import numpy as np
from scipy.sparse.linalg import spsolve
from scipy.sparse import coo_matrix, csr_matrix, spdiags, bmat
from scipy.sparse.linalg import spsolve, cg, LinearOperator, spilu
from fealpy.pde.linear_elasticity_model import HuangModel2d as PDE
from fealpy.pde.linear_elasticity_model import PolyModel3d as PDE
from fealpy.functionspace import LagrangeFiniteElementSpace
from fealpy.boundarycondition import DirichletBC
n = 5
pde = PDE(lam=1.0, mu=1.0)
mesh = pde.init_mesh(n=n)
space = LagrangeFiniteElementSpace(mesh, p=1, q=3)
uh = space.function(dim=3)
#A = space.recovery_linear_elasticity_matrix(pde.lam, pde.mu)
A = space.linear_elasticity_matrix(pde.lam, pde.mu)
F = space.source_vector(pde.source, dim=3)
bc = DirichletBC(space, pde.dirichlet)
A, F = bc.apply(A, F, uh)
uh.T.flat[:], info = cg(A, F, tol=1e-10)
error = space.integralalg.L2_error(pde.displacement, uh)
print('error:\n', error)
class WaterFloodingModelSolver():
"""
Notes
-----
"""
def __init__(self, model, T=800 * 3600 * 24, NS=32, NT=800 * 24):
self.model = model
self.mesh = model.space_mesh(n=NS)
self.timeline = model.time_mesh(T=T, n=NT)
self.GD = model.GD
if self.GD == 2:
self.vspace = RaviartThomasFiniteElementSpace2d(self.mesh,
p=0) # 速度空间
elif self.GD == 3:
self.vspace = RaviartThomasFiniteElementSpace3d(self.mesh, p=0)
self.pspace = self.vspace.smspace # 压强和饱和度所属的空间, 分片常数
self.cspace = LagrangeFiniteElementSpace(self.mesh, p=1) # 位移空间
# 上一时刻物理量的值
self.v = self.vspace.function() # 速度函数
self.p = self.pspace.function() # 压强函数
self.s = self.pspace.function() # 水的饱和度函数 默认为0, 初始时刻区域内水的饱和度为0
self.u = self.cspace.function(dim=self.GD) # 位移函数
self.phi = self.pspace.function() # 孔隙度函数, 分片常数
# 当前时刻物理量的值, 用于保存临时计算出的值, 模型中系数的计算由当前时刻
# 的物理量的值决定
self.cv = self.vspace.function() # 速度函数
self.cp = self.pspace.function() # 压强函数
self.cs = self.pspace.function() # 水的饱和度函数 默认为0, 初始时刻区域内水的饱和度为0
self.cu = self.cspace.function(dim=self.GD) # 位移函数
self.cphi = self.pspace.function() # 孔隙度函数, 分片常数
# 初值
self.p[:] = model.rock['initial pressure'] # MPa
self.phi[:] = model.rock['porosity'] # 初始孔隙度
self.cp[:] = model.rock['initial pressure'] # 初始地层压强
self.cphi[:] = model.rock['porosity'] # 当前孔隙度系数
# 源项, TODO: 注意这里假设用的是结构网格, 换其它的网格需要修改代码
self.fo = self.cspace.function()
self.fo[-1] = -self.model.oil['production rate'] # 产出
self.fw = self.cspace.function()
self.fw[0] = self.model.water['injection rate'] # 注入
# 一些常数矩阵和向量
# 速度散度矩阵, 速度方程对应的散度矩阵, (\nabla\cdot v, w)
self.B = self.vspace.div_matrix()
# 压强方程对应的位移散度矩阵, (\nabla\cdot u, w) 位移散度矩阵
# * 注意这里利用了压强空间分片常数, 线性函数导数也是分片常数的事实
c = self.mesh.entity_measure('cell')
c *= self.model.rock['biot']
val = self.mesh.grad_lambda() # (NC, TD+1, GD)
val *= c[:, None, None]
pc2d = self.pspace.cell_to_dof()
cc2d = self.cspace.cell_to_dof()
pgdof = self.pspace.number_of_global_dofs()
cgdof = self.cspace.number_of_global_dofs()
I = np.broadcast_to(pc2d, shape=cc2d.shape)
J = cc2d
self.PU0 = csr_matrix((val[..., 0].flat, (I.flat, J.flat)),
shape=(pgdof, cgdof))
self.PU1 = csr_matrix((val[..., 1].flat, (I.flat, J.flat)),
shape=(pgdof, cgdof))
if self.GD == 3:
self.PU2 = csr_matrix((val[..., 2].flat, (I.flat, J.flat)),
shape=(pgdof, cgdof))
# 线弹性矩阵的右端向量
sigma0 = self.pspace.function()
sigma0[:] = self.model.rock['initial stress']
self.FU = np.zeros(self.GD * cgdof, dtype=np.float64)
self.FU[0 * cgdof:1 * cgdof] -= self.p @ self.PU0
self.FU[1 * cgdof:2 * cgdof] -= self.p @ self.PU1
if self.GD == 3:
self.FU[2 * cgdof:3 * cgdof] -= self.p @ self.PU2
# 初始应力和等效应力项
self.FU[0 * cgdof:1 * cgdof] -= sigma0 @ self.PU0
self.FU[1 * cgdof:2 * cgdof] -= sigma0 @ self.PU1
if self.GD == 3:
self.FU[2 * cgdof:3 * cgdof] -= sigma0 @ self.PU2
# vtk 文件输出
node, cell, cellType, NC = self.mesh.to_vtk()
self.points = vtk.vtkPoints()
self.points.SetData(vnp.numpy_to_vtk(node))
self.cells = vtk.vtkCellArray()
self.cells.SetCells(NC, vnp.numpy_to_vtkIdTypeArray(cell))
self.cellType = cellType
def recover(self, val):
"""
Notes
-----
给定一个分片常数的量, 恢复为分片连续的量
"""
mesh = self.mesh
cell = self.mesh.entity('cell')
NC = mesh.number_of_cells()
NN = mesh.number_of_nodes()
w = 1 / self.mesh.entity_measure('cell') # 恢复权重
w = np.broadcast_to(w[:, None], shape=cell.shape)
r = np.zeros(NN, dtype=np.float64)
d = np.zeros(NN, dtype=np.float64)
np.add.at(d, cell, w)
np.add.at(r, cell, val[:, None] * w)
return r / d
def pressure_coefficient(self):
"""
Notes
-----
计算当前物理量下的压强质量矩阵系数
"""
b = self.model.rock['biot']
Ks = self.model.rock['solid grain stiffness']
cw = self.model.water['compressibility']
co = self.model.oil['compressibility']
Sw = self.cs[:].copy() # 当前的水饱和度 (NQ, NC)
phi = self.cphi[:].copy() # 当前的孔隙度
val = phi * Sw * cw
val += phi * (1 - Sw) * co # 注意这里的 co 是常数, 但在水气混合物条件下应该依赖于压力
val += (b - phi) / Ks
return val
def saturation_pressure_coefficient(self):
"""
Notes
-----
计算当前物理量下, 饱和度方程中, 压强项对应的系数
"""
b = self.model.rock['biot']
Ks = self.model.rock['solid grain stiffness']
cw = self.model.water['compressibility']
val = self.cs[:].copy() # 当前水饱和度
phi = self.cphi[:].copy() # 当前孔隙度
val *= (b - phi) / Ks + phi * cw
return val
def flux_coefficient(self):
"""
Notes
-----
计算**当前**物理量下, 速度方程对应的系数
计算通量的系数, 流体的相对渗透率除以粘性系数得到流动性系数
krg/mu_g 是气体的流动性系数
kro/mu_o 是油的流动性系数
krw/mu_w 是水的流动性系数
这里假设压强的单位是 MPa
1 d = 9.869 233e-13 m^2 = 1000 md
1 cp = 1 mPa s = 1e-9 MPa.s
"""
muw = self.model.water['viscosity'] # 单位是 1 cp = 1 mPa.s
muo = self.model.oil['viscosity'] # 单位是 1 cp = 1 mPa.s
# 岩石的绝对渗透率, 这里考虑了量纲的一致性, 压强是 MPa
k = self.model.rock['permeability'] * 9.869233e-4
Sw = self.cs[:].copy() # 当前水的饱和度系数
lamw = self.model.water_relative_permeability(Sw)
lamw /= muw
lamo = self.model.oil_relative_permeability(Sw)
lamo /= muo
val = 1 / (lamw + lamo) / k #
return val
def water_fractional_flow_coefficient(self):
"""
Notes
-----
计算**当前**物理量下, 饱和度方程中, 水的流动性系数
"""
Sw = self.cs[:].copy() # 当前水的饱和度系数
lamw = self.model.water_relative_permeability(Sw)
lamw /= self.model.water['viscosity']
lamo = self.model.oil_relative_permeability(Sw)
lamo /= self.model.oil['viscosity']
val = lamw / (lamw + lamo)
return val
def get_total_system(self):
"""
Notes
-----
构造整个系统
二维情形：
x = [v, p, s, u0, u1]
A = [[ V, VP, None, None, None]
[ PV, P, None, PU0, PU1]
[None, SP, S, SU0, SU1]
[None, UP0, None, U00, U01]
[None, UP1, None, U10, U11]
F = [FV, FP, FS, FU0, FU1]
三维情形：
x = [v, p, s, u0, u1, u2]
A = [[ V, VP, None, None, None, None]
[ PV, P, None, PU0, PU1, PU2]
[None, SP, S, SU0, SU1, SU2]
[None, UP0, None, U00, U01, U02]
[None, UP1, None, U10, U11, U12]
[None, UP2, None, U20, U21, U22]]
F = [FV, FP, FS, FU0, FU1, FU2]
FS 中考虑的迎风格式
"""
GD = self.GD
A0, FV, isBdDof0 = self.get_velocity_system(q=2)
A1, FP, isBdDof1 = self.get_pressure_system(q=2)
A2, FS, isBdDof2 = self.get_saturation_system(q=2)
if GD == 2:
A3, A4, FU, isBdDof3 = self.get_dispacement_system(q=2)
A = bmat([A0, A1, A2, A3, A4], format='csr')
F = np.r_['0', FV, FP, FS, FU]
elif GD == 3:
A3, A4, A5, FU, isBdDof3 = self.get_dispacement_system(q=2)
A = bmat([A0, A1, A2, A3, A4, A5], format='csr')
F = np.r_['0', FV, FP, FS, FU]
isBdDof = np.r_['0', isBdDof0, isBdDof1, isBdDof2, isBdDof3]
return A, F, isBdDof
def get_velocity_system(self, q=2):
"""
Notes
-----
计算速度方程对应的离散系统.
if GD == 2:
[ V, VP, None, None, None]
elif GD == 3:
[ V, VP, None, None, None, None]
"""
GD = self.GD
dt = self.timeline.current_time_step_length()
cellmeasure = self.mesh.entity_measure('cell')
qf = self.mesh.integrator(q, etype='cell')
bcs, ws = qf.get_quadrature_points_and_weights()
# 速度对应的矩阵 V
c = self.flux_coefficient()
c *= cellmeasure
phi = self.vspace.basis(bcs)
V = np.einsum('q, qcin, qcjn, c->cij', ws, phi, phi, c, optimize=True)
c2d = self.vspace.cell_to_dof()
I = np.broadcast_to(c2d[:, :, None], shape=V.shape)
J = np.broadcast_to(c2d[:, None, :], shape=V.shape)
gdof = self.vspace.number_of_global_dofs()
V = csr_matrix((V.flat, (I.flat, J.flat)), shape=(gdof, gdof))
# 压强矩阵
VP = -self.B
# 右端向量, 0 向量
FV = np.zeros(gdof, dtype=np.float64)
isBdDof = self.vspace.dof.is_boundary_dof()
if GD == 2:
return [V, VP, None, None, None], FV, isBdDof
elif GD == 3:
return [V, VP, None, None, None, None], FV, isBdDof
def get_pressure_system(self, q=2):
"""
Notes
-----
计算压强方程对应的离散系统
这里组装矩阵时, 利用了压强是分片常数的特殊性
[ PV, P, None, PU0, PU1]
[ PV, P, None, PU0, PU1, PU2]
"""
GD = self.GD
dt = self.timeline.current_time_step_length()
cellmeasure = self.mesh.entity_measure('cell')
qf = self.mesh.integrator(q, etype='cell')
bcs, ws = qf.get_quadrature_points_and_weights()
PV = dt * self.B.T
# P 是对角矩阵, 利用分片常数的
c = self.pressure_coefficient() # (NQ, NC)
c *= cellmeasure
P = diags(c, 0)
# 组装压强方程的右端向量
# * 这里利用了压强空间基是分片常数
FP = self.fo.value(bcs) + self.fw.value(bcs) # (NQ, NC)
FP *= ws[:, None]
FP = np.sum(FP, axis=0)
FP *= cellmeasure
FP *= dt
FP += P @ self.p # 上一步的压强向量
FP += self.PU0 @ self.u[:, 0]
FP += self.PU1 @ self.u[:, 1]
if GD == 3:
FP += self.PU2 @ self.u[:, 2]
gdof = self.pspace.number_of_global_dofs()
isBdDof = np.zeros(gdof, dtype=np.bool_)
if GD == 2:
return [PV, P, None, self.PU0, self.PU1], FP, isBdDof
elif GD == 3:
return [PV, P, None, self.PU0, self.PU1, self.PU2], FP, isBdDof
def get_saturation_system(self, q=2):
"""
Notes
----
计算饱和度方程对应的离散系统
[ None, SP, S, SU0, SU1]
"""
GD = self.GD
qf = self.mesh.integrator(q, etype='cell')
bcs, ws = qf.get_quadrature_points_and_weights()
cellmeasure = self.mesh.entity_measure('cell')
# SP 是对角矩阵
c = self.saturation_pressure_coefficient() # (NC, )
c *= cellmeasure
SP = diags(c, 0)
# S 质量矩阵组装, 对角矩阵
c = self.cphi[:] * cellmeasure
S = diags(c, 0)
# SU0, SU1, 饱和度方程中的位移散度对应的矩阵
val = self.mesh.grad_lambda() # (NC, TD+1, GD)
c = self.cs[:] * self.model.rock['biot'] # (NC, ), 注意用当前的水饱和度
c *= cellmeasure
val *= c[:, None, None]
pgdof = self.pspace.number_of_global_dofs() # 压力空间自由度个数
cgdof = self.cspace.number_of_global_dofs() # 连续空间自由度个数
pc2d = self.pspace.cell_to_dof()
cc2d = self.cspace.cell_to_dof()
I = np.broadcast_to(pc2d, shape=cc2d.shape)
J = cc2d
SU0 = csr_matrix((val[..., 0].flat, (I.flat, J.flat)),
shape=(pgdof, cgdof))
SU1 = csr_matrix((val[..., 1].flat, (I.flat, J.flat)),
shape=(pgdof, cgdof))
if GD == 3:
SU2 = csr_matrix((val[..., 2].flat, (I.flat, J.flat)),
shape=(pgdof, cgdof))
# 右端矩阵
dt = self.timeline.current_time_step_length()
FS = self.fw.value(bcs) # (NQ, NC)
FS *= ws[:, None]
FS = np.sum(FS, axis=0)
FS *= cellmeasure
FS *= dt
FS += SP @ self.p # 上一时间步的压强
FS += S @ self.s # 上一时间步的饱和度
FS += SU0 @ self.u[:, 0] # 上一时间步的位移 x 分量
FS += SU1 @ self.u[:, 1] # 上一个时间步的位移 y 分量
if GD == 3:
FS += SU2 @ self.u[:, 2] # 上一个时间步的位移 y 分量
# 用当前时刻的速度场, 构造非线性迎风格式
face2cell = self.mesh.ds.face_to_cell()
isBdFace = face2cell[:, 0] == face2cell[:, 1]
qf = self.mesh.integrator(2, 'face') # 边上的积分公式
bcs, ws = qf.get_quadrature_points_and_weights()
facemeasure = self.mesh.entity_measure('face')
# 边的定向法线，它是左边单元的外法线，右边单元内法线。
fn = self.mesh.face_unit_normal()
val0 = np.einsum('qfm, fm->qf', self.cv.face_value(bcs),
fn) # 当前速度和法线的内积
# 水的流动分数, 与水的饱和度有关, 如果饱和度为0, 则为 0
Fw = self.water_fractional_flow_coefficient()
val1 = Fw[face2cell[:, 0]] # 边的左边单元的水流动分数
val2 = Fw[face2cell[:, 1]] # 边的右边单元的水流动分数
val2[isBdFace] = 0.0 # 边界的贡献是 0
flag = val0 < 0.0 # 对于左边单元来说，是流出项
# 对于右边单元来说，是流入项
# 左右单元流入流出的绝对量是一样的
val = val0 * val1[None, :] # val0 >= 0.0, 左边单元是流出
# val0 < 0.0, 左边单元是流入
val[flag] = (val0 * val2[None, :])[flag] # val0 >= 0, 右边单元是流入
# val0 < 0, 右边单元是流出
b = np.einsum('q, qf, f->f', ws, val, facemeasure)
b *= dt
np.subtract.at(FS, face2cell[:, 0], b)
isInFace = ~isBdFace # 只处理内部边
np.add.at(FS, face2cell[isInFace, 1], b[isInFace])
isBdDof = np.zeros(pgdof, dtype=np.bool_)
if GD == 2:
return [None, SP, S, SU0, SU1], FS, isBdDof
elif GD == 3:
return [None, SP, S, SU0, SU1, SU2], FS, isBdDof
def get_dispacement_system(self, q=2):
"""
Notes
-----
计算位移方程对应的离散系统, 即线弹性方程系统.
GD == 2:
[None, UP0, None, U00, U01]
[None, UP1, None, U10, U11]
GD == 3:
[None, UP0, None, U00, U01, U02]
[None, UP1, None, U10, U11, U12]
[None, UP2, None, U20, U21, U22]
"""
GD = self.GD
# 拉梅参数 (lambda, mu)
lam, mu = self.model.rock['lame']
U = self.cspace.linear_elasticity_matrix(lam, mu, format='list')
isBdDof = self.cspace.dof.is_boundary_dof()
if GD == 2:
return ([None, -self.PU0.T, None, U[0][0],
U[0][1]], [None, -self.PU1.T, None, U[1][0],
U[1][1]], self.FU, np.r_['0', isBdDof,
isBdDof])
elif GD == 3:
return ([None, -self.PU0.T, None, U[0][0], U[0][1], U[0][2]
], [None, -self.PU1.T, None, U[1][0], U[1][1], U[1][2]],
[None, -self.PU2.T, None, U[2][0], U[2][1],
U[2][2]], self.FU, np.r_['0', isBdDof, isBdDof, isBdDof])
def picard_iteration(self, maxit=10):
GD = self.GD
e0 = 1.0
k = 0
while e0 > 1e-10:
# 构建总系统
A, F, isBdDof = self.get_total_system()
# 处理边界条件, 这里是 0 边界
gdof = len(isBdDof)
bdIdx = np.zeros(gdof, dtype=np.int_)
bdIdx[isBdDof] = 1
Tbd = diags(bdIdx)
T = diags(1 - bdIdx)
A = T @ A @ T + Tbd
F[isBdDof] = 0.0
# 求解
x = spsolve(A, F)
vgdof = self.vspace.number_of_global_dofs()
pgdof = self.pspace.number_of_global_dofs()
cgdof = self.cspace.number_of_global_dofs()
e0 = 0.0
start = 0
end = vgdof
self.cv[:] = x[start:end]
start = end
end += pgdof
e0 += np.sum((self.cp - x[start:end])**2)
self.cp[:] = x[start:end]
start = end
end += pgdof
e0 += np.sum((self.cs - x[start:end])**2)
self.cs[:] = x[start:end]
e0 = np.sqrt(e0) # 误差的 l2 norm
k += 1
for i in range(GD):
start = end
end += cgdof
self.cu[:, i] = x[start:end]
print(e0)
if k >= maxit:
print('picard iteration arrive max iteration with error:', e0)
break
def update_solution(self):
self.v[:] = self.cv
self.p[:] = self.cp
self.s[:] = self.cs
#flag = self.s < 0.0
#self.s[flag] = 0.0
self.u[:] = self.cu
def solve(self, step=24):
"""
Notes
-----
计算所有的时间层。
"""
timeline = self.timeline
dt = timeline.current_time_step_length()
timeline.reset() # 时间置零
n = timeline.current
fname = 'test_' + str(n).zfill(10) + '.vtu'
self.write_to_vtk(fname)
while not timeline.stop():
ct = timeline.current_time_level() / 3600 / 24 # 天为单位
print('当前时刻为第', ct, '天')
self.picard_iteration()
self.update_solution()
timeline.current += 1
if timeline.current % step == 0:
n = timeline.current
fname = 'test_' + str(n).zfill(10) + '.vtu'
self.write_to_vtk(fname)
timeline.reset()
def write_to_vtk(self, fname):
# 重心处的值
mesh = self.mesh
GD = self.GD
bc = np.array((GD + 1) * [1 / (GD + 1)], dtype=np.float64)
ps = self.mesh.bc_to_point(bc)
vmesh = vtk.vtkUnstructuredGrid()
vmesh.SetPoints(self.points)
vmesh.SetCells(self.cellType, self.cells)
cdata = vmesh.GetCellData()
pdata = vmesh.GetPointData()
v = self.v
p = self.p
s = self.s
u = self.u
val = v.value(bc)
if GD == 2:
val = np.concatenate(
(val, np.zeros((val.shape[0], 1), dtype=val.dtype)), axis=1)
val = vnp.numpy_to_vtk(val)
val.SetName('velocity')
cdata.AddArray(val)
val = self.recover(p[:])
val = vnp.numpy_to_vtk(val)
val.SetName('pressure')
pdata.AddArray(val)
val = self.recover(s[:])
val = vnp.numpy_to_vtk(val)
val.SetName('saturation')
pdata.AddArray(val)
if GD == 2:
val = np.concatenate((u[:], np.zeros(
(u.shape[0], 1), dtype=u.dtype)),
axis=1)
val = vnp.numpy_to_vtk(val)
val.SetName('displacement')
pdata.AddArray(val)
writer = vtk.vtkXMLUnstructuredGridWriter()
writer.SetFileName(fname)
writer.SetInputData(vmesh)
writer.Write()
scale = float(sys.argv[3])
pde = BoxDomain2DData()
mesh = pde.init_mesh(n=n)
area = mesh.entity_measure('cell')
space = LagrangeFiniteElementSpace(mesh, p=p)
bc0 = DirichletBC(space, pde.dirichlet, threshold=pde.is_dirichlet_boundary)
bc1 = NeumannBC(space, pde.neumann, threshold=pde.is_neumann_boundary)
uh = space.function(dim=2) # (gdof, 2) and vector fem function uh[i, j]
P = space.stiff_matrix(c=2 * pde.mu)
A = space.linear_elasticity_matrix(pde.mu, pde.lam) # (2*gdof, 2*gdof)
F = space.source_vector(pde.source, dim=2)
F = bc1.apply(F)
A, F = bc0.apply(A, F, uh)
print(A.shape)
if False:
uh.T.flat[:] = spsolve(A, F) # (2, gdof ).flat
elif False:
N = len(F)
print(N)
ilu = spilu(A.tocsc(), drop_tol=1e-6, fill_factor=40)
M = LinearOperator((N, N), lambda x: ilu.solve(x))
start = timer()
uh.T.flat[:], info = cg(A, F, tol=1e-8, M=M) # solve with CG
p = int(sys.argv[2])
scale = float(sys.argv[3])
E = 1e+5
nu = 0.3
pde = LShapeDomainData2d(E=E, nu=nu)
mu = pde.mu
lam = pde.lam
mesh = pde.init_mesh(n=n)
space = LagrangeFiniteElementSpace(mesh, p=p)
bc = BoundaryCondition(space, dirichlet=pde.dirichlet, neuman=pde.neuman)
uh = space.function(dim=2)
A = space.linear_elasticity_matrix(mu, lam)
F = space.source_vector(pde.source, dim=2)
bc.apply_neuman_bc(F, is_neuman_boundary=pde.is_neuman_boundary)
A, F = bc.apply_dirichlet_bc(A,
F,
uh,
is_dirichlet_boundary=pde.is_dirichlet_boundary)
uh.T.flat[:] = spsolve(A, F)
error = space.integralalg.L2_error(pde.displacement, uh)
bc = mesh.entity_barycenter('edge')
isBdEdge = mesh.ds.boundary_edge_flag()
from fealpy.functionspace import LagrangeFiniteElementSpace
from fealpy.boundarycondition import BoundaryCondition
n = int(sys.argv[1])
p = int(sys.argv[2])
scale = float(sys.argv[3])
pde = BoxDomainData3d()
mu = pde.mu
lam = pde.lam
mesh = pde.init_mesh(n=n)
space = LagrangeFiniteElementSpace(mesh, p=p)
bc = BoundaryCondition(space, dirichlet=pde.dirichlet)
uh = space.function(dim=3)
A = space.linear_elasticity_matrix(lam, mu)
F = space.source_vector(pde.source, dim=3)
A, F = bc.apply_dirichlet_bc(A,
F,
uh,
is_dirichlet_boundary=pde.is_dirichlet_boundary)
uh.T.flat[:] = spsolve(A, F)
node = mesh.node.copy()
mesh.node = node + scale * uh
fig = plt.figure()
axes = Axes3D(fig)
mesh.add_plot(axes)
plt.show()
