.0 c_{y} \mu^{2}}{1.0 \lambda^{2} + 9.0 \lambda \mu + 10.0 \mu^{2}}\\\frac{- 1.0 L^{2} f \lambda - 2.5 L^{2} f \mu + 0.5 c_{x} \lambda^{2} + 4.5 c_{x} \lambda \mu + 5.0 c_{x} \mu^{2} + 0.5 i_{x} \lambda^{2} + 4.5 i_{x} \lambda \mu + 5.0 i_{x} \mu^{2}}{\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}}\\\frac{0.5 L^{2} f \lambda + 1.0 c_{y} \lambda^{2} + 9.0 c_{y} \lambda \mu + 10.0 c_{y} \mu^{2}}{\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}}\\\frac{- 0.5 L^{2} f \lambda - 1.25 L^{2} f \mu + 0.25 c_{x} \lambda^{2} + 2.25 c_{x} \lambda \mu + 2.5 c_{x} \mu^{2} + 0.75 i_{x} \lambda^{2} + 6.75 i_{x} \lambda \mu + 7.5 i_{x} \mu^{2}}{1.0 \lambda^{2} + 9.0 \lambda \mu + 10.0 \mu^{2}}\\\frac{0.75 L^{2} f \lambda + 1.0 c_{y} \lambda^{2} + 9.0 c_{y} \lambda \mu + 10.0 c_{y} \mu^{2}}{\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}}\\1.0 i_{x}\\\frac{1.0 L^{2} f \lambda + 1.0 c_{y} \lambda^{2} + 9.0 c_{y} \lambda \mu + 10.0 c_{y} \mu^{2}}{\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}}\\1.0 c_{x}\\\frac{0.5 L^{2} f \lambda^{2} + 1.0 L^{2} f \lambda \mu + 0.25 c_{x} \lambda^{3} + 2.25 c_{x} \lambda^{2} \mu + 2.5 c_{x} \lambda \mu^{2} + 1.0 c_{y} \lambda^{3} + 11.0 c_{y} \lambda^{2} \mu + 28.0 c_{y} \lambda \mu^{2} + 20.0 c_{y} \mu^{3} - 0.25 i_{x} \lambda^{3} - 2.25 i_{x} \lambda^{2} \mu - 2.5 i_{x} \lambda \mu^{2}}{1.0 \lambda^{3} + 11.0 \lambda^{2} \mu + 28.0 \lambda \mu^{2} + 20.0 \mu^{3}}\\\frac{- 0.5 L^{2} f \lambda - 1.25 L^{2} f \mu + 0.75 c_{x} \lambda^{2} + 6.75 c_{x} \lambda \mu + 7.5 c_{x} \mu^{2} + 0.25 i_{x} \lambda^{2} + 2.25 i_{x} \lambda \mu + 2.5 i_{x} \mu^{2}}{1.0 \lambda^{2} + 9.0 \lambda \mu + 10.0 \mu^{2}}\\\frac{0.25 L^{2} f \lambda^{2} + 0.5 L^{2} f \lambda \mu + 0.25 c_{x} \lambda^{3} + 2.25 c_{x} \lambda^{2} \mu + 2.5 c_{x} \lambda \mu^{2} + 1.0 c_{y} \lambda^{3} + 11.0 c_{y} \lambda^{2} \mu + 28.0 c_{y} \lambda \mu^{2} + 20.0 c_{y} \mu^{3} - 0.25 i_{x} \lambda^{3} - 2.25 i_{x} \lambda^{2} \mu - 2.5 i_{x} \lambda \mu^{2}}{1.0 \lambda^{3} + 11.0 \lambda^{2} \mu + 28.0 \lambda \mu^{2} + 20.0 \mu^{3}}\\\frac{- 1.0 L^{2} f \lambda - 2.5 L^{2} f \mu + 0.5 c_{x} \lambda^{2} + 4.5 c_{x} \lambda \mu + 5.0 c_{x} \mu^{2} + 0.5 i_{x} \lambda^{2} + 4.5 i_{x} \lambda \mu + 5.0 i_{x} \mu^{2}}{\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}}\\\frac{0.5 L^{2} f \lambda^{2} + 1.0 L^{2} f \lambda \mu + 0.25 c_{x} \lambda^{3} + 2.25 c_{x} \lambda^{2} \mu + 2.5 c_{x} \lambda \mu^{2} + 1.0 c_{y} \lambda^{3} + 11.0 c_{y} \lambda^{2} \mu + 28.0 c_{y} \lambda \mu^{2} + 20.0 c_{y} \mu^{3} - 0.25 i_{x} \lambda^{3} - 2.25 i_{x} \lambda^{2} \mu - 2.5 i_{x} \lambda \mu^{2}}{1.0 \lambda^{3} + 11.0 \lambda^{2} \mu + 28.0 \lambda \mu^{2} + 20.0 \mu^{3}}\\\frac{- 0.5 L^{2} f \lambda - 1.25 L^{2} f \mu + 0.25 c_{x} \lambda^{2} + 2.25 c_{x} \lambda \mu + 2.5 c_{x} \mu^{2} + 0.75 i_{x} \lambda^{2} + 6.75 i_{x} \lambda \mu + 7.5 i_{x} \mu^{2}}{1.0 \lambda^{2} + 9.0 \lambda \mu + 10.0 \mu^{2}}\\\frac{\left(\lambda + 2 \mu\right) \left(0.25 L^{2} f \lambda + 0.5 c_{y} \lambda^{2} + 4.5 c_{y} \lambda \mu + 5.0 c_{y} \mu^{2}\right) + \left(\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}\right) \left(0.25 c_{x} \lambda + 0.5 c_{y} \lambda + 1.0 c_{y} \mu - 0.25 i_{x} \lambda\right)}{\left(\lambda + 2 \mu\right) \left(\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}\right)}\\1.0 i_{x}\\\frac{\left(\lambda + 2 \mu\right) \left(0.5 L^{2} f \lambda + 0.5 c_{y} \lambda^{2} + 4.5 c_{y} \lambda \mu + 5.0 c_{y} \mu^{2}\right) + \left(\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}\right) \left(0.25 c_{x} \lambda + 0.5 c_{y} \lambda + 1.0 c_{y} \mu - 0.25 i_{x} \lambda\right)}{\left(\lambda + 2 \mu\right) \left(\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}\right)}\\1.0 c_{x}\\\frac{1.0 L^{2} f \lambda^{2} + 2.0 L^{2} f \lambda \mu + 0.5 c_{x} \lambda^{3} + 4.5 c_{x} \lambda^{2} \mu + 5.0 c_{x} \lambda \mu^{2} + 1.0 c_{y} \lambda^{3} + 11.0 c_{y} \lambda^{2} \mu + 28.0 c_{y} \lambda \mu^{2} + 20.0 c_{y} \mu^{3} - 0.5 i_{x} \lambda^{3} - 4.5 i_{x} \lambda^{2} \mu - 5.0 i_{x} \lambda \mu^{2}}{\lambda^{3} + 11 \lambda^{2} \mu + 28 \lambda \mu^{2} + 20 \mu^{3}}\\\frac{- 0.5 L^{2} f \lambda - 1.25 L^{2} f \mu + 0.75 c_{x} \lambda^{2} + 6.75 c_{x} \lambda \mu + 7.5 c_{x} \mu^{2} + 0.25 i_{x} \lambda^{2} + 2.25 i_{x} \lambda \mu + 2.5 i_{x} \mu^{2}}{1.0 \lambda^{2} + 9.0 \lambda \mu + 10.0 \mu^{2}}\\\frac{0.75 L^{2} f \lambda^{2} + 1.5 L^{2} f \lambda \mu + 0.5 c_{x} \lambda^{3} + 4.5 c_{x} \lambda^{2} \mu + 5.0 c_{x} \lambda \mu^{2} + 1.0 c_{y} \lambda^{3} + 11.0 c_{y} \lambda^{2} \mu + 28.0 c_{y} \lambda \mu^{2} + 20.0 c_{y} \mu^{3} - 0.5 i_{x} \lambda^{3} - 4.5 i_{x} \lambda^{2} \mu - 5.0 i_{x} \lambda \mu^{2}}{\lambda^{3} + 11 \lambda^{2} \mu + 28 \lambda \mu^{2} + 20 \mu^{3}}\\\frac{- 1.0 L^{2} f \lambda - 2.5 L^{2} f \mu + 0.5 c_{x} \lambda^{2} + 4.5 c_{x} \lambda \mu + 5.0 c_{x} \mu^{2} + 0.5 i_{x} \lambda^{2} + 4.5 i_{x} \lambda \mu + 5.0 i_{x} \mu^{2}}{\lambda^{2} + 9 \lambda \mu + 10 \mu^{2}}\\\frac{0.5 L^{2} f \lambda^{2} + 1.0 L^{2} f \lambda \mu + 0.5 c_{x} \lambda^{3} + 4.5 c_{x} \lambda^{2} \mu + 5.0 c_{x} \lambda \mu^{2} + 1.0 c_{y} \lambda^{3} + 11.0 c_{y} \lambda^{2} \mu + 28.0 c_{y} \lambda \mu^{2} + 20.0 c_{y} \mu^{3} - 0.5 i_{x} \lambda^{3} - 4.5 i_{x} \lambda^{2} \mu - 5.0 i_{x} \lambda \mu^{2}}{\lambda^{3} + 11 \lambda^{2} \mu + 28 \lambda \mu^{2} + 20 \mu^{3}}\\\frac{- 0.5 L^{2} f \lambda - 1.25 L^{2} f \mu + 0.25 c_{x} \lambda^{2} + 2.25 c_{x} \lambda \mu + 2.5 c_{x} \mu^{2} + 0.75 i_{x} \lambda^{2} + 6.75 i_{x} \lambda \mu + 7.5 i_{x} \mu^{2}}{1.0 \lambda^{2} + 9.0 \lambda \mu + 10.0 \mu^{2}}\\\frac{0.25 L^{2} f \lambda^{2} + 0.5 L^{2} f \lambda \mu + 0.5 c_{x} \lambda^{3} + 4.5 c_{x} \lambda^{2} \mu + 5.0 c_{x} \lambda \mu^{2} + 1.0 c_{y} \lambda^{3} + 11.0 c_{y} \lambda^{2} \mu + 28.0 c_{y} \lambda \mu^{2} + 20.0 c_{y} \mu^{3} - 0.5 i_{x} \lambda^{3} - 4.5 i_{x} \lambda^{2} \mu - 5.0 i_{x} \lambda \mu^{2}}{1.0 \lambda^{3} + 11.0 \lambda^{2} \mu + 28.0 \lambda \mu^{2} + 20.0 \mu^{3}}\\1.0 i_{x}\\\frac{0.5 c_{x} \lambda + 1.0 c_{y} \lambda + 2.0 c_{y} \mu - 0.5 i_{x} \lambda}{\lambda + 2 \mu}\end{matrix}\right]\end{split}\]
which is naturally not equal to \(S_F\) when \(f\neq 0\).
With TS library the FEniCSx computation at coarse scale is used by projectStdCoarse method but results must be stored in enriched field with monolithic strategy.
%%px
if not nested_strategy:
coarse_field.x.array[std_to_mix_np]=disp_ce.x.array
And finaly the projection is done using projectStdCoarse method of cm instance
%%px
Sts0=fem.Function(space)
cm.projectStdCoarse(coarse_field,Sts0)
#Sts0.x.scatter_reverse(la.InsertMode.add)
Sts0.x.petsc_vec.view()
Compare to symbolic computation the TS library provides the same projection (see below).
Patch resolution#
\(Sts_0\) provides information to impose BC to patches
Here the sympbolic resolution of the first patch(in drop down).
Has can be seen, it is to complex to use symbolic expression. We are going to switch to symbolic variables for patch solution and thus for enrichement function but for some computation we will use the solution coresponding to numerical application:
Symbolic dofs used for patch computation:
Patch problem for enriched dof (12,13): free/BC [0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23] TS BC [4, 5, 14, 15, 24, 25] shift idx [0, 1]
Patch problem for enriched dof (14,15): free/BC [2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 26, 27, 28, 29] TS BC [0, 1, 12, 13, 24, 25] shift idx [2, 3]
Patch problem for enriched dof (16,17): free/BC [6, 7, 8, 9, 18, 19] TS BC [4, 5, 16, 17, 28, 29] shift idx [2, 3]
Patch problem for enriched dof (18,19): free/BC [10, 11, 20, 21, 22, 23] TS BC [0, 1, 12, 13, 24, 25] shift idx [2, 3]
Patch problem for enriched dof (20,21): free/BC [0, 1, 2, 3, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26, 27] TS BC [4, 5, 16, 17, 28, 29] shift idx [14, 15]
Patch problem for enriched dof (22,23): free/BC [6, 7, 8, 9, 16, 17, 18, 19, 26, 27, 28, 29] TS BC [4, 5, 14, 15, 24, 25] shift idx [10, 11]
With TS library/FEniCSx patches are managed by an instance of patchManager class create by generatePatchManager:
%%px
pm =core.generatePatchManager(sj,space)
This instance is responsible of the creation of the patches problem extracted from \(AD\),\(BD\) matrices. Note that patch problem generation expect that Dirichlet boundary condition are already imposed to the fine scale matrices.
%%px
pm.generateProblems(AD,BD)
As soon as patches problem are created they can be solved imposinging current TS approximation (i.e.\(Sts_0\)):
%%px
pm.solveProblems(Sts0)
At this stage patch solutions from TS library can be checked. Normally every symbolic solution can be found in one computing sequence that groupe resolution of 1 or more patches. In this example we have 4/5 sequences that computes 6 patches:
Symbolic solution of patch 0 is identified in TS library sequence 2
Symbolic solution of patch 1 is identified in TS library sequence 1
Symbolic solution of patch 2 is identified in TS library sequence 2
Symbolic solution of patch 3 is identified in TS library sequence 3
Symbolic solution of patch 4 is identified in TS library sequence 0
Symbolic solution of patch 5 is identified in TS library sequence 3
Fine to coarse operator in the general case#
We use the following for the symbolic computation
\(\theta_i\): enriched values at shift dof (so we can easily turn them to zero when using shift enrichment function)
\(\alpha_i^j\): a priori non null enriched function values for pathch related to DOF (i) and for fine DOF (j)
and set \(PG\) (G for general case) using \(PS\) and those variables:
\[\begin{split}\displaystyle PG = \left[\begin{array}{cccccccccccccccccccccccc}1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{2}_{0} & 0 & \alpha^{2}_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{3}_{1} & 0 & \alpha^{3}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{6}_{2} & 0 & \alpha^{6}_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{7}_{3} & 0 & \alpha^{7}_{5} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{5} & 0 & 0 & 0 & 0 & 0 & 0\\0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{0} & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} & 0 & 0 & 0 & 0 & 0\\0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{11}_{1} & 0 & 0 & 0 & 0 & 0 & \alpha^{11}_{7} & 0 & 0 & 0 & 0\\0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{12}_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{12}_{8} & 0 & 0 & 0\\0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{13}_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{13}_{9} & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{14}_{2} & 0 & 0 & 0 & 0 & 0 & \alpha^{14}_{8} & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{15}_{3} & 0 & 0 & 0 & 0 & 0 & \alpha^{15}_{9} & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{16}_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{16}_{10} & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{17}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{17}_{11}\\0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{4} & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} & 0\\0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{19}_{5} & 0 & 0 & 0 & 0 & 0 & \alpha^{19}_{11}\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{6} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{7} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{22}_{6} & 0 & \alpha^{22}_{8} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{23}_{7} & 0 & \alpha^{23}_{9} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{8} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{9} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{26}_{8} & 0 & \alpha^{26}_{10} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{27}_{9} & 0 & \alpha^{27}_{11}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{10} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \theta_{11}\end{array}\right]\end{split}\]
TS approch (I) in general case#
Applying \(PG\) to \(A\) matrix lead to the following expression of the coarse enriched matrix:
\[\begin{split}\displaystyle PG^t.A.PG = \left[\begin{array}{cccccccccccccccccccccccc}0.5 \lambda + 1.5 \mu & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0 & 0 & - 0.5 \mu & 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0 & \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) + \theta_{0} \left(0.25 \lambda + 0.75 \mu\right) & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) + 0.5 \alpha^{3}_{1} \lambda & - \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{2} \left(0.25 \lambda + 0.5 \mu\right) & \lambda \left(0.5 \alpha^{3}_{3} + 0.25 \theta_{3}\right) & 0 & 0 & \mu \left(- 0.5 \alpha^{22}_{6} - 0.25 \theta_{6}\right) & \mu \left(0.5 \alpha^{11}_{7} + 0.25 \theta_{7}\right) & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{22}_{8} \mu & \left(- 0.5 \alpha^{13}_{9} - 0.25 \theta_{9}\right) \left(\lambda + \mu\right) & 0 & 0\\0 & 0.5 \lambda + 1.5 \mu & 0.5 \mu & - 0.5 \mu & 0 & 0 & 0.5 \lambda & - 0.5 \lambda - 1.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0.5 \alpha^{10}_{0} \lambda - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) + 0.5 \alpha^{2}_{0} \mu & \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \theta_{1} \left(0.25 \lambda + 0.75 \mu\right) & \mu \left(0.5 \alpha^{2}_{2} + 0.25 \theta_{2}\right) & \mu \left(- 0.5 \alpha^{15}_{3} - 0.25 \theta_{3}\right) & 0 & 0 & \lambda \left(0.5 \alpha^{10}_{6} + 0.25 \theta_{6}\right) & - \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{7} \left(0.25 \lambda + 0.5 \mu\right) & \left(- 0.5 \alpha^{12}_{8} - 0.25 \theta_{8}\right) \left(\lambda + \mu\right) & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{15}_{9} \mu - \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0\\- 0.5 \lambda - 1.0 \mu & 0.5 \mu & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{0} \mu - \theta_{0} \left(0.25 \lambda + 0.5 \mu\right) & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda + 0.25 \mu \theta_{1} & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{2} \mu + \theta_{2} \left(0.5 \lambda + 1.5 \mu\right) & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda - 0.25 \theta_{3} \left(\lambda + \mu\right) & - \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{4} \left(0.25 \lambda + 0.5 \mu\right) & \lambda \left(0.5 \alpha^{7}_{5} + 0.25 \theta_{5}\right) & 0 & 0 & - \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{8} \mu - 0.5 \mu \theta_{8} & \left(0.5 \alpha^{13}_{9} + 0.25 \theta_{9}\right) \left(\lambda + \mu\right) & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{10} \mu & \left(- 0.5 \alpha^{17}_{11} - 0.25 \theta_{11}\right) \left(\lambda + \mu\right)\\0.5 \lambda & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0.5 \mu & - 0.5 \mu & 0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu + 0.25 \lambda \theta_{0} & - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{1} & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu - 0.25 \theta_{2} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) + \theta_{3} \left(0.5 \lambda + 1.5 \mu\right) & \mu \left(0.5 \alpha^{6}_{4} + 0.25 \theta_{4}\right) & \mu \left(- 0.5 \alpha^{19}_{5} - 0.25 \theta_{5}\right) & 0 & 0 & \left(0.5 \alpha^{12}_{8} + 0.25 \theta_{8}\right) \left(\lambda + \mu\right) & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{9} \left(0.5 \lambda + 1.0 \mu\right) & \left(- 0.5 \alpha^{16}_{10} - 0.25 \theta_{10}\right) \left(\lambda + \mu\right) & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{19}_{11} \mu - \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & 0.5 \lambda + 1.5 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0 & - 0.5 \mu & 0.5 \lambda & 0 & 0 & - \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{6}_{2} \mu - \theta_{2} \left(0.25 \lambda + 0.5 \mu\right) & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{7}_{3} \lambda + 0.25 \mu \theta_{3} & \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{6}_{4} \mu + \theta_{4} \left(0.25 \lambda + 0.75 \mu\right) & - 0.5 \alpha^{19}_{5} \mu - 0.5 \alpha^{7}_{5} \lambda - 0.25 \theta_{5} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0 & - \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{10} & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{19}_{11} \mu + 0.25 \lambda \theta_{11}\\0 & 0 & 0.5 \lambda & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu + 0.25 \lambda \theta_{2} & - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{3} & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu - 0.25 \theta_{4} \left(\lambda + \mu\right) & 0.5 \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) + \theta_{5} \left(0.25 \lambda + 0.75 \mu\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{18}_{10} \lambda + 0.25 \mu \theta_{10} & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{19}_{11} \mu - \theta_{11} \left(0.25 \lambda + 0.5 \mu\right)\\- 0.5 \mu & 0.5 \lambda & 0 & 0 & 0 & 0 & 0.5 \lambda + 1.5 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & 0 & 0 & \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.25 \mu \theta_{0} & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) + 0.25 \lambda \theta_{1} & 0 & 0 & 0 & 0 & \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{22}_{6} \mu + \theta_{6} \left(0.25 \lambda + 0.75 \mu\right) & - 0.5 \alpha^{11}_{7} \mu - 0.5 \alpha^{23}_{7} \lambda - 0.25 \theta_{7} \left(\lambda + \mu\right) & - \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{22}_{8} \mu - \theta_{8} \left(0.25 \lambda + 0.5 \mu\right) & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) - 0.5 \alpha^{23}_{9} \lambda + 0.25 \mu \theta_{9} & 0 & 0\\0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0 & 0 & - 0.5 \lambda - 0.5 \mu & 0.5 \lambda + 1.5 \mu & 0.5 \lambda & - 0.5 \mu & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) + 0.25 \mu \theta_{0} & 0.5 \alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) - \theta_{1} \left(0.25 \lambda + 0.5 \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu - 0.25 \theta_{6} \left(\lambda + \mu\right) & 0.5 \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) + \theta_{7} \left(0.25 \lambda + 0.75 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu + 0.25 \lambda \theta_{8} & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{9} & 0 & 0\\0 & - 0.5 \lambda - 0.5 \mu & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & - \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{0} \mu & \left(- 0.5 \alpha^{13}_{1} - 0.25 \theta_{1}\right) \left(\lambda + \mu\right) & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{2} \mu - 0.5 \mu \theta_{2} & \left(0.5 \alpha^{17}_{3} + 0.25 \theta_{3}\right) \left(\lambda + \mu\right) & 0 & 0 & - \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{6} \left(0.25 \lambda + 0.5 \mu\right) & \lambda \left(0.5 \alpha^{23}_{7} + 0.25 \theta_{7}\right) & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{26}_{8} \mu + \theta_{8} \left(0.5 \lambda + 1.5 \mu\right) & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda - 0.25 \theta_{9} \left(\lambda + \mu\right) & - \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{26}_{10} \mu - \theta_{10} \left(0.25 \lambda + 0.5 \mu\right) & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda + 0.25 \mu \theta_{11}\\- 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0.5 \lambda & - 0.5 \mu & \left(- 0.5 \alpha^{12}_{0} - 0.25 \theta_{0}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & \left(0.5 \alpha^{16}_{2} + 0.25 \theta_{2}\right) \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & \mu \left(0.5 \alpha^{22}_{6} + 0.25 \theta_{6}\right) & \mu \left(- 0.5 \alpha^{11}_{7} - 0.25 \theta_{7}\right) & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu - 0.25 \theta_{8} \left(\lambda + \mu\right) & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) + \theta_{9} \left(0.5 \lambda + 1.5 \mu\right) & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu + 0.25 \lambda \theta_{10} & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{11}\\0 & 0 & 0 & - 0.5 \lambda - 0.5 \mu & - 0.5 \mu & 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & - \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{6}_{2} \mu & \left(- 0.5 \alpha^{17}_{3} - 0.25 \theta_{3}\right) \left(\lambda + \mu\right) & \mu \left(- 0.5 \alpha^{6}_{4} - 0.25 \theta_{4}\right) & \mu \left(0.5 \alpha^{19}_{5} + 0.25 \theta_{5}\right) & 0 & 0 & - \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{8} \left(0.25 \lambda + 0.5 \mu\right) & \lambda \left(0.5 \alpha^{27}_{9} + 0.25 \theta_{9}\right) & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + \theta_{10} \left(0.25 \lambda + 0.75 \mu\right) & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) + 0.5 \alpha^{19}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & \left(- 0.5 \alpha^{16}_{2} - 0.25 \theta_{2}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & \lambda \left(0.5 \alpha^{18}_{4} + 0.25 \theta_{4}\right) & - \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{5} \left(0.25 \lambda + 0.5 \mu\right) & 0 & 0 & \mu \left(0.5 \alpha^{26}_{8} + 0.25 \theta_{8}\right) & \mu \left(- 0.5 \alpha^{15}_{9} - 0.25 \theta_{9}\right) & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + \theta_{11} \left(0.25 \lambda + 0.75 \mu\right)\\0.5 \alpha^{12}_{0} \lambda + 1.5 \alpha^{12}_{0} \mu + 0.25 \lambda \theta_{0} + 0.75 \mu \theta_{0} & 0.5 \alpha^{10}_{0} \lambda - 0.5 \alpha^{12}_{0} \lambda - 0.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \mu - 0.25 \lambda \theta_{0} - 0.5 \mu \theta_{0} & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu + 0.25 \lambda \theta_{0} & 0 & 0 & 0.5 \alpha^{10}_{0} \lambda + 1.0 \alpha^{10}_{0} \mu - 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu - 0.25 \mu \theta_{0} & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) + 0.25 \mu \theta_{0} & - 0.5 \alpha^{10}_{0} \lambda - 1.0 \alpha^{10}_{0} \mu + 0.5 \alpha^{12}_{0} \lambda + 1.5 \alpha^{12}_{0} \mu - 0.5 \alpha^{2}_{0} \mu & \left(- 0.5 \alpha^{12}_{0} - 0.25 \theta_{0}\right) \left(\lambda + \mu\right) & 0 & 0 & - \frac{\alpha^{10}_{0} \left(- 2 \alpha^{10}_{0} \left(\lambda + 3 \mu\right) + 2 \alpha^{12}_{0} \left(\lambda + 2 \mu\right) + \mu \theta_{0}\right)}{2} - \alpha^{12}_{0} \left(\alpha^{10}_{0} \left(\lambda + 2 \mu\right) - 2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) + \alpha^{2}_{0} \mu\right) - \frac{\alpha^{2}_{0} \left(2 \alpha^{12}_{0} \mu - 2 \alpha^{2}_{0} \left(\lambda + 3 \mu\right) + \theta_{0} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{0} \left(\alpha^{10}_{0} \mu + \alpha^{2}_{0} \left(\lambda + 2 \mu\right) - \theta_{0} \left(\lambda + 3 \mu\right)\right)}{2} & \frac{\alpha^{11}_{1} \mu \theta_{0}}{2} - \frac{\alpha^{13}_{1} \theta_{0} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{3}_{1} \lambda \theta_{0}}{2} + \frac{\theta_{1} \left(\alpha^{10}_{0} \lambda - \alpha^{12}_{0} \left(\lambda + \mu\right) + \alpha^{2}_{0} \mu\right)}{2} & - \alpha^{12}_{0} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \frac{\alpha^{2}_{0} \theta_{2} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{2}_{2} \left(2 \alpha^{12}_{0} \mu - 2 \alpha^{2}_{0} \left(\lambda + 3 \mu\right) + \theta_{0} \left(\lambda + 2 \mu\right)\right)}{2} & \frac{\alpha^{3}_{3} \lambda \theta_{0}}{2} + \frac{\theta_{3} \left(\alpha^{12}_{0} \left(\lambda + \mu\right) - \alpha^{2}_{0} \mu\right)}{2} & 0 & 0 & - \frac{\alpha^{10}_{0} \mu \theta_{6}}{2} - \alpha^{12}_{0} \alpha^{22}_{6} \mu - \frac{\alpha^{10}_{6} \left(- 2 \alpha^{10}_{0} \left(\lambda + 3 \mu\right) + 2 \alpha^{12}_{0} \left(\lambda + 2 \mu\right) + \mu \theta_{0}\right)}{2} & \frac{\alpha^{11}_{7} \mu \theta_{0}}{2} - \frac{\theta_{7} \left(\alpha^{10}_{0} \lambda - \alpha^{12}_{0} \left(\lambda + \mu\right)\right)}{2} & - \alpha^{12}_{0} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \alpha^{22}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{10}_{0} \left(\lambda + 2 \mu\right) - 2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) + \alpha^{2}_{0} \mu\right) & \frac{\left(\lambda + \mu\right) \left(- \alpha^{12}_{0} \theta_{9} - \alpha^{13}_{9} \theta_{0}\right)}{2} & 0 & 0\\0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \lambda - 0.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \lambda & 0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu + 0.25 \lambda \theta_{1} + 0.75 \mu \theta_{1} & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda + 0.25 \mu \theta_{1} & - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \lambda + 1.0 \alpha^{3}_{1} \mu - 0.25 \mu \theta_{1} & 0 & 0 & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) + 0.25 \lambda \theta_{1} & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu - 0.25 \lambda \theta_{1} - 0.5 \mu \theta_{1} & \left(- 0.5 \alpha^{13}_{1} - 0.25 \theta_{1}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu - 0.5 \alpha^{3}_{1} \lambda - 1.0 \alpha^{3}_{1} \mu & 0 & 0 & \frac{\alpha^{10}_{0} \lambda \theta_{1}}{2} - \frac{\alpha^{12}_{0} \theta_{1} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{2}_{0} \mu \theta_{1}}{2} + \frac{\theta_{0} \left(\alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(\lambda + \mu\right) + \alpha^{3}_{1} \lambda\right)}{2} & - \frac{\alpha^{11}_{1} \left(- 2 \alpha^{11}_{1} \left(\lambda + 3 \mu\right) + 2 \alpha^{13}_{1} \mu + \theta_{1} \left(\lambda + 2 \mu\right)\right)}{2} - \alpha^{13}_{1} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) - \frac{\alpha^{3}_{1} \left(2 \alpha^{13}_{1} \left(\lambda + 2 \mu\right) - 2 \alpha^{3}_{1} \left(\lambda + 3 \mu\right) + \mu \theta_{1}\right)}{2} - \frac{\theta_{1} \left(\alpha^{11}_{1} \left(\lambda + 2 \mu\right) + \alpha^{3}_{1} \mu - \theta_{1} \left(\lambda + 3 \mu\right)\right)}{2} & \frac{\alpha^{2}_{2} \mu \theta_{1}}{2} + \frac{\theta_{2} \left(\alpha^{13}_{1} \left(\lambda + \mu\right) - \alpha^{3}_{1} \lambda\right)}{2} & - \alpha^{13}_{1} \alpha^{15}_{3} \mu - \frac{\alpha^{3}_{1} \mu \theta_{3}}{2} - \frac{\alpha^{3}_{3} \left(2 \alpha^{13}_{1} \left(\lambda + 2 \mu\right) - 2 \alpha^{3}_{1} \left(\lambda + 3 \mu\right) + \mu \theta_{1}\right)}{2} & 0 & 0 & \frac{\alpha^{10}_{6} \lambda \theta_{1}}{2} - \frac{\theta_{6} \left(\alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(\lambda + \mu\right)\right)}{2} & - \frac{\alpha^{11}_{1} \theta_{7} \left(\lambda + 2 \mu\right)}{2} - \alpha^{13}_{1} \alpha^{23}_{7} \left(\lambda + 2 \mu\right) - \frac{\alpha^{11}_{7} \left(- 2 \alpha^{11}_{1} \left(\lambda + 3 \mu\right) + 2 \alpha^{13}_{1} \mu + \theta_{1} \left(\lambda + 2 \mu\right)\right)}{2} & \frac{\left(\lambda + \mu\right) \left(- \alpha^{13}_{1} \theta_{8} - \alpha^{12}_{8} \theta_{1}\right)}{2} & - \alpha^{13}_{1} \alpha^{15}_{9} \mu - \alpha^{13}_{1} \alpha^{23}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\- 0.5 \alpha^{14}_{2} \lambda - 1.0 \alpha^{14}_{2} \mu - 0.25 \lambda \theta_{2} - 0.5 \mu \theta_{2} & \mu \left(0.5 \alpha^{2}_{2} + 0.25 \theta_{2}\right) & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{2}_{2} \mu + 0.5 \lambda \theta_{2} + 1.5 \mu \theta_{2} & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu - 0.25 \theta_{2} \left(\lambda + \mu\right) & - 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{6}_{2} \mu - 0.25 \lambda \theta_{2} - 0.5 \mu \theta_{2} & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu + 0.25 \lambda \theta_{2} & 0 & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu - 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{2}_{2} \mu - 0.5 \mu \theta_{2} & \left(0.5 \alpha^{16}_{2} + 0.25 \theta_{2}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{14}_{2} \lambda - 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{6}_{2} \mu & \left(- 0.5 \alpha^{16}_{2} - 0.25 \theta_{2}\right) \left(\lambda + \mu\right) & - \alpha^{12}_{0} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \frac{\alpha^{2}_{0} \left(2 \alpha^{2}_{2} \left(\lambda + 3 \mu\right) - \theta_{2} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{2}_{2} \theta_{0} \left(\lambda + 2 \mu\right)}{2} & \frac{\alpha^{13}_{1} \theta_{2} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{3}_{1} \lambda \theta_{2}}{2} + \frac{\alpha^{2}_{2} \mu \theta_{1}}{2} & - \alpha^{14}_{2} \left(- 2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) + \alpha^{16}_{2} \left(\lambda + 2 \mu\right) + \mu \theta_{2}\right) - \alpha^{16}_{2} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) + \frac{\alpha^{2}_{2} \left(2 \alpha^{2}_{2} \left(\lambda + 3 \mu\right) - \theta_{2} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{6}_{2} \left(2 \alpha^{16}_{2} \mu - 2 \alpha^{6}_{2} \left(\lambda + 3 \mu\right) + \theta_{2} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{2} \left(2 \alpha^{14}_{2} \mu + \alpha^{2}_{2} \left(\lambda + 2 \mu\right) + \alpha^{6}_{2} \left(\lambda + 2 \mu\right) - 2 \theta_{2} \left(\lambda + 3 \mu\right)\right)}{2} & - \frac{\alpha^{17}_{3} \theta_{2} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{3}_{3} \lambda \theta_{2}}{2} + \frac{\alpha^{7}_{3} \lambda \theta_{2}}{2} - \frac{\theta_{3} \left(\alpha^{16}_{2} \left(\lambda + \mu\right) + \alpha^{2}_{2} \mu - \alpha^{6}_{2} \mu\right)}{2} & - \alpha^{16}_{2} \alpha^{18}_{4} \left(\lambda + 2 \mu\right) - \frac{\alpha^{6}_{2} \theta_{4} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{6}_{4} \left(2 \alpha^{16}_{2} \mu - 2 \alpha^{6}_{2} \left(\lambda + 3 \mu\right) + \theta_{2} \left(\lambda + 2 \mu\right)\right)}{2} & \frac{\alpha^{7}_{5} \lambda \theta_{2}}{2} + \frac{\theta_{5} \left(\alpha^{16}_{2} \left(\lambda + \mu\right) - \alpha^{6}_{2} \mu\right)}{2} & 0 & 0 & - \alpha^{14}_{2} \mu \theta_{8} - \alpha^{16}_{2} \alpha^{26}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) - \alpha^{14}_{8} \left(- 2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) + \alpha^{16}_{2} \left(\lambda + 2 \mu\right) + \mu \theta_{2}\right) & \frac{\left(\lambda + \mu\right) \left(\alpha^{16}_{2} \theta_{9} + \alpha^{13}_{9} \theta_{2}\right)}{2} & - \alpha^{16}_{10} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) - \alpha^{18}_{10} \alpha^{16}_{2} \left(\lambda + 2 \mu\right) - \alpha^{26}_{10} \alpha^{16}_{2} \mu & \frac{\left(\lambda + \mu\right) \left(- \alpha^{17}_{11} \theta_{2} - \alpha^{16}_{2} \theta_{11}\right)}{2}\\\lambda \left(0.5 \alpha^{3}_{3} + 0.25 \theta_{3}\right) & \mu \left(- 0.5 \alpha^{15}_{3} - 0.25 \theta_{3}\right) & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda - 0.25 \theta_{3} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{3}_{3} \lambda + 1.0 \alpha^{3}_{3} \mu + 0.5 \lambda \theta_{3} + 1.5 \mu \theta_{3} & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{7}_{3} \lambda + 0.25 \mu \theta_{3} & - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{7}_{3} \lambda + 1.0 \alpha^{7}_{3} \mu - 0.25 \mu \theta_{3} & 0 & 0 & \left(0.5 \alpha^{17}_{3} + 0.25 \theta_{3}\right) \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{3}_{3} \lambda - 1.0 \alpha^{3}_{3} \mu - 0.5 \lambda \theta_{3} - 1.0 \mu \theta_{3} & \left(- 0.5 \alpha^{17}_{3} - 0.25 \theta_{3}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{7}_{3} \lambda - 1.0 \alpha^{7}_{3} \mu & \frac{\alpha^{12}_{0} \theta_{3} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{2}_{0} \mu \theta_{3}}{2} + \frac{\alpha^{3}_{3} \lambda \theta_{0}}{2} & - \alpha^{13}_{1} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{3}_{1} \left(2 \alpha^{3}_{3} \left(\lambda + 3 \mu\right) - \mu \theta_{3}\right)}{2} - \frac{\alpha^{3}_{3} \mu \theta_{1}}{2} & - \frac{\alpha^{16}_{2} \theta_{3} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{2}_{2} \mu \theta_{3}}{2} + \frac{\alpha^{6}_{2} \mu \theta_{3}}{2} - \frac{\theta_{2} \left(\alpha^{17}_{3} \left(\lambda + \mu\right) + \alpha^{3}_{3} \lambda - \alpha^{7}_{3} \lambda\right)}{2} & - \alpha^{15}_{3} \left(- 2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) + \alpha^{17}_{3} \mu + \theta_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{17}_{3} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{3}_{3} \left(2 \alpha^{3}_{3} \left(\lambda + 3 \mu\right) - \mu \theta_{3}\right)}{2} - \frac{\alpha^{7}_{3} \left(2 \alpha^{17}_{3} \left(\lambda + 2 \mu\right) - 2 \alpha^{7}_{3} \left(\lambda + 3 \mu\right) + \mu \theta_{3}\right)}{2} - \frac{\theta_{3} \left(2 \alpha^{15}_{3} \left(\lambda + 2 \mu\right) + \alpha^{3}_{3} \mu + \alpha^{7}_{3} \mu - 2 \theta_{3} \left(\lambda + 3 \mu\right)\right)}{2} & \frac{\alpha^{6}_{4} \mu \theta_{3}}{2} + \frac{\theta_{4} \left(\alpha^{17}_{3} \left(\lambda + \mu\right) - \alpha^{7}_{3} \lambda\right)}{2} & - \alpha^{17}_{3} \alpha^{19}_{5} \mu - \frac{\alpha^{7}_{3} \mu \theta_{5}}{2} - \frac{\alpha^{7}_{5} \left(2 \alpha^{17}_{3} \left(\lambda + 2 \mu\right) - 2 \alpha^{7}_{3} \left(\lambda + 3 \mu\right) + \mu \theta_{3}\right)}{2} & 0 & 0 & \frac{\left(\lambda + \mu\right) \left(\alpha^{17}_{3} \theta_{8} + \alpha^{12}_{8} \theta_{3}\right)}{2} & - \alpha^{15}_{3} \theta_{9} \left(\lambda + 2 \mu\right) - \alpha^{17}_{3} \alpha^{27}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{15}_{9} \left(- 2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) + \alpha^{17}_{3} \mu + \theta_{3} \left(\lambda + 2 \mu\right)\right) & \frac{\left(\lambda + \mu\right) \left(- \alpha^{16}_{10} \theta_{3} - \alpha^{17}_{3} \theta_{10}\right)}{2} & - \alpha^{17}_{11} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \alpha^{17}_{3} \mu - \alpha^{27}_{11} \alpha^{17}_{3} \left(\lambda + 2 \mu\right)\\0 & 0 & - 0.5 \alpha^{18}_{4} \lambda - 1.0 \alpha^{18}_{4} \mu - 0.25 \lambda \theta_{4} - 0.5 \mu \theta_{4} & \mu \left(0.5 \alpha^{6}_{4} + 0.25 \theta_{4}\right) & 0.5 \alpha^{18}_{4} \lambda + 1.0 \alpha^{18}_{4} \mu + 0.5 \alpha^{6}_{4} \mu + 0.25 \lambda \theta_{4} + 0.75 \mu \theta_{4} & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu - 0.25 \lambda \theta_{4} - 0.25 \mu \theta_{4} & 0 & 0 & 0 & 0 & \mu \left(- 0.5 \alpha^{6}_{4} - 0.25 \theta_{4}\right) & \lambda \left(0.5 \alpha^{18}_{4} + 0.25 \theta_{4}\right) & 0 & 0 & - \alpha^{16}_{2} \left(\alpha^{18}_{4} \left(\lambda + 2 \mu\right) + \alpha^{6}_{4} \mu\right) + \frac{\alpha^{6}_{2} \left(2 \alpha^{6}_{4} \left(\lambda + 3 \mu\right) - \theta_{4} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{6}_{4} \theta_{2} \left(\lambda + 2 \mu\right)}{2} & \frac{\alpha^{17}_{3} \theta_{4} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{7}_{3} \lambda \theta_{4}}{2} + \frac{\alpha^{6}_{4} \mu \theta_{3}}{2} & \frac{\alpha^{18}_{4} \left(2 \alpha^{18}_{4} \left(\lambda + 3 \mu\right) - \mu \theta_{4}\right)}{2} + \frac{\alpha^{6}_{4} \left(2 \alpha^{6}_{4} \left(\lambda + 3 \mu\right) - \theta_{4} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{4} \left(\alpha^{18}_{4} \mu + \alpha^{6}_{4} \left(\lambda + 2 \mu\right) - \theta_{4} \left(\lambda + 3 \mu\right)\right)}{2} & - \frac{\alpha^{19}_{5} \mu \theta_{4}}{2} - \frac{\alpha^{7}_{5} \lambda \theta_{4}}{2} - \frac{\theta_{5} \left(\alpha^{18}_{4} \lambda + \alpha^{6}_{4} \mu\right)}{2} & 0 & 0 & 0 & 0 & - \alpha^{16}_{10} \left(\alpha^{18}_{4} \left(\lambda + 2 \mu\right) + \alpha^{6}_{4} \mu\right) + \frac{\alpha^{18}_{10} \left(2 \alpha^{18}_{4} \left(\lambda + 3 \mu\right) - \mu \theta_{4}\right)}{2} - \frac{\alpha^{18}_{4} \mu \theta_{10}}{2} & \frac{\alpha^{17}_{11} \theta_{4} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{19}_{11} \mu \theta_{4}}{2} + \frac{\alpha^{18}_{4} \lambda \theta_{11}}{2}\\0 & 0 & \lambda \left(0.5 \alpha^{7}_{5} + 0.25 \theta_{5}\right) & \mu \left(- 0.5 \alpha^{19}_{5} - 0.25 \theta_{5}\right) & - 0.5 \alpha^{19}_{5} \mu - 0.5 \alpha^{7}_{5} \lambda - 0.25 \lambda \theta_{5} - 0.25 \mu \theta_{5} & 0.5 \alpha^{19}_{5} \mu + 0.5 \alpha^{7}_{5} \lambda + 1.0 \alpha^{7}_{5} \mu + 0.25 \lambda \theta_{5} + 0.75 \mu \theta_{5} & 0 & 0 & 0 & 0 & \mu \left(0.5 \alpha^{19}_{5} + 0.25 \theta_{5}\right) & - 0.5 \alpha^{7}_{5} \lambda - 1.0 \alpha^{7}_{5} \mu - 0.25 \lambda \theta_{5} - 0.5 \mu \theta_{5} & 0 & 0 & \frac{\alpha^{16}_{2} \theta_{5} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{6}_{2} \mu \theta_{5}}{2} + \frac{\alpha^{7}_{5} \lambda \theta_{2}}{2} & - \alpha^{17}_{3} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{7}_{3} \left(2 \alpha^{7}_{5} \left(\lambda + 3 \mu\right) - \mu \theta_{5}\right)}{2} - \frac{\alpha^{7}_{5} \mu \theta_{3}}{2} & - \frac{\alpha^{18}_{4} \lambda \theta_{5}}{2} - \frac{\alpha^{6}_{4} \mu \theta_{5}}{2} - \frac{\theta_{4} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \lambda\right)}{2} & \frac{\alpha^{19}_{5} \left(2 \alpha^{19}_{5} \left(\lambda + 3 \mu\right) - \theta_{5} \left(\lambda + 2 \mu\right)\right)}{2} + \frac{\alpha^{7}_{5} \left(2 \alpha^{7}_{5} \left(\lambda + 3 \mu\right) - \mu \theta_{5}\right)}{2} - \frac{\theta_{5} \left(\alpha^{19}_{5} \left(\lambda + 2 \mu\right) + \alpha^{7}_{5} \mu - \theta_{5} \left(\lambda + 3 \mu\right)\right)}{2} & 0 & 0 & 0 & 0 & \frac{\alpha^{16}_{10} \theta_{5} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{18}_{10} \lambda \theta_{5}}{2} + \frac{\alpha^{19}_{5} \mu \theta_{10}}{2} & - \alpha^{17}_{11} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{19}_{11} \left(2 \alpha^{19}_{5} \left(\lambda + 3 \mu\right) - \theta_{5} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{19}_{5} \theta_{11} \left(\lambda + 2 \mu\right)}{2}\\\mu \left(- 0.5 \alpha^{22}_{6} - 0.25 \theta_{6}\right) & \lambda \left(0.5 \alpha^{10}_{6} + 0.25 \theta_{6}\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{6} \lambda + 1.0 \alpha^{10}_{6} \mu + 0.5 \alpha^{22}_{6} \mu + 0.25 \lambda \theta_{6} + 0.75 \mu \theta_{6} & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu - 0.25 \lambda \theta_{6} - 0.25 \mu \theta_{6} & - 0.5 \alpha^{10}_{6} \lambda - 1.0 \alpha^{10}_{6} \mu - 0.25 \lambda \theta_{6} - 0.5 \mu \theta_{6} & \mu \left(0.5 \alpha^{22}_{6} + 0.25 \theta_{6}\right) & 0 & 0 & \frac{\alpha^{10}_{0} \left(2 \alpha^{10}_{6} \left(\lambda + 3 \mu\right) - \mu \theta_{6}\right)}{2} - \alpha^{12}_{0} \left(\alpha^{10}_{6} \left(\lambda + 2 \mu\right) + \alpha^{22}_{6} \mu\right) - \frac{\alpha^{10}_{6} \mu \theta_{0}}{2} & - \frac{\alpha^{11}_{1} \mu \theta_{6}}{2} + \frac{\alpha^{13}_{1} \theta_{6} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{10}_{6} \lambda \theta_{1}}{2} & 0 & 0 & 0 & 0 & \frac{\alpha^{10}_{6} \left(2 \alpha^{10}_{6} \left(\lambda + 3 \mu\right) - \mu \theta_{6}\right)}{2} + \frac{\alpha^{22}_{6} \left(2 \alpha^{22}_{6} \left(\lambda + 3 \mu\right) - \theta_{6} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{6} \left(\alpha^{10}_{6} \mu + \alpha^{22}_{6} \left(\lambda + 2 \mu\right) - \theta_{6} \left(\lambda + 3 \mu\right)\right)}{2} & - \frac{\alpha^{11}_{7} \mu \theta_{6}}{2} - \frac{\alpha^{23}_{7} \lambda \theta_{6}}{2} - \frac{\theta_{7} \left(\alpha^{10}_{6} \lambda + \alpha^{22}_{6} \mu\right)}{2} & - \frac{\alpha^{22}_{6} \theta_{8} \left(\lambda + 2 \mu\right)}{2} - \alpha^{12}_{8} \left(\alpha^{10}_{6} \left(\lambda + 2 \mu\right) + \alpha^{22}_{6} \mu\right) + \frac{\alpha^{22}_{8} \left(2 \alpha^{22}_{6} \left(\lambda + 3 \mu\right) - \theta_{6} \left(\lambda + 2 \mu\right)\right)}{2} & \frac{\alpha^{22}_{6} \mu \theta_{9}}{2} + \frac{\alpha^{13}_{9} \theta_{6} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{23}_{9} \lambda \theta_{6}}{2} & 0 & 0\\\mu \left(0.5 \alpha^{11}_{7} + 0.25 \theta_{7}\right) & - 0.5 \alpha^{23}_{7} \lambda - 1.0 \alpha^{23}_{7} \mu - 0.25 \lambda \theta_{7} - 0.5 \mu \theta_{7} & 0 & 0 & 0 & 0 & - 0.5 \alpha^{11}_{7} \mu - 0.5 \alpha^{23}_{7} \lambda - 0.25 \lambda \theta_{7} - 0.25 \mu \theta_{7} & 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{23}_{7} \lambda + 1.0 \alpha^{23}_{7} \mu + 0.25 \lambda \theta_{7} + 0.75 \mu \theta_{7} & \lambda \left(0.5 \alpha^{23}_{7} + 0.25 \theta_{7}\right) & \mu \left(- 0.5 \alpha^{11}_{7} - 0.25 \theta_{7}\right) & 0 & 0 & - \frac{\alpha^{10}_{0} \lambda \theta_{7}}{2} + \frac{\alpha^{12}_{0} \theta_{7} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{11}_{7} \mu \theta_{0}}{2} & \frac{\alpha^{11}_{1} \left(2 \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - \theta_{7} \left(\lambda + 2 \mu\right)\right)}{2} - \alpha^{13}_{1} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) - \frac{\alpha^{11}_{7} \theta_{1} \left(\lambda + 2 \mu\right)}{2} & 0 & 0 & 0 & 0 & - \frac{\alpha^{10}_{6} \lambda \theta_{7}}{2} - \frac{\alpha^{22}_{6} \mu \theta_{7}}{2} - \frac{\theta_{6} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \lambda\right)}{2} & \frac{\alpha^{11}_{7} \left(2 \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - \theta_{7} \left(\lambda + 2 \mu\right)\right)}{2} + \frac{\alpha^{23}_{7} \left(2 \alpha^{23}_{7} \left(\lambda + 3 \mu\right) - \mu \theta_{7}\right)}{2} - \frac{\theta_{7} \left(\alpha^{11}_{7} \left(\lambda + 2 \mu\right) + \alpha^{23}_{7} \mu - \theta_{7} \left(\lambda + 3 \mu\right)\right)}{2} & \frac{\alpha^{23}_{7} \lambda \theta_{8}}{2} + \frac{\alpha^{12}_{8} \theta_{7} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{22}_{8} \mu \theta_{7}}{2} & - \frac{\alpha^{23}_{7} \mu \theta_{9}}{2} - \alpha^{13}_{9} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{23}_{9} \left(2 \alpha^{23}_{7} \left(\lambda + 3 \mu\right) - \mu \theta_{7}\right)}{2} & 0 & 0\\0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu - 0.5 \alpha^{14}_{8} \lambda - 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{22}_{8} \mu & \left(- 0.5 \alpha^{12}_{8} - 0.25 \theta_{8}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{26}_{8} \mu - 0.5 \mu \theta_{8} & \left(0.5 \alpha^{12}_{8} + 0.25 \theta_{8}\right) \left(\lambda + \mu\right) & 0 & 0 & - 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{22}_{8} \mu - 0.25 \lambda \theta_{8} - 0.5 \mu \theta_{8} & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu + 0.25 \lambda \theta_{8} & 0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu + 0.5 \alpha^{26}_{8} \mu + 0.5 \lambda \theta_{8} + 1.5 \mu \theta_{8} & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu - 0.25 \theta_{8} \left(\lambda + \mu\right) & - 0.5 \alpha^{14}_{8} \lambda - 1.0 \alpha^{14}_{8} \mu - 0.25 \lambda \theta_{8} - 0.5 \mu \theta_{8} & \mu \left(0.5 \alpha^{26}_{8} + 0.25 \theta_{8}\right) & - \alpha^{10}_{0} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{2}_{0} \alpha^{12}_{8} \mu & \frac{\left(\lambda + \mu\right) \left(- \alpha^{13}_{1} \theta_{8} - \alpha^{12}_{8} \theta_{1}\right)}{2} & - \alpha^{14}_{2} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right) + \mu \theta_{8}\right) - \alpha^{16}_{2} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) - \alpha^{2}_{2} \alpha^{12}_{8} \mu - \alpha^{14}_{8} \mu \theta_{2} & \frac{\left(\lambda + \mu\right) \left(\alpha^{17}_{3} \theta_{8} + \alpha^{12}_{8} \theta_{3}\right)}{2} & 0 & 0 & - \alpha^{10}_{6} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) - \frac{\alpha^{22}_{6} \left(2 \alpha^{12}_{8} \mu - 2 \alpha^{22}_{8} \left(\lambda + 3 \mu\right) + \theta_{8} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{22}_{8} \theta_{6} \left(\lambda + 2 \mu\right)}{2} & \frac{\alpha^{23}_{7} \lambda \theta_{8}}{2} + \frac{\theta_{7} \left(\alpha^{12}_{8} \left(\lambda + \mu\right) - \alpha^{22}_{8} \mu\right)}{2} & - \alpha^{12}_{8} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{14}_{8} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right) + \mu \theta_{8}\right) - \frac{\alpha^{22}_{8} \left(2 \alpha^{12}_{8} \mu - 2 \alpha^{22}_{8} \left(\lambda + 3 \mu\right) + \theta_{8} \left(\lambda + 2 \mu\right)\right)}{2} + \frac{\alpha^{26}_{8} \left(2 \alpha^{26}_{8} \left(\lambda + 3 \mu\right) - \theta_{8} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{8} \left(2 \alpha^{14}_{8} \mu + \alpha^{22}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \left(\lambda + 2 \mu\right) - 2 \theta_{8} \left(\lambda + 3 \mu\right)\right)}{2} & - \frac{\alpha^{13}_{9} \theta_{8} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{23}_{9} \lambda \theta_{8}}{2} - \frac{\alpha^{27}_{9} \lambda \theta_{8}}{2} - \frac{\theta_{9} \left(\alpha^{12}_{8} \left(\lambda + \mu\right) - \alpha^{22}_{8} \mu + \alpha^{26}_{8} \mu\right)}{2} & - \alpha^{16}_{10} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) + \frac{\alpha^{26}_{10} \left(2 \alpha^{26}_{8} \left(\lambda + 3 \mu\right) - \theta_{8} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{26}_{8} \theta_{10} \left(\lambda + 2 \mu\right)}{2} & \frac{\alpha^{17}_{11} \theta_{8} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{27}_{11} \lambda \theta_{8}}{2} + \frac{\alpha^{26}_{8} \mu \theta_{11}}{2}\\\left(- 0.5 \alpha^{13}_{9} - 0.25 \theta_{9}\right) \left(\lambda + \mu\right) & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu - 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{23}_{9} \lambda - 1.0 \alpha^{23}_{9} \mu & \left(0.5 \alpha^{13}_{9} + 0.25 \theta_{9}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{27}_{9} \lambda - 1.0 \alpha^{27}_{9} \mu - 0.5 \lambda \theta_{9} - 1.0 \mu \theta_{9} & 0 & 0 & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) - 0.5 \alpha^{23}_{9} \lambda + 0.25 \mu \theta_{9} & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{23}_{9} \lambda + 1.0 \alpha^{23}_{9} \mu - 0.25 \mu \theta_{9} & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda - 0.25 \theta_{9} \left(\lambda + \mu\right) & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu + 0.5 \alpha^{27}_{9} \lambda + 1.0 \alpha^{27}_{9} \mu + 0.5 \lambda \theta_{9} + 1.5 \mu \theta_{9} & \lambda \left(0.5 \alpha^{27}_{9} + 0.25 \theta_{9}\right) & \mu \left(- 0.5 \alpha^{15}_{9} - 0.25 \theta_{9}\right) & \frac{\left(\lambda + \mu\right) \left(- \alpha^{12}_{0} \theta_{9} - \alpha^{13}_{9} \theta_{0}\right)}{2} & - \alpha^{11}_{1} \alpha^{13}_{9} \mu - \alpha^{13}_{1} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & \frac{\left(\lambda + \mu\right) \left(\alpha^{16}_{2} \theta_{9} + \alpha^{13}_{9} \theta_{2}\right)}{2} & - \alpha^{15}_{3} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right) + \theta_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{17}_{3} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{3} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{15}_{9} \theta_{3} \left(\lambda + 2 \mu\right) & 0 & 0 & \frac{\alpha^{22}_{6} \mu \theta_{9}}{2} + \frac{\theta_{6} \left(\alpha^{13}_{9} \left(\lambda + \mu\right) - \alpha^{23}_{9} \lambda\right)}{2} & - \alpha^{11}_{7} \alpha^{13}_{9} \mu - \frac{\alpha^{23}_{7} \left(2 \alpha^{13}_{9} \left(\lambda + 2 \mu\right) - 2 \alpha^{23}_{9} \left(\lambda + 3 \mu\right) + \mu \theta_{9}\right)}{2} - \frac{\alpha^{23}_{9} \mu \theta_{7}}{2} & - \frac{\alpha^{12}_{8} \theta_{9} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{22}_{8} \mu \theta_{9}}{2} - \frac{\alpha^{26}_{8} \mu \theta_{9}}{2} - \frac{\theta_{8} \left(\alpha^{13}_{9} \left(\lambda + \mu\right) - \alpha^{23}_{9} \lambda + \alpha^{27}_{9} \lambda\right)}{2} & - \alpha^{13}_{9} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{15}_{9} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right) + \theta_{9} \left(\lambda + 2 \mu\right)\right) - \frac{\alpha^{23}_{9} \left(2 \alpha^{13}_{9} \left(\lambda + 2 \mu\right) - 2 \alpha^{23}_{9} \left(\lambda + 3 \mu\right) + \mu \theta_{9}\right)}{2} + \frac{\alpha^{27}_{9} \left(2 \alpha^{27}_{9} \left(\lambda + 3 \mu\right) - \mu \theta_{9}\right)}{2} - \frac{\theta_{9} \left(2 \alpha^{15}_{9} \left(\lambda + 2 \mu\right) + \alpha^{23}_{9} \mu + \alpha^{27}_{9} \mu - 2 \theta_{9} \left(\lambda + 3 \mu\right)\right)}{2} & \frac{\alpha^{16}_{10} \theta_{9} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{26}_{10} \mu \theta_{9}}{2} + \frac{\alpha^{27}_{9} \lambda \theta_{10}}{2} & - \alpha^{17}_{11} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{27}_{11} \left(2 \alpha^{27}_{9} \left(\lambda + 3 \mu\right) - \mu \theta_{9}\right)}{2} - \frac{\alpha^{27}_{9} \mu \theta_{11}}{2}\\0 & 0 & 0.5 \alpha^{16}_{10} \lambda + 1.5 \alpha^{16}_{10} \mu - 0.5 \alpha^{18}_{10} \lambda - 1.0 \alpha^{18}_{10} \mu - 0.5 \alpha^{26}_{10} \mu & \left(- 0.5 \alpha^{16}_{10} - 0.25 \theta_{10}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} \lambda + 1.0 \alpha^{18}_{10} \mu - 0.25 \mu \theta_{10} & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{18}_{10} \lambda + 0.25 \mu \theta_{10} & 0 & 0 & - 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{26}_{10} \mu - 0.25 \lambda \theta_{10} - 0.5 \mu \theta_{10} & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu + 0.25 \lambda \theta_{10} & 0.5 \alpha^{16}_{10} \lambda + 1.5 \alpha^{16}_{10} \mu + 0.25 \lambda \theta_{10} + 0.75 \mu \theta_{10} & - 0.5 \alpha^{16}_{10} \lambda - 0.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{16}_{10} \alpha^{6}_{2} \mu - \alpha^{16}_{2} \left(- 2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10} \left(\lambda + 2 \mu\right) + \alpha^{26}_{10} \mu\right) & \frac{\left(\lambda + \mu\right) \left(- \alpha^{16}_{10} \theta_{3} - \alpha^{17}_{3} \theta_{10}\right)}{2} & - \alpha^{16}_{10} \alpha^{6}_{4} \mu - \frac{\alpha^{18}_{10} \mu \theta_{4}}{2} - \frac{\alpha^{18}_{4} \left(2 \alpha^{16}_{10} \left(\lambda + 2 \mu\right) - 2 \alpha^{18}_{10} \left(\lambda + 3 \mu\right) + \mu \theta_{10}\right)}{2} & \frac{\alpha^{19}_{5} \mu \theta_{10}}{2} + \frac{\theta_{5} \left(\alpha^{16}_{10} \left(\lambda + \mu\right) - \alpha^{18}_{10} \lambda\right)}{2} & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \frac{\alpha^{26}_{10} \theta_{8} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{26}_{8} \left(2 \alpha^{16}_{10} \mu - 2 \alpha^{26}_{10} \left(\lambda + 3 \mu\right) + \theta_{10} \left(\lambda + 2 \mu\right)\right)}{2} & \frac{\alpha^{27}_{9} \lambda \theta_{10}}{2} + \frac{\theta_{9} \left(\alpha^{16}_{10} \left(\lambda + \mu\right) - \alpha^{26}_{10} \mu\right)}{2} & - \alpha^{16}_{10} \left(- 2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10} \left(\lambda + 2 \mu\right) + \alpha^{26}_{10} \mu\right) - \frac{\alpha^{18}_{10} \left(2 \alpha^{16}_{10} \left(\lambda + 2 \mu\right) - 2 \alpha^{18}_{10} \left(\lambda + 3 \mu\right) + \mu \theta_{10}\right)}{2} - \frac{\alpha^{26}_{10} \left(2 \alpha^{16}_{10} \mu - 2 \alpha^{26}_{10} \left(\lambda + 3 \mu\right) + \theta_{10} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{10} \left(\alpha^{18}_{10} \mu + \alpha^{26}_{10} \left(\lambda + 2 \mu\right) - \theta_{10} \left(\lambda + 3 \mu\right)\right)}{2} & - \frac{\alpha^{17}_{11} \theta_{10} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{19}_{11} \mu \theta_{10}}{2} + \frac{\alpha^{27}_{11} \lambda \theta_{10}}{2} + \frac{\theta_{11} \left(- \alpha^{16}_{10} \left(\lambda + \mu\right) + \alpha^{18}_{10} \lambda + \alpha^{26}_{10} \mu\right)}{2}\\0 & 0 & \left(- 0.5 \alpha^{17}_{11} - 0.25 \theta_{11}\right) \left(\lambda + \mu\right) & 0.5 \alpha^{17}_{11} \lambda + 1.5 \alpha^{17}_{11} \mu - 0.5 \alpha^{19}_{11} \mu - 0.5 \alpha^{27}_{11} \lambda - 1.0 \alpha^{27}_{11} \mu & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{19}_{11} \mu + 0.25 \lambda \theta_{11} & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{19}_{11} \mu - 0.25 \lambda \theta_{11} - 0.5 \mu \theta_{11} & 0 & 0 & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda + 0.25 \mu \theta_{11} & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda + 1.0 \alpha^{27}_{11} \mu - 0.25 \mu \theta_{11} & - 0.5 \alpha^{17}_{11} \lambda - 0.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{19}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda & 0.5 \alpha^{17}_{11} \lambda + 1.5 \alpha^{17}_{11} \mu + 0.25 \lambda \theta_{11} + 0.75 \mu \theta_{11} & 0 & 0 & \frac{\left(\lambda + \mu\right) \left(- \alpha^{17}_{11} \theta_{2} - \alpha^{16}_{2} \theta_{11}\right)}{2} & - \alpha^{17}_{11} \alpha^{15}_{3} \mu - \alpha^{17}_{11} \alpha^{7}_{3} \left(\lambda + 2 \mu\right) - \alpha^{17}_{3} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) & \frac{\alpha^{18}_{4} \lambda \theta_{11}}{2} + \frac{\theta_{4} \left(\alpha^{17}_{11} \left(\lambda + \mu\right) - \alpha^{19}_{11} \mu\right)}{2} & - \alpha^{17}_{11} \alpha^{7}_{5} \left(\lambda + 2 \mu\right) - \frac{\alpha^{19}_{11} \theta_{5} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{19}_{5} \left(2 \alpha^{17}_{11} \mu - 2 \alpha^{19}_{11} \left(\lambda + 3 \mu\right) + \theta_{11} \left(\lambda + 2 \mu\right)\right)}{2} & 0 & 0 & \frac{\alpha^{26}_{8} \mu \theta_{11}}{2} + \frac{\theta_{8} \left(\alpha^{17}_{11} \left(\lambda + \mu\right) - \alpha^{27}_{11} \lambda\right)}{2} & - \alpha^{17}_{11} \alpha^{15}_{9} \mu - \frac{\alpha^{27}_{11} \mu \theta_{9}}{2} - \frac{\alpha^{27}_{9} \left(2 \alpha^{17}_{11} \left(\lambda + 2 \mu\right) - 2 \alpha^{27}_{11} \left(\lambda + 3 \mu\right) + \mu \theta_{11}\right)}{2} & - \frac{\alpha^{16}_{10} \theta_{11} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{18}_{10} \lambda \theta_{11}}{2} + \frac{\alpha^{26}_{10} \mu \theta_{11}}{2} + \frac{\theta_{10} \left(- \alpha^{17}_{11} \left(\lambda + \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \lambda\right)}{2} & - \alpha^{17}_{11} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) - \frac{\alpha^{19}_{11} \left(2 \alpha^{17}_{11} \mu - 2 \alpha^{19}_{11} \left(\lambda + 3 \mu\right) + \theta_{11} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{27}_{11} \left(2 \alpha^{17}_{11} \left(\lambda + 2 \mu\right) - 2 \alpha^{27}_{11} \left(\lambda + 3 \mu\right) + \mu \theta_{11}\right)}{2} - \frac{\theta_{11} \left(\alpha^{19}_{11} \left(\lambda + 2 \mu\right) + \alpha^{27}_{11} \mu - \theta_{11} \left(\lambda + 3 \mu\right)\right)}{2}\end{array}\right]\end{split}\]
it’s rank is to difficult to obtain symbolicaly so we try to get information by passing by numerical equivalent:
\[\displaystyle rank(PG^t.A.PG) = 24\]
With sympy rank method it is not rank deficient ?
Translating symbolic numerical values into numpy float, it is then possible to use matrix_rank function of numpy. We obtain :
\[\displaystyle rank(PG^t.A.PG) = \mathtt{\text{21}}\]
As the standard part naturaly exibite 3 rigides body modes and as deficiancy is of 3 we can conclude that the only rigide body modes to block are the one coresponding to standard part. Thus in our case we use Dirichlet boundary condition for that with the natural choice to impose the same condiditons at both scale.
Thus imposing Dirichlet at coarse scale lead to add equations of the from:
\(x_k+\theta_i\times x_i=c\)
with \((k,i)\in\{(0,12),(1,13),(4,16),(6,18),(10,22)\}\)
Which coresponds to the use of a prolongation matrix \(R\) and fixed value \(RF\) if we eliminate (k) standard dofs:
\[\begin{split}\displaystyle R = \left[\begin{array}{ccccccccccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & - \theta_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \theta_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \theta_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \theta_{6} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \theta_{10} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right] ~ RF= \left[\begin{matrix}c_{x}\\c_{y}\\0\\0\\i_{x}\\0\\c_{x}\\0\\0\\0\\i_{x}\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\end{matrix}\right]\end{split}\]
The coarse enriched matrix is in its general forme:
\[\begin{split}\displaystyle AG_C=R^t.PG^t.A.PG.R = \left[\begin{array}{ccccccccccccccccccc}1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0.5 \lambda & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & - 0.5 \lambda - 0.5 \mu & - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{0} \mu - \theta_{0} \left(- 0.5 \lambda - 1.0 \mu\right) - \theta_{0} \left(0.25 \lambda + 0.5 \mu\right) & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda - 0.25 \mu \theta_{1} & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{2} \mu + \theta_{2} \left(0.5 \lambda + 1.5 \mu\right) & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda - 0.25 \theta_{3} \left(\lambda + \mu\right) & - \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{4} \left(- 0.5 \lambda - 1.0 \mu\right) - \theta_{4} \left(0.25 \lambda + 0.5 \mu\right) & \lambda \left(0.5 \alpha^{7}_{5} + 0.25 \theta_{5}\right) & 0 & 0 & - \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{8} \mu - 0.5 \mu \theta_{8} & \left(0.5 \alpha^{13}_{9} + 0.25 \theta_{9}\right) \left(\lambda + \mu\right) & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{10} \mu & \left(- 0.5 \alpha^{17}_{11} - 0.25 \theta_{11}\right) \left(\lambda + \mu\right)\\- 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & - 0.5 \mu & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu - 0.25 \lambda \theta_{0} & - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) + 0.25 \mu \theta_{1} & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu - 0.25 \theta_{2} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) + \theta_{3} \left(0.5 \lambda + 1.5 \mu\right) & - 0.5 \mu \theta_{4} + \mu \left(0.5 \alpha^{6}_{4} + 0.25 \theta_{4}\right) & \mu \left(- 0.5 \alpha^{19}_{5} - 0.25 \theta_{5}\right) & 0 & 0 & \left(0.5 \alpha^{12}_{8} + 0.25 \theta_{8}\right) \left(\lambda + \mu\right) & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{9} \left(0.5 \lambda + 1.0 \mu\right) & - \theta_{10} \left(- 0.5 \lambda - 0.5 \mu\right) + \left(- 0.5 \alpha^{16}_{10} - 0.25 \theta_{10}\right) \left(\lambda + \mu\right) & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{19}_{11} \mu - \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0.5 \lambda & - 0.5 \mu & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu + 0.25 \lambda \theta_{2} & - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{3} & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu - \theta_{4} \left(- 0.5 \lambda - 0.5 \mu\right) - 0.25 \theta_{4} \left(\lambda + \mu\right) & 0.5 \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) + \theta_{5} \left(0.25 \lambda + 0.75 \mu\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{18}_{10} \lambda - 0.25 \mu \theta_{10} & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{19}_{11} \mu - \theta_{11} \left(0.25 \lambda + 0.5 \mu\right)\\0 & 0 & 0 & 0.5 \lambda + 1.5 \mu & 0.5 \lambda & - 0.5 \mu & 0 & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.25 \mu \theta_{0} & 0.5 \alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) - \theta_{1} \left(- 0.5 \lambda - 1.0 \mu\right) - \theta_{1} \left(0.25 \lambda + 0.5 \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu - \theta_{6} \left(- 0.5 \lambda - 0.5 \mu\right) - 0.25 \theta_{6} \left(\lambda + \mu\right) & 0.5 \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) + \theta_{7} \left(0.25 \lambda + 0.75 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu + 0.25 \lambda \theta_{8} & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{9} & 0 & 0\\- 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & 0.5 \lambda & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0.5 \mu & - \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{0} \mu & - \theta_{1} \left(- 0.5 \lambda - 0.5 \mu\right) + \left(- 0.5 \alpha^{13}_{1} - 0.25 \theta_{1}\right) \left(\lambda + \mu\right) & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{2} \mu - 0.5 \mu \theta_{2} & \left(0.5 \alpha^{17}_{3} + 0.25 \theta_{3}\right) \left(\lambda + \mu\right) & 0 & 0 & - \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{6} \left(- 0.5 \lambda - 1.0 \mu\right) - \theta_{6} \left(0.25 \lambda + 0.5 \mu\right) & \lambda \left(0.5 \alpha^{23}_{7} + 0.25 \theta_{7}\right) & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{26}_{8} \mu + \theta_{8} \left(0.5 \lambda + 1.5 \mu\right) & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda - 0.25 \theta_{9} \left(\lambda + \mu\right) & - \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{26}_{10} \mu - \theta_{10} \left(- 0.5 \lambda - 1.0 \mu\right) - \theta_{10} \left(0.25 \lambda + 0.5 \mu\right) & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda + 0.25 \mu \theta_{11}\\0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & - 0.5 \mu & - \theta_{0} \left(- 0.5 \lambda - 0.5 \mu\right) + \left(- 0.5 \alpha^{12}_{0} - 0.25 \theta_{0}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & \left(0.5 \alpha^{16}_{2} + 0.25 \theta_{2}\right) \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & - 0.5 \mu \theta_{6} + \mu \left(0.5 \alpha^{22}_{6} + 0.25 \theta_{6}\right) & \mu \left(- 0.5 \alpha^{11}_{7} - 0.25 \theta_{7}\right) & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu - 0.25 \theta_{8} \left(\lambda + \mu\right) & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) + \theta_{9} \left(0.5 \lambda + 1.5 \mu\right) & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu - 0.25 \lambda \theta_{10} & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{11}\\- 0.5 \lambda - 0.5 \mu & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0.5 \mu & - 0.5 \mu & 0.5 \lambda + 1.5 \mu & 0 & 0 & \left(- 0.5 \alpha^{16}_{2} - 0.25 \theta_{2}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \lambda \theta_{4} + \lambda \left(0.5 \alpha^{18}_{4} + 0.25 \theta_{4}\right) & - \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{5} \left(0.25 \lambda + 0.5 \mu\right) & 0 & 0 & \mu \left(0.5 \alpha^{26}_{8} + 0.25 \theta_{8}\right) & \mu \left(- 0.5 \alpha^{15}_{9} - 0.25 \theta_{9}\right) & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + \theta_{11} \left(0.25 \lambda + 0.75 \mu\right)\\- 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \mu - 0.25 \lambda \theta_{0} - 0.5 \mu \theta_{0} - \theta_{0} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu - 0.25 \lambda \theta_{0} & 0 & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.25 \mu \theta_{0} & - 0.5 \alpha^{10}_{0} \lambda - 1.0 \alpha^{10}_{0} \mu + 0.5 \alpha^{12}_{0} \lambda + 1.5 \alpha^{12}_{0} \mu - 0.5 \alpha^{2}_{0} \mu & - \theta_{0} \left(- 0.5 \lambda - 0.5 \mu\right) + \left(- 0.5 \alpha^{12}_{0} - 0.25 \theta_{0}\right) \left(\lambda + \mu\right) & 0 & - \frac{\alpha^{10}_{0} \left(- 2 \alpha^{10}_{0} \left(\lambda + 3 \mu\right) + 2 \alpha^{12}_{0} \left(\lambda + 2 \mu\right) + \mu \theta_{0}\right)}{2} - \alpha^{12}_{0} \left(\alpha^{10}_{0} \left(\lambda + 2 \mu\right) - 2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) + \alpha^{2}_{0} \mu\right) - \frac{\alpha^{2}_{0} \left(2 \alpha^{12}_{0} \mu - 2 \alpha^{2}_{0} \left(\lambda + 3 \mu\right) + \theta_{0} \left(\lambda + 2 \mu\right)\right)}{2} - \theta_{0} \left(\alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) + \theta_{0} \left(0.25 \lambda + 0.75 \mu\right)\right) - \frac{\theta_{0} \left(\alpha^{10}_{0} \mu + \alpha^{2}_{0} \left(\lambda + 2 \mu\right) - \theta_{0} \left(\lambda + 3 \mu\right)\right)}{2} - \theta_{0} \left(0.5 \alpha^{12}_{0} \lambda + 1.5 \alpha^{12}_{0} \mu + 0.25 \lambda \theta_{0} + 0.75 \mu \theta_{0} - \theta_{0} \left(0.5 \lambda + 1.5 \mu\right)\right) & \frac{\alpha^{11}_{1} \mu \theta_{0}}{2} - \frac{\alpha^{13}_{1} \theta_{0} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{3}_{1} \lambda \theta_{0}}{2} - \theta_{0} \left(0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) + 0.5 \alpha^{3}_{1} \lambda\right) + \frac{\theta_{1} \left(\alpha^{10}_{0} \lambda - \alpha^{12}_{0} \left(\lambda + \mu\right) + \alpha^{2}_{0} \mu\right)}{2} - \theta_{1} \left(0.5 \alpha^{10}_{0} \lambda - 0.5 \alpha^{12}_{0} \lambda - 0.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \mu\right) & - \alpha^{12}_{0} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \frac{\alpha^{2}_{0} \theta_{2} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{2}_{2} \left(2 \alpha^{12}_{0} \mu - 2 \alpha^{2}_{0} \left(\lambda + 3 \mu\right) + \theta_{0} \left(\lambda + 2 \mu\right)\right)}{2} - \theta_{0} \left(- \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{2} \left(0.25 \lambda + 0.5 \mu\right)\right) & \frac{\alpha^{3}_{3} \lambda \theta_{0}}{2} - \lambda \theta_{0} \left(0.5 \alpha^{3}_{3} + 0.25 \theta_{3}\right) + \frac{\theta_{3} \left(\alpha^{12}_{0} \left(\lambda + \mu\right) - \alpha^{2}_{0} \mu\right)}{2} & 0 & 0 & - \frac{\alpha^{10}_{0} \mu \theta_{6}}{2} - \alpha^{12}_{0} \alpha^{22}_{6} \mu - \frac{\alpha^{10}_{6} \left(- 2 \alpha^{10}_{0} \left(\lambda + 3 \mu\right) + 2 \alpha^{12}_{0} \left(\lambda + 2 \mu\right) + \mu \theta_{0}\right)}{2} - \mu \theta_{0} \left(- 0.5 \alpha^{22}_{6} - 0.25 \theta_{6}\right) - \theta_{6} \left(0.5 \alpha^{10}_{0} \lambda + 1.0 \alpha^{10}_{0} \mu - 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu + 0.25 \mu \theta_{0}\right) & \frac{\alpha^{11}_{7} \mu \theta_{0}}{2} - \mu \theta_{0} \left(0.5 \alpha^{11}_{7} + 0.25 \theta_{7}\right) - \frac{\theta_{7} \left(\alpha^{10}_{0} \lambda - \alpha^{12}_{0} \left(\lambda + \mu\right)\right)}{2} & - \alpha^{12}_{0} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \alpha^{22}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{10}_{0} \left(\lambda + 2 \mu\right) - 2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) + \alpha^{2}_{0} \mu\right) - \theta_{0} \left(\alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{22}_{8} \mu\right) & - \theta_{0} \left(- 0.5 \alpha^{13}_{9} - 0.25 \theta_{9}\right) \left(\lambda + \mu\right) + \frac{\left(\lambda + \mu\right) \left(- \alpha^{12}_{0} \theta_{9} - \alpha^{13}_{9} \theta_{0}\right)}{2} & 0 & 0\\0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda - 0.25 \mu \theta_{1} & - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \lambda + 1.0 \alpha^{3}_{1} \mu + 0.25 \mu \theta_{1} & 0 & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu - 0.25 \lambda \theta_{1} - 0.5 \mu \theta_{1} - \theta_{1} \left(- 0.5 \lambda - 1.0 \mu\right) & - \theta_{1} \left(- 0.5 \lambda - 0.5 \mu\right) + \left(- 0.5 \alpha^{13}_{1} - 0.25 \theta_{1}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu - 0.5 \alpha^{3}_{1} \lambda - 1.0 \alpha^{3}_{1} \mu & 0 & \frac{\alpha^{10}_{0} \lambda \theta_{1}}{2} - \frac{\alpha^{12}_{0} \theta_{1} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{2}_{0} \mu \theta_{1}}{2} + \frac{\theta_{0} \left(\alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(\lambda + \mu\right) + \alpha^{3}_{1} \lambda\right)}{2} - \theta_{0} \left(0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \lambda - 0.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \lambda\right) - \theta_{1} \left(0.5 \alpha^{10}_{0} \lambda - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) + 0.5 \alpha^{2}_{0} \mu\right) & - \frac{\alpha^{11}_{1} \left(- 2 \alpha^{11}_{1} \left(\lambda + 3 \mu\right) + 2 \alpha^{13}_{1} \mu + \theta_{1} \left(\lambda + 2 \mu\right)\right)}{2} - \alpha^{13}_{1} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) - \frac{\alpha^{3}_{1} \left(2 \alpha^{13}_{1} \left(\lambda + 2 \mu\right) - 2 \alpha^{3}_{1} \left(\lambda + 3 \mu\right) + \mu \theta_{1}\right)}{2} - \theta_{1} \left(\alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \theta_{1} \left(0.25 \lambda + 0.75 \mu\right)\right) - \frac{\theta_{1} \left(\alpha^{11}_{1} \left(\lambda + 2 \mu\right) + \alpha^{3}_{1} \mu - \theta_{1} \left(\lambda + 3 \mu\right)\right)}{2} - \theta_{1} \left(0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu + 0.25 \lambda \theta_{1} + 0.75 \mu \theta_{1} - \theta_{1} \left(0.5 \lambda + 1.5 \mu\right)\right) & \frac{\alpha^{2}_{2} \mu \theta_{1}}{2} - \mu \theta_{1} \left(0.5 \alpha^{2}_{2} + 0.25 \theta_{2}\right) + \frac{\theta_{2} \left(\alpha^{13}_{1} \left(\lambda + \mu\right) - \alpha^{3}_{1} \lambda\right)}{2} & - \alpha^{13}_{1} \alpha^{15}_{3} \mu - \frac{\alpha^{3}_{1} \mu \theta_{3}}{2} - \frac{\alpha^{3}_{3} \left(2 \alpha^{13}_{1} \left(\lambda + 2 \mu\right) - 2 \alpha^{3}_{1} \left(\lambda + 3 \mu\right) + \mu \theta_{1}\right)}{2} - \mu \theta_{1} \left(- 0.5 \alpha^{15}_{3} - 0.25 \theta_{3}\right) & 0 & 0 & \frac{\alpha^{10}_{6} \lambda \theta_{1}}{2} - \lambda \theta_{1} \left(0.5 \alpha^{10}_{6} + 0.25 \theta_{6}\right) - \frac{\theta_{6} \left(\alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(\lambda + \mu\right)\right)}{2} - \theta_{6} \left(- 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.25 \lambda \theta_{1}\right) & - \frac{\alpha^{11}_{1} \theta_{7} \left(\lambda + 2 \mu\right)}{2} - \alpha^{13}_{1} \alpha^{23}_{7} \left(\lambda + 2 \mu\right) - \frac{\alpha^{11}_{7} \left(- 2 \alpha^{11}_{1} \left(\lambda + 3 \mu\right) + 2 \alpha^{13}_{1} \mu + \theta_{1} \left(\lambda + 2 \mu\right)\right)}{2} - \theta_{1} \left(- \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{7} \left(0.25 \lambda + 0.5 \mu\right)\right) & - \theta_{1} \left(- 0.5 \alpha^{12}_{8} - 0.25 \theta_{8}\right) \left(\lambda + \mu\right) + \frac{\left(\lambda + \mu\right) \left(- \alpha^{13}_{1} \theta_{8} - \alpha^{12}_{8} \theta_{1}\right)}{2} & - \alpha^{13}_{1} \alpha^{15}_{9} \mu - \alpha^{13}_{1} \alpha^{23}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) - \theta_{1} \left(\alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{15}_{9} \mu - \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right)\right) & 0 & 0\\0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{2}_{2} \mu + 0.5 \lambda \theta_{2} + 1.5 \mu \theta_{2} & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu - 0.25 \theta_{2} \left(\lambda + \mu\right) & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu + 0.25 \lambda \theta_{2} & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu - 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{2}_{2} \mu - 0.5 \mu \theta_{2} & \left(0.5 \alpha^{16}_{2} + 0.25 \theta_{2}\right) \left(\lambda + \mu\right) & \left(- 0.5 \alpha^{16}_{2} - 0.25 \theta_{2}\right) \left(\lambda + \mu\right) & - \alpha^{12}_{0} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \frac{\alpha^{2}_{0} \left(2 \alpha^{2}_{2} \left(\lambda + 3 \mu\right) - \theta_{2} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{2}_{2} \theta_{0} \left(\lambda + 2 \mu\right)}{2} - \theta_{0} \left(- 0.5 \alpha^{14}_{2} \lambda - 1.0 \alpha^{14}_{2} \mu - 0.25 \lambda \theta_{2} - 0.5 \mu \theta_{2}\right) & \frac{\alpha^{13}_{1} \theta_{2} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{3}_{1} \lambda \theta_{2}}{2} + \frac{\alpha^{2}_{2} \mu \theta_{1}}{2} - \mu \theta_{1} \left(0.5 \alpha^{2}_{2} + 0.25 \theta_{2}\right) & - \alpha^{14}_{2} \left(- 2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) + \alpha^{16}_{2} \left(\lambda + 2 \mu\right) + \mu \theta_{2}\right) - \alpha^{16}_{2} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) + \frac{\alpha^{2}_{2} \left(2 \alpha^{2}_{2} \left(\lambda + 3 \mu\right) - \theta_{2} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{6}_{2} \left(2 \alpha^{16}_{2} \mu - 2 \alpha^{6}_{2} \left(\lambda + 3 \mu\right) + \theta_{2} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{2} \left(2 \alpha^{14}_{2} \mu + \alpha^{2}_{2} \left(\lambda + 2 \mu\right) + \alpha^{6}_{2} \left(\lambda + 2 \mu\right) - 2 \theta_{2} \left(\lambda + 3 \mu\right)\right)}{2} & - \frac{\alpha^{17}_{3} \theta_{2} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{3}_{3} \lambda \theta_{2}}{2} + \frac{\alpha^{7}_{3} \lambda \theta_{2}}{2} - \frac{\theta_{3} \left(\alpha^{16}_{2} \left(\lambda + \mu\right) + \alpha^{2}_{2} \mu - \alpha^{6}_{2} \mu\right)}{2} & - \alpha^{16}_{2} \alpha^{18}_{4} \left(\lambda + 2 \mu\right) - \frac{\alpha^{6}_{2} \theta_{4} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{6}_{4} \left(2 \alpha^{16}_{2} \mu - 2 \alpha^{6}_{2} \left(\lambda + 3 \mu\right) + \theta_{2} \left(\lambda + 2 \mu\right)\right)}{2} - \theta_{4} \left(- 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{6}_{2} \mu - 0.25 \lambda \theta_{2} - 0.5 \mu \theta_{2}\right) & \frac{\alpha^{7}_{5} \lambda \theta_{2}}{2} + \frac{\theta_{5} \left(\alpha^{16}_{2} \left(\lambda + \mu\right) - \alpha^{6}_{2} \mu\right)}{2} & 0 & 0 & - \alpha^{14}_{2} \mu \theta_{8} - \alpha^{16}_{2} \alpha^{26}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) - \alpha^{14}_{8} \left(- 2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) + \alpha^{16}_{2} \left(\lambda + 2 \mu\right) + \mu \theta_{2}\right) & \frac{\left(\lambda + \mu\right) \left(\alpha^{16}_{2} \theta_{9} + \alpha^{13}_{9} \theta_{2}\right)}{2} & - \alpha^{16}_{10} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) - \alpha^{18}_{10} \alpha^{16}_{2} \left(\lambda + 2 \mu\right) - \alpha^{26}_{10} \alpha^{16}_{2} \mu - \theta_{10} \left(- 0.5 \alpha^{14}_{2} \lambda - 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{6}_{2} \mu\right) & \frac{\left(\lambda + \mu\right) \left(- \alpha^{17}_{11} \theta_{2} - \alpha^{16}_{2} \theta_{11}\right)}{2}\\- 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda - 0.25 \theta_{3} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{3}_{3} \lambda + 1.0 \alpha^{3}_{3} \mu + 0.5 \lambda \theta_{3} + 1.5 \mu \theta_{3} & - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{7}_{3} \lambda + 1.0 \alpha^{7}_{3} \mu - 0.25 \mu \theta_{3} & 0 & \left(0.5 \alpha^{17}_{3} + 0.25 \theta_{3}\right) \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{3}_{3} \lambda - 1.0 \alpha^{3}_{3} \mu - 0.5 \lambda \theta_{3} - 1.0 \mu \theta_{3} & - 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{7}_{3} \lambda - 1.0 \alpha^{7}_{3} \mu & \frac{\alpha^{12}_{0} \theta_{3} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{2}_{0} \mu \theta_{3}}{2} + \frac{\alpha^{3}_{3} \lambda \theta_{0}}{2} - \lambda \theta_{0} \left(0.5 \alpha^{3}_{3} + 0.25 \theta_{3}\right) & - \alpha^{13}_{1} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{3}_{1} \left(2 \alpha^{3}_{3} \left(\lambda + 3 \mu\right) - \mu \theta_{3}\right)}{2} - \frac{\alpha^{3}_{3} \mu \theta_{1}}{2} - \mu \theta_{1} \left(- 0.5 \alpha^{15}_{3} - 0.25 \theta_{3}\right) & - \frac{\alpha^{16}_{2} \theta_{3} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{2}_{2} \mu \theta_{3}}{2} + \frac{\alpha^{6}_{2} \mu \theta_{3}}{2} - \frac{\theta_{2} \left(\alpha^{17}_{3} \left(\lambda + \mu\right) + \alpha^{3}_{3} \lambda - \alpha^{7}_{3} \lambda\right)}{2} & - \alpha^{15}_{3} \left(- 2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) + \alpha^{17}_{3} \mu + \theta_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{17}_{3} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{3}_{3} \left(2 \alpha^{3}_{3} \left(\lambda + 3 \mu\right) - \mu \theta_{3}\right)}{2} - \frac{\alpha^{7}_{3} \left(2 \alpha^{17}_{3} \left(\lambda + 2 \mu\right) - 2 \alpha^{7}_{3} \left(\lambda + 3 \mu\right) + \mu \theta_{3}\right)}{2} - \frac{\theta_{3} \left(2 \alpha^{15}_{3} \left(\lambda + 2 \mu\right) + \alpha^{3}_{3} \mu + \alpha^{7}_{3} \mu - 2 \theta_{3} \left(\lambda + 3 \mu\right)\right)}{2} & \frac{\alpha^{6}_{4} \mu \theta_{3}}{2} + \frac{\theta_{4} \left(\alpha^{17}_{3} \left(\lambda + \mu\right) - \alpha^{7}_{3} \lambda\right)}{2} - \theta_{4} \left(0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{7}_{3} \lambda + 0.25 \mu \theta_{3}\right) & - \alpha^{17}_{3} \alpha^{19}_{5} \mu - \frac{\alpha^{7}_{3} \mu \theta_{5}}{2} - \frac{\alpha^{7}_{5} \left(2 \alpha^{17}_{3} \left(\lambda + 2 \mu\right) - 2 \alpha^{7}_{3} \left(\lambda + 3 \mu\right) + \mu \theta_{3}\right)}{2} & 0 & 0 & \frac{\left(\lambda + \mu\right) \left(\alpha^{17}_{3} \theta_{8} + \alpha^{12}_{8} \theta_{3}\right)}{2} & - \alpha^{15}_{3} \theta_{9} \left(\lambda + 2 \mu\right) - \alpha^{17}_{3} \alpha^{27}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{15}_{9} \left(- 2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) + \alpha^{17}_{3} \mu + \theta_{3} \left(\lambda + 2 \mu\right)\right) & - \theta_{10} \left(- 0.5 \alpha^{17}_{3} - 0.25 \theta_{3}\right) \left(\lambda + \mu\right) + \frac{\left(\lambda + \mu\right) \left(- \alpha^{16}_{10} \theta_{3} - \alpha^{17}_{3} \theta_{10}\right)}{2} & - \alpha^{17}_{11} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \alpha^{17}_{3} \mu - \alpha^{27}_{11} \alpha^{17}_{3} \left(\lambda + 2 \mu\right)\\- 0.5 \alpha^{18}_{4} \lambda - 1.0 \alpha^{18}_{4} \mu - 0.25 \lambda \theta_{4} - 0.5 \mu \theta_{4} - \theta_{4} \left(- 0.5 \lambda - 1.0 \mu\right) & - 0.5 \mu \theta_{4} + \mu \left(0.5 \alpha^{6}_{4} + 0.25 \theta_{4}\right) & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu - 0.25 \lambda \theta_{4} - 0.25 \mu \theta_{4} - \theta_{4} \left(- 0.5 \lambda - 0.5 \mu\right) & 0 & 0 & 0 & - 0.5 \lambda \theta_{4} + \lambda \left(0.5 \alpha^{18}_{4} + 0.25 \theta_{4}\right) & 0 & 0 & - \alpha^{16}_{2} \left(\alpha^{18}_{4} \left(\lambda + 2 \mu\right) + \alpha^{6}_{4} \mu\right) + \frac{\alpha^{6}_{2} \left(2 \alpha^{6}_{4} \left(\lambda + 3 \mu\right) - \theta_{4} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{6}_{4} \theta_{2} \left(\lambda + 2 \mu\right)}{2} - \theta_{4} \left(- \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{6}_{2} \mu - \theta_{2} \left(0.25 \lambda + 0.5 \mu\right)\right) & \frac{\alpha^{17}_{3} \theta_{4} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{7}_{3} \lambda \theta_{4}}{2} + \frac{\alpha^{6}_{4} \mu \theta_{3}}{2} - \theta_{4} \left(0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{7}_{3} \lambda + 0.25 \mu \theta_{3}\right) & \frac{\alpha^{18}_{4} \left(2 \alpha^{18}_{4} \left(\lambda + 3 \mu\right) - \mu \theta_{4}\right)}{2} + \frac{\alpha^{6}_{4} \left(2 \alpha^{6}_{4} \left(\lambda + 3 \mu\right) - \theta_{4} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{4} \left(\alpha^{18}_{4} \mu + \alpha^{6}_{4} \left(\lambda + 2 \mu\right) - \theta_{4} \left(\lambda + 3 \mu\right)\right)}{2} - \theta_{4} \left(\alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{6}_{4} \mu + \theta_{4} \left(0.25 \lambda + 0.75 \mu\right)\right) - \theta_{4} \left(0.5 \alpha^{18}_{4} \lambda + 1.0 \alpha^{18}_{4} \mu + 0.5 \alpha^{6}_{4} \mu + 0.25 \lambda \theta_{4} + 0.75 \mu \theta_{4} - \theta_{4} \left(0.5 \lambda + 1.5 \mu\right)\right) & - \frac{\alpha^{19}_{5} \mu \theta_{4}}{2} - \frac{\alpha^{7}_{5} \lambda \theta_{4}}{2} - \theta_{4} \left(- 0.5 \alpha^{19}_{5} \mu - 0.5 \alpha^{7}_{5} \lambda - 0.25 \theta_{5} \left(\lambda + \mu\right)\right) - \frac{\theta_{5} \left(\alpha^{18}_{4} \lambda + \alpha^{6}_{4} \mu\right)}{2} & 0 & 0 & 0 & 0 & - \alpha^{16}_{10} \left(\alpha^{18}_{4} \left(\lambda + 2 \mu\right) + \alpha^{6}_{4} \mu\right) + \frac{\alpha^{18}_{10} \left(2 \alpha^{18}_{4} \left(\lambda + 3 \mu\right) - \mu \theta_{4}\right)}{2} - \frac{\alpha^{18}_{4} \mu \theta_{10}}{2} - \theta_{10} \left(0.5 \mu \theta_{4} + \mu \left(- 0.5 \alpha^{6}_{4} - 0.25 \theta_{4}\right)\right) - \theta_{4} \left(- \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) - 0.25 \mu \theta_{10}\right) & \frac{\alpha^{17}_{11} \theta_{4} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{19}_{11} \mu \theta_{4}}{2} + \frac{\alpha^{18}_{4} \lambda \theta_{11}}{2} - \theta_{4} \left(0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{19}_{11} \mu + 0.25 \lambda \theta_{11}\right)\\\lambda \left(0.5 \alpha^{7}_{5} + 0.25 \theta_{5}\right) & \mu \left(- 0.5 \alpha^{19}_{5} - 0.25 \theta_{5}\right) & 0.5 \alpha^{19}_{5} \mu + 0.5 \alpha^{7}_{5} \lambda + 1.0 \alpha^{7}_{5} \mu + 0.25 \lambda \theta_{5} + 0.75 \mu \theta_{5} & 0 & 0 & 0 & - 0.5 \alpha^{7}_{5} \lambda - 1.0 \alpha^{7}_{5} \mu - 0.25 \lambda \theta_{5} - 0.5 \mu \theta_{5} & 0 & 0 & \frac{\alpha^{16}_{2} \theta_{5} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{6}_{2} \mu \theta_{5}}{2} + \frac{\alpha^{7}_{5} \lambda \theta_{2}}{2} & - \alpha^{17}_{3} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{7}_{3} \left(2 \alpha^{7}_{5} \left(\lambda + 3 \mu\right) - \mu \theta_{5}\right)}{2} - \frac{\alpha^{7}_{5} \mu \theta_{3}}{2} & - \frac{\alpha^{18}_{4} \lambda \theta_{5}}{2} - \frac{\alpha^{6}_{4} \mu \theta_{5}}{2} - \frac{\theta_{4} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \lambda\right)}{2} - \theta_{4} \left(- 0.5 \alpha^{19}_{5} \mu - 0.5 \alpha^{7}_{5} \lambda - 0.25 \lambda \theta_{5} - 0.25 \mu \theta_{5}\right) & \frac{\alpha^{19}_{5} \left(2 \alpha^{19}_{5} \left(\lambda + 3 \mu\right) - \theta_{5} \left(\lambda + 2 \mu\right)\right)}{2} + \frac{\alpha^{7}_{5} \left(2 \alpha^{7}_{5} \left(\lambda + 3 \mu\right) - \mu \theta_{5}\right)}{2} - \frac{\theta_{5} \left(\alpha^{19}_{5} \left(\lambda + 2 \mu\right) + \alpha^{7}_{5} \mu - \theta_{5} \left(\lambda + 3 \mu\right)\right)}{2} & 0 & 0 & 0 & 0 & \frac{\alpha^{16}_{10} \theta_{5} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{18}_{10} \lambda \theta_{5}}{2} + \frac{\alpha^{19}_{5} \mu \theta_{10}}{2} - \mu \theta_{10} \left(0.5 \alpha^{19}_{5} + 0.25 \theta_{5}\right) & - \alpha^{17}_{11} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{19}_{11} \left(2 \alpha^{19}_{5} \left(\lambda + 3 \mu\right) - \theta_{5} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{19}_{5} \theta_{11} \left(\lambda + 2 \mu\right)}{2}\\0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu - 0.25 \lambda \theta_{6} - 0.25 \mu \theta_{6} - \theta_{6} \left(- 0.5 \lambda - 0.5 \mu\right) & - 0.5 \alpha^{10}_{6} \lambda - 1.0 \alpha^{10}_{6} \mu - 0.25 \lambda \theta_{6} - 0.5 \mu \theta_{6} - \theta_{6} \left(- 0.5 \lambda - 1.0 \mu\right) & - 0.5 \mu \theta_{6} + \mu \left(0.5 \alpha^{22}_{6} + 0.25 \theta_{6}\right) & 0 & \frac{\alpha^{10}_{0} \left(2 \alpha^{10}_{6} \left(\lambda + 3 \mu\right) - \mu \theta_{6}\right)}{2} - \alpha^{12}_{0} \left(\alpha^{10}_{6} \left(\lambda + 2 \mu\right) + \alpha^{22}_{6} \mu\right) - \frac{\alpha^{10}_{6} \mu \theta_{0}}{2} - \theta_{0} \left(0.5 \mu \theta_{6} + \mu \left(- 0.5 \alpha^{22}_{6} - 0.25 \theta_{6}\right)\right) - \theta_{6} \left(\alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.25 \mu \theta_{0}\right) & - \frac{\alpha^{11}_{1} \mu \theta_{6}}{2} + \frac{\alpha^{13}_{1} \theta_{6} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{10}_{6} \lambda \theta_{1}}{2} - \theta_{1} \left(- 0.5 \lambda \theta_{6} + \lambda \left(0.5 \alpha^{10}_{6} + 0.25 \theta_{6}\right)\right) - \theta_{6} \left(- 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) + 0.25 \lambda \theta_{1}\right) & 0 & 0 & 0 & 0 & \frac{\alpha^{10}_{6} \left(2 \alpha^{10}_{6} \left(\lambda + 3 \mu\right) - \mu \theta_{6}\right)}{2} + \frac{\alpha^{22}_{6} \left(2 \alpha^{22}_{6} \left(\lambda + 3 \mu\right) - \theta_{6} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{6} \left(\alpha^{10}_{6} \mu + \alpha^{22}_{6} \left(\lambda + 2 \mu\right) - \theta_{6} \left(\lambda + 3 \mu\right)\right)}{2} - \theta_{6} \left(\alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{22}_{6} \mu + \theta_{6} \left(0.25 \lambda + 0.75 \mu\right)\right) - \theta_{6} \left(0.5 \alpha^{10}_{6} \lambda + 1.0 \alpha^{10}_{6} \mu + 0.5 \alpha^{22}_{6} \mu + 0.25 \lambda \theta_{6} + 0.75 \mu \theta_{6} - \theta_{6} \left(0.5 \lambda + 1.5 \mu\right)\right) & - \frac{\alpha^{11}_{7} \mu \theta_{6}}{2} - \frac{\alpha^{23}_{7} \lambda \theta_{6}}{2} - \theta_{6} \left(- 0.5 \alpha^{11}_{7} \mu - 0.5 \alpha^{23}_{7} \lambda - 0.25 \theta_{7} \left(\lambda + \mu\right)\right) - \frac{\theta_{7} \left(\alpha^{10}_{6} \lambda + \alpha^{22}_{6} \mu\right)}{2} & - \frac{\alpha^{22}_{6} \theta_{8} \left(\lambda + 2 \mu\right)}{2} - \alpha^{12}_{8} \left(\alpha^{10}_{6} \left(\lambda + 2 \mu\right) + \alpha^{22}_{6} \mu\right) + \frac{\alpha^{22}_{8} \left(2 \alpha^{22}_{6} \left(\lambda + 3 \mu\right) - \theta_{6} \left(\lambda + 2 \mu\right)\right)}{2} - \theta_{6} \left(- \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{22}_{8} \mu - \theta_{8} \left(0.25 \lambda + 0.5 \mu\right)\right) & \frac{\alpha^{22}_{6} \mu \theta_{9}}{2} + \frac{\alpha^{13}_{9} \theta_{6} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{23}_{9} \lambda \theta_{6}}{2} - \theta_{6} \left(0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) - 0.5 \alpha^{23}_{9} \lambda + 0.25 \mu \theta_{9}\right) & 0 & 0\\0 & 0 & 0 & 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{23}_{7} \lambda + 1.0 \alpha^{23}_{7} \mu + 0.25 \lambda \theta_{7} + 0.75 \mu \theta_{7} & \lambda \left(0.5 \alpha^{23}_{7} + 0.25 \theta_{7}\right) & \mu \left(- 0.5 \alpha^{11}_{7} - 0.25 \theta_{7}\right) & 0 & - \frac{\alpha^{10}_{0} \lambda \theta_{7}}{2} + \frac{\alpha^{12}_{0} \theta_{7} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{11}_{7} \mu \theta_{0}}{2} - \mu \theta_{0} \left(0.5 \alpha^{11}_{7} + 0.25 \theta_{7}\right) & \frac{\alpha^{11}_{1} \left(2 \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - \theta_{7} \left(\lambda + 2 \mu\right)\right)}{2} - \alpha^{13}_{1} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) - \frac{\alpha^{11}_{7} \theta_{1} \left(\lambda + 2 \mu\right)}{2} - \theta_{1} \left(- 0.5 \alpha^{23}_{7} \lambda - 1.0 \alpha^{23}_{7} \mu - 0.25 \lambda \theta_{7} - 0.5 \mu \theta_{7}\right) & 0 & 0 & 0 & 0 & - \frac{\alpha^{10}_{6} \lambda \theta_{7}}{2} - \frac{\alpha^{22}_{6} \mu \theta_{7}}{2} - \frac{\theta_{6} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \lambda\right)}{2} - \theta_{6} \left(- 0.5 \alpha^{11}_{7} \mu - 0.5 \alpha^{23}_{7} \lambda - 0.25 \lambda \theta_{7} - 0.25 \mu \theta_{7}\right) & \frac{\alpha^{11}_{7} \left(2 \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - \theta_{7} \left(\lambda + 2 \mu\right)\right)}{2} + \frac{\alpha^{23}_{7} \left(2 \alpha^{23}_{7} \left(\lambda + 3 \mu\right) - \mu \theta_{7}\right)}{2} - \frac{\theta_{7} \left(\alpha^{11}_{7} \left(\lambda + 2 \mu\right) + \alpha^{23}_{7} \mu - \theta_{7} \left(\lambda + 3 \mu\right)\right)}{2} & \frac{\alpha^{23}_{7} \lambda \theta_{8}}{2} + \frac{\alpha^{12}_{8} \theta_{7} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{22}_{8} \mu \theta_{7}}{2} & - \frac{\alpha^{23}_{7} \mu \theta_{9}}{2} - \alpha^{13}_{9} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{23}_{9} \left(2 \alpha^{23}_{7} \left(\lambda + 3 \mu\right) - \mu \theta_{7}\right)}{2} & 0 & 0\\- 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{26}_{8} \mu - 0.5 \mu \theta_{8} & \left(0.5 \alpha^{12}_{8} + 0.25 \theta_{8}\right) \left(\lambda + \mu\right) & 0 & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu + 0.25 \lambda \theta_{8} & 0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu + 0.5 \alpha^{26}_{8} \mu + 0.5 \lambda \theta_{8} + 1.5 \mu \theta_{8} & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu - 0.25 \theta_{8} \left(\lambda + \mu\right) & \mu \left(0.5 \alpha^{26}_{8} + 0.25 \theta_{8}\right) & - \alpha^{10}_{0} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{2}_{0} \alpha^{12}_{8} \mu - \theta_{0} \left(0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu - 0.5 \alpha^{14}_{8} \lambda - 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{22}_{8} \mu\right) & - \theta_{1} \left(- 0.5 \alpha^{12}_{8} - 0.25 \theta_{8}\right) \left(\lambda + \mu\right) + \frac{\left(\lambda + \mu\right) \left(- \alpha^{13}_{1} \theta_{8} - \alpha^{12}_{8} \theta_{1}\right)}{2} & - \alpha^{14}_{2} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right) + \mu \theta_{8}\right) - \alpha^{16}_{2} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) - \alpha^{2}_{2} \alpha^{12}_{8} \mu - \alpha^{14}_{8} \mu \theta_{2} & \frac{\left(\lambda + \mu\right) \left(\alpha^{17}_{3} \theta_{8} + \alpha^{12}_{8} \theta_{3}\right)}{2} & 0 & 0 & - \alpha^{10}_{6} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) - \frac{\alpha^{22}_{6} \left(2 \alpha^{12}_{8} \mu - 2 \alpha^{22}_{8} \left(\lambda + 3 \mu\right) + \theta_{8} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{22}_{8} \theta_{6} \left(\lambda + 2 \mu\right)}{2} - \theta_{6} \left(- 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{22}_{8} \mu - 0.25 \lambda \theta_{8} - 0.5 \mu \theta_{8}\right) & \frac{\alpha^{23}_{7} \lambda \theta_{8}}{2} + \frac{\theta_{7} \left(\alpha^{12}_{8} \left(\lambda + \mu\right) - \alpha^{22}_{8} \mu\right)}{2} & - \alpha^{12}_{8} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{14}_{8} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right) + \mu \theta_{8}\right) - \frac{\alpha^{22}_{8} \left(2 \alpha^{12}_{8} \mu - 2 \alpha^{22}_{8} \left(\lambda + 3 \mu\right) + \theta_{8} \left(\lambda + 2 \mu\right)\right)}{2} + \frac{\alpha^{26}_{8} \left(2 \alpha^{26}_{8} \left(\lambda + 3 \mu\right) - \theta_{8} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\theta_{8} \left(2 \alpha^{14}_{8} \mu + \alpha^{22}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \left(\lambda + 2 \mu\right) - 2 \theta_{8} \left(\lambda + 3 \mu\right)\right)}{2} & - \frac{\alpha^{13}_{9} \theta_{8} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{23}_{9} \lambda \theta_{8}}{2} - \frac{\alpha^{27}_{9} \lambda \theta_{8}}{2} - \frac{\theta_{9} \left(\alpha^{12}_{8} \left(\lambda + \mu\right) - \alpha^{22}_{8} \mu + \alpha^{26}_{8} \mu\right)}{2} & - \alpha^{16}_{10} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) + \frac{\alpha^{26}_{10} \left(2 \alpha^{26}_{8} \left(\lambda + 3 \mu\right) - \theta_{8} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{26}_{8} \theta_{10} \left(\lambda + 2 \mu\right)}{2} - \theta_{10} \left(- 0.5 \alpha^{14}_{8} \lambda - 1.0 \alpha^{14}_{8} \mu - 0.25 \lambda \theta_{8} - 0.5 \mu \theta_{8}\right) & \frac{\alpha^{17}_{11} \theta_{8} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{27}_{11} \lambda \theta_{8}}{2} + \frac{\alpha^{26}_{8} \mu \theta_{11}}{2}\\\left(0.5 \alpha^{13}_{9} + 0.25 \theta_{9}\right) \left(\lambda + \mu\right) & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{27}_{9} \lambda - 1.0 \alpha^{27}_{9} \mu - 0.5 \lambda \theta_{9} - 1.0 \mu \theta_{9} & 0 & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{23}_{9} \lambda + 1.0 \alpha^{23}_{9} \mu - 0.25 \mu \theta_{9} & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda - 0.25 \theta_{9} \left(\lambda + \mu\right) & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu + 0.5 \alpha^{27}_{9} \lambda + 1.0 \alpha^{27}_{9} \mu + 0.5 \lambda \theta_{9} + 1.5 \mu \theta_{9} & \mu \left(- 0.5 \alpha^{15}_{9} - 0.25 \theta_{9}\right) & - \theta_{0} \left(- 0.5 \alpha^{13}_{9} - 0.25 \theta_{9}\right) \left(\lambda + \mu\right) + \frac{\left(\lambda + \mu\right) \left(- \alpha^{12}_{0} \theta_{9} - \alpha^{13}_{9} \theta_{0}\right)}{2} & - \alpha^{11}_{1} \alpha^{13}_{9} \mu - \alpha^{13}_{1} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \theta_{1} \left(0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu - 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{23}_{9} \lambda - 1.0 \alpha^{23}_{9} \mu\right) & \frac{\left(\lambda + \mu\right) \left(\alpha^{16}_{2} \theta_{9} + \alpha^{13}_{9} \theta_{2}\right)}{2} & - \alpha^{15}_{3} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right) + \theta_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{17}_{3} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{3} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{15}_{9} \theta_{3} \left(\lambda + 2 \mu\right) & 0 & 0 & \frac{\alpha^{22}_{6} \mu \theta_{9}}{2} + \frac{\theta_{6} \left(\alpha^{13}_{9} \left(\lambda + \mu\right) - \alpha^{23}_{9} \lambda\right)}{2} - \theta_{6} \left(0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) - 0.5 \alpha^{23}_{9} \lambda + 0.25 \mu \theta_{9}\right) & - \alpha^{11}_{7} \alpha^{13}_{9} \mu - \frac{\alpha^{23}_{7} \left(2 \alpha^{13}_{9} \left(\lambda + 2 \mu\right) - 2 \alpha^{23}_{9} \left(\lambda + 3 \mu\right) + \mu \theta_{9}\right)}{2} - \frac{\alpha^{23}_{9} \mu \theta_{7}}{2} & - \frac{\alpha^{12}_{8} \theta_{9} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{22}_{8} \mu \theta_{9}}{2} - \frac{\alpha^{26}_{8} \mu \theta_{9}}{2} - \frac{\theta_{8} \left(\alpha^{13}_{9} \left(\lambda + \mu\right) - \alpha^{23}_{9} \lambda + \alpha^{27}_{9} \lambda\right)}{2} & - \alpha^{13}_{9} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{15}_{9} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right) + \theta_{9} \left(\lambda + 2 \mu\right)\right) - \frac{\alpha^{23}_{9} \left(2 \alpha^{13}_{9} \left(\lambda + 2 \mu\right) - 2 \alpha^{23}_{9} \left(\lambda + 3 \mu\right) + \mu \theta_{9}\right)}{2} + \frac{\alpha^{27}_{9} \left(2 \alpha^{27}_{9} \left(\lambda + 3 \mu\right) - \mu \theta_{9}\right)}{2} - \frac{\theta_{9} \left(2 \alpha^{15}_{9} \left(\lambda + 2 \mu\right) + \alpha^{23}_{9} \mu + \alpha^{27}_{9} \mu - 2 \theta_{9} \left(\lambda + 3 \mu\right)\right)}{2} & \frac{\alpha^{16}_{10} \theta_{9} \left(\lambda + \mu\right)}{2} - \frac{\alpha^{26}_{10} \mu \theta_{9}}{2} + \frac{\alpha^{27}_{9} \lambda \theta_{10}}{2} - \lambda \theta_{10} \left(0.5 \alpha^{27}_{9} + 0.25 \theta_{9}\right) & - \alpha^{17}_{11} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) + \frac{\alpha^{27}_{11} \left(2 \alpha^{27}_{9} \left(\lambda + 3 \mu\right) - \mu \theta_{9}\right)}{2} - \frac{\alpha^{27}_{9} \mu \theta_{11}}{2}\\0.5 \alpha^{16}_{10} \lambda + 1.5 \alpha^{16}_{10} \mu - 0.5 \alpha^{18}_{10} \lambda - 1.0 \alpha^{18}_{10} \mu - 0.5 \alpha^{26}_{10} \mu & - \theta_{10} \left(- 0.5 \lambda - 0.5 \mu\right) + \left(- 0.5 \alpha^{16}_{10} - 0.25 \theta_{10}\right) \left(\lambda + \mu\right) & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{18}_{10} \lambda - 0.25 \mu \theta_{10} & 0 & - 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{26}_{10} \mu - 0.25 \lambda \theta_{10} - 0.5 \mu \theta_{10} - \theta_{10} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu - 0.25 \lambda \theta_{10} & - 0.5 \alpha^{16}_{10} \lambda - 0.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{16}_{10} \alpha^{6}_{2} \mu - \alpha^{16}_{2} \left(- 2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10} \left(\lambda + 2 \mu\right) + \alpha^{26}_{10} \mu\right) - \theta_{10} \left(- \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{6}_{2} \mu\right) & - \theta_{10} \left(- 0.5 \alpha^{17}_{3} - 0.25 \theta_{3}\right) \left(\lambda + \mu\right) + \frac{\left(\lambda + \mu\right) \left(- \alpha^{16}_{10} \theta_{3} - \alpha^{17}_{3} \theta_{10}\right)}{2} & - \alpha^{16}_{10} \alpha^{6}_{4} \mu - \frac{\alpha^{18}_{10} \mu \theta_{4}}{2} - \frac{\alpha^{18}_{4} \left(2 \alpha^{16}_{10} \left(\lambda + 2 \mu\right) - 2 \alpha^{18}_{10} \left(\lambda + 3 \mu\right) + \mu \theta_{10}\right)}{2} - \mu \theta_{10} \left(- 0.5 \alpha^{6}_{4} - 0.25 \theta_{4}\right) - \theta_{4} \left(- 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} \lambda + 1.0 \alpha^{18}_{10} \mu + 0.25 \mu \theta_{10}\right) & \frac{\alpha^{19}_{5} \mu \theta_{10}}{2} - \mu \theta_{10} \left(0.5 \alpha^{19}_{5} + 0.25 \theta_{5}\right) + \frac{\theta_{5} \left(\alpha^{16}_{10} \left(\lambda + \mu\right) - \alpha^{18}_{10} \lambda\right)}{2} & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \frac{\alpha^{26}_{10} \theta_{8} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{26}_{8} \left(2 \alpha^{16}_{10} \mu - 2 \alpha^{26}_{10} \left(\lambda + 3 \mu\right) + \theta_{10} \left(\lambda + 2 \mu\right)\right)}{2} - \theta_{10} \left(- \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - \theta_{8} \left(0.25 \lambda + 0.5 \mu\right)\right) & \frac{\alpha^{27}_{9} \lambda \theta_{10}}{2} - \lambda \theta_{10} \left(0.5 \alpha^{27}_{9} + 0.25 \theta_{9}\right) + \frac{\theta_{9} \left(\alpha^{16}_{10} \left(\lambda + \mu\right) - \alpha^{26}_{10} \mu\right)}{2} & - \alpha^{16}_{10} \left(- 2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10} \left(\lambda + 2 \mu\right) + \alpha^{26}_{10} \mu\right) - \frac{\alpha^{18}_{10} \left(2 \alpha^{16}_{10} \left(\lambda + 2 \mu\right) - 2 \alpha^{18}_{10} \left(\lambda + 3 \mu\right) + \mu \theta_{10}\right)}{2} - \frac{\alpha^{26}_{10} \left(2 \alpha^{16}_{10} \mu - 2 \alpha^{26}_{10} \left(\lambda + 3 \mu\right) + \theta_{10} \left(\lambda + 2 \mu\right)\right)}{2} - \theta_{10} \left(\alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + \theta_{10} \left(0.25 \lambda + 0.75 \mu\right)\right) - \frac{\theta_{10} \left(\alpha^{18}_{10} \mu + \alpha^{26}_{10} \left(\lambda + 2 \mu\right) - \theta_{10} \left(\lambda + 3 \mu\right)\right)}{2} - \theta_{10} \left(0.5 \alpha^{16}_{10} \lambda + 1.5 \alpha^{16}_{10} \mu + 0.25 \lambda \theta_{10} + 0.75 \mu \theta_{10} - \theta_{10} \left(0.5 \lambda + 1.5 \mu\right)\right) & - \frac{\alpha^{17}_{11} \theta_{10} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{19}_{11} \mu \theta_{10}}{2} + \frac{\alpha^{27}_{11} \lambda \theta_{10}}{2} - \theta_{10} \left(- 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) + 0.5 \alpha^{19}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda\right) + \frac{\theta_{11} \left(- \alpha^{16}_{10} \left(\lambda + \mu\right) + \alpha^{18}_{10} \lambda + \alpha^{26}_{10} \mu\right)}{2}\\\left(- 0.5 \alpha^{17}_{11} - 0.25 \theta_{11}\right) \left(\lambda + \mu\right) & 0.5 \alpha^{17}_{11} \lambda + 1.5 \alpha^{17}_{11} \mu - 0.5 \alpha^{19}_{11} \mu - 0.5 \alpha^{27}_{11} \lambda - 1.0 \alpha^{27}_{11} \mu & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{19}_{11} \mu - 0.25 \lambda \theta_{11} - 0.5 \mu \theta_{11} & 0 & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda + 0.25 \mu \theta_{11} & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda + 1.0 \alpha^{27}_{11} \mu - 0.25 \mu \theta_{11} & 0.5 \alpha^{17}_{11} \lambda + 1.5 \alpha^{17}_{11} \mu + 0.25 \lambda \theta_{11} + 0.75 \mu \theta_{11} & 0 & 0 & \frac{\left(\lambda + \mu\right) \left(- \alpha^{17}_{11} \theta_{2} - \alpha^{16}_{2} \theta_{11}\right)}{2} & - \alpha^{17}_{11} \alpha^{15}_{3} \mu - \alpha^{17}_{11} \alpha^{7}_{3} \left(\lambda + 2 \mu\right) - \alpha^{17}_{3} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) & \frac{\alpha^{18}_{4} \lambda \theta_{11}}{2} + \frac{\theta_{4} \left(\alpha^{17}_{11} \left(\lambda + \mu\right) - \alpha^{19}_{11} \mu\right)}{2} - \theta_{4} \left(0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{19}_{11} \mu + 0.25 \lambda \theta_{11}\right) & - \alpha^{17}_{11} \alpha^{7}_{5} \left(\lambda + 2 \mu\right) - \frac{\alpha^{19}_{11} \theta_{5} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{19}_{5} \left(2 \alpha^{17}_{11} \mu - 2 \alpha^{19}_{11} \left(\lambda + 3 \mu\right) + \theta_{11} \left(\lambda + 2 \mu\right)\right)}{2} & 0 & 0 & \frac{\alpha^{26}_{8} \mu \theta_{11}}{2} + \frac{\theta_{8} \left(\alpha^{17}_{11} \left(\lambda + \mu\right) - \alpha^{27}_{11} \lambda\right)}{2} & - \alpha^{17}_{11} \alpha^{15}_{9} \mu - \frac{\alpha^{27}_{11} \mu \theta_{9}}{2} - \frac{\alpha^{27}_{9} \left(2 \alpha^{17}_{11} \left(\lambda + 2 \mu\right) - 2 \alpha^{27}_{11} \left(\lambda + 3 \mu\right) + \mu \theta_{11}\right)}{2} & - \frac{\alpha^{16}_{10} \theta_{11} \left(\lambda + \mu\right)}{2} + \frac{\alpha^{18}_{10} \lambda \theta_{11}}{2} + \frac{\alpha^{26}_{10} \mu \theta_{11}}{2} + \frac{\theta_{10} \left(- \alpha^{17}_{11} \left(\lambda + \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \lambda\right)}{2} - \theta_{10} \left(- 0.5 \alpha^{17}_{11} \lambda - 0.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{19}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda\right) & - \alpha^{17}_{11} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) - \frac{\alpha^{19}_{11} \left(2 \alpha^{17}_{11} \mu - 2 \alpha^{19}_{11} \left(\lambda + 3 \mu\right) + \theta_{11} \left(\lambda + 2 \mu\right)\right)}{2} - \frac{\alpha^{27}_{11} \left(2 \alpha^{17}_{11} \left(\lambda + 2 \mu\right) - 2 \alpha^{27}_{11} \left(\lambda + 3 \mu\right) + \mu \theta_{11}\right)}{2} - \frac{\theta_{11} \left(\alpha^{19}_{11} \left(\lambda + 2 \mu\right) + \alpha^{27}_{11} \mu - \theta_{11} \left(\lambda + 3 \mu\right)\right)}{2}\end{array}\right]\end{split}\]
\[\displaystyle rank(AG_C) = 19\]
Again with sympy, rank looks correct numericaly. But we check with numpy to be sure:
\[\displaystyle rank(AG_C) = \mathtt{\text{19}}\]
Thus explicite enriched dof elimination does not look mandatory and for now we do note introduce any.
The rank is correct and we can now create the system.
So the matrix \(AG_C\) and \(BG_C\) are given by applying \(PG\) and \(R\),\(RF\):
\[\begin{split}\displaystyle BG_C = R^t.PG^t.B-R^t.PG^t.A.PG.RF=\left[\begin{matrix}- 0.5 L^{2} f\\0\\0\\0\\- 0.5 L^{2} f\\0\\0\\- \frac{L^{2} f \left(\alpha^{10}_{0} - 0.5 \theta_{0}\right)}{12} - \frac{L^{2} f \left(\alpha^{12}_{0} - 0.5 \theta_{0}\right)}{3} - \frac{L^{2} f \left(\alpha^{2}_{0} - 0.5 \theta_{0}\right)}{12}\\0\\- \frac{L^{2} \alpha^{14}_{2} f}{6} - \frac{L^{2} \alpha^{16}_{2} f}{3} - \frac{L^{2} \alpha^{2}_{2} f}{12} - \frac{L^{2} \alpha^{6}_{2} f}{12} - \frac{L^{2} f \theta_{2}}{6}\\0\\- \frac{L^{2} f \left(\alpha^{18}_{4} - 0.5 \theta_{4}\right)}{12} - \frac{L^{2} f \left(\alpha^{6}_{4} - 0.5 \theta_{4}\right)}{12}\\0\\- \frac{L^{2} f \left(\alpha^{10}_{6} - 0.5 \theta_{6}\right)}{12} - \frac{L^{2} f \left(\alpha^{22}_{6} - 0.5 \theta_{6}\right)}{12}\\0\\- \frac{L^{2} \alpha^{12}_{8} f}{3} - \frac{L^{2} \alpha^{14}_{8} f}{6} - \frac{L^{2} \alpha^{22}_{8} f}{12} - \frac{L^{2} \alpha^{26}_{8} f}{12} - \frac{L^{2} f \theta_{8}}{6}\\0\\- \frac{L^{2} f \left(\alpha^{16}_{10} - 0.5 \theta_{10}\right)}{3} - \frac{L^{2} f \left(\alpha^{18}_{10} - 0.5 \theta_{10}\right)}{12} - \frac{L^{2} f \left(\alpha^{26}_{10} - 0.5 \theta_{10}\right)}{12}\\0\end{matrix}\right]-\left[\begin{matrix}- c_{x} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 c_{y} \mu - i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\0.5 c_{x} \lambda - 0.5 c_{y} \mu - 0.5 i_{x} \lambda\\- 0.5 i_{x} \lambda\\- 0.5 c_{x} \lambda - 0.5 c_{y} \lambda - 1.0 c_{y} \mu\\- c_{x} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 c_{y} \left(\lambda + \mu\right) - i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\0.5 \lambda \left(- c_{x} + i_{x}\right)\\0.5 i_{x} \lambda\\0.5 \alpha^{10}_{0} c_{x} \lambda + 1.0 \alpha^{10}_{0} c_{x} \mu + 0.5 \alpha^{10}_{0} c_{y} \lambda - 0.5 \alpha^{12}_{0} c_{y} \lambda - 0.5 \alpha^{12}_{0} c_{y} \mu + 0.5 \alpha^{2}_{0} c_{y} \mu - 0.25 c_{x} \lambda \theta_{0} - 0.5 c_{x} \mu \theta_{0}\\0.5 \alpha^{13}_{1} c_{y} \lambda + 1.5 \alpha^{13}_{1} c_{y} \mu + 0.5 \alpha^{3}_{1} c_{x} \lambda - 0.25 c_{x} \lambda \theta_{1} - 0.25 c_{y} \lambda \theta_{1} - 0.75 c_{y} \mu \theta_{1}\\- 0.5 \alpha^{14}_{2} c_{x} \lambda - 1.0 \alpha^{14}_{2} c_{x} \mu - 0.5 \alpha^{14}_{2} i_{x} \lambda - 1.0 \alpha^{14}_{2} i_{x} \mu + 0.5 \alpha^{2}_{2} c_{y} \mu - 0.25 c_{x} \lambda \theta_{2} - 0.5 c_{x} \mu \theta_{2} + 0.25 c_{y} \mu \theta_{2} - 0.25 i_{x} \lambda \theta_{2} - 0.5 i_{x} \mu \theta_{2}\\- 0.5 \alpha^{15}_{3} c_{y} \mu + 0.5 \alpha^{3}_{3} c_{x} \lambda - 0.5 \alpha^{7}_{3} i_{x} \lambda + 0.25 c_{x} \lambda \theta_{3} - 0.25 c_{y} \mu \theta_{3} - 0.25 i_{x} \lambda \theta_{3}\\i_{x} \left(0.5 \alpha^{18}_{4} \lambda + 1.0 \alpha^{18}_{4} \mu - 0.25 \lambda \theta_{4} - 0.5 \mu \theta_{4}\right)\\i_{x} \lambda \left(- 0.5 \alpha^{7}_{5} - 0.25 \theta_{5}\right)\\0.5 \alpha^{10}_{6} c_{x} \lambda + 1.0 \alpha^{10}_{6} c_{x} \mu + 0.5 \alpha^{10}_{6} c_{y} \lambda - 0.25 c_{x} \lambda \theta_{6} - 0.5 c_{x} \mu \theta_{6} - 0.25 c_{y} \lambda \theta_{6}\\- 0.5 \alpha^{23}_{7} c_{x} \lambda - 0.5 \alpha^{23}_{7} c_{y} \lambda - 1.0 \alpha^{23}_{7} c_{y} \mu - 0.25 c_{x} \lambda \theta_{7} - 0.25 c_{y} \lambda \theta_{7} - 0.5 c_{y} \mu \theta_{7}\\- 0.5 \alpha^{12}_{8} c_{y} \lambda - 0.5 \alpha^{12}_{8} c_{y} \mu - 0.5 \alpha^{14}_{8} c_{x} \lambda - 1.0 \alpha^{14}_{8} c_{x} \mu - 0.5 \alpha^{14}_{8} i_{x} \lambda - 1.0 \alpha^{14}_{8} i_{x} \mu - 0.25 c_{x} \lambda \theta_{8} - 0.5 c_{x} \mu \theta_{8} - 0.25 c_{y} \lambda \theta_{8} - 0.25 c_{y} \mu \theta_{8} - 0.25 i_{x} \lambda \theta_{8} - 0.5 i_{x} \mu \theta_{8}\\0.5 \alpha^{13}_{9} c_{y} \lambda + 1.5 \alpha^{13}_{9} c_{y} \mu - 0.5 \alpha^{15}_{9} c_{y} \mu - 0.5 \alpha^{23}_{9} c_{x} \lambda - 0.5 \alpha^{23}_{9} c_{y} \lambda - 1.0 \alpha^{23}_{9} c_{y} \mu + 0.5 \alpha^{27}_{9} i_{x} \lambda - 0.25 c_{x} \lambda \theta_{9} + 0.25 i_{x} \lambda \theta_{9}\\i_{x} \left(0.5 \alpha^{18}_{10} \lambda + 1.0 \alpha^{18}_{10} \mu - 0.25 \lambda \theta_{10} - 0.5 \mu \theta_{10}\right)\\i_{x} \lambda \left(0.5 \alpha^{27}_{11} + 0.25 \theta_{11}\right)\end{matrix}\right]=\left[\begin{matrix}- 0.5 L^{2} f + c_{x} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 c_{y} \mu + i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\- 0.5 c_{x} \lambda + 0.5 c_{y} \mu + 0.5 i_{x} \lambda\\0.5 i_{x} \lambda\\0.5 c_{x} \lambda + 0.5 c_{y} \lambda + 1.0 c_{y} \mu\\- 0.5 L^{2} f + c_{x} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 c_{y} \left(\lambda + \mu\right) + i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\0.5 \lambda \left(c_{x} - i_{x}\right)\\- 0.5 i_{x} \lambda\\- 0.0833333333333333 L^{2} \alpha^{10}_{0} f - 0.333333333333333 L^{2} \alpha^{12}_{0} f - 0.0833333333333333 L^{2} \alpha^{2}_{0} f + 0.25 L^{2} f \theta_{0} - 0.5 \alpha^{10}_{0} c_{x} \lambda - 1.0 \alpha^{10}_{0} c_{x} \mu - 0.5 \alpha^{10}_{0} c_{y} \lambda + 0.5 \alpha^{12}_{0} c_{y} \lambda + 0.5 \alpha^{12}_{0} c_{y} \mu - 0.5 \alpha^{2}_{0} c_{y} \mu + 0.25 c_{x} \lambda \theta_{0} + 0.5 c_{x} \mu \theta_{0}\\- 0.5 \alpha^{13}_{1} c_{y} \lambda - 1.5 \alpha^{13}_{1} c_{y} \mu - 0.5 \alpha^{3}_{1} c_{x} \lambda + 0.25 c_{x} \lambda \theta_{1} + 0.25 c_{y} \lambda \theta_{1} + 0.75 c_{y} \mu \theta_{1}\\- 0.166666666666667 L^{2} \alpha^{14}_{2} f - 0.333333333333333 L^{2} \alpha^{16}_{2} f - 0.0833333333333333 L^{2} \alpha^{2}_{2} f - 0.0833333333333333 L^{2} \alpha^{6}_{2} f - 0.166666666666667 L^{2} f \theta_{2} + 0.5 \alpha^{14}_{2} c_{x} \lambda + 1.0 \alpha^{14}_{2} c_{x} \mu + 0.5 \alpha^{14}_{2} i_{x} \lambda + 1.0 \alpha^{14}_{2} i_{x} \mu - 0.5 \alpha^{2}_{2} c_{y} \mu + 0.25 c_{x} \lambda \theta_{2} + 0.5 c_{x} \mu \theta_{2} - 0.25 c_{y} \mu \theta_{2} + 0.25 i_{x} \lambda \theta_{2} + 0.5 i_{x} \mu \theta_{2}\\0.5 \alpha^{15}_{3} c_{y} \mu - 0.5 \alpha^{3}_{3} c_{x} \lambda + 0.5 \alpha^{7}_{3} i_{x} \lambda - 0.25 c_{x} \lambda \theta_{3} + 0.25 c_{y} \mu \theta_{3} + 0.25 i_{x} \lambda \theta_{3}\\- 0.0833333333333333 L^{2} \alpha^{18}_{4} f - 0.0833333333333333 L^{2} \alpha^{6}_{4} f + 0.0833333333333333 L^{2} f \theta_{4} - 0.5 \alpha^{18}_{4} i_{x} \lambda - 1.0 \alpha^{18}_{4} i_{x} \mu + 0.25 i_{x} \lambda \theta_{4} + 0.5 i_{x} \mu \theta_{4}\\i_{x} \lambda \left(0.5 \alpha^{7}_{5} + 0.25 \theta_{5}\right)\\- 0.0833333333333333 L^{2} \alpha^{10}_{6} f - 0.0833333333333333 L^{2} \alpha^{22}_{6} f + 0.0833333333333333 L^{2} f \theta_{6} - 0.5 \alpha^{10}_{6} c_{x} \lambda - 1.0 \alpha^{10}_{6} c_{x} \mu - 0.5 \alpha^{10}_{6} c_{y} \lambda + 0.25 c_{x} \lambda \theta_{6} + 0.5 c_{x} \mu \theta_{6} + 0.25 c_{y} \lambda \theta_{6}\\0.5 \alpha^{23}_{7} c_{x} \lambda + 0.5 \alpha^{23}_{7} c_{y} \lambda + 1.0 \alpha^{23}_{7} c_{y} \mu + 0.25 c_{x} \lambda \theta_{7} + 0.25 c_{y} \lambda \theta_{7} + 0.5 c_{y} \mu \theta_{7}\\- 0.333333333333333 L^{2} \alpha^{12}_{8} f - 0.166666666666667 L^{2} \alpha^{14}_{8} f - 0.0833333333333333 L^{2} \alpha^{22}_{8} f - 0.0833333333333333 L^{2} \alpha^{26}_{8} f - 0.166666666666667 L^{2} f \theta_{8} + 0.5 \alpha^{12}_{8} c_{y} \lambda + 0.5 \alpha^{12}_{8} c_{y} \mu + 0.5 \alpha^{14}_{8} c_{x} \lambda + 1.0 \alpha^{14}_{8} c_{x} \mu + 0.5 \alpha^{14}_{8} i_{x} \lambda + 1.0 \alpha^{14}_{8} i_{x} \mu + 0.25 c_{x} \lambda \theta_{8} + 0.5 c_{x} \mu \theta_{8} + 0.25 c_{y} \lambda \theta_{8} + 0.25 c_{y} \mu \theta_{8} + 0.25 i_{x} \lambda \theta_{8} + 0.5 i_{x} \mu \theta_{8}\\- 0.5 \alpha^{13}_{9} c_{y} \lambda - 1.5 \alpha^{13}_{9} c_{y} \mu + 0.5 \alpha^{15}_{9} c_{y} \mu + 0.5 \alpha^{23}_{9} c_{x} \lambda + 0.5 \alpha^{23}_{9} c_{y} \lambda + 1.0 \alpha^{23}_{9} c_{y} \mu - 0.5 \alpha^{27}_{9} i_{x} \lambda + 0.25 c_{x} \lambda \theta_{9} - 0.25 i_{x} \lambda \theta_{9}\\- 0.333333333333333 L^{2} \alpha^{16}_{10} f - 0.0833333333333333 L^{2} \alpha^{18}_{10} f - 0.0833333333333333 L^{2} \alpha^{26}_{10} f + 0.25 L^{2} f \theta_{10} - 0.5 \alpha^{18}_{10} i_{x} \lambda - 1.0 \alpha^{18}_{10} i_{x} \mu + 0.25 i_{x} \lambda \theta_{10} + 0.5 i_{x} \mu \theta_{10}\\i_{x} \lambda \left(- 0.5 \alpha^{27}_{11} - 0.25 \theta_{11}\right)\end{matrix}\right]\end{split}\]
Reduced solution \(SG_r\) at coarse scale is:
\[\displaystyle SG_r=inv(AG_C).BG_C = ~to ~long\]
And the full solution \(SG\) at the coarse scale is:
\[\displaystyle SG_C=R.SG_r+RF = ~to ~ long\]
Fine to coarse operator in the shift case#
If the enriched function is a shifted version of the patch solutions all \(\theta_i\) are null by construction which lead to the following particular operator \(P\) :
\[\begin{split}\displaystyle P = \left[\begin{array}{cccccccccccccccccccccccc}1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{2}_{0} & 0 & \alpha^{2}_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{3}_{1} & 0 & \alpha^{3}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{6}_{2} & 0 & \alpha^{6}_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{7}_{3} & 0 & \alpha^{7}_{5} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{0} & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} & 0 & 0 & 0 & 0 & 0\\0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{11}_{1} & 0 & 0 & 0 & 0 & 0 & \alpha^{11}_{7} & 0 & 0 & 0 & 0\\0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{12}_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{12}_{8} & 0 & 0 & 0\\0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{13}_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{13}_{9} & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{14}_{2} & 0 & 0 & 0 & 0 & 0 & \alpha^{14}_{8} & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{15}_{3} & 0 & 0 & 0 & 0 & 0 & \alpha^{15}_{9} & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{16}_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{16}_{10} & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{17}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{17}_{11}\\0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{4} & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} & 0\\0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{19}_{5} & 0 & 0 & 0 & 0 & 0 & \alpha^{19}_{11}\\0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{22}_{6} & 0 & \alpha^{22}_{8} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{23}_{7} & 0 & \alpha^{23}_{9} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{26}_{8} & 0 & \alpha^{26}_{10} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{27}_{9} & 0 & \alpha^{27}_{11}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
TS approach (I) in shift case#
Applying \(P\) to \(A\) matrix lead to the following expression of the coarse enriched matrix:
\[\begin{split}\displaystyle AP_C=P^t.A.P = \left[\begin{array}{cccccccccccccccccccccccc}0.5 \lambda + 1.5 \mu & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0 & 0 & - 0.5 \mu & 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0 & \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) + 0.5 \alpha^{3}_{1} \lambda & - \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{3}_{3} \lambda & 0 & 0 & - 0.5 \alpha^{22}_{6} \mu & 0.5 \alpha^{11}_{7} \mu & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{22}_{8} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & 0 & 0\\0 & 0.5 \lambda + 1.5 \mu & 0.5 \mu & - 0.5 \mu & 0 & 0 & 0.5 \lambda & - 0.5 \lambda - 1.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0.5 \alpha^{10}_{0} \lambda - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) + 0.5 \alpha^{2}_{0} \mu & \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) & 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{15}_{3} \mu & 0 & 0 & 0.5 \alpha^{10}_{6} \lambda & - \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{15}_{9} \mu - \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0\\- 0.5 \lambda - 1.0 \mu & 0.5 \mu & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{0} \mu & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & - \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{7}_{5} \lambda & 0 & 0 & - \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{8} \mu & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{10} \mu & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right)\\0.5 \lambda & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0.5 \mu & - 0.5 \mu & 0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{6}_{4} \mu & - 0.5 \alpha^{19}_{5} \mu & 0 & 0 & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{19}_{11} \mu - \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & 0.5 \lambda + 1.5 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0 & 0 & 0 & - 0.5 \mu & 0.5 \lambda & 0 & 0 & - \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{6}_{2} \mu & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{7}_{3} \lambda & \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{6}_{4} \mu & - 0.5 \alpha^{19}_{5} \mu - 0.5 \alpha^{7}_{5} \lambda & 0 & 0 & 0 & 0 & - \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{19}_{11} \mu\\0 & 0 & 0.5 \lambda & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu & - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{18}_{10} \lambda & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{19}_{11} \mu\\- 0.5 \mu & 0.5 \lambda & 0 & 0 & 0 & 0 & 0.5 \lambda + 1.5 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & 0 & 0 & \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0 & \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{22}_{6} \mu & - 0.5 \alpha^{11}_{7} \mu - 0.5 \alpha^{23}_{7} \lambda & - \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{22}_{8} \mu & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) - 0.5 \alpha^{23}_{9} \lambda & 0 & 0\\0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0 & 0 & - 0.5 \lambda - 0.5 \mu & 0.5 \lambda + 1.5 \mu & 0.5 \lambda & - 0.5 \mu & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0.5 \alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu & 0.5 \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0\\0 & - 0.5 \lambda - 0.5 \mu & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & - 0.5 \lambda - 1.0 \mu & 0.5 \mu & - \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0 & 0 & - \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{23}_{7} \lambda & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & - \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{26}_{10} \mu & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda\\- 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0.5 \lambda & - 0.5 \mu & - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{22}_{6} \mu & - 0.5 \alpha^{11}_{7} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & 0 & - 0.5 \lambda - 0.5 \mu & - 0.5 \mu & 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0.5 \lambda & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & - \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{6}_{2} \mu & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & - 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{19}_{5} \mu & 0 & 0 & - \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{27}_{9} \lambda & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) + 0.5 \alpha^{19}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{18}_{4} \lambda & - \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{15}_{9} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right)\\\alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) & 0.5 \alpha^{10}_{0} \lambda - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) + 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \mu & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & 0 & 0 & 0.5 \alpha^{10}_{0} \lambda + 1.0 \alpha^{10}_{0} \mu - 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & - 0.5 \alpha^{10}_{0} \lambda - 1.0 \alpha^{10}_{0} \mu + 0.5 \alpha^{12}_{0} \lambda + 1.5 \alpha^{12}_{0} \mu - 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0 & 0 & \alpha^{10}_{0} \left(\alpha^{10}_{0} \left(\lambda + 3 \mu\right) - \alpha^{12}_{0} \left(\lambda + 2 \mu\right)\right) - \alpha^{12}_{0} \left(\alpha^{10}_{0} \left(\lambda + 2 \mu\right) - 2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) + \alpha^{2}_{0} \mu\right) - \alpha^{2}_{0} \left(\alpha^{12}_{0} \mu - \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{12}_{0} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{2}_{2} \left(\alpha^{12}_{0} \mu - \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{12}_{0} \alpha^{22}_{6} \mu + \alpha^{10}_{6} \left(\alpha^{10}_{0} \left(\lambda + 3 \mu\right) - \alpha^{12}_{0} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{12}_{0} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \alpha^{22}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{10}_{0} \left(\lambda + 2 \mu\right) - 2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) + \alpha^{2}_{0} \mu\right) & 0 & 0 & 0\\0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) + 0.5 \alpha^{3}_{1} \lambda & \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \lambda + 1.0 \alpha^{3}_{1} \mu & 0 & 0 & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu & - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu - 0.5 \alpha^{3}_{1} \lambda - 1.0 \alpha^{3}_{1} \mu & 0 & 0 & 0 & \alpha^{11}_{1} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) - \alpha^{13}_{1} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \left(\alpha^{13}_{1} \left(\lambda + 2 \mu\right) - \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{3} \mu - \alpha^{3}_{3} \left(\alpha^{13}_{1} \left(\lambda + 2 \mu\right) - \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{13}_{1} \alpha^{23}_{7} \left(\lambda + 2 \mu\right) + \alpha^{11}_{7} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{9} \mu - \alpha^{13}_{1} \alpha^{23}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\\alpha^{14}_{2} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & - 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu & 0 & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu - 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - 0.5 \alpha^{14}_{2} \lambda - 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{6}_{2} \mu & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - \alpha^{12}_{0} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \alpha^{2}_{0} \alpha^{2}_{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{14}_{2} \left(2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) - \alpha^{16}_{2} \left(\lambda + 2 \mu\right)\right) - \alpha^{16}_{2} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) + \left(\alpha^{2}_{2}\right)^{2} \left(\lambda + 3 \mu\right) - \alpha^{6}_{2} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{16}_{2} \alpha^{18}_{4} \left(\lambda + 2 \mu\right) - \alpha^{6}_{4} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{16}_{2} \alpha^{26}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \alpha^{14}_{8} \left(2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) - \alpha^{16}_{2} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{16}_{10} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) - \alpha^{18}_{10} \alpha^{16}_{2} \left(\lambda + 2 \mu\right) - \alpha^{26}_{10} \alpha^{16}_{2} \mu & 0\\0.5 \alpha^{3}_{3} \lambda & - 0.5 \alpha^{15}_{3} \mu & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{3}_{3} \lambda + 1.0 \alpha^{3}_{3} \mu & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{7}_{3} \lambda & - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{7}_{3} \lambda + 1.0 \alpha^{7}_{3} \mu & 0 & 0 & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{3}_{3} \lambda - 1.0 \alpha^{3}_{3} \mu & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & - 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{7}_{3} \lambda - 1.0 \alpha^{7}_{3} \mu & 0 & - \alpha^{13}_{1} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \alpha^{3}_{1} \alpha^{3}_{3} \left(\lambda + 3 \mu\right) & 0 & \alpha^{15}_{3} \left(2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) - \alpha^{17}_{3} \mu\right) - \alpha^{17}_{3} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) + \left(\alpha^{3}_{3}\right)^{2} \left(\lambda + 3 \mu\right) - \alpha^{7}_{3} \left(\alpha^{17}_{3} \left(\lambda + 2 \mu\right) - \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{3} \alpha^{19}_{5} \mu - \alpha^{7}_{5} \left(\alpha^{17}_{3} \left(\lambda + 2 \mu\right) - \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{3} \alpha^{27}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \alpha^{15}_{9} \left(2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) - \alpha^{17}_{3} \mu\right) & 0 & - \alpha^{17}_{11} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \alpha^{17}_{3} \mu - \alpha^{27}_{11} \alpha^{17}_{3} \left(\lambda + 2 \mu\right)\\0 & 0 & \alpha^{18}_{4} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{18}_{4} \lambda + 1.0 \alpha^{18}_{4} \mu + 0.5 \alpha^{6}_{4} \mu & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu & 0 & 0 & 0 & 0 & - 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & - \alpha^{16}_{2} \left(\alpha^{18}_{4} \left(\lambda + 2 \mu\right) + \alpha^{6}_{4} \mu\right) + \alpha^{6}_{2} \alpha^{6}_{4} \left(\lambda + 3 \mu\right) & 0 & \left(\left(\alpha^{18}_{4}\right)^{2} + \left(\alpha^{6}_{4}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - \alpha^{16}_{10} \left(\alpha^{18}_{4} \left(\lambda + 2 \mu\right) + \alpha^{6}_{4} \mu\right) + \alpha^{18}_{10} \alpha^{18}_{4} \left(\lambda + 3 \mu\right) & 0\\0 & 0 & 0.5 \alpha^{7}_{5} \lambda & - 0.5 \alpha^{19}_{5} \mu & - 0.5 \alpha^{19}_{5} \mu - 0.5 \alpha^{7}_{5} \lambda & 0.5 \alpha^{19}_{5} \mu + 0.5 \alpha^{7}_{5} \lambda + 1.0 \alpha^{7}_{5} \mu & 0 & 0 & 0 & 0 & 0.5 \alpha^{19}_{5} \mu & \alpha^{7}_{5} \left(- 0.5 \lambda - 1.0 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{3} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \alpha^{7}_{3} \alpha^{7}_{5} \left(\lambda + 3 \mu\right) & 0 & \left(\left(\alpha^{19}_{5}\right)^{2} + \left(\alpha^{7}_{5}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - \alpha^{17}_{11} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \alpha^{19}_{11} \alpha^{19}_{5} \left(\lambda + 3 \mu\right)\\- 0.5 \alpha^{22}_{6} \mu & 0.5 \alpha^{10}_{6} \lambda & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{6} \lambda + 1.0 \alpha^{10}_{6} \mu + 0.5 \alpha^{22}_{6} \mu & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu & \alpha^{10}_{6} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{22}_{6} \mu & 0 & 0 & \alpha^{10}_{0} \alpha^{10}_{6} \left(\lambda + 3 \mu\right) - \alpha^{12}_{0} \left(\alpha^{10}_{6} \left(\lambda + 2 \mu\right) + \alpha^{22}_{6} \mu\right) & 0 & 0 & 0 & 0 & 0 & \left(\left(\alpha^{10}_{6}\right)^{2} + \left(\alpha^{22}_{6}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & \alpha^{22}_{6} \alpha^{22}_{8} \left(\lambda + 3 \mu\right) - \alpha^{12}_{8} \left(\alpha^{10}_{6} \left(\lambda + 2 \mu\right) + \alpha^{22}_{6} \mu\right) & 0 & 0 & 0\\0.5 \alpha^{11}_{7} \mu & \alpha^{23}_{7} \left(- 0.5 \lambda - 1.0 \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{11}_{7} \mu - 0.5 \alpha^{23}_{7} \lambda & 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{23}_{7} \lambda + 1.0 \alpha^{23}_{7} \mu & 0.5 \alpha^{23}_{7} \lambda & - 0.5 \alpha^{11}_{7} \mu & 0 & 0 & 0 & \alpha^{11}_{1} \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) & 0 & 0 & 0 & 0 & 0 & \left(\left(\alpha^{11}_{7}\right)^{2} + \left(\alpha^{23}_{7}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & \alpha^{23}_{7} \alpha^{23}_{9} \left(\lambda + 3 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu - 0.5 \alpha^{14}_{8} \lambda - 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{22}_{8} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & - 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & 0 & 0 & - 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{22}_{8} \mu & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu & 0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu + 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & \alpha^{14}_{8} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{26}_{8} \mu & - \alpha^{10}_{0} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{2}_{0} \alpha^{12}_{8} \mu & 0 & - \alpha^{14}_{2} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right)\right) - \alpha^{16}_{2} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) - \alpha^{2}_{2} \alpha^{12}_{8} \mu & 0 & 0 & 0 & - \alpha^{10}_{6} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) - \alpha^{22}_{6} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{12}_{8} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{14}_{8} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right)\right) - \alpha^{22}_{8} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{26}_{8}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{16}_{10} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) + \alpha^{26}_{10} \alpha^{26}_{8} \left(\lambda + 3 \mu\right) & 0\\- 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu - 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{23}_{9} \lambda - 1.0 \alpha^{23}_{9} \mu & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{27}_{9} \lambda - 1.0 \alpha^{27}_{9} \mu & 0 & 0 & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) - 0.5 \alpha^{23}_{9} \lambda & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{23}_{9} \lambda + 1.0 \alpha^{23}_{9} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu + 0.5 \alpha^{27}_{9} \lambda + 1.0 \alpha^{27}_{9} \mu & 0.5 \alpha^{27}_{9} \lambda & - 0.5 \alpha^{15}_{9} \mu & 0 & - \alpha^{11}_{1} \alpha^{13}_{9} \mu - \alpha^{13}_{1} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & 0 & - \alpha^{15}_{3} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right)\right) - \alpha^{17}_{3} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{3} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & - \alpha^{11}_{7} \alpha^{13}_{9} \mu - \alpha^{23}_{7} \left(\alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{9} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{15}_{9} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right)\right) - \alpha^{23}_{9} \left(\alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{27}_{9}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{17}_{11} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) + \alpha^{27}_{11} \alpha^{27}_{9} \left(\lambda + 3 \mu\right)\\0 & 0 & 0.5 \alpha^{16}_{10} \lambda + 1.5 \alpha^{16}_{10} \mu - 0.5 \alpha^{18}_{10} \lambda - 1.0 \alpha^{18}_{10} \mu - 0.5 \alpha^{26}_{10} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & - 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} \lambda + 1.0 \alpha^{18}_{10} \mu & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{18}_{10} \lambda & 0 & 0 & - 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{26}_{10} \mu & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{16}_{10} \alpha^{6}_{2} \mu - \alpha^{16}_{2} \left(- 2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10} \left(\lambda + 2 \mu\right) + \alpha^{26}_{10} \mu\right) & 0 & - \alpha^{16}_{10} \alpha^{6}_{4} \mu - \alpha^{18}_{4} \left(\alpha^{16}_{10} \left(\lambda + 2 \mu\right) - \alpha^{18}_{10} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{26}_{8} \left(\alpha^{16}_{10} \mu - \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{16}_{10} \left(- 2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10} \left(\lambda + 2 \mu\right) + \alpha^{26}_{10} \mu\right) - \alpha^{18}_{10} \left(\alpha^{16}_{10} \left(\lambda + 2 \mu\right) - \alpha^{18}_{10} \left(\lambda + 3 \mu\right)\right) - \alpha^{26}_{10} \left(\alpha^{16}_{10} \mu - \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0\\0 & 0 & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) & 0.5 \alpha^{17}_{11} \lambda + 1.5 \alpha^{17}_{11} \mu - 0.5 \alpha^{19}_{11} \mu - 0.5 \alpha^{27}_{11} \lambda - 1.0 \alpha^{27}_{11} \mu & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{19}_{11} \mu & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{19}_{11} \mu & 0 & 0 & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda + 1.0 \alpha^{27}_{11} \mu & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) + 0.5 \alpha^{19}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{3} \mu - \alpha^{17}_{11} \alpha^{7}_{3} \left(\lambda + 2 \mu\right) - \alpha^{17}_{3} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{17}_{11} \alpha^{7}_{5} \left(\lambda + 2 \mu\right) - \alpha^{19}_{5} \left(\alpha^{17}_{11} \mu - \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(\alpha^{17}_{11} \left(\lambda + 2 \mu\right) - \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{11} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \left(\alpha^{17}_{11} \mu - \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) - \alpha^{27}_{11} \left(\alpha^{17}_{11} \left(\lambda + 2 \mu\right) - \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right)\end{array}\right]\end{split}\]
\[\displaystyle rank(P^t.A.P) = 24\]
which is again full rank numerically with sympy. With numpy:
\[\displaystyle rank(P^t.A.P) = \mathtt{\text{21}}\]
which is again full rank-deficient with 3 modes.
Again use of similar Dirichlet boundary condition at both scale leads to imposing, at the coarse scale, the following equations:
\(x_k=c\)
with \(k \in\{0,1,4,6,10\}\)
that can be treated by simple Dirichlet boundary conditions operator without restriction operator as is done by FEniCSx/PETSc.
For the enriched sub bloc, like in the general case no extra Dirichlet boundary condition are imposed.
The Dirichlet operator are then:
\[\begin{split}\displaystyle D_C = \left[\begin{array}{cccccccccccccccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle U_C = \left[\begin{array}{cccccccccccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle XD_C = \left[\begin{matrix}c_{x}\\c_{y}\\0\\0\\i_{x}\\0\\c_{x}\\0\\0\\0\\i_{x}\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\end{matrix}\right]\end{split}\]
Applying those operator to \(P^t.A.P\) gives the final \(A_C\) matrix:
\[\begin{split}\displaystyle A_C=D_C^t.P^t.A.P.D_C+U_C= \left[\begin{array}{cccccccccccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda & 0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{0} \mu & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & - \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{7}_{5} \lambda & 0 & 0 & - \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{8} \mu & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{10} \mu & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right)\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0 & - 0.5 \mu & 0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{6}_{4} \mu & - 0.5 \alpha^{19}_{5} \mu & 0 & 0 & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{19}_{11} \mu - \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 \lambda & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu & - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{18}_{10} \lambda & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{19}_{11} \mu\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \lambda + 1.5 \mu & 0.5 \lambda & - 0.5 \mu & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0.5 \alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu & 0.5 \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0\\0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & 0 & 0 & 0.5 \lambda & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \mu & - \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0 & 0 & - \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{23}_{7} \lambda & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & - \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{26}_{10} \mu & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda\\0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0 & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0 & - 0.5 \mu & - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{22}_{6} \mu & - 0.5 \alpha^{11}_{7} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{18}_{4} \lambda & - \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{15}_{9} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right)\\0 & 0 & - 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \mu & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & 0 & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & - 0.5 \alpha^{10}_{0} \lambda - 1.0 \alpha^{10}_{0} \mu + 0.5 \alpha^{12}_{0} \lambda + 1.5 \alpha^{12}_{0} \mu - 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0 & 0 & \alpha^{10}_{0} \left(\alpha^{10}_{0} \left(\lambda + 3 \mu\right) - \alpha^{12}_{0} \left(\lambda + 2 \mu\right)\right) - \alpha^{12}_{0} \left(\alpha^{10}_{0} \left(\lambda + 2 \mu\right) - 2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) + \alpha^{2}_{0} \mu\right) - \alpha^{2}_{0} \left(\alpha^{12}_{0} \mu - \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{12}_{0} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{2}_{2} \left(\alpha^{12}_{0} \mu - \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{12}_{0} \alpha^{22}_{6} \mu + \alpha^{10}_{6} \left(\alpha^{10}_{0} \left(\lambda + 3 \mu\right) - \alpha^{12}_{0} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{12}_{0} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \alpha^{22}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{10}_{0} \left(\lambda + 2 \mu\right) - 2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) + \alpha^{2}_{0} \mu\right) & 0 & 0 & 0\\0 & 0 & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \lambda + 1.0 \alpha^{3}_{1} \mu & 0 & 0 & 0 & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu & - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu - 0.5 \alpha^{3}_{1} \lambda - 1.0 \alpha^{3}_{1} \mu & 0 & 0 & 0 & \alpha^{11}_{1} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) - \alpha^{13}_{1} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \left(\alpha^{13}_{1} \left(\lambda + 2 \mu\right) - \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{3} \mu - \alpha^{3}_{3} \left(\alpha^{13}_{1} \left(\lambda + 2 \mu\right) - \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{13}_{1} \alpha^{23}_{7} \left(\lambda + 2 \mu\right) + \alpha^{11}_{7} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{9} \mu - \alpha^{13}_{1} \alpha^{23}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\0 & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0 & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu & 0 & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu - 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & 0 & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - \alpha^{12}_{0} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \alpha^{2}_{0} \alpha^{2}_{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{14}_{2} \left(2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) - \alpha^{16}_{2} \left(\lambda + 2 \mu\right)\right) - \alpha^{16}_{2} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) + \left(\alpha^{2}_{2}\right)^{2} \left(\lambda + 3 \mu\right) - \alpha^{6}_{2} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{16}_{2} \alpha^{18}_{4} \left(\lambda + 2 \mu\right) - \alpha^{6}_{4} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{16}_{2} \alpha^{26}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \alpha^{14}_{8} \left(2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) - \alpha^{16}_{2} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{16}_{10} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) - \alpha^{18}_{10} \alpha^{16}_{2} \left(\lambda + 2 \mu\right) - \alpha^{26}_{10} \alpha^{16}_{2} \mu & 0\\0 & 0 & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{3}_{3} \lambda + 1.0 \alpha^{3}_{3} \mu & 0 & - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{7}_{3} \lambda + 1.0 \alpha^{7}_{3} \mu & 0 & 0 & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{3}_{3} \lambda - 1.0 \alpha^{3}_{3} \mu & 0 & - 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{7}_{3} \lambda - 1.0 \alpha^{7}_{3} \mu & 0 & - \alpha^{13}_{1} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \alpha^{3}_{1} \alpha^{3}_{3} \left(\lambda + 3 \mu\right) & 0 & \alpha^{15}_{3} \left(2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) - \alpha^{17}_{3} \mu\right) - \alpha^{17}_{3} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) + \left(\alpha^{3}_{3}\right)^{2} \left(\lambda + 3 \mu\right) - \alpha^{7}_{3} \left(\alpha^{17}_{3} \left(\lambda + 2 \mu\right) - \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{3} \alpha^{19}_{5} \mu - \alpha^{7}_{5} \left(\alpha^{17}_{3} \left(\lambda + 2 \mu\right) - \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{3} \alpha^{27}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \alpha^{15}_{9} \left(2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) - \alpha^{17}_{3} \mu\right) & 0 & - \alpha^{17}_{11} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \alpha^{17}_{3} \mu - \alpha^{27}_{11} \alpha^{17}_{3} \left(\lambda + 2 \mu\right)\\0 & 0 & \alpha^{18}_{4} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{6}_{4} \mu & 0 & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & - \alpha^{16}_{2} \left(\alpha^{18}_{4} \left(\lambda + 2 \mu\right) + \alpha^{6}_{4} \mu\right) + \alpha^{6}_{2} \alpha^{6}_{4} \left(\lambda + 3 \mu\right) & 0 & \left(\left(\alpha^{18}_{4}\right)^{2} + \left(\alpha^{6}_{4}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - \alpha^{16}_{10} \left(\alpha^{18}_{4} \left(\lambda + 2 \mu\right) + \alpha^{6}_{4} \mu\right) + \alpha^{18}_{10} \alpha^{18}_{4} \left(\lambda + 3 \mu\right) & 0\\0 & 0 & 0.5 \alpha^{7}_{5} \lambda & - 0.5 \alpha^{19}_{5} \mu & 0 & 0.5 \alpha^{19}_{5} \mu + 0.5 \alpha^{7}_{5} \lambda + 1.0 \alpha^{7}_{5} \mu & 0 & 0 & 0 & 0 & 0 & \alpha^{7}_{5} \left(- 0.5 \lambda - 1.0 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{3} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \alpha^{7}_{3} \alpha^{7}_{5} \left(\lambda + 3 \mu\right) & 0 & \left(\left(\alpha^{19}_{5}\right)^{2} + \left(\alpha^{7}_{5}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - \alpha^{17}_{11} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \alpha^{19}_{11} \alpha^{19}_{5} \left(\lambda + 3 \mu\right)\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu & \alpha^{10}_{6} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{22}_{6} \mu & 0 & 0 & \alpha^{10}_{0} \alpha^{10}_{6} \left(\lambda + 3 \mu\right) - \alpha^{12}_{0} \left(\alpha^{10}_{6} \left(\lambda + 2 \mu\right) + \alpha^{22}_{6} \mu\right) & 0 & 0 & 0 & 0 & 0 & \left(\left(\alpha^{10}_{6}\right)^{2} + \left(\alpha^{22}_{6}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & \alpha^{22}_{6} \alpha^{22}_{8} \left(\lambda + 3 \mu\right) - \alpha^{12}_{8} \left(\alpha^{10}_{6} \left(\lambda + 2 \mu\right) + \alpha^{22}_{6} \mu\right) & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{23}_{7} \lambda + 1.0 \alpha^{23}_{7} \mu & 0.5 \alpha^{23}_{7} \lambda & - 0.5 \alpha^{11}_{7} \mu & 0 & 0 & 0 & \alpha^{11}_{1} \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) & 0 & 0 & 0 & 0 & 0 & \left(\left(\alpha^{11}_{7}\right)^{2} + \left(\alpha^{23}_{7}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & \alpha^{23}_{7} \alpha^{23}_{9} \left(\lambda + 3 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\0 & 0 & - 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu & 0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu + 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & 0 & 0.5 \alpha^{26}_{8} \mu & - \alpha^{10}_{0} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{2}_{0} \alpha^{12}_{8} \mu & 0 & - \alpha^{14}_{2} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right)\right) - \alpha^{16}_{2} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) - \alpha^{2}_{2} \alpha^{12}_{8} \mu & 0 & 0 & 0 & - \alpha^{10}_{6} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) - \alpha^{22}_{6} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{12}_{8} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{14}_{8} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right)\right) - \alpha^{22}_{8} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{26}_{8}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{16}_{10} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) + \alpha^{26}_{10} \alpha^{26}_{8} \left(\lambda + 3 \mu\right) & 0\\0 & 0 & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{27}_{9} \lambda - 1.0 \alpha^{27}_{9} \mu & 0 & 0 & 0 & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{23}_{9} \lambda + 1.0 \alpha^{23}_{9} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu + 0.5 \alpha^{27}_{9} \lambda + 1.0 \alpha^{27}_{9} \mu & 0 & - 0.5 \alpha^{15}_{9} \mu & 0 & - \alpha^{11}_{1} \alpha^{13}_{9} \mu - \alpha^{13}_{1} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & 0 & - \alpha^{15}_{3} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right)\right) - \alpha^{17}_{3} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{3} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & - \alpha^{11}_{7} \alpha^{13}_{9} \mu - \alpha^{23}_{7} \left(\alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{9} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{15}_{9} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right)\right) - \alpha^{23}_{9} \left(\alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{27}_{9}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{17}_{11} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) + \alpha^{27}_{11} \alpha^{27}_{9} \left(\lambda + 3 \mu\right)\\0 & 0 & 0.5 \alpha^{16}_{10} \lambda + 1.5 \alpha^{16}_{10} \mu - 0.5 \alpha^{18}_{10} \lambda - 1.0 \alpha^{18}_{10} \mu - 0.5 \alpha^{26}_{10} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & 0 & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{18}_{10} \lambda & 0 & 0 & - 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{26}_{10} \mu & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu & 0 & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{16}_{10} \alpha^{6}_{2} \mu - \alpha^{16}_{2} \left(- 2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10} \left(\lambda + 2 \mu\right) + \alpha^{26}_{10} \mu\right) & 0 & - \alpha^{16}_{10} \alpha^{6}_{4} \mu - \alpha^{18}_{4} \left(\alpha^{16}_{10} \left(\lambda + 2 \mu\right) - \alpha^{18}_{10} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{26}_{8} \left(\alpha^{16}_{10} \mu - \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{16}_{10} \left(- 2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10} \left(\lambda + 2 \mu\right) + \alpha^{26}_{10} \mu\right) - \alpha^{18}_{10} \left(\alpha^{16}_{10} \left(\lambda + 2 \mu\right) - \alpha^{18}_{10} \left(\lambda + 3 \mu\right)\right) - \alpha^{26}_{10} \left(\alpha^{16}_{10} \mu - \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0\\0 & 0 & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) & 0.5 \alpha^{17}_{11} \lambda + 1.5 \alpha^{17}_{11} \mu - 0.5 \alpha^{19}_{11} \mu - 0.5 \alpha^{27}_{11} \lambda - 1.0 \alpha^{27}_{11} \mu & 0 & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{19}_{11} \mu & 0 & 0 & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda + 1.0 \alpha^{27}_{11} \mu & 0 & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{3} \mu - \alpha^{17}_{11} \alpha^{7}_{3} \left(\lambda + 2 \mu\right) - \alpha^{17}_{3} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{17}_{11} \alpha^{7}_{5} \left(\lambda + 2 \mu\right) - \alpha^{19}_{5} \left(\alpha^{17}_{11} \mu - \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(\alpha^{17}_{11} \left(\lambda + 2 \mu\right) - \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{11} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \left(\alpha^{17}_{11} \mu - \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) - \alpha^{27}_{11} \left(\alpha^{17}_{11} \left(\lambda + 2 \mu\right) - \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right)\end{array}\right]\end{split}\]
\[\displaystyle rank(A_C)=24~(sympy)~\mathtt{\text{24}}~(numpy)\]
Which has a correct rank both with sympy and numpy.
Applying \(P\) operator and Dirichlet operator gives the following rhs \(B_C\):
\[\begin{split}\displaystyle B_C = XD_C+D_C^t.P^t.B-D_C^t.P^t.A.P.XD_C=\left[\begin{matrix}c_{x}\\c_{y}\\0\\0\\i_{x}\\0\\c_{x}\\0\\0\\0\\i_{x}\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\end{matrix}\right]+\left[\begin{matrix}0\\0\\- 0.5 L^{2} f\\0\\0\\0\\0\\0\\- 0.5 L^{2} f\\0\\0\\0\\\frac{L^{2} f \left(- \alpha^{10}_{0} - 4 \alpha^{12}_{0} - \alpha^{2}_{0}\right)}{12}\\0\\\frac{L^{2} f \left(- 2 \alpha^{14}_{2} - 4 \alpha^{16}_{2} - \alpha^{2}_{2} - \alpha^{6}_{2}\right)}{12}\\0\\\frac{L^{2} f \left(- \alpha^{18}_{4} - \alpha^{6}_{4}\right)}{12}\\0\\\frac{L^{2} f \left(- \alpha^{10}_{6} - \alpha^{22}_{6}\right)}{12}\\0\\\frac{L^{2} f \left(- 4 \alpha^{12}_{8} - 2 \alpha^{14}_{8} - \alpha^{22}_{8} - \alpha^{26}_{8}\right)}{12}\\0\\\frac{L^{2} f \left(- 4 \alpha^{16}_{10} - \alpha^{18}_{10} - \alpha^{26}_{10}\right)}{12}\\0\end{matrix}\right]-\left[\begin{matrix}0\\0\\- c_{x} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 c_{y} \mu - i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\0.5 c_{x} \lambda - 0.5 c_{y} \mu - 0.5 i_{x} \lambda\\0\\- 0.5 i_{x} \lambda\\0\\- 0.5 c_{x} \lambda - 0.5 c_{y} \lambda - 1.0 c_{y} \mu\\- c_{x} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 c_{y} \left(\lambda + \mu\right) - i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\0.5 \lambda \left(- c_{x} + i_{x}\right)\\0\\0.5 i_{x} \lambda\\0.5 \alpha^{10}_{0} c_{x} \lambda + 1.0 \alpha^{10}_{0} c_{x} \mu + 0.5 \alpha^{10}_{0} c_{y} \lambda - 0.5 \alpha^{12}_{0} c_{y} \lambda - 0.5 \alpha^{12}_{0} c_{y} \mu + 0.5 \alpha^{2}_{0} c_{y} \mu\\0.5 \alpha^{13}_{1} c_{y} \lambda + 1.5 \alpha^{13}_{1} c_{y} \mu + 0.5 \alpha^{3}_{1} c_{x} \lambda\\- 0.5 \alpha^{14}_{2} c_{x} \lambda - 1.0 \alpha^{14}_{2} c_{x} \mu - 0.5 \alpha^{14}_{2} i_{x} \lambda - 1.0 \alpha^{14}_{2} i_{x} \mu + 0.5 \alpha^{2}_{2} c_{y} \mu\\- 0.5 \alpha^{15}_{3} c_{y} \mu + 0.5 \alpha^{3}_{3} c_{x} \lambda - 0.5 \alpha^{7}_{3} i_{x} \lambda\\\alpha^{18}_{4} i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\- 0.5 \alpha^{7}_{5} i_{x} \lambda\\\alpha^{10}_{6} \left(0.5 c_{x} \lambda + 1.0 c_{x} \mu + 0.5 c_{y} \lambda\right)\\\alpha^{23}_{7} \left(- 0.5 c_{x} \lambda - 0.5 c_{y} \lambda - 1.0 c_{y} \mu\right)\\- 0.5 \alpha^{12}_{8} c_{y} \lambda - 0.5 \alpha^{12}_{8} c_{y} \mu - 0.5 \alpha^{14}_{8} c_{x} \lambda - 1.0 \alpha^{14}_{8} c_{x} \mu - 0.5 \alpha^{14}_{8} i_{x} \lambda - 1.0 \alpha^{14}_{8} i_{x} \mu\\0.5 \alpha^{13}_{9} c_{y} \lambda + 1.5 \alpha^{13}_{9} c_{y} \mu - 0.5 \alpha^{15}_{9} c_{y} \mu - 0.5 \alpha^{23}_{9} c_{x} \lambda - 0.5 \alpha^{23}_{9} c_{y} \lambda - 1.0 \alpha^{23}_{9} c_{y} \mu + 0.5 \alpha^{27}_{9} i_{x} \lambda\\\alpha^{18}_{10} i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\0.5 \alpha^{27}_{11} i_{x} \lambda\end{matrix}\right]=\left[\begin{matrix}c_{x}\\c_{y}\\- 0.5 L^{2} f + c_{x} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 c_{y} \mu + i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\- 0.5 c_{x} \lambda + 0.5 c_{y} \mu + 0.5 i_{x} \lambda\\i_{x}\\0.5 i_{x} \lambda\\c_{x}\\0.5 c_{x} \lambda + 0.5 c_{y} \lambda + 1.0 c_{y} \mu\\- 0.5 L^{2} f + c_{x} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 c_{y} \left(\lambda + \mu\right) + i_{x} \left(0.5 \lambda + 1.0 \mu\right)\\0.5 \lambda \left(c_{x} - i_{x}\right)\\i_{x}\\- 0.5 i_{x} \lambda\\- 0.0833333333333333 L^{2} \alpha^{10}_{0} f - 0.333333333333333 L^{2} \alpha^{12}_{0} f - 0.0833333333333333 L^{2} \alpha^{2}_{0} f - 0.5 \alpha^{10}_{0} c_{x} \lambda - 1.0 \alpha^{10}_{0} c_{x} \mu - 0.5 \alpha^{10}_{0} c_{y} \lambda + 0.5 \alpha^{12}_{0} c_{y} \lambda + 0.5 \alpha^{12}_{0} c_{y} \mu - 0.5 \alpha^{2}_{0} c_{y} \mu\\- 0.5 \alpha^{13}_{1} c_{y} \lambda - 1.5 \alpha^{13}_{1} c_{y} \mu - 0.5 \alpha^{3}_{1} c_{x} \lambda\\- 0.166666666666667 L^{2} \alpha^{14}_{2} f - 0.333333333333333 L^{2} \alpha^{16}_{2} f - 0.0833333333333333 L^{2} \alpha^{2}_{2} f - 0.0833333333333333 L^{2} \alpha^{6}_{2} f + 0.5 \alpha^{14}_{2} c_{x} \lambda + 1.0 \alpha^{14}_{2} c_{x} \mu + 0.5 \alpha^{14}_{2} i_{x} \lambda + 1.0 \alpha^{14}_{2} i_{x} \mu - 0.5 \alpha^{2}_{2} c_{y} \mu\\0.5 \alpha^{15}_{3} c_{y} \mu - 0.5 \alpha^{3}_{3} c_{x} \lambda + 0.5 \alpha^{7}_{3} i_{x} \lambda\\- 0.0833333333333333 L^{2} \alpha^{18}_{4} f - 0.0833333333333333 L^{2} \alpha^{6}_{4} f - 0.5 \alpha^{18}_{4} i_{x} \lambda - 1.0 \alpha^{18}_{4} i_{x} \mu\\0.5 \alpha^{7}_{5} i_{x} \lambda\\- 0.0833333333333333 L^{2} \alpha^{10}_{6} f - 0.0833333333333333 L^{2} \alpha^{22}_{6} f - 0.5 \alpha^{10}_{6} c_{x} \lambda - 1.0 \alpha^{10}_{6} c_{x} \mu - 0.5 \alpha^{10}_{6} c_{y} \lambda\\\alpha^{23}_{7} \left(0.5 c_{x} \lambda + 0.5 c_{y} \lambda + 1.0 c_{y} \mu\right)\\- 0.333333333333333 L^{2} \alpha^{12}_{8} f - 0.166666666666667 L^{2} \alpha^{14}_{8} f - 0.0833333333333333 L^{2} \alpha^{22}_{8} f - 0.0833333333333333 L^{2} \alpha^{26}_{8} f + 0.5 \alpha^{12}_{8} c_{y} \lambda + 0.5 \alpha^{12}_{8} c_{y} \mu + 0.5 \alpha^{14}_{8} c_{x} \lambda + 1.0 \alpha^{14}_{8} c_{x} \mu + 0.5 \alpha^{14}_{8} i_{x} \lambda + 1.0 \alpha^{14}_{8} i_{x} \mu\\- 0.5 \alpha^{13}_{9} c_{y} \lambda - 1.5 \alpha^{13}_{9} c_{y} \mu + 0.5 \alpha^{15}_{9} c_{y} \mu + 0.5 \alpha^{23}_{9} c_{x} \lambda + 0.5 \alpha^{23}_{9} c_{y} \lambda + 1.0 \alpha^{23}_{9} c_{y} \mu - 0.5 \alpha^{27}_{9} i_{x} \lambda\\- 0.333333333333333 L^{2} \alpha^{16}_{10} f - 0.0833333333333333 L^{2} \alpha^{18}_{10} f - 0.0833333333333333 L^{2} \alpha^{26}_{10} f - 0.5 \alpha^{18}_{10} i_{x} \lambda - 1.0 \alpha^{18}_{10} i_{x} \mu\\- 0.5 \alpha^{27}_{11} i_{x} \lambda\end{matrix}\right]\end{split}\]
Solution \(S_C\) at coarse scale is then:
\[\displaystyle S_C=inv(A_C).B_C = ~too ~ long\]
\[\begin{split}\displaystyle S_C=inv(A_C).B_C = \left[\begin{matrix}0\\0\\0.00614326189494154\\0.018081452395792\\0.1\\0.0380781929445188\\0\\0.00724250911379751\\0.00396377827487203\\-0.00769897479386017\\0.1\\-0.0247091505966851\\1.00708135110216\\-2.6666448981649\\0.732917367077309\\0.749406352678771\\0.823322159966413\\1.44319691101113\\-0.625790157446757\\1.60722211707393\\0.82004774021788\\0.498104754777559\\0.865983109176288\\1.01705784933148\end{matrix}\right] <> SG_C=\left[\begin{matrix}0\\0\\-0.0027050143182574\\0.0181453708682113\\0.0176677840033592\\0.0167810163063337\\0\\0.0237053745491031\\-0.00770399228565553\\0.00373160917788545\\0.0134016890823716\\0.00987838156444837\\1.00708135110217\\-2.66664489816481\\0.732917367077305\\0.749406352678779\\0.823322159966408\\1.44319691101113\\-0.625790157446831\\1.60722211707393\\0.820047740217874\\0.498104754777565\\0.865983109176284\\1.01705784933149\end{matrix}\right] => S_C-SG_C = \left[\begin{matrix}0\\0\\0.00884827621319895\\-6.39184724192977 \cdot 10^{-5}\\0.0823322159966408\\0.0212971766381851\\0\\-0.0164628654353056\\0.0116677705605276\\-0.0114305839717456\\0.0865983109176284\\-0.0345875321611335\\-8.65973959207622 \cdot 10^{-15}\\-8.97060203897126 \cdot 10^{-14}\\3.99680288865056 \cdot 10^{-15}\\-7.99360577730113 \cdot 10^{-15}\\4.88498130835069 \cdot 10^{-15}\\1.55431223447522 \cdot 10^{-15}\\7.39408534400354 \cdot 10^{-14}\\4.88498130835069 \cdot 10^{-15}\\6.43929354282591 \cdot 10^{-15}\\-6.10622663543836 \cdot 10^{-15}\\4.32986979603811 \cdot 10^{-15}\\-4.88498130835069 \cdot 10^{-15}\end{matrix}\right]\end{split}\]
Note that \(Sc\) an \(SG_C\) are note the same numericaly as enrichment function are different.
With shift enrichment standard dofs represent the physical displacement which is not the case in general case. Nevertheless enriched dofs are the same. Why ? Euhhh … Erichement functions have the same shape and are just a translation of one another. So we can imagine that only the shape impose the solution of enriched dofs ….
Any way those two solutions projected with their operators gives the same approximates fine scale field:
\[\begin{split}\displaystyle Sts_1 = \left[\begin{matrix}0\\0\\-0.000916510248535682\\0.00374772269118444\\0.00614326189494154\\0.018081452395792\\0.0477047963892676\\0.031738345073715\\0.1\\0.0380781929445188\\0\\0.00392113289309548\\-0.0109066050936931\\0.00215902409527212\\0.00852213428228484\\0.00607048597299588\\0.0384613101043138\\0.00868008971325389\\0.1\\0.00705161509602248\\0\\0.00724250911379751\\-0.00354610576246846\\0.00253203966139497\\0.00396377827487203\\-0.00769897479386017\\0.0463201267033249\\-0.0134931274931402\\0.1\\-0.0247091505966851\end{matrix}\right] <> SGts_1 = \left[\begin{matrix}0\\0\\-0.000916510248535611\\0.00374772269118456\\0.00614326189494164\\0.018081452395792\\0.0477047963892677\\0.0317383450737151\\0.1\\0.0380781929445188\\0\\0.00392113289309546\\-0.010906605093693\\0.0021590240952722\\0.0085221342822849\\0.00607048597299595\\0.0384613101043139\\0.00868008971325392\\0.1\\0.00705161509602254\\0\\0.00724250911379728\\-0.00354610576246848\\0.00253203966139495\\0.00396377827487211\\-0.00769897479386007\\0.0463201267033249\\-0.0134931274931401\\0.1\\-0.0247091505966852\end{matrix}\right] <> Sts_1-SGts_1=\left[\begin{matrix}0\\0\\-7.0906822080552 \cdot 10^{-17}\\-1.20129600711394 \cdot 10^{-16}\\-1.02348685082632 \cdot 10^{-16}\\-2.42861286636753 \cdot 10^{-17}\\-8.32667268468867 \cdot 10^{-17}\\-6.93889390390723 \cdot 10^{-17}\\0\\0\\0\\1.56125112837913 \cdot 10^{-17}\\-6.76542155630955 \cdot 10^{-17}\\-8.15320033709099 \cdot 10^{-17}\\-6.93889390390723 \cdot 10^{-17}\\-7.71951946809679 \cdot 10^{-17}\\-1.2490009027033 \cdot 10^{-16}\\-2.42861286636753 \cdot 10^{-17}\\0\\-6.41847686111419 \cdot 10^{-17}\\0\\2.30718222304915 \cdot 10^{-16}\\1.38777878078145 \cdot 10^{-17}\\1.30104260698261 \cdot 10^{-17}\\-7.97972798949331 \cdot 10^{-17}\\-9.71445146547012 \cdot 10^{-17}\\-2.77555756156289 \cdot 10^{-17}\\-2.42861286636753 \cdot 10^{-17}\\0\\5.20417042793042 \cdot 10^{-17}\end{matrix}\right]\end{split}\]
Compare inital and new approximation with fine solution give:
\[\begin{split}\displaystyle Sts_1-S_f=\left[\begin{matrix}0\\0\\0.00160116348762545\\-0.00461562264695081\\-7.92271006566994 \cdot 10^{-5}\\-0.00093948931422538\\0.000222470125428746\\0.00205980699181559\\0\\3.63095244841419 \cdot 10^{-5}\\0\\-0.00259980881692186\\0.00141499021109549\\-0.00436191761474522\\0.0015793571714405\\-0.000450455737021464\\0.000782905409102377\\0.00215914800323655\\0\\0.000530673386005136\\0\\-0.00579937430623717\\-0.00102843202630733\\-0.00214649842050446\\-0.00225871072072621\\-0.00171991650387751\\-0.00116219956051399\\0.00314352716872458\\0\\0.000290849403314884\end{matrix}\right] ~~Sts_0-S_f=\left[\begin{matrix}0\\0\\0.00876767373616113\\-0.00419667867146859\\0.00627751100440176\\-0.010687608376684\\0.00876767373616114\\-0.0171785380818994\\0\\-0.021375216753368\\0\\-0.010687608376684\\0.0185715953047886\\-0.0148542750433507\\0.00555722288915567\\-0.010687608376684\\0.0185715953047886\\-0.0148542750433507\\0\\-0.010687608376684\\0\\-0.021375216753368\\0.00876767373616113\\-0.0171785380818994\\0.00627751100440176\\-0.010687608376684\\0.00876767373616114\\-0.00419667867146859\\0\\0\end{matrix}\right]\end{split}\]
\[\displaystyle norm(Sts_1-S_f)=0.0111250318987568 ~~norm(Sts_0-S_f)=0.060486741655273\]
The residual norm of those approximation are:
\[\displaystyle norm(AD.Sts_0-BD)/norm(BD)=0.461736312807732\]
\[\displaystyle norm(AD.Sts_1-BD)/norm(BD)=0.0585475777430638\]
Doing more up-down symbolic Two Scale resolution gives:
\[\displaystyle norm(AD.Sts_2-BD)/norm(BD)=0.00928585388296698\]
\[\displaystyle norm(AD.Sts_3-BD)/norm(BD)=0.0024904359828855\]
\[\displaystyle norm(AD.Sts_4-BD)/norm(BD)=0.00034554800393268\]
\[\displaystyle norm(AD.Sts_5-BD)/norm(BD)=6.81137405848145 \cdot 10^{-5}\]
\[\displaystyle norm(AD.Sts_6-BD)/norm(BD)=1.32608407543472 \cdot 10^{-5}\]
\[\displaystyle norm(AD.Sts_7-BD)/norm(BD)=2.36932874445921 \cdot 10^{-6}\]
\[\displaystyle norm(AD.Sts_8-BD)/norm(BD)=4.72126125814533 \cdot 10^{-7}\]
\[\displaystyle norm(AD.Sts_9-BD)/norm(BD)=8.89794132129813 \cdot 10^{-8}\]
Text(0, 0.5, '$norm(AD.Sts_i-BD)/norm(BD)$')
Comparatively the direct fine scale resolution gives with direct numeric(simpy) resolution:
\[\displaystyle norm(AD.S_F-BD)/norm(BD)=1.62430021152726 \cdot 10^{-16}\]
TS approach (II) in shift case#
From an implementation point of view it would be efficient if the TwoScale solver use \(AD\) and \(BD\) directly avoiding creation/storing \(A\),\(B\).
Thus on this specific case we check here that this approch does not work.
First apply \(P\) (or \(PG\), but here we will check only the shift case):
\(AP_C^2=P^t.AD.P\)
\(BP_C^2=P^t.BD\)
\[\begin{split}\displaystyle AP_C^2 = \left[\begin{array}{cccccccccccccccccccccccc}0.75 \lambda + 1.75 \mu + 1.25 & 0 & - 0.25 \lambda - 0.5 \mu & 0.25 \lambda & 0 & 0 & 0.25 - 0.25 \mu & 0.25 \lambda + 0.25 \mu & 0.25 \lambda + 0.5 \mu & - 0.25 \lambda - 0.25 \mu & 0 & 0 & 0.5 \alpha^{10}_{0} + \alpha^{12}_{0} \left(1.0 \lambda + 2.5 \mu\right) + \alpha^{2}_{0} \left(0.5 \lambda + 1.0 \mu\right) & 0 & \left(- \alpha^{14}_{2} + \alpha^{2}_{2}\right) \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0.5 \alpha^{10}_{6} - 0.5 \alpha^{22}_{6} \mu & 0 & \alpha^{12}_{8} \left(1.0 \lambda + 2.5 \mu\right) - \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{22}_{8} \mu & 0 & 0 & 0\\0 & 0.5 \lambda + 1.5 \mu + 1.0 & 0.25 \mu & - 0.25 \mu & 0 & 0 & 0 & - 0.25 \lambda - 0.5 \mu & - 0.25 \lambda - 0.25 \mu & 0 & 0 & 0 & 0 & \alpha^{11}_{1} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{3}_{1} \mu & 0 & 0.5 \mu \left(- \alpha^{15}_{3} + \alpha^{3}_{3}\right) & 0 & 0 & 0 & \left(\alpha^{11}_{7} - \alpha^{23}_{7}\right) \left(0.5 \lambda + 1.0 \mu\right) & 0 & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{15}_{9} \mu - \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0\\- 0.25 \lambda - 0.5 \mu & 0.25 \mu & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda & 0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0.25 \lambda + 0.5 \mu & - 0.5 \lambda - 0.5 \mu & - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{0} \mu & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & 0 & 0.5 \alpha^{7}_{5} \lambda & 0 & 0 & - \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{8} \mu & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{26}_{10} \mu & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right)\\0.25 \lambda & - 0.25 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0.25 \mu & - 0.5 \mu & 0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & - 0.25 \lambda - 0.25 \mu & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{6}_{4} \mu & - 0.5 \alpha^{19}_{5} \mu & 0 & 0 & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{19}_{11} \mu - \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & 0 & 0.25 \mu & 0.25 \lambda + 0.75 \mu + 1.25 & - 0.25 \mu & 0 & 0 & 0 & 0 & 0.25 - 0.25 \mu & 0 & 0 & 0 & - 0.5 \alpha^{16}_{2} \mu + \alpha^{6}_{2} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0.5 \alpha^{18}_{4} + \alpha^{6}_{4} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} & 0\\0 & 0 & 0.5 \lambda & - 0.5 \mu & - 0.25 \mu & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0.25 \lambda + 0.25 \mu & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu & - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{19}_{11} \mu\\0.25 - 0.25 \mu & 0 & 0 & 0 & 0 & 0 & 0.25 \lambda + 0.75 \mu + 1.25 & - 0.25 \mu & 0 & 0.25 \mu & 0 & 0 & 0.5 \alpha^{10}_{0} - 0.5 \alpha^{12}_{0} \mu & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{6} + \alpha^{22}_{6} \left(0.5 \lambda + 1.5 \mu\right) & 0 & - 0.5 \alpha^{12}_{8} \mu + \alpha^{22}_{8} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0\\0.25 \lambda + 0.25 \mu & - 0.25 \lambda - 0.5 \mu & 0 & 0 & 0 & 0 & - 0.25 \mu & 0.5 \lambda + 1.5 \mu & 0.5 \lambda & - 0.5 \mu & 0 & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0.5 \alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{22}_{6} \mu & 0.5 \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0\\0.25 \lambda + 0.5 \mu & - 0.25 \lambda - 0.25 \mu & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & 0 & 0 & 0.5 \lambda & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & - 0.25 \lambda - 0.5 \mu & 0.5 \mu & \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0.5 \alpha^{23}_{7} \lambda & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & - \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{26}_{10} \mu & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda\\- 0.25 \lambda - 0.25 \mu & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0.25 \mu & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0.25 \lambda & - 0.5 \mu & - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{22}_{6} \mu & - 0.5 \alpha^{11}_{7} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & 0.25 \lambda + 0.5 \mu & - 0.25 \lambda - 0.25 \mu & 0.25 - 0.25 \mu & 0.25 \lambda + 0.25 \mu & 0 & 0 & - 0.25 \lambda - 0.5 \mu & 0.25 \lambda & 0.75 \lambda + 1.75 \mu + 1.25 & - 0.25 \lambda & 0 & 0 & - \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(1.0 \lambda + 2.5 \mu\right) - 0.5 \alpha^{6}_{2} \mu & 0 & 0.5 \alpha^{18}_{4} - 0.5 \alpha^{6}_{4} \mu & 0 & 0 & 0 & \left(- \alpha^{14}_{8} + \alpha^{26}_{8}\right) \left(0.5 \lambda + 1.0 \mu\right) & 0 & \alpha^{16}_{10} \left(1.0 \lambda + 2.5 \mu\right) + 0.5 \alpha^{18}_{10} + \alpha^{26}_{10} \left(0.5 \lambda + 1.0 \mu\right) & 0\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & - 0.25 \lambda & 0.5 \lambda + 1.5 \mu & 0 & 0 & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0 & - \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{15}_{9} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right)\\0.5 \alpha^{10}_{0} + 1.0 \alpha^{12}_{0} \lambda + 2.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \lambda + 1.0 \alpha^{2}_{0} \mu & 0 & - 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \mu & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & 0 & 0 & 0.5 \alpha^{10}_{0} - 0.5 \alpha^{12}_{0} \mu & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0.5 \alpha^{12}_{0} \lambda + 1.5 \alpha^{12}_{0} \mu - 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0 & 0 & \left(\alpha^{10}_{0}\right)^{2} + \alpha^{12}_{0} \left(2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) - \alpha^{2}_{0} \mu\right) - \alpha^{2}_{0} \left(\alpha^{12}_{0} \mu - \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{12}_{0} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{2}_{2} \left(\alpha^{12}_{0} \mu - \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & \alpha^{10}_{0} \alpha^{10}_{6} - \alpha^{12}_{0} \alpha^{22}_{6} \mu & 0 & - \alpha^{12}_{0} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \alpha^{22}_{8} \mu + \alpha^{12}_{8} \left(2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) - \alpha^{2}_{0} \mu\right) & 0 & 0 & 0\\0 & 0.5 \alpha^{11}_{1} \lambda + 1.0 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \mu & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \lambda + 1.0 \alpha^{3}_{1} \mu & 0 & 0 & 0 & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu & - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu - 0.5 \alpha^{3}_{1} \lambda - 1.0 \alpha^{3}_{1} \mu & 0 & 0 & 0 & \alpha^{11}_{1} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) - \alpha^{13}_{1} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \left(\alpha^{13}_{1} \left(\lambda + 2 \mu\right) - \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{3} \mu - \alpha^{3}_{3} \left(\alpha^{13}_{1} \left(\lambda + 2 \mu\right) - \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{13}_{1} \alpha^{23}_{7} \left(\lambda + 2 \mu\right) + \alpha^{11}_{7} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{9} \mu - \alpha^{13}_{1} \alpha^{23}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\- 0.5 \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{2}_{2} \left(\lambda + 3 \mu\right) & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & - 0.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{6}_{2} \left(\lambda + 3 \mu\right) & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu & 0 & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu - 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - 0.5 \alpha^{14}_{2} \lambda - 1.0 \alpha^{14}_{2} \mu + 1.0 \alpha^{16}_{2} \lambda + 2.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{6}_{2} \mu & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - \alpha^{12}_{0} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \alpha^{2}_{0} \alpha^{2}_{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{14}_{2} \left(2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) - \alpha^{16}_{2} \left(\lambda + 2 \mu\right)\right) - \alpha^{16}_{2} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) + \left(\alpha^{2}_{2}\right)^{2} \left(\lambda + 3 \mu\right) - \alpha^{6}_{2} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{6}_{4} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{16}_{2} \alpha^{26}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \alpha^{14}_{8} \left(2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) - \alpha^{16}_{2} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{16}_{10} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) - \alpha^{26}_{10} \alpha^{16}_{2} \mu & 0\\0 & 0.5 \mu \left(- \alpha^{15}_{3} + \alpha^{3}_{3}\right) & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{3}_{3} \lambda + 1.0 \alpha^{3}_{3} \mu & 0 & - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{7}_{3} \lambda + 1.0 \alpha^{7}_{3} \mu & 0 & 0 & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{3}_{3} \lambda - 1.0 \alpha^{3}_{3} \mu & 0 & - 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{7}_{3} \lambda - 1.0 \alpha^{7}_{3} \mu & 0 & - \alpha^{13}_{1} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \alpha^{3}_{1} \alpha^{3}_{3} \left(\lambda + 3 \mu\right) & 0 & \alpha^{15}_{3} \left(2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) - \alpha^{17}_{3} \mu\right) - \alpha^{17}_{3} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) + \left(\alpha^{3}_{3}\right)^{2} \left(\lambda + 3 \mu\right) - \alpha^{7}_{3} \left(\alpha^{17}_{3} \left(\lambda + 2 \mu\right) - \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{3} \alpha^{19}_{5} \mu - \alpha^{7}_{5} \left(\alpha^{17}_{3} \left(\lambda + 2 \mu\right) - \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{3} \alpha^{27}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \alpha^{15}_{9} \left(2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) - \alpha^{17}_{3} \mu\right) & 0 & - \alpha^{17}_{11} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \alpha^{17}_{3} \mu - \alpha^{27}_{11} \alpha^{17}_{3} \left(\lambda + 2 \mu\right)\\0 & 0 & 0 & 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{18}_{4} + 0.5 \alpha^{6}_{4} \left(\lambda + 3 \mu\right) & - 0.5 \alpha^{6}_{4} \mu & 0 & 0 & 0 & 0 & 0.5 \alpha^{18}_{4} - 0.5 \alpha^{6}_{4} \mu & 0 & 0 & 0 & \alpha^{6}_{4} \left(- \alpha^{16}_{2} \mu + \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & \left(\alpha^{18}_{4}\right)^{2} + \left(\alpha^{6}_{4}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - \alpha^{16}_{10} \alpha^{6}_{4} \mu + \alpha^{18}_{10} \alpha^{18}_{4} & 0\\0 & 0 & 0.5 \alpha^{7}_{5} \lambda & - 0.5 \alpha^{19}_{5} \mu & 0 & 0.5 \alpha^{19}_{5} \mu + 0.5 \alpha^{7}_{5} \lambda + 1.0 \alpha^{7}_{5} \mu & 0 & 0 & 0 & 0 & 0 & \alpha^{7}_{5} \left(- 0.5 \lambda - 1.0 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{3} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \alpha^{7}_{3} \alpha^{7}_{5} \left(\lambda + 3 \mu\right) & 0 & \left(\left(\alpha^{19}_{5}\right)^{2} + \left(\alpha^{7}_{5}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - \alpha^{17}_{11} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \alpha^{19}_{11} \alpha^{19}_{5} \left(\lambda + 3 \mu\right)\\0.5 \alpha^{10}_{6} - 0.5 \alpha^{22}_{6} \mu & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{6} + 0.5 \alpha^{22}_{6} \left(\lambda + 3 \mu\right) & - 0.5 \alpha^{22}_{6} \mu & 0 & 0.5 \alpha^{22}_{6} \mu & 0 & 0 & \alpha^{10}_{0} \alpha^{10}_{6} - \alpha^{12}_{0} \alpha^{22}_{6} \mu & 0 & 0 & 0 & 0 & 0 & \left(\alpha^{10}_{6}\right)^{2} + \left(\alpha^{22}_{6}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{22}_{6} \left(- \alpha^{12}_{8} \mu + \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0\\0 & - 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - 0.5 \alpha^{23}_{7} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{23}_{7} \lambda + 1.0 \alpha^{23}_{7} \mu & 0.5 \alpha^{23}_{7} \lambda & - 0.5 \alpha^{11}_{7} \mu & 0 & 0 & 0 & \alpha^{11}_{1} \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) & 0 & 0 & 0 & 0 & 0 & \left(\left(\alpha^{11}_{7}\right)^{2} + \left(\alpha^{23}_{7}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & \alpha^{23}_{7} \alpha^{23}_{9} \left(\lambda + 3 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\1.0 \alpha^{12}_{8} \lambda + 2.5 \alpha^{12}_{8} \mu - 0.5 \alpha^{14}_{8} \lambda - 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{22}_{8} \mu & 0 & - 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & 0 & 0 & - 0.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{22}_{8} \left(\lambda + 3 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu & 0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu + 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - 0.5 \alpha^{26}_{8} \mu + 0.5 \alpha^{26}_{8} \left(\lambda + 3 \mu\right) & 0.5 \alpha^{26}_{8} \mu & - \alpha^{12}_{0} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{2}_{0} \alpha^{12}_{8} \mu & 0 & - \alpha^{14}_{2} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right)\right) - \alpha^{16}_{2} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) - \alpha^{2}_{2} \alpha^{12}_{8} \mu & 0 & 0 & 0 & - \alpha^{22}_{6} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{12}_{8} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{14}_{8} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right)\right) - \alpha^{22}_{8} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{26}_{8}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{16}_{10} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) + \alpha^{26}_{10} \alpha^{26}_{8} \left(\lambda + 3 \mu\right) & 0\\0 & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu - 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{23}_{9} \lambda - 1.0 \alpha^{23}_{9} \mu & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{27}_{9} \lambda - 1.0 \alpha^{27}_{9} \mu & 0 & 0 & 0 & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{23}_{9} \lambda + 1.0 \alpha^{23}_{9} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu + 0.5 \alpha^{27}_{9} \lambda + 1.0 \alpha^{27}_{9} \mu & 0 & - 0.5 \alpha^{15}_{9} \mu & 0 & - \alpha^{11}_{1} \alpha^{13}_{9} \mu - \alpha^{13}_{1} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & 0 & - \alpha^{15}_{3} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right)\right) - \alpha^{17}_{3} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{3} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & - \alpha^{11}_{7} \alpha^{13}_{9} \mu - \alpha^{23}_{7} \left(\alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{9} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{15}_{9} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right)\right) - \alpha^{23}_{9} \left(\alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{27}_{9}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{17}_{11} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) + \alpha^{27}_{11} \alpha^{27}_{9} \left(\lambda + 3 \mu\right)\\0 & 0 & 0.5 \alpha^{16}_{10} \lambda + 1.5 \alpha^{16}_{10} \mu - 0.5 \alpha^{26}_{10} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & - 0.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & 0 & 0 & - 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{26}_{10} \mu & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu & 1.0 \alpha^{16}_{10} \lambda + 2.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} + 0.5 \alpha^{26}_{10} \lambda + 1.0 \alpha^{26}_{10} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{26}_{10} \mu & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{16}_{10} \alpha^{6}_{2} \mu + \alpha^{16}_{2} \left(2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) - \alpha^{26}_{10} \mu\right) & 0 & - \alpha^{16}_{10} \alpha^{6}_{4} \mu + \alpha^{18}_{10} \alpha^{18}_{4} & 0 & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{26}_{8} \left(\alpha^{16}_{10} \mu - \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0 & \alpha^{16}_{10} \left(2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) - \alpha^{26}_{10} \mu\right) + \left(\alpha^{18}_{10}\right)^{2} - \alpha^{26}_{10} \left(\alpha^{16}_{10} \mu - \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0\\0 & 0 & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) & 0.5 \alpha^{17}_{11} \lambda + 1.5 \alpha^{17}_{11} \mu - 0.5 \alpha^{19}_{11} \mu - 0.5 \alpha^{27}_{11} \lambda - 1.0 \alpha^{27}_{11} \mu & 0 & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{19}_{11} \mu & 0 & 0 & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda + 1.0 \alpha^{27}_{11} \mu & 0 & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{3} \mu - \alpha^{17}_{11} \alpha^{7}_{3} \left(\lambda + 2 \mu\right) - \alpha^{17}_{3} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{17}_{11} \alpha^{7}_{5} \left(\lambda + 2 \mu\right) - \alpha^{19}_{5} \left(\alpha^{17}_{11} \mu - \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(\alpha^{17}_{11} \left(\lambda + 2 \mu\right) - \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{11} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \left(\alpha^{17}_{11} \mu - \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) - \alpha^{27}_{11} \left(\alpha^{17}_{11} \left(\lambda + 2 \mu\right) - \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right)\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle BP_C^2 = \left[\begin{matrix}- 0.208333333333333 L^{2} f + 0.75 c_{x} \lambda + 1.5 c_{x} \mu + 1.5 c_{x} + 0.25 c_{y} \lambda\\- 0.25 c_{x} \lambda + 0.25 c_{y} \lambda + 0.75 c_{y} \mu + 1.0 c_{y}\\- 0.5 L^{2} f + 0.25 c_{x} \left(\lambda + 2 \mu\right) - 0.25 c_{y} \mu + 0.75 i_{x} \left(\lambda + 2 \mu\right)\\- 0.25 c_{x} \lambda + 0.25 c_{y} \mu + 0.25 i_{x} \lambda\\- 0.0416666666666667 L^{2} f + 0.25 i_{x} \left(\lambda + 2 \mu\right) + 1.5 i_{x}\\0.75 i_{x} \lambda\\- 0.0416666666666667 L^{2} f + 0.25 c_{x} \left(\lambda + 2 \mu\right) + 1.5 c_{x}\\0.75 c_{x} \lambda + 0.25 c_{y} \left(\lambda + 2 \mu\right)\\- 0.5 L^{2} f + 0.75 c_{x} \left(\lambda + 2 \mu\right) + 0.25 c_{y} \left(\lambda + \mu\right) + 0.25 i_{x} \left(\lambda + 2 \mu\right)\\0.25 \lambda \left(c_{x} - i_{x}\right)\\- 0.208333333333333 L^{2} f + 0.75 i_{x} \left(\lambda + 2 \mu\right) + 1.5 i_{x}\\- 0.75 i_{x} \lambda\\\alpha^{10}_{0} c_{x} + \frac{\alpha^{12}_{0} \left(- 2 L^{2} f + 6 c_{x} \left(\lambda + 2 \mu\right) + 3 c_{y} \left(\lambda + \mu\right)\right)}{6} - \frac{\alpha^{2}_{0} \left(L^{2} f - 6 c_{x} \left(\lambda + 2 \mu\right) + 6 c_{y} \mu\right)}{12}\\\frac{\alpha^{11}_{1} c_{y} \left(\lambda + 2 \mu\right)}{2} - \frac{\alpha^{3}_{1} \left(c_{x} \lambda - c_{y} \mu\right)}{2}\\- \frac{L^{2} \alpha^{14}_{2} f}{6} - \frac{\alpha^{16}_{2} \left(L^{2} f - 3 i_{x} \left(\lambda + 2 \mu\right)\right)}{3} - \frac{\alpha^{2}_{2} \left(L^{2} f - 6 c_{x} \left(\lambda + 2 \mu\right) + 6 c_{y} \mu\right)}{12} - \frac{\alpha^{6}_{2} \left(L^{2} f - 6 i_{x} \left(\lambda + 2 \mu\right)\right)}{12}\\- \frac{\alpha^{3}_{3} \left(c_{x} \lambda - c_{y} \mu\right)}{2} + \frac{\alpha^{7}_{3} i_{x} \lambda}{2}\\\alpha^{18}_{4} i_{x} - \frac{\alpha^{6}_{4} \left(L^{2} f - 6 i_{x} \left(\lambda + 2 \mu\right)\right)}{12}\\\frac{\alpha^{7}_{5} i_{x} \lambda}{2}\\\alpha^{10}_{6} c_{x} - \frac{\alpha^{22}_{6} \left(L^{2} f - 6 c_{x} \left(\lambda + 2 \mu\right)\right)}{12}\\\frac{\alpha^{11}_{7} c_{y} \left(\lambda + 2 \mu\right)}{2} + \frac{\alpha^{23}_{7} c_{x} \lambda}{2}\\- \frac{L^{2} \alpha^{14}_{8} f}{6} + \frac{\alpha^{12}_{8} \left(- 2 L^{2} f + 6 c_{x} \left(\lambda + 2 \mu\right) + 3 c_{y} \left(\lambda + \mu\right)\right)}{6} - \frac{\alpha^{22}_{8} \left(L^{2} f - 6 c_{x} \left(\lambda + 2 \mu\right)\right)}{12} - \frac{\alpha^{26}_{8} \left(L^{2} f - 6 i_{x} \left(\lambda + 2 \mu\right)\right)}{12}\\\frac{\lambda \left(\alpha^{23}_{9} c_{x} - \alpha^{27}_{9} i_{x}\right)}{2}\\- \frac{\alpha^{16}_{10} \left(L^{2} f - 3 i_{x} \left(\lambda + 2 \mu\right)\right)}{3} + \alpha^{18}_{10} i_{x} - \frac{\alpha^{26}_{10} \left(L^{2} f - 6 i_{x} \left(\lambda + 2 \mu\right)\right)}{12}\\- \frac{\alpha^{27}_{11} i_{x} \lambda}{2}\end{matrix}\right]\end{split}\]
\[\displaystyle rank(AP_C^2)=24~(sympy)~\mathtt{\text{24}}~(numpy)\]
Matrix is numericaly full rank with clearly wrong terms comming from the fact that some row/columns in \(AD\) has been replace by identity diagonal sub matrix:
\[\begin{split}\displaystyle AP_C^2-AP_C = \left[\begin{array}{cccccccccccccccccccccccc}0.25 \lambda + 0.25 \mu + 1.25 & 0 & 0.25 \lambda + 0.5 \mu & - 0.25 \lambda & 0 & 0 & 0.25 \mu + 0.25 & 0.25 \lambda - 0.25 \mu & 0.25 \lambda + 0.5 \mu & 0.25 \lambda + 0.25 \mu & 0 & 0 & 0.5 \alpha^{10}_{0} + 0.5 \alpha^{12}_{0} \lambda + 1.0 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \lambda + 1.0 \alpha^{2}_{0} \mu & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & \alpha^{2}_{2} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{3}_{3} \lambda & 0 & 0 & 0.5 \alpha^{10}_{6} & - 0.5 \alpha^{11}_{7} \mu & \alpha^{12}_{8} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & 0 & 0\\0 & 1.0 & - 0.25 \mu & 0.25 \mu & 0 & 0 & - 0.5 \lambda & 0.25 \lambda + 0.5 \mu & 0.25 \lambda + 0.25 \mu & 0 & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & \alpha^{11}_{1} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{3}_{1} \mu & - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{3}_{3} \mu & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda & \alpha^{11}_{7} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & 0 & 0 & 0\\0.25 \lambda + 0.5 \mu & - 0.25 \mu & 0 & 0 & 0.5 \lambda + 1.0 \mu & 0 & 0 & 0 & 0 & 0 & 0.25 \lambda + 0.5 \mu & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) & 0\\- 0.25 \lambda & 0.25 \mu & 0 & 0 & - 0.25 \mu & 0 & 0 & 0 & 0 & 0 & 0.25 \lambda + 0.25 \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 \lambda + 1.0 \mu & - 0.25 \mu & - 0.25 \lambda - 0.75 \mu + 1.25 & 0.5 \lambda + 0.25 \mu & 0 & 0 & 0 & 0 & 0.25 \mu + 0.25 & - 0.5 \lambda & 0 & 0 & 0.5 \alpha^{16}_{2} \lambda + 1.0 \alpha^{16}_{2} \mu + 0.5 \alpha^{6}_{2} \lambda + 1.0 \alpha^{6}_{2} \mu & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) + 0.5 \alpha^{7}_{3} \lambda & - 0.5 \alpha^{18}_{4} \lambda - 1.0 \alpha^{18}_{4} \mu + 0.5 \alpha^{18}_{4} + 0.5 \alpha^{6}_{4} \lambda + 1.0 \alpha^{6}_{4} \mu & 0.5 \alpha^{19}_{5} \mu + 0.5 \alpha^{7}_{5} \lambda & 0 & 0 & 0 & 0 & 0.5 \alpha^{16}_{10} \lambda + 1.0 \alpha^{16}_{10} \mu - 0.5 \alpha^{18}_{10} \lambda - 1.0 \alpha^{18}_{10} \mu + 0.5 \alpha^{18}_{10} & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) + 0.5 \alpha^{19}_{11} \mu\\0 & 0 & 0 & 0 & 0.5 \lambda + 0.25 \mu & 0 & 0 & 0 & 0 & 0 & 0.25 \lambda - 0.25 \mu & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{18}_{10} \lambda & 0\\0.25 \mu + 0.25 & - 0.5 \lambda & 0 & 0 & 0 & 0 & - 0.25 \lambda - 0.75 \mu + 1.25 & 0.5 \lambda + 0.25 \mu & 0.5 \lambda + 1.0 \mu & - 0.25 \mu & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda - 1.0 \alpha^{10}_{0} \mu + 0.5 \alpha^{10}_{0} + 0.5 \alpha^{12}_{0} \lambda + 1.0 \alpha^{12}_{0} \mu & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 1.0 \alpha^{10}_{6} \mu + 0.5 \alpha^{10}_{6} + 0.5 \alpha^{22}_{6} \lambda + 1.0 \alpha^{22}_{6} \mu & 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{23}_{7} \lambda & 0.5 \alpha^{12}_{8} \lambda + 1.0 \alpha^{12}_{8} \mu + 0.5 \alpha^{22}_{8} \lambda + 1.0 \alpha^{22}_{8} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda & 0 & 0\\0.25 \lambda - 0.25 \mu & 0.25 \lambda + 0.5 \mu & 0 & 0 & 0 & 0 & 0.5 \lambda + 0.25 \mu & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{0} \lambda & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{6} \lambda & 0 & 0 & 0 & 0 & 0\\0.25 \lambda + 0.5 \mu & 0.25 \lambda + 0.25 \mu & 0 & 0 & 0 & 0 & 0.5 \lambda + 1.0 \mu & 0 & 0 & 0 & 0.25 \lambda + 0.5 \mu & 0 & \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0\\0.25 \lambda + 0.25 \mu & 0 & 0 & 0 & 0 & 0 & - 0.25 \mu & 0 & 0 & 0 & - 0.25 \lambda & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.25 \lambda + 0.5 \mu & 0.25 \lambda + 0.25 \mu & 0.25 \mu + 0.25 & 0.25 \lambda - 0.25 \mu & 0 & 0 & 0.25 \lambda + 0.5 \mu & - 0.25 \lambda & 0.25 \lambda + 0.25 \mu + 1.25 & - 0.25 \lambda & 0 & 0 & \alpha^{16}_{2} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0.5 \alpha^{18}_{4} & - 0.5 \alpha^{19}_{5} \mu & 0 & 0 & \alpha^{26}_{8} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{27}_{9} \lambda & 0.5 \alpha^{16}_{10} \lambda + 1.0 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} + 0.5 \alpha^{26}_{10} \lambda + 1.0 \alpha^{26}_{10} \mu & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{19}_{11} \mu - 0.5 \alpha^{27}_{11} \lambda\\0 & 0 & 0 & 0 & - 0.5 \lambda & 0 & 0 & 0 & 0 & 0 & - 0.25 \lambda & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{18}_{10} \lambda & 0\\0.5 \alpha^{10}_{0} + 0.5 \alpha^{12}_{0} \lambda + 1.0 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \lambda + 1.0 \alpha^{2}_{0} \mu & - 0.5 \alpha^{10}_{0} \lambda + 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda - 1.0 \alpha^{10}_{0} \mu + 0.5 \alpha^{10}_{0} + 0.5 \alpha^{12}_{0} \lambda + 1.0 \alpha^{12}_{0} \mu & 0.5 \alpha^{10}_{0} \lambda & \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & \alpha^{10}_{0} \left(- \alpha^{10}_{0} \lambda - 3 \alpha^{10}_{0} \mu + \alpha^{10}_{0} + 2 \alpha^{12}_{0} \lambda + 4 \alpha^{12}_{0} \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} \left(- \alpha^{10}_{0} \left(\lambda + 3 \mu\right) + \alpha^{10}_{0} + \alpha^{12}_{0} \left(\lambda + 2 \mu\right)\right) & 0 & \alpha^{10}_{0} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) & 0 & 0 & 0\\- 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & 0.5 \alpha^{11}_{1} \lambda + 1.0 \alpha^{11}_{1} \mu + 0.5 \alpha^{3}_{1} \mu & 0 & 0 & 0 & 0 & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\alpha^{2}_{2} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{2}_{2} \mu & 0 & 0 & 0.5 \alpha^{16}_{2} \lambda + 1.0 \alpha^{16}_{2} \mu + 0.5 \alpha^{6}_{2} \lambda + 1.0 \alpha^{6}_{2} \mu & 0 & 0 & 0 & 0 & 0 & \alpha^{16}_{2} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{16}_{2} \alpha^{18}_{4} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} \alpha^{16}_{2} \left(\lambda + 2 \mu\right) & 0\\- 0.5 \alpha^{3}_{3} \lambda & 0.5 \alpha^{3}_{3} \mu & 0 & 0 & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) + 0.5 \alpha^{7}_{3} \lambda & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) & 0 & - 0.5 \alpha^{18}_{4} \lambda - 1.0 \alpha^{18}_{4} \mu + 0.5 \alpha^{18}_{4} + 0.5 \alpha^{6}_{4} \lambda + 1.0 \alpha^{6}_{4} \mu & 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & 0 & 0 & 0.5 \alpha^{18}_{4} & - 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & \alpha^{16}_{2} \alpha^{18}_{4} \left(\lambda + 2 \mu\right) & 0 & \left(\alpha^{18}_{4}\right)^{2} \left(- \lambda - 3 \mu + 1\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{4} \left(\alpha^{16}_{10} \lambda + 2 \alpha^{16}_{10} \mu - \alpha^{18}_{10} \lambda - 3 \alpha^{18}_{10} \mu + \alpha^{18}_{10}\right) & 0\\0 & 0 & 0 & 0 & 0.5 \alpha^{19}_{5} \mu + 0.5 \alpha^{7}_{5} \lambda & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{19}_{5} \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0.5 \alpha^{10}_{6} & - 0.5 \alpha^{10}_{6} \lambda & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 1.0 \alpha^{10}_{6} \mu + 0.5 \alpha^{10}_{6} + 0.5 \alpha^{22}_{6} \lambda + 1.0 \alpha^{22}_{6} \mu & 0.5 \alpha^{10}_{6} \lambda & \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & \alpha^{10}_{6} \left(- \alpha^{10}_{0} \lambda - 3 \alpha^{10}_{0} \mu + \alpha^{10}_{0} + \alpha^{12}_{0} \lambda + 2 \alpha^{12}_{0} \mu\right) & 0 & 0 & 0 & 0 & 0 & \left(\alpha^{10}_{6}\right)^{2} \left(- \lambda - 3 \mu + 1\right) & 0 & \alpha^{10}_{6} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) & 0 & 0 & 0\\- 0.5 \alpha^{11}_{7} \mu & \alpha^{11}_{7} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{23}_{7} \lambda & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\alpha^{12}_{8} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{12}_{8} \lambda + 1.0 \alpha^{12}_{8} \mu + 0.5 \alpha^{22}_{8} \lambda + 1.0 \alpha^{22}_{8} \mu & 0 & 0 & 0 & \alpha^{26}_{8} \left(0.5 \lambda + 1.0 \mu\right) & 0 & \alpha^{10}_{0} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & 0\\0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda & 0 & 0 & 0 & - 0.5 \alpha^{27}_{9} \lambda & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0.5 \alpha^{16}_{10} \lambda + 1.0 \alpha^{16}_{10} \mu - 0.5 \alpha^{18}_{10} \lambda - 1.0 \alpha^{18}_{10} \mu + 0.5 \alpha^{18}_{10} & 0.5 \alpha^{18}_{10} \lambda & 0 & 0 & 0 & 0 & 0.5 \alpha^{16}_{10} \lambda + 1.0 \alpha^{16}_{10} \mu + 0.5 \alpha^{18}_{10} + 0.5 \alpha^{26}_{10} \lambda + 1.0 \alpha^{26}_{10} \mu & - 0.5 \alpha^{18}_{10} \lambda & 0 & 0 & \alpha^{18}_{10} \alpha^{16}_{2} \left(\lambda + 2 \mu\right) & 0 & \alpha^{18}_{4} \left(\alpha^{16}_{10} \left(\lambda + 2 \mu\right) - \alpha^{18}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10}\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} \left(2 \alpha^{16}_{10} \lambda + 4 \alpha^{16}_{10} \mu - \alpha^{18}_{10} \lambda - 3 \alpha^{18}_{10} \mu + \alpha^{18}_{10}\right) & 0\\0 & 0 & 0 & 0 & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) + 0.5 \alpha^{19}_{11} \mu & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{19}_{11} \mu - 0.5 \alpha^{27}_{11} \lambda & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
Applying bluntely Dirichlet operator
\(A_C^2=DC^t.AP_C^2.DC+IDC\)
\(B_C^2=XD_C+DC^t.BP_C^2-DC^t.AP_C^2.XD_C\)
gives:
\[\begin{split}\displaystyle A_C^2 = \left[\begin{array}{cccccccccccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda & 0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & - \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{0} \mu & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & 0 & 0.5 \alpha^{7}_{5} \lambda & 0 & 0 & - \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{8} \mu & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{26}_{10} \mu & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right)\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0 & - 0.5 \mu & 0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{6}_{4} \mu & - 0.5 \alpha^{19}_{5} \mu & 0 & 0 & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{19}_{11} \mu - \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 \lambda & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu & - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{19}_{11} \mu\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \lambda + 1.5 \mu & 0.5 \lambda & - 0.5 \mu & 0 & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0.5 \alpha^{11}_{1} \mu - \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{22}_{6} \mu & 0.5 \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu & - \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0\\0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & 0 & 0 & 0.5 \lambda & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \mu & \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) - \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0.5 \alpha^{23}_{7} \lambda & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & - \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{26}_{10} \mu & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda\\0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0 & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0 & - 0.5 \mu & - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{22}_{6} \mu & - 0.5 \alpha^{11}_{7} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu & - \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) - \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0 & - \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{15}_{9} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right)\\0 & 0 & - 0.5 \alpha^{12}_{0} \lambda - 1.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{2}_{0} \mu & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{0} \mu & 0 & 0 & 0 & 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0.5 \alpha^{12}_{0} \lambda + 1.5 \alpha^{12}_{0} \mu - 0.5 \alpha^{2}_{0} \mu & - 0.5 \alpha^{12}_{0} \left(\lambda + \mu\right) & 0 & 0 & \left(\alpha^{10}_{0}\right)^{2} + \alpha^{12}_{0} \left(2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) - \alpha^{2}_{0} \mu\right) - \alpha^{2}_{0} \left(\alpha^{12}_{0} \mu - \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{12}_{0} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{2}_{2} \left(\alpha^{12}_{0} \mu - \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & \alpha^{10}_{0} \alpha^{10}_{6} - \alpha^{12}_{0} \alpha^{22}_{6} \mu & 0 & - \alpha^{12}_{0} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{12}_{0} \alpha^{22}_{8} \mu + \alpha^{12}_{8} \left(2 \alpha^{12}_{0} \left(\lambda + 3 \mu\right) - \alpha^{2}_{0} \mu\right) & 0 & 0 & 0\\0 & 0 & 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{1} \lambda & - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{3}_{1} \lambda + 1.0 \alpha^{3}_{1} \mu & 0 & 0 & 0 & 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \lambda - 1.5 \alpha^{13}_{1} \mu & - 0.5 \alpha^{13}_{1} \left(\lambda + \mu\right) & - 0.5 \alpha^{11}_{1} \mu + 0.5 \alpha^{13}_{1} \lambda + 1.5 \alpha^{13}_{1} \mu - 0.5 \alpha^{3}_{1} \lambda - 1.0 \alpha^{3}_{1} \mu & 0 & 0 & 0 & \alpha^{11}_{1} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) - \alpha^{13}_{1} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \left(\alpha^{13}_{1} \left(\lambda + 2 \mu\right) - \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{3} \mu - \alpha^{3}_{3} \left(\alpha^{13}_{1} \left(\lambda + 2 \mu\right) - \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{13}_{1} \alpha^{23}_{7} \left(\lambda + 2 \mu\right) + \alpha^{11}_{7} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{9} \mu - \alpha^{13}_{1} \alpha^{23}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{1} \mu - 2 \alpha^{13}_{1} \left(\lambda + 3 \mu\right) + \alpha^{3}_{1} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\0 & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{16}_{2} \lambda + 1.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{2}_{2} \mu & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0 & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) - 0.5 \alpha^{6}_{2} \mu & 0 & 0 & 0.5 \alpha^{14}_{2} \lambda + 1.0 \alpha^{14}_{2} \mu - 0.5 \alpha^{16}_{2} \lambda - 1.5 \alpha^{16}_{2} \mu - 0.5 \alpha^{2}_{2} \mu & 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & 0 & - 0.5 \alpha^{16}_{2} \left(\lambda + \mu\right) & - \alpha^{12}_{0} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \alpha^{2}_{0} \alpha^{2}_{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{14}_{2} \left(2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) - \alpha^{16}_{2} \left(\lambda + 2 \mu\right)\right) - \alpha^{16}_{2} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) + \left(\alpha^{2}_{2}\right)^{2} \left(\lambda + 3 \mu\right) - \alpha^{6}_{2} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{6}_{4} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{16}_{2} \alpha^{26}_{8} \mu - \alpha^{12}_{8} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) + \alpha^{2}_{2} \mu\right) + \alpha^{14}_{8} \left(2 \alpha^{14}_{2} \left(\lambda + 3 \mu\right) - \alpha^{16}_{2} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{16}_{10} \left(\alpha^{14}_{2} \left(\lambda + 2 \mu\right) - 2 \alpha^{16}_{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \mu\right) - \alpha^{26}_{10} \alpha^{16}_{2} \mu & 0\\0 & 0 & - 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{3}_{3} \lambda + 1.0 \alpha^{3}_{3} \mu & 0 & - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{7}_{3} \lambda + 1.0 \alpha^{7}_{3} \mu & 0 & 0 & 0.5 \alpha^{17}_{3} \left(\lambda + \mu\right) & 0.5 \alpha^{15}_{3} \mu - 0.5 \alpha^{17}_{3} \lambda - 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{3}_{3} \lambda - 1.0 \alpha^{3}_{3} \mu & 0 & - 0.5 \alpha^{15}_{3} \mu + 0.5 \alpha^{17}_{3} \lambda + 1.5 \alpha^{17}_{3} \mu - 0.5 \alpha^{7}_{3} \lambda - 1.0 \alpha^{7}_{3} \mu & 0 & - \alpha^{13}_{1} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \alpha^{3}_{1} \alpha^{3}_{3} \left(\lambda + 3 \mu\right) & 0 & \alpha^{15}_{3} \left(2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) - \alpha^{17}_{3} \mu\right) - \alpha^{17}_{3} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) + \left(\alpha^{3}_{3}\right)^{2} \left(\lambda + 3 \mu\right) - \alpha^{7}_{3} \left(\alpha^{17}_{3} \left(\lambda + 2 \mu\right) - \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{3} \alpha^{19}_{5} \mu - \alpha^{7}_{5} \left(\alpha^{17}_{3} \left(\lambda + 2 \mu\right) - \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{3} \alpha^{27}_{9} \left(\lambda + 2 \mu\right) - \alpha^{13}_{9} \left(\alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\lambda + 2 \mu\right)\right) + \alpha^{15}_{9} \left(2 \alpha^{15}_{3} \left(\lambda + 3 \mu\right) - \alpha^{17}_{3} \mu\right) & 0 & - \alpha^{17}_{11} \left(\alpha^{15}_{3} \mu - 2 \alpha^{17}_{3} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \alpha^{17}_{3} \mu - \alpha^{27}_{11} \alpha^{17}_{3} \left(\lambda + 2 \mu\right)\\0 & 0 & 0 & 0.5 \alpha^{6}_{4} \mu & 0 & - 0.5 \alpha^{6}_{4} \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \alpha^{6}_{4} \left(\alpha^{16}_{2} \mu - \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & \left(\alpha^{18}_{4}\right)^{2} + \left(\alpha^{6}_{4}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - \alpha^{16}_{10} \alpha^{6}_{4} \mu + \alpha^{18}_{10} \alpha^{18}_{4} & 0\\0 & 0 & 0.5 \alpha^{7}_{5} \lambda & - 0.5 \alpha^{19}_{5} \mu & 0 & 0.5 \alpha^{19}_{5} \mu + 0.5 \alpha^{7}_{5} \lambda + 1.0 \alpha^{7}_{5} \mu & 0 & 0 & 0 & 0 & 0 & - \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{3} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \alpha^{7}_{3} \alpha^{7}_{5} \left(\lambda + 3 \mu\right) & 0 & \left(\left(\alpha^{19}_{5}\right)^{2} + \left(\alpha^{7}_{5}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - \alpha^{17}_{11} \left(\alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\lambda + 2 \mu\right)\right) + \alpha^{19}_{11} \alpha^{19}_{5} \left(\lambda + 3 \mu\right)\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{22}_{6} \mu & 0 & 0.5 \alpha^{22}_{6} \mu & 0 & 0 & \alpha^{10}_{0} \alpha^{10}_{6} - \alpha^{12}_{0} \alpha^{22}_{6} \mu & 0 & 0 & 0 & 0 & 0 & \left(\alpha^{10}_{6}\right)^{2} + \left(\alpha^{22}_{6}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{22}_{6} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{11}_{7} \mu + 0.5 \alpha^{23}_{7} \lambda + 1.0 \alpha^{23}_{7} \mu & 0.5 \alpha^{23}_{7} \lambda & - 0.5 \alpha^{11}_{7} \mu & 0 & 0 & 0 & \alpha^{11}_{1} \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) & 0 & 0 & 0 & 0 & 0 & \left(\left(\alpha^{11}_{7}\right)^{2} + \left(\alpha^{23}_{7}\right)^{2}\right) \left(\lambda + 3 \mu\right) & 0 & \alpha^{23}_{7} \alpha^{23}_{9} \left(\lambda + 3 \mu\right) - \alpha^{13}_{9} \left(\alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(\lambda + 2 \mu\right)\right) & 0 & 0\\0 & 0 & - 0.5 \alpha^{12}_{8} \lambda - 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) & 0 & 0 & 0 & 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) - 0.5 \alpha^{22}_{8} \mu & 0.5 \alpha^{12}_{8} \lambda + 1.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{14}_{8} \lambda + 1.0 \alpha^{14}_{8} \mu + 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{12}_{8} \left(\lambda + \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & 0 & 0.5 \alpha^{26}_{8} \mu & - \alpha^{12}_{0} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{2}_{0} \alpha^{12}_{8} \mu & 0 & - \alpha^{14}_{2} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right)\right) - \alpha^{16}_{2} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) - \alpha^{2}_{2} \alpha^{12}_{8} \mu & 0 & 0 & 0 & - \alpha^{22}_{6} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{12}_{8} \left(- 2 \alpha^{12}_{8} \left(\lambda + 3 \mu\right) + \alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{22}_{8} \mu\right) - \alpha^{14}_{8} \left(\alpha^{12}_{8} \left(\lambda + 2 \mu\right) - 2 \alpha^{14}_{8} \left(\lambda + 3 \mu\right)\right) - \alpha^{22}_{8} \left(\alpha^{12}_{8} \mu - \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{26}_{8}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{16}_{10} \left(\alpha^{14}_{8} \left(\lambda + 2 \mu\right) + \alpha^{26}_{8} \mu\right) + \alpha^{26}_{10} \alpha^{26}_{8} \left(\lambda + 3 \mu\right) & 0\\0 & 0 & 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{27}_{9} \lambda - 1.0 \alpha^{27}_{9} \mu & 0 & 0 & 0 & - 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{23}_{9} \lambda + 1.0 \alpha^{23}_{9} \mu & - 0.5 \alpha^{13}_{9} \left(\lambda + \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & 0.5 \alpha^{13}_{9} \lambda + 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu + 0.5 \alpha^{27}_{9} \lambda + 1.0 \alpha^{27}_{9} \mu & 0 & - 0.5 \alpha^{15}_{9} \mu & 0 & - \alpha^{11}_{1} \alpha^{13}_{9} \mu - \alpha^{13}_{1} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{1} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & 0 & - \alpha^{15}_{3} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right)\right) - \alpha^{17}_{3} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{3}_{3} \alpha^{13}_{9} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & - \alpha^{11}_{7} \alpha^{13}_{9} \mu - \alpha^{23}_{7} \left(\alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{9} \left(- 2 \alpha^{13}_{9} \left(\lambda + 3 \mu\right) + \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(\lambda + 2 \mu\right)\right) - \alpha^{15}_{9} \left(\alpha^{13}_{9} \mu - 2 \alpha^{15}_{9} \left(\lambda + 3 \mu\right)\right) - \alpha^{23}_{9} \left(\alpha^{13}_{9} \left(\lambda + 2 \mu\right) - \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{27}_{9}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{17}_{11} \left(\alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\lambda + 2 \mu\right)\right) + \alpha^{27}_{11} \alpha^{27}_{9} \left(\lambda + 3 \mu\right)\\0 & 0 & 0.5 \alpha^{16}_{10} \lambda + 1.5 \alpha^{16}_{10} \mu - 0.5 \alpha^{26}_{10} \mu & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & 0 & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) & 0 & 0 & - 0.5 \alpha^{16}_{10} \lambda - 1.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{26}_{10} \mu & 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) - 0.5 \alpha^{26}_{10} \mu & 0 & - 0.5 \alpha^{16}_{10} \left(\lambda + \mu\right) + 0.5 \alpha^{26}_{10} \mu & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{2} \left(\lambda + 2 \mu\right) - \alpha^{16}_{10} \alpha^{6}_{2} \mu + \alpha^{16}_{2} \left(2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) - \alpha^{26}_{10} \mu\right) & 0 & - \alpha^{16}_{10} \alpha^{6}_{4} \mu + \alpha^{18}_{10} \alpha^{18}_{4} & 0 & 0 & 0 & - \alpha^{16}_{10} \alpha^{14}_{8} \left(\lambda + 2 \mu\right) - \alpha^{26}_{8} \left(\alpha^{16}_{10} \mu - \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0 & \alpha^{16}_{10} \left(2 \alpha^{16}_{10} \left(\lambda + 3 \mu\right) - \alpha^{26}_{10} \mu\right) + \left(\alpha^{18}_{10}\right)^{2} - \alpha^{26}_{10} \left(\alpha^{16}_{10} \mu - \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0\\0 & 0 & - 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) & 0.5 \alpha^{17}_{11} \lambda + 1.5 \alpha^{17}_{11} \mu - 0.5 \alpha^{19}_{11} \mu - 0.5 \alpha^{27}_{11} \lambda - 1.0 \alpha^{27}_{11} \mu & 0 & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{19}_{11} \mu & 0 & 0 & 0.5 \alpha^{17}_{11} \left(\lambda + \mu\right) - 0.5 \alpha^{27}_{11} \lambda & - 0.5 \alpha^{17}_{11} \lambda - 1.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{27}_{11} \lambda + 1.0 \alpha^{27}_{11} \mu & 0 & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{3} \mu - \alpha^{17}_{11} \alpha^{7}_{3} \left(\lambda + 2 \mu\right) - \alpha^{17}_{3} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) & 0 & - \alpha^{17}_{11} \alpha^{7}_{5} \left(\lambda + 2 \mu\right) - \alpha^{19}_{5} \left(\alpha^{17}_{11} \mu - \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{9} \mu - \alpha^{27}_{9} \left(\alpha^{17}_{11} \left(\lambda + 2 \mu\right) - \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{11} \left(- 2 \alpha^{17}_{11} \left(\lambda + 3 \mu\right) + \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(\lambda + 2 \mu\right)\right) - \alpha^{19}_{11} \left(\alpha^{17}_{11} \mu - \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) - \alpha^{27}_{11} \left(\alpha^{17}_{11} \left(\lambda + 2 \mu\right) - \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right)\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle B_C^2 = \left[\begin{matrix}c_{x}\\c_{y}\\- 0.5 L^{2} f + 0.5 c_{x} \lambda + 1.0 c_{x} \mu - 0.5 c_{y} \mu + 0.5 i_{x} \lambda + 1.0 i_{x} \mu\\- 0.5 c_{x} \lambda + 0.5 c_{y} \mu + 0.5 i_{x} \lambda\\i_{x}\\0.5 i_{x} \lambda\\c_{x}\\0.5 c_{x} \lambda + 0.5 c_{y} \lambda + 1.0 c_{y} \mu\\- 0.5 L^{2} f + 0.5 c_{x} \lambda + 1.0 c_{x} \mu + 0.5 c_{y} \lambda + 0.5 c_{y} \mu + 0.5 i_{x} \lambda + 1.0 i_{x} \mu\\0.5 \lambda \left(c_{x} - i_{x}\right)\\i_{x}\\- 0.5 i_{x} \lambda\\- \frac{L^{2} \alpha^{12}_{0} f}{3} - \frac{L^{2} \alpha^{2}_{0} f}{12} + \frac{\alpha^{12}_{0} c_{y} \lambda}{2} + \frac{\alpha^{12}_{0} c_{y} \mu}{2} - \frac{\alpha^{2}_{0} c_{y} \mu}{2}\\- 0.5 \alpha^{13}_{1} c_{y} \lambda - 1.5 \alpha^{13}_{1} c_{y} \mu - 0.5 \alpha^{3}_{1} c_{x} \lambda\\- 0.166666666666667 L^{2} \alpha^{14}_{2} f - 0.333333333333333 L^{2} \alpha^{16}_{2} f - 0.0833333333333333 L^{2} \alpha^{2}_{2} f - 0.0833333333333333 L^{2} \alpha^{6}_{2} f + 0.5 \alpha^{14}_{2} c_{x} \lambda + 1.0 \alpha^{14}_{2} c_{x} \mu + 0.5 \alpha^{14}_{2} i_{x} \lambda + 1.0 \alpha^{14}_{2} i_{x} \mu - 0.5 \alpha^{2}_{2} c_{y} \mu\\0.5 \alpha^{15}_{3} c_{y} \mu - 0.5 \alpha^{3}_{3} c_{x} \lambda + 0.5 \alpha^{7}_{3} i_{x} \lambda\\- \frac{L^{2} \alpha^{6}_{4} f}{12}\\\frac{\alpha^{7}_{5} i_{x} \lambda}{2}\\- \frac{L^{2} \alpha^{22}_{6} f}{12}\\\alpha^{23}_{7} \left(0.5 c_{x} \lambda + 0.5 c_{y} \lambda + 1.0 c_{y} \mu\right)\\- 0.333333333333333 L^{2} \alpha^{12}_{8} f - 0.166666666666667 L^{2} \alpha^{14}_{8} f - 0.0833333333333333 L^{2} \alpha^{22}_{8} f - 0.0833333333333333 L^{2} \alpha^{26}_{8} f + 0.5 \alpha^{12}_{8} c_{y} \lambda + 0.5 \alpha^{12}_{8} c_{y} \mu + 0.5 \alpha^{14}_{8} c_{x} \lambda + 1.0 \alpha^{14}_{8} c_{x} \mu + 0.5 \alpha^{14}_{8} i_{x} \lambda + 1.0 \alpha^{14}_{8} i_{x} \mu\\c_{y} \left(- 0.5 \alpha^{13}_{9} \lambda - 1.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{15}_{9} \mu + 0.5 \alpha^{23}_{9} \lambda + 1.0 \alpha^{23}_{9} \mu\right) + \frac{\lambda \left(\alpha^{23}_{9} c_{x} - \alpha^{27}_{9} i_{x}\right)}{2}\\\frac{L^{2} f \left(- 4 \alpha^{16}_{10} - \alpha^{26}_{10}\right)}{12}\\- \frac{\alpha^{27}_{11} i_{x} \lambda}{2}\end{matrix}\right]\end{split}\]
Compaire to \(A_C\), \(A_C^2\) gives almost the same matrix because the wrong terms are eliminated by \(DC\) operator in standard part but unforunately not in enriched and coupled parts as already analysys in theoric section.
For \(B_C^2\) it is the same, terms are not the same.
\[\begin{split}\displaystyle A_C^2 - A_C = \left[\begin{array}{cccccccccccccccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{18}_{10} \lambda & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{0} \lambda & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{6} \lambda & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{18}_{10} \lambda & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{0} \lambda & \alpha^{10}_{0} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & \alpha^{10}_{0} \left(- \alpha^{10}_{0} \lambda - 3 \alpha^{10}_{0} \mu + \alpha^{10}_{0} + 2 \alpha^{12}_{0} \lambda + 4 \alpha^{12}_{0} \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} \left(- \alpha^{10}_{0} \left(\lambda + 3 \mu\right) + \alpha^{10}_{0} + \alpha^{12}_{0} \left(\lambda + 2 \mu\right)\right) & 0 & \alpha^{10}_{0} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{16}_{2} \alpha^{18}_{4} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} \alpha^{16}_{2} \left(\lambda + 2 \mu\right) & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \alpha^{18}_{4} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & \alpha^{16}_{2} \alpha^{18}_{4} \left(\lambda + 2 \mu\right) & 0 & \left(\alpha^{18}_{4}\right)^{2} \left(- \lambda - 3 \mu + 1\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{4} \left(\alpha^{16}_{10} \lambda + 2 \alpha^{16}_{10} \mu - \alpha^{18}_{10} \lambda - 3 \alpha^{18}_{10} \mu + \alpha^{18}_{10}\right) & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{10}_{6} \lambda & \alpha^{10}_{6} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & \alpha^{10}_{6} \left(- \alpha^{10}_{0} \lambda - 3 \alpha^{10}_{0} \mu + \alpha^{10}_{0} + \alpha^{12}_{0} \lambda + 2 \alpha^{12}_{0} \mu\right) & 0 & 0 & 0 & 0 & 0 & \left(\alpha^{10}_{6}\right)^{2} \left(- \lambda - 3 \mu + 1\right) & 0 & \alpha^{10}_{6} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{0} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} \alpha^{12}_{8} \left(\lambda + 2 \mu\right) & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \alpha^{18}_{10} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{18}_{10} \lambda & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{18}_{10} \lambda & 0 & 0 & \alpha^{18}_{10} \alpha^{16}_{2} \left(\lambda + 2 \mu\right) & 0 & \alpha^{18}_{4} \left(\alpha^{16}_{10} \left(\lambda + 2 \mu\right) - \alpha^{18}_{10} \left(\lambda + 3 \mu\right) + \alpha^{18}_{10}\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} \left(2 \alpha^{16}_{10} \lambda + 4 \alpha^{16}_{10} \mu - \alpha^{18}_{10} \lambda - 3 \alpha^{18}_{10} \mu + \alpha^{18}_{10}\right) & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle B_C^2 - B_C = \left[\begin{matrix}0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\\alpha^{10}_{0} \left(0.0833333333333333 L^{2} f + 0.5 c_{x} \lambda + 1.0 c_{x} \mu + 0.5 c_{y} \lambda\right)\\0\\0\\0\\\alpha^{18}_{4} \left(0.0833333333333333 L^{2} f + 0.5 i_{x} \lambda + 1.0 i_{x} \mu\right)\\0\\\alpha^{10}_{6} \left(0.0833333333333333 L^{2} f + 0.5 c_{x} \lambda + 1.0 c_{x} \mu + 0.5 c_{y} \lambda\right)\\0\\0\\0\\\alpha^{18}_{10} \left(0.0833333333333333 L^{2} f + 0.5 i_{x} \lambda + 1.0 i_{x} \mu\right)\\0\end{matrix}\right]\end{split}\]
TS approach (III)#
As mentioned in theoric section
this approach only proposed in the shift case, use:
\(Q=P.D_C\)
\(W_S=P_S.X_{DC}\)
\(Z_S=A.W_S\)
under the condition that \(X_{DCE}=0\) which is the case here.
\[\begin{split}\displaystyle Q = \left[\begin{array}{cccccccccccccccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{2}_{0} & 0 & \alpha^{2}_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{3}_{1} & 0 & \alpha^{3}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{6}_{2} & 0 & \alpha^{6}_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{7}_{3} & 0 & \alpha^{7}_{5} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{0} & 0 & 0 & 0 & 0 & 0 & \alpha^{10}_{6} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{11}_{1} & 0 & 0 & 0 & 0 & 0 & \alpha^{11}_{7} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{12}_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{12}_{8} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{13}_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{13}_{9} & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{14}_{2} & 0 & 0 & 0 & 0 & 0 & \alpha^{14}_{8} & 0 & 0 & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{15}_{3} & 0 & 0 & 0 & 0 & 0 & \alpha^{15}_{9} & 0 & 0\\0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{16}_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{16}_{10} & 0\\0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & \alpha^{17}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{17}_{11}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{4} & 0 & 0 & 0 & 0 & 0 & \alpha^{18}_{10} & 0\\0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & \alpha^{19}_{5} & 0 & 0 & 0 & 0 & 0 & \alpha^{19}_{11}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{22}_{6} & 0 & \alpha^{22}_{8} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{23}_{7} & 0 & \alpha^{23}_{9} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{26}_{8} & 0 & \alpha^{26}_{10} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \alpha^{27}_{9} & 0 & \alpha^{27}_{11}\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle W_S = \left[\begin{matrix}1.0 c_{x}\\1.0 c_{y}\\0.5 c_{x}\\0.5 c_{y}\\0\\0\\0.5 i_{x}\\0\\1.0 i_{x}\\0\\1.0 c_{x}\\0.5 c_{y}\\0.5 c_{x}\\0.5 c_{y}\\0\\0\\0.5 i_{x}\\0\\1.0 i_{x}\\0\\1.0 c_{x}\\0\\0.5 c_{x}\\0\\0\\0\\0.5 i_{x}\\0\\1.0 i_{x}\\0\end{matrix}\right] Z_S = \left[\begin{matrix}- 0.5 c_{x} \mu + 0.5 c_{x} \left(- \frac{\lambda}{2} - \mu\right) + 1.0 c_{x} \left(\frac{\lambda}{2} + \frac{3 \mu}{2}\right) + 0.25 c_{y} \lambda + 0.25 c_{y} \mu + 0.5 c_{y} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right)\\0.5 c_{x} \lambda + 0.25 c_{x} \mu + 0.5 c_{x} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) - 0.25 c_{y} \mu + 0.5 c_{y} \left(- \frac{\lambda}{2} - \mu\right) + 1.0 c_{y} \left(\frac{\lambda}{2} + \frac{3 \mu}{2}\right)\\- 0.5 c_{x} \mu + 1.0 c_{x} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 c_{x} \left(\lambda + 3 \mu\right) + 0.5 c_{y} \mu\\0.5 c_{x} \lambda - 0.5 c_{y} \mu + 0.5 c_{y} \left(- \lambda - 2 \mu\right) + 0.5 c_{y} \left(\lambda + 3 \mu\right)\\0.5 c_{x} \left(- \frac{\lambda}{2} - \mu\right) - 0.25 c_{y} \lambda + 0.5 c_{y} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) + 0.5 i_{x} \left(- \frac{\lambda}{2} - \mu\right)\\- 0.25 c_{x} \mu + 0.5 c_{x} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.25 c_{y} \mu + 0.25 i_{x} \mu + 0.5 i_{x} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right)\\- 0.5 i_{x} \mu + 1.0 i_{x} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 i_{x} \left(\lambda + 3 \mu\right)\\- 0.5 i_{x} \lambda\\- 0.5 i_{x} \mu + 0.5 i_{x} \left(- \frac{\lambda}{2} - \mu\right) + 1.0 i_{x} \left(\frac{\lambda}{2} + \frac{3 \mu}{2}\right)\\- 0.5 i_{x} \lambda - 0.25 i_{x} \mu + 0.5 i_{x} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right)\\- 1.0 c_{x} \mu + 0.5 c_{x} \left(- \lambda - 2 \mu\right) + 1.0 c_{x} \left(\lambda + 3 \mu\right) + 0.5 c_{y} \lambda\\- 0.5 c_{y} \mu + 1.0 c_{y} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 c_{y} \left(\lambda + 3 \mu\right)\\- 1.0 c_{x} \mu + 1.0 c_{x} \left(- \lambda - 2 \mu\right) + 0.5 c_{x} \left(2 \lambda + 6 \mu\right) + 1.0 c_{y} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right)\\1.0 c_{x} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) + 1.0 c_{x} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.5 c_{y} \mu + 0.5 c_{y} \left(- \lambda - 2 \mu\right) + 0.5 c_{y} \left(2 \lambda + 6 \mu\right)\\0.5 c_{x} \left(- \lambda - 2 \mu\right) + 0.5 i_{x} \left(- \lambda - 2 \mu\right)\\- 0.5 c_{y} \mu\\- 1.0 i_{x} \mu + 1.0 i_{x} \left(- \lambda - 2 \mu\right) + 0.5 i_{x} \left(2 \lambda + 6 \mu\right)\\1.0 i_{x} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) + 1.0 i_{x} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right)\\- 1.0 i_{x} \mu + 0.5 i_{x} \left(- \lambda - 2 \mu\right) + 1.0 i_{x} \left(\lambda + 3 \mu\right)\\0\\- 0.5 c_{x} \mu + 0.5 c_{x} \left(- \frac{\lambda}{2} - \mu\right) + 1.0 c_{x} \left(\frac{\lambda}{2} + \frac{3 \mu}{2}\right) - 0.25 c_{y} \mu + 0.5 c_{y} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right)\\- 0.5 c_{x} \lambda - 0.25 c_{x} \mu + 0.5 c_{x} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) + 0.5 c_{y} \left(- \frac{\lambda}{2} - \mu\right)\\- 0.5 c_{x} \mu + 1.0 c_{x} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 c_{x} \left(\lambda + 3 \mu\right)\\- 0.5 c_{x} \lambda + 0.5 c_{y} \left(- \lambda - 2 \mu\right)\\0.5 c_{x} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 c_{y} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) + 0.5 i_{x} \left(- \frac{\lambda}{2} - \mu\right)\\0.25 c_{x} \mu + 0.5 c_{x} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) - 0.25 i_{x} \mu + 0.5 i_{x} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right)\\- 0.5 i_{x} \mu + 1.0 i_{x} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 i_{x} \left(\lambda + 3 \mu\right)\\0.5 i_{x} \lambda\\- 0.5 i_{x} \mu + 0.5 i_{x} \left(- \frac{\lambda}{2} - \mu\right) + 1.0 i_{x} \left(\frac{\lambda}{2} + \frac{3 \mu}{2}\right)\\0.5 i_{x} \lambda + 0.25 i_{x} \mu + 0.5 i_{x} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right)\end{matrix}\right]\end{split}\]
The assocated system is then
\[\begin{split}\displaystyle A_C^3 = Q^t.A.Q+U_C =\left[\begin{array}{cccccccccccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \lambda & 0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & - 0.5 \lambda - 0.5 \mu & \alpha^{12}_{0} \left(- 0.5 \lambda - 1.5 \mu\right) + 0.5 \alpha^{2}_{0} \mu & \alpha^{13}_{1} \left(0.5 \lambda + 0.5 \mu\right) - 0.5 \alpha^{3}_{1} \lambda & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{2}_{2} \mu & \alpha^{17}_{3} \left(- 0.5 \lambda - 0.5 \mu\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & \alpha^{18}_{4} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{7}_{5} \lambda & 0 & 0 & \alpha^{12}_{8} \left(- 0.5 \lambda - 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) - 0.5 \alpha^{26}_{8} \mu & \alpha^{13}_{9} \left(0.5 \lambda + 0.5 \mu\right) & \alpha^{16}_{10} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{18}_{10} \left(- 0.5 \lambda - 1.0 \mu\right) - 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(- 0.5 \lambda - 0.5 \mu\right)\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0 & - 0.5 \mu & 0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & \alpha^{12}_{0} \left(0.5 \lambda + 0.5 \mu\right) - 0.5 \alpha^{2}_{0} \mu & \alpha^{13}_{1} \left(- 0.5 \lambda - 1.5 \mu\right) + \alpha^{3}_{1} \left(0.5 \lambda + 1.0 \mu\right) & \alpha^{16}_{2} \left(- 0.5 \lambda - 0.5 \mu\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{3} \left(0.5 \lambda + 1.0 \mu\right) & 0.5 \alpha^{6}_{4} \mu & - 0.5 \alpha^{19}_{5} \mu & 0 & 0 & \alpha^{12}_{8} \left(0.5 \lambda + 0.5 \mu\right) & \alpha^{13}_{9} \left(- 0.5 \lambda - 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(- 0.5 \lambda - 1.0 \mu\right) & \alpha^{16}_{10} \left(- 0.5 \lambda - 0.5 \mu\right) & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(- 0.5 \lambda - 1.0 \mu\right)\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.5 \lambda & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & 0 & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0 & \alpha^{16}_{2} \left(0.5 \lambda + 0.5 \mu\right) - 0.5 \alpha^{6}_{2} \mu & \alpha^{17}_{3} \left(- 0.5 \lambda - 1.5 \mu\right) + \alpha^{7}_{3} \left(0.5 \lambda + 1.0 \mu\right) & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu & 0.5 \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0 & 0 & 0 & \alpha^{16}_{10} \left(0.5 \lambda + 0.5 \mu\right) - 0.5 \alpha^{18}_{10} \lambda & \alpha^{17}_{11} \left(- 0.5 \lambda - 1.5 \mu\right) + 0.5 \alpha^{19}_{11} \mu\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 \lambda + 1.5 \mu & 0.5 \lambda & - 0.5 \mu & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda + \alpha^{12}_{0} \left(0.5 \lambda + 0.5 \mu\right) & 0.5 \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(- 0.5 \lambda - 1.5 \mu\right) & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu & 0.5 \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(0.5 \lambda + 1.0 \mu\right) & \alpha^{12}_{8} \left(0.5 \lambda + 0.5 \mu\right) - 0.5 \alpha^{22}_{8} \mu & \alpha^{13}_{9} \left(- 0.5 \lambda - 1.5 \mu\right) + \alpha^{23}_{9} \left(0.5 \lambda + 1.0 \mu\right) & 0 & 0\\0 & 0 & - 1.0 \mu & 0.5 \lambda + 0.5 \mu & 0 & 0 & 0 & 0.5 \lambda & 1.0 \lambda + 3.0 \mu & - 0.5 \lambda - 0.5 \mu & 0 & 0.5 \mu & \alpha^{10}_{0} \left(- 0.5 \lambda - 1.0 \mu\right) + \alpha^{12}_{0} \left(0.5 \lambda + 1.5 \mu\right) - 0.5 \alpha^{2}_{0} \mu & \alpha^{13}_{1} \left(- 0.5 \lambda - 0.5 \mu\right) & \alpha^{14}_{2} \left(0.5 \lambda + 1.0 \mu\right) + \alpha^{16}_{2} \left(- 0.5 \lambda - 1.5 \mu\right) - 0.5 \alpha^{2}_{2} \mu & \alpha^{17}_{3} \left(0.5 \lambda + 0.5 \mu\right) & 0 & 0 & \alpha^{10}_{6} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{23}_{7} \lambda & \alpha^{12}_{8} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{14}_{8} \left(0.5 \lambda + 1.0 \mu\right) + 0.5 \alpha^{26}_{8} \mu & \alpha^{13}_{9} \left(- 0.5 \lambda - 0.5 \mu\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & \alpha^{16}_{10} \left(- 0.5 \lambda - 1.5 \mu\right) + 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(0.5 \lambda + 0.5 \mu\right) - 0.5 \alpha^{27}_{11} \lambda\\0 & 0 & 0.5 \lambda + 0.5 \mu & - 1.0 \lambda - 2.0 \mu & 0 & 0 & 0 & - 0.5 \mu & - 0.5 \lambda - 0.5 \mu & 1.0 \lambda + 3.0 \mu & 0 & - 0.5 \mu & \alpha^{12}_{0} \left(- 0.5 \lambda - 0.5 \mu\right) & - 0.5 \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{3}_{1} \left(- 0.5 \lambda - 1.0 \mu\right) & \alpha^{16}_{2} \left(0.5 \lambda + 0.5 \mu\right) & 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(- 0.5 \lambda - 1.5 \mu\right) + \alpha^{3}_{3} \left(- 0.5 \lambda - 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{22}_{6} \mu & - 0.5 \alpha^{11}_{7} \mu & \alpha^{12}_{8} \left(- 0.5 \lambda - 0.5 \mu\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & \alpha^{13}_{9} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(0.5 \lambda + 1.0 \mu\right) & \alpha^{16}_{10} \left(0.5 \lambda + 0.5 \mu\right) - 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(- 0.5 \lambda - 1.5 \mu\right) + \alpha^{27}_{11} \left(0.5 \lambda + 1.0 \mu\right)\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - 0.5 \lambda - 0.5 \mu & 0 & 0 & - 0.5 \lambda - 1.0 \mu & 0 & 0 & 0.5 \mu & - 0.5 \mu & 0 & 0.5 \lambda + 1.5 \mu & 0 & 0 & \alpha^{16}_{2} \left(- 0.5 \lambda - 0.5 \mu\right) & - 0.5 \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(0.5 \lambda + 1.5 \mu\right) + \alpha^{7}_{3} \left(- 0.5 \lambda - 1.0 \mu\right) & 0.5 \alpha^{18}_{4} \lambda & \alpha^{7}_{5} \left(- 0.5 \lambda - 1.0 \mu\right) & 0 & 0 & 0.5 \alpha^{26}_{8} \mu & - 0.5 \alpha^{15}_{9} \mu & \alpha^{16}_{10} \left(- 0.5 \lambda - 0.5 \mu\right) + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & \alpha^{17}_{11} \left(0.5 \lambda + 1.5 \mu\right)\\0 & 0 & - 0.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{12}_{0} \left(- \lambda - 2 \mu\right) + 1.0 \alpha^{2}_{0} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{2}_{0} \left(\lambda + 3 \mu\right) & 1.0 \alpha^{12}_{0} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.5 \alpha^{2}_{0} \mu & 0 & 0 & 0 & - 0.5 \alpha^{10}_{0} \lambda + 1.0 \alpha^{12}_{0} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) & 0.5 \alpha^{10}_{0} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{12}_{0} \mu + 0.5 \alpha^{12}_{0} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{12}_{0} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{2}_{0} \mu & 1.0 \alpha^{12}_{0} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) & 0 & 0 & \alpha^{10}_{0} \left(\alpha^{10}_{0} \left(\lambda + 3 \mu\right) + \alpha^{12}_{0} \left(- \lambda - 2 \mu\right)\right) + \alpha^{12}_{0} \left(\alpha^{10}_{0} \left(- \lambda - 2 \mu\right) + \alpha^{12}_{0} \left(2 \lambda + 6 \mu\right) - \alpha^{2}_{0} \mu\right) + \alpha^{2}_{0} \left(- \alpha^{12}_{0} \mu + \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & \alpha^{12}_{0} \alpha^{14}_{2} \left(- \lambda - 2 \mu\right) + \alpha^{2}_{2} \left(- \alpha^{12}_{0} \mu + \alpha^{2}_{0} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{12}_{0} \alpha^{22}_{6} \mu + \alpha^{10}_{6} \left(\alpha^{10}_{0} \left(\lambda + 3 \mu\right) + \alpha^{12}_{0} \left(- \lambda - 2 \mu\right)\right) & 0 & \alpha^{12}_{0} \alpha^{14}_{8} \left(- \lambda - 2 \mu\right) - \alpha^{12}_{0} \alpha^{22}_{8} \mu + \alpha^{12}_{8} \left(\alpha^{10}_{0} \left(- \lambda - 2 \mu\right) + \alpha^{12}_{0} \left(2 \lambda + 6 \mu\right) - \alpha^{2}_{0} \mu\right) & 0 & 0 & 0\\0 & 0 & 1.0 \alpha^{13}_{1} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.5 \alpha^{3}_{1} \lambda & - 0.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{13}_{1} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{3}_{1} \mu + 0.5 \alpha^{3}_{1} \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 1.0 \alpha^{11}_{1} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{11}_{1} \left(\lambda + 3 \mu\right) - 0.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{13}_{1} \left(- \lambda - 2 \mu\right) & 1.0 \alpha^{13}_{1} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) & - 0.5 \alpha^{11}_{1} \mu - 0.5 \alpha^{13}_{1} \mu + 0.5 \alpha^{13}_{1} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{13}_{1} \left(2 \lambda + 6 \mu\right) + 0.5 \alpha^{3}_{1} \left(- \lambda - 2 \mu\right) & 0 & 0 & 0 & \alpha^{11}_{1} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) + \alpha^{13}_{1} \left(- \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(2 \lambda + 6 \mu\right) + \alpha^{3}_{1} \left(- \lambda - 2 \mu\right)\right) + \alpha^{3}_{1} \left(\alpha^{13}_{1} \left(- \lambda - 2 \mu\right) + \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(\alpha^{13}_{1} \left(- \lambda - 2 \mu\right) + \alpha^{3}_{1} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & \alpha^{13}_{1} \alpha^{23}_{7} \left(- \lambda - 2 \mu\right) + \alpha^{11}_{7} \left(\alpha^{11}_{1} \left(\lambda + 3 \mu\right) - \alpha^{13}_{1} \mu\right) & 0 & - \alpha^{13}_{1} \alpha^{15}_{9} \mu + \alpha^{13}_{1} \alpha^{23}_{9} \left(- \lambda - 2 \mu\right) + \alpha^{13}_{9} \left(- \alpha^{11}_{1} \mu + \alpha^{13}_{1} \left(2 \lambda + 6 \mu\right) + \alpha^{3}_{1} \left(- \lambda - 2 \mu\right)\right) & 0 & 0\\0 & 0 & - 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{14}_{2} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{14}_{2} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{16}_{2} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{16}_{2} \left(2 \lambda + 6 \mu\right) + 1.0 \alpha^{2}_{2} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{2}_{2} \left(\lambda + 3 \mu\right) - 0.5 \alpha^{6}_{2} \mu + 1.0 \alpha^{6}_{2} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{6}_{2} \left(\lambda + 3 \mu\right) & 1.0 \alpha^{16}_{2} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) - 0.5 \alpha^{2}_{2} \mu + 0.5 \alpha^{6}_{2} \mu & 0 & 1.0 \alpha^{16}_{2} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.5 \alpha^{6}_{2} \mu & 0 & 0 & - 1.0 \alpha^{14}_{2} \mu + 0.5 \alpha^{14}_{2} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{14}_{2} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{16}_{2} \mu + 0.5 \alpha^{16}_{2} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{2}_{2} \mu & 1.0 \alpha^{16}_{2} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) & 0 & 1.0 \alpha^{16}_{2} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) & \alpha^{12}_{0} \left(\alpha^{14}_{2} \left(- \lambda - 2 \mu\right) - \alpha^{2}_{2} \mu\right) + \alpha^{2}_{0} \alpha^{2}_{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{14}_{2} \left(\alpha^{14}_{2} \left(2 \lambda + 6 \mu\right) + \alpha^{16}_{2} \left(- \lambda - 2 \mu\right)\right) + \alpha^{16}_{2} \left(\alpha^{14}_{2} \left(- \lambda - 2 \mu\right) + \alpha^{16}_{2} \left(2 \lambda + 6 \mu\right) - \alpha^{6}_{2} \mu\right) + \left(\alpha^{2}_{2}\right)^{2} \left(\lambda + 3 \mu\right) + \alpha^{6}_{2} \left(- \alpha^{16}_{2} \mu + \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & \alpha^{16}_{2} \alpha^{18}_{4} \left(- \lambda - 2 \mu\right) + \alpha^{6}_{4} \left(- \alpha^{16}_{2} \mu + \alpha^{6}_{2} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{16}_{2} \alpha^{26}_{8} \mu + \alpha^{12}_{8} \left(\alpha^{14}_{2} \left(- \lambda - 2 \mu\right) - \alpha^{2}_{2} \mu\right) + \alpha^{14}_{8} \left(\alpha^{14}_{2} \left(2 \lambda + 6 \mu\right) + \alpha^{16}_{2} \left(- \lambda - 2 \mu\right)\right) & 0 & \alpha^{16}_{10} \left(\alpha^{14}_{2} \left(- \lambda - 2 \mu\right) + \alpha^{16}_{2} \left(2 \lambda + 6 \mu\right) - \alpha^{6}_{2} \mu\right) + \alpha^{18}_{10} \alpha^{16}_{2} \left(- \lambda - 2 \mu\right) - \alpha^{26}_{10} \alpha^{16}_{2} \mu & 0\\0 & 0 & 1.0 \alpha^{17}_{3} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) - 0.5 \alpha^{3}_{3} \lambda + 0.5 \alpha^{7}_{3} \lambda & - 0.5 \alpha^{15}_{3} \mu + 1.0 \alpha^{15}_{3} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{15}_{3} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{17}_{3} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{17}_{3} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{3}_{3} \mu + 0.5 \alpha^{3}_{3} \left(\lambda + 3 \mu\right) - 0.5 \alpha^{7}_{3} \mu + 0.5 \alpha^{7}_{3} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{7}_{3} \left(\lambda + 3 \mu\right) & 0 & - 0.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{17}_{3} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{7}_{3} \mu + 0.5 \alpha^{7}_{3} \left(\lambda + 3 \mu\right) & 0 & 0 & 1.0 \alpha^{17}_{3} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) & - 0.5 \alpha^{15}_{3} \mu + 1.0 \alpha^{15}_{3} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{15}_{3} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{17}_{3} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{3}_{3} \left(- \lambda - 2 \mu\right) & 0 & - 0.5 \alpha^{15}_{3} \mu - 0.5 \alpha^{17}_{3} \mu + 0.5 \alpha^{17}_{3} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{17}_{3} \left(2 \lambda + 6 \mu\right) + 0.5 \alpha^{7}_{3} \left(- \lambda - 2 \mu\right) & 0 & \alpha^{13}_{1} \left(- \alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(- \lambda - 2 \mu\right)\right) + \alpha^{3}_{1} \alpha^{3}_{3} \left(\lambda + 3 \mu\right) & 0 & \alpha^{15}_{3} \left(\alpha^{15}_{3} \left(2 \lambda + 6 \mu\right) - \alpha^{17}_{3} \mu\right) + \alpha^{17}_{3} \left(- \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(2 \lambda + 6 \mu\right) + \alpha^{7}_{3} \left(- \lambda - 2 \mu\right)\right) + \left(\alpha^{3}_{3}\right)^{2} \left(\lambda + 3 \mu\right) + \alpha^{7}_{3} \left(\alpha^{17}_{3} \left(- \lambda - 2 \mu\right) + \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & - \alpha^{17}_{3} \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(\alpha^{17}_{3} \left(- \lambda - 2 \mu\right) + \alpha^{7}_{3} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & \alpha^{17}_{3} \alpha^{27}_{9} \left(- \lambda - 2 \mu\right) + \alpha^{13}_{9} \left(- \alpha^{15}_{3} \mu + \alpha^{3}_{3} \left(- \lambda - 2 \mu\right)\right) + \alpha^{15}_{9} \left(\alpha^{15}_{3} \left(2 \lambda + 6 \mu\right) - \alpha^{17}_{3} \mu\right) & 0 & \alpha^{17}_{11} \left(- \alpha^{15}_{3} \mu + \alpha^{17}_{3} \left(2 \lambda + 6 \mu\right) + \alpha^{7}_{3} \left(- \lambda - 2 \mu\right)\right) - \alpha^{19}_{11} \alpha^{17}_{3} \mu + \alpha^{27}_{11} \alpha^{17}_{3} \left(- \lambda - 2 \mu\right)\\0 & 0 & 0.5 \alpha^{18}_{4} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{6}_{4} \mu + 1.0 \alpha^{6}_{4} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{6}_{4} \left(\lambda + 3 \mu\right) & 0.5 \alpha^{6}_{4} \mu & 0 & - 0.5 \alpha^{18}_{4} \lambda - 0.5 \alpha^{6}_{4} \mu & 0 & 0 & 0 & 0 & 0 & 0.5 \alpha^{18}_{4} \lambda & 0 & 0 & \alpha^{16}_{2} \left(\alpha^{18}_{4} \left(- \lambda - 2 \mu\right) - \alpha^{6}_{4} \mu\right) + \alpha^{6}_{2} \alpha^{6}_{4} \left(\lambda + 3 \mu\right) & 0 & \left(\alpha^{18}_{4}\right)^{2} \left(\lambda + 3 \mu\right) + \left(\alpha^{6}_{4}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{16}_{10} \left(\alpha^{18}_{4} \left(- \lambda - 2 \mu\right) - \alpha^{6}_{4} \mu\right) + \alpha^{18}_{10} \alpha^{18}_{4} \left(\lambda + 3 \mu\right) & 0\\0 & 0 & 0.5 \alpha^{7}_{5} \lambda & - 0.5 \alpha^{19}_{5} \mu - 0.5 \alpha^{7}_{5} \mu + 0.5 \alpha^{7}_{5} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{7}_{5} \left(\lambda + 3 \mu\right) & 0 & 1.0 \alpha^{19}_{5} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{19}_{5} \left(\lambda + 3 \mu\right) - 0.5 \alpha^{7}_{5} \mu + 0.5 \alpha^{7}_{5} \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{19}_{5} \mu + 1.0 \alpha^{19}_{5} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{19}_{5} \left(\lambda + 3 \mu\right) + 0.5 \alpha^{7}_{5} \left(- \lambda - 2 \mu\right) & 0 & 0 & 0 & \alpha^{17}_{3} \left(- \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(- \lambda - 2 \mu\right)\right) + \alpha^{7}_{3} \alpha^{7}_{5} \left(\lambda + 3 \mu\right) & 0 & \left(\alpha^{19}_{5}\right)^{2} \left(\lambda + 3 \mu\right) + \left(\alpha^{7}_{5}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & 0 & 0 & \alpha^{17}_{11} \left(- \alpha^{19}_{5} \mu + \alpha^{7}_{5} \left(- \lambda - 2 \mu\right)\right) + \alpha^{19}_{11} \alpha^{19}_{5} \left(\lambda + 3 \mu\right)\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - 0.5 \alpha^{10}_{6} \lambda - 0.5 \alpha^{22}_{6} \mu & 0.5 \alpha^{10}_{6} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{22}_{6} \mu + 1.0 \alpha^{22}_{6} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{22}_{6} \left(\lambda + 3 \mu\right) & 0.5 \alpha^{22}_{6} \mu & 0 & 0 & \alpha^{10}_{0} \alpha^{10}_{6} \left(\lambda + 3 \mu\right) + \alpha^{12}_{0} \left(\alpha^{10}_{6} \left(- \lambda - 2 \mu\right) - \alpha^{22}_{6} \mu\right) & 0 & 0 & 0 & 0 & 0 & \left(\alpha^{10}_{6}\right)^{2} \left(\lambda + 3 \mu\right) + \left(\alpha^{22}_{6}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{22}_{6} \alpha^{22}_{8} \left(\lambda + 3 \mu\right) + \alpha^{12}_{8} \left(\alpha^{10}_{6} \left(- \lambda - 2 \mu\right) - \alpha^{22}_{6} \mu\right) & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0 \alpha^{11}_{7} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{11}_{7} \left(\lambda + 3 \mu\right) - 0.5 \alpha^{23}_{7} \mu + 0.5 \alpha^{23}_{7} \left(\lambda + 3 \mu\right) & 0.5 \alpha^{23}_{7} \lambda & - 0.5 \alpha^{11}_{7} \mu - 0.5 \alpha^{23}_{7} \mu + 0.5 \alpha^{23}_{7} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{23}_{7} \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & \alpha^{11}_{1} \alpha^{11}_{7} \left(\lambda + 3 \mu\right) + \alpha^{13}_{1} \left(- \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(- \lambda - 2 \mu\right)\right) & 0 & 0 & 0 & 0 & 0 & \left(\alpha^{11}_{7}\right)^{2} \left(\lambda + 3 \mu\right) + \left(\alpha^{23}_{7}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{23}_{7} \alpha^{23}_{9} \left(\lambda + 3 \mu\right) + \alpha^{13}_{9} \left(- \alpha^{11}_{7} \mu + \alpha^{23}_{7} \left(- \lambda - 2 \mu\right)\right) & 0 & 0\\0 & 0 & - 0.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{12}_{8} \left(- \lambda - 2 \mu\right) - 1.0 \alpha^{14}_{8} \mu + 0.5 \alpha^{14}_{8} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{14}_{8} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{26}_{8} \mu & 1.0 \alpha^{12}_{8} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) & 0 & 0 & 0 & 1.0 \alpha^{12}_{8} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.5 \alpha^{22}_{8} \mu & - 0.5 \alpha^{12}_{8} \mu + 0.5 \alpha^{12}_{8} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{12}_{8} \left(2 \lambda + 6 \mu\right) - 1.0 \alpha^{14}_{8} \mu + 0.5 \alpha^{14}_{8} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{14}_{8} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{22}_{8} \mu + 1.0 \alpha^{22}_{8} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{22}_{8} \left(\lambda + 3 \mu\right) + 1.0 \alpha^{26}_{8} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{26}_{8} \left(\lambda + 3 \mu\right) & 1.0 \alpha^{12}_{8} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) + 0.5 \alpha^{22}_{8} \mu - 0.5 \alpha^{26}_{8} \mu & 0 & 0.5 \alpha^{26}_{8} \mu & \alpha^{10}_{0} \alpha^{12}_{8} \left(- \lambda - 2 \mu\right) + \alpha^{12}_{0} \left(\alpha^{12}_{8} \left(2 \lambda + 6 \mu\right) + \alpha^{14}_{8} \left(- \lambda - 2 \mu\right) - \alpha^{22}_{8} \mu\right) - \alpha^{2}_{0} \alpha^{12}_{8} \mu & 0 & \alpha^{14}_{2} \left(\alpha^{12}_{8} \left(- \lambda - 2 \mu\right) + \alpha^{14}_{8} \left(2 \lambda + 6 \mu\right)\right) + \alpha^{16}_{2} \left(\alpha^{14}_{8} \left(- \lambda - 2 \mu\right) - \alpha^{26}_{8} \mu\right) - \alpha^{2}_{2} \alpha^{12}_{8} \mu & 0 & 0 & 0 & \alpha^{10}_{6} \alpha^{12}_{8} \left(- \lambda - 2 \mu\right) + \alpha^{22}_{6} \left(- \alpha^{12}_{8} \mu + \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) & 0 & \alpha^{12}_{8} \left(\alpha^{12}_{8} \left(2 \lambda + 6 \mu\right) + \alpha^{14}_{8} \left(- \lambda - 2 \mu\right) - \alpha^{22}_{8} \mu\right) + \alpha^{14}_{8} \left(\alpha^{12}_{8} \left(- \lambda - 2 \mu\right) + \alpha^{14}_{8} \left(2 \lambda + 6 \mu\right)\right) + \alpha^{22}_{8} \left(- \alpha^{12}_{8} \mu + \alpha^{22}_{8} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{26}_{8}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{16}_{10} \left(\alpha^{14}_{8} \left(- \lambda - 2 \mu\right) - \alpha^{26}_{8} \mu\right) + \alpha^{26}_{10} \alpha^{26}_{8} \left(\lambda + 3 \mu\right) & 0\\0 & 0 & 1.0 \alpha^{13}_{9} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) & - 0.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{13}_{9} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{15}_{9} \mu + 1.0 \alpha^{15}_{9} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{15}_{9} \left(2 \lambda + 6 \mu\right) + 0.5 \alpha^{27}_{9} \left(- \lambda - 2 \mu\right) & 0 & 0 & 0 & - 0.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{13}_{9} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{23}_{9} \mu + 0.5 \alpha^{23}_{9} \left(\lambda + 3 \mu\right) & 1.0 \alpha^{13}_{9} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) + 0.5 \alpha^{23}_{9} \lambda - 0.5 \alpha^{27}_{9} \lambda & - 0.5 \alpha^{13}_{9} \mu + 0.5 \alpha^{13}_{9} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{13}_{9} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{15}_{9} \mu + 1.0 \alpha^{15}_{9} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{15}_{9} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{23}_{9} \mu + 0.5 \alpha^{23}_{9} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{23}_{9} \left(\lambda + 3 \mu\right) - 0.5 \alpha^{27}_{9} \mu + 0.5 \alpha^{27}_{9} \left(\lambda + 3 \mu\right) & 0 & - 0.5 \alpha^{15}_{9} \mu - 0.5 \alpha^{27}_{9} \mu + 0.5 \alpha^{27}_{9} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{27}_{9} \left(\lambda + 3 \mu\right) & 0 & - \alpha^{11}_{1} \alpha^{13}_{9} \mu + \alpha^{13}_{1} \left(\alpha^{13}_{9} \left(2 \lambda + 6 \mu\right) - \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(- \lambda - 2 \mu\right)\right) + \alpha^{3}_{1} \alpha^{13}_{9} \left(- \lambda - 2 \mu\right) & 0 & \alpha^{15}_{3} \left(- \alpha^{13}_{9} \mu + \alpha^{15}_{9} \left(2 \lambda + 6 \mu\right)\right) + \alpha^{17}_{3} \left(- \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(- \lambda - 2 \mu\right)\right) + \alpha^{3}_{3} \alpha^{13}_{9} \left(- \lambda - 2 \mu\right) & 0 & 0 & 0 & - \alpha^{11}_{7} \alpha^{13}_{9} \mu + \alpha^{23}_{7} \left(\alpha^{13}_{9} \left(- \lambda - 2 \mu\right) + \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) & 0 & \alpha^{13}_{9} \left(\alpha^{13}_{9} \left(2 \lambda + 6 \mu\right) - \alpha^{15}_{9} \mu + \alpha^{23}_{9} \left(- \lambda - 2 \mu\right)\right) + \alpha^{15}_{9} \left(- \alpha^{13}_{9} \mu + \alpha^{15}_{9} \left(2 \lambda + 6 \mu\right)\right) + \alpha^{23}_{9} \left(\alpha^{13}_{9} \left(- \lambda - 2 \mu\right) + \alpha^{23}_{9} \left(\lambda + 3 \mu\right)\right) + \left(\alpha^{27}_{9}\right)^{2} \left(\lambda + 3 \mu\right) & 0 & \alpha^{17}_{11} \left(- \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(- \lambda - 2 \mu\right)\right) + \alpha^{27}_{11} \alpha^{27}_{9} \left(\lambda + 3 \mu\right)\\0 & 0 & - 0.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{16}_{10} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{16}_{10} \left(2 \lambda + 6 \mu\right) + 0.5 \alpha^{18}_{10} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{26}_{10} \mu & 1.0 \alpha^{16}_{10} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) & 0 & 1.0 \alpha^{16}_{10} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.5 \alpha^{18}_{10} \lambda & 0 & 0 & - 0.5 \alpha^{16}_{10} \mu + 0.5 \alpha^{16}_{10} \left(- \lambda - 2 \mu\right) + 1.0 \alpha^{26}_{10} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{26}_{10} \left(\lambda + 3 \mu\right) & 1.0 \alpha^{16}_{10} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.5 \alpha^{26}_{10} \mu & 0 & 1.0 \alpha^{16}_{10} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) + 0.5 \alpha^{18}_{10} \lambda + 0.5 \alpha^{26}_{10} \mu & 0 & 0 & \alpha^{16}_{10} \alpha^{14}_{2} \left(- \lambda - 2 \mu\right) - \alpha^{16}_{10} \alpha^{6}_{2} \mu + \alpha^{16}_{2} \left(\alpha^{16}_{10} \left(2 \lambda + 6 \mu\right) + \alpha^{18}_{10} \left(- \lambda - 2 \mu\right) - \alpha^{26}_{10} \mu\right) & 0 & - \alpha^{16}_{10} \alpha^{6}_{4} \mu + \alpha^{18}_{4} \left(\alpha^{16}_{10} \left(- \lambda - 2 \mu\right) + \alpha^{18}_{10} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & \alpha^{16}_{10} \alpha^{14}_{8} \left(- \lambda - 2 \mu\right) + \alpha^{26}_{8} \left(- \alpha^{16}_{10} \mu + \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0 & \alpha^{16}_{10} \left(\alpha^{16}_{10} \left(2 \lambda + 6 \mu\right) + \alpha^{18}_{10} \left(- \lambda - 2 \mu\right) - \alpha^{26}_{10} \mu\right) + \alpha^{18}_{10} \left(\alpha^{16}_{10} \left(- \lambda - 2 \mu\right) + \alpha^{18}_{10} \left(\lambda + 3 \mu\right)\right) + \alpha^{26}_{10} \left(- \alpha^{16}_{10} \mu + \alpha^{26}_{10} \left(\lambda + 3 \mu\right)\right) & 0\\0 & 0 & 1.0 \alpha^{17}_{11} \left(- \frac{\lambda}{2} - \frac{\mu}{2}\right) & - 0.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{17}_{11} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{17}_{11} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{19}_{11} \mu + 0.5 \alpha^{27}_{11} \left(- \lambda - 2 \mu\right) & 0 & - 0.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{17}_{11} \left(- \lambda - 2 \mu\right) + 1.0 \alpha^{19}_{11} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{19}_{11} \left(\lambda + 3 \mu\right) & 0 & 0 & 1.0 \alpha^{17}_{11} \left(\frac{\lambda}{2} + \frac{\mu}{2}\right) - 0.5 \alpha^{27}_{11} \lambda & - 0.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{17}_{11} \left(- \lambda - 2 \mu\right) - 0.5 \alpha^{27}_{11} \mu + 0.5 \alpha^{27}_{11} \left(\lambda + 3 \mu\right) & 0 & - 0.5 \alpha^{17}_{11} \mu + 0.5 \alpha^{17}_{11} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{17}_{11} \left(2 \lambda + 6 \mu\right) - 0.5 \alpha^{19}_{11} \mu + 1.0 \alpha^{19}_{11} \left(- \frac{\lambda}{2} - \mu\right) + 0.5 \alpha^{19}_{11} \left(\lambda + 3 \mu\right) - 0.5 \alpha^{27}_{11} \mu + 0.5 \alpha^{27}_{11} \left(- \lambda - 2 \mu\right) + 0.5 \alpha^{27}_{11} \left(\lambda + 3 \mu\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{3} \mu + \alpha^{17}_{11} \alpha^{7}_{3} \left(- \lambda - 2 \mu\right) + \alpha^{17}_{3} \left(\alpha^{17}_{11} \left(2 \lambda + 6 \mu\right) - \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(- \lambda - 2 \mu\right)\right) & 0 & \alpha^{17}_{11} \alpha^{7}_{5} \left(- \lambda - 2 \mu\right) + \alpha^{19}_{5} \left(- \alpha^{17}_{11} \mu + \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & 0 & 0 & - \alpha^{17}_{11} \alpha^{15}_{9} \mu + \alpha^{27}_{9} \left(\alpha^{17}_{11} \left(- \lambda - 2 \mu\right) + \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right) & 0 & \alpha^{17}_{11} \left(\alpha^{17}_{11} \left(2 \lambda + 6 \mu\right) - \alpha^{19}_{11} \mu + \alpha^{27}_{11} \left(- \lambda - 2 \mu\right)\right) + \alpha^{19}_{11} \left(- \alpha^{17}_{11} \mu + \alpha^{19}_{11} \left(\lambda + 3 \mu\right)\right) + \alpha^{27}_{11} \left(\alpha^{17}_{11} \left(- \lambda - 2 \mu\right) + \alpha^{27}_{11} \left(\lambda + 3 \mu\right)\right)\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle B_C^3 = Q^t.(B-Z_S)+XD_C=\left[\begin{matrix}c_{x}\\c_{y}\\- 0.5 L^{2} f + 0.5 c_{x} \lambda + 1.0 c_{x} \mu - 0.5 c_{y} \mu + 0.5 i_{x} \lambda + 1.0 i_{x} \mu\\- 0.5 c_{x} \lambda + 0.5 c_{y} \mu + 0.5 i_{x} \lambda\\i_{x}\\0.5 i_{x} \lambda\\c_{x}\\0.5 c_{x} \lambda + 0.5 c_{y} \lambda + 1.0 c_{y} \mu\\- 0.5 L^{2} f + 0.5 c_{x} \lambda + 1.0 c_{x} \mu + 0.5 c_{y} \lambda + 0.5 c_{y} \mu + 0.5 i_{x} \lambda + 1.0 i_{x} \mu\\0.5 \lambda \left(c_{x} - i_{x}\right)\\i_{x}\\- 0.5 i_{x} \lambda\\- 0.0833333333333333 L^{2} \alpha^{10}_{0} f - 0.333333333333333 L^{2} \alpha^{12}_{0} f - 0.0833333333333333 L^{2} \alpha^{2}_{0} f - 0.5 \alpha^{10}_{0} c_{x} \lambda - 1.0 \alpha^{10}_{0} c_{x} \mu - 0.5 \alpha^{10}_{0} c_{y} \lambda + 0.5 \alpha^{12}_{0} c_{y} \lambda + 0.5 \alpha^{12}_{0} c_{y} \mu - 0.5 \alpha^{2}_{0} c_{y} \mu\\- 0.5 \alpha^{13}_{1} c_{y} \lambda - 1.5 \alpha^{13}_{1} c_{y} \mu - 0.5 \alpha^{3}_{1} c_{x} \lambda\\- 0.166666666666667 L^{2} \alpha^{14}_{2} f - 0.333333333333333 L^{2} \alpha^{16}_{2} f - 0.0833333333333333 L^{2} \alpha^{2}_{2} f - 0.0833333333333333 L^{2} \alpha^{6}_{2} f + 0.5 \alpha^{14}_{2} c_{x} \lambda + 1.0 \alpha^{14}_{2} c_{x} \mu + 0.5 \alpha^{14}_{2} i_{x} \lambda + 1.0 \alpha^{14}_{2} i_{x} \mu - 0.5 \alpha^{2}_{2} c_{y} \mu\\0.5 \alpha^{15}_{3} c_{y} \mu - 0.5 \alpha^{3}_{3} c_{x} \lambda + 0.5 \alpha^{7}_{3} i_{x} \lambda\\- 0.0833333333333333 L^{2} \alpha^{18}_{4} f - 0.0833333333333333 L^{2} \alpha^{6}_{4} f - 0.5 \alpha^{18}_{4} i_{x} \lambda - 1.0 \alpha^{18}_{4} i_{x} \mu\\0.5 \alpha^{7}_{5} i_{x} \lambda\\- 0.0833333333333333 L^{2} \alpha^{10}_{6} f - 0.0833333333333333 L^{2} \alpha^{22}_{6} f - 0.5 \alpha^{10}_{6} c_{x} \lambda - 1.0 \alpha^{10}_{6} c_{x} \mu - 0.5 \alpha^{10}_{6} c_{y} \lambda\\0.5 \alpha^{23}_{7} \left(c_{x} \lambda + c_{y} \left(\lambda + 2 \mu\right)\right)\\- 0.333333333333333 L^{2} \alpha^{12}_{8} f - 0.166666666666667 L^{2} \alpha^{14}_{8} f - 0.0833333333333333 L^{2} \alpha^{22}_{8} f - 0.0833333333333333 L^{2} \alpha^{26}_{8} f + 0.5 \alpha^{12}_{8} c_{y} \lambda + 0.5 \alpha^{12}_{8} c_{y} \mu + 0.5 \alpha^{14}_{8} c_{x} \lambda + 1.0 \alpha^{14}_{8} c_{x} \mu + 0.5 \alpha^{14}_{8} i_{x} \lambda + 1.0 \alpha^{14}_{8} i_{x} \mu\\- \alpha^{13}_{9} c_{y} \left(0.5 \lambda + 1.5 \mu\right) + 0.5 \alpha^{15}_{9} c_{y} \mu + 0.5 \alpha^{23}_{9} \left(c_{x} \lambda + c_{y} \left(\lambda + 2 \mu\right)\right) - 0.5 \alpha^{27}_{9} i_{x} \lambda\\- 0.333333333333333 L^{2} \alpha^{16}_{10} f - 0.0833333333333333 L^{2} \alpha^{18}_{10} f - 0.0833333333333333 L^{2} \alpha^{26}_{10} f - 0.5 \alpha^{18}_{10} i_{x} \lambda - 1.0 \alpha^{18}_{10} i_{x} \mu\\- 0.5 \alpha^{27}_{11} i_{x} \lambda\end{matrix}\right]\end{split}\]
Compare to approch (I) we have
\[\begin{split}\displaystyle A_C^3 -A_C =\left[\begin{array}{cccccccccccccccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\end{split}\]
\[\begin{split}\displaystyle B_C^3 -B_C=\left[\begin{matrix}0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\end{matrix}\right]\end{split}\]
So the solution of this system will be the same as the one of the approach (I)
It is this approach that as been retained for the implementation of the TS library. Thus the coarse system is generated by the cm instance as follows. Before looping, \(Z_C\) and the standard constant part of \(A_C^3\) and \(B_C^3\) are computed out of \(A\),\(B\):
%%px
cm.setStdCoarse(A,B)
To continue we must create the enriched function
%px enriched_shift=core.generateEnrichedShiftFunction(sf)
And then use the last computed solution of the patches to set the enriched part of the \(Q\) operator:
%px cm.updateEnrichedOperator(pm,enriched_shift)
At this stage \(Q\) is fully created and \(A_C^3\),\(B_C^3\) can be finalised by computing all missing blocks from \(A\),\(B\) and \(Q\)
%%px
#cm.resetCoarseToStd()
cm.updateEnrichCoarse(A,B)
The system \(A_C^3\),\(B_C^3\) is now created and can be solved. The library updates the TS approximation in the same call.
%%px
Sts1=fem.Function(space)
cm.solve(Sts1)
The residual norm of \(AD\),\(BD\) system is given for the first two approximation by:
%%px
nb=BD.norm()
Sts0p=Sts0.x.petsc_vec
Sts1p=Sts1.x.petsc_vec
#nb=linear.residual(AD,BD,Sts0p,1)
normsts0=linear.residual(AD,BD,Sts0p,nb)
normsts1=linear.residual(AD,BD,Sts1p,nb)
[output:0]
' norm(AD.Sts_0-BD)/norm(BD)=0.4617363128077323'
[output:0]
' norm(AD.Sts_1-BD)/norm(BD)=0.058547577746052605'
Then loop on level gives
%%px
convTS_curv=[normsts0,normsts1]
Stsi=fem.Function(space)
Stsi.x.array[:]=Sts1.x.array
for i in range(2,30):
pm.solveProblems(Stsi)
cm.resetCoarseToStd()
cm.updateEnrichedOperator(pm,enriched_shift)
cm.updateEnrichCoarse(A,B)
cm.solve(Stsi)
normstsiN=linear.residual(AD,BD,Stsi.x.petsc_vec,nb)
if i<10 :
if MPI.COMM_WORLD.rank<1:
display(f' norm(AD.Sts_{i}-BD)/norm(BD)={normstsiN}')
convTS_curv.append(normstsiN)
mplt.plot(conv_curv,label='Symbolic computation')
mplt.plot(view['convTS_curv'][0],label='Library computation')
mplt.legend()
mplt.yscale('log')
mplt.xlabel('TS iterations')
mplt.ylabel('$norm(AD.Sts_i-BD)/norm(BD)$')
Text(0, 0.5, '$norm(AD.Sts_i-BD)/norm(BD)$')
With TS library/FEniCSx all the steps above are grouped in linearBasicLoop function that use two criterions to stop the loop:
a maximum number of iterations
a threshold under which relative system residual is considered as small as needed. Relativeness is either against rhs norm (hrb) or first residual value (hrr).
%%px
[dispf, r,nm, it,hrb,hrr]=linear.linearBasicLoop(sj,space,A,AD,B,BD,enriched_shift,coarse_field,None,bcc,30,1e-13)
Text(0, 0.5, 'Criterion')
