def alg_3_4(self, maxit=None):
"""
1. 最粗网格上求解最小特征特征值问题，得到最小特征值 d_H 和特征向量 u_H
2. 自适应求解 - \Delta u_h = u_H
* u_H 插值到下一层网格上做为新 u_H
3. 最新网格层上求出的 uh 做为一个基函数，加入到最粗网格的有限元空间中，
求解最小特征值问题。
自适应 maxit， picard:2
"""
print("算法 3.4")
if maxit is None:
maxit = self.maxit
start = timer()
if self.step == 0:
idx = []
else:
idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1]
mesh = self.pde.init_mesh(n=self.numrefine)
integrator = mesh.integrator(self.q)
# 1. 粗网格上求解最小特征值问题
space = LagrangeFiniteElementSpace(mesh, 1)
AH = self.get_stiff_matrix(space)
MH = self.get_mass_matrix(space)
isFreeHDof = ~(space.boundary_dof())
gdof = space.number_of_global_dofs()
print("initial mesh :", gdof)
uH = np.zeros(gdof, dtype=np.float)
A = AH[isFreeHDof, :][:, isFreeHDof].tocsr()
M = MH[isFreeHDof, :][:, isFreeHDof].tocsr()
if self.matlab is False:
uH[isFreeHDof], d = self.eig(A, M)
else:
uH[isFreeHDof], d = self.meigs(A, M)
uh = space.function()
uh[:] = uH
GD = mesh.geo_dimension()
if (self.step > 0) and (0 in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_4_0_' + str(NN) + '.pdf')
plt.close()
self.savemesh(mesh,
self.resultdir + 'mesh_3_4_0_' + str(NN) + '.mat')
# 2. 以 u_H 为右端项自适应求解 -\Deta u = u_H
I = eye(gdof)
final = 0
for i in range(maxit):
eta = self.residual_estimate(uh)
markedCell = mark(eta, self.theta)
IM = mesh.bisect(markedCell, returnim=True)
print(i + 1, "refine :", mesh.number_of_nodes())
if (self.step > 0) and (i in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_4_' + str(i + 1) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_4_' + str(i + 1) + '_' +
str(NN) + '.mat')
I = IM @ I
uH = IM @ uH
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
b = M @ uH
uh = space.function()
if self.matlab is False:
uh[isFreeDof] = self.psolve(
A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof],
M[isFreeDof, :][:, isFreeDof].tocsr())
else:
uh[isFreeDof] = self.msolve(
A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof])
final = i + 1
if gdof > self.maxdof:
break
if self.step > 0:
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_4_' + str(final) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_4_' + str(final) + '_' +
str(NN) + '.mat')
# 3. 把 uh 加入粗网格空间, 组装刚度和质量矩阵
w0 = uh @ A
w1 = w0 @ uh
w2 = w0 @ I
AA = bmat([[AH, w2.reshape(-1, 1)], [w2, w1]], format='csr')
w0 = uh @ M
w1 = w0 @ uh
w2 = w0 @ I
MM = bmat([[MH, w2.reshape(-1, 1)], [w2, w1]], format='csr')
isFreeDof = np.r_[isFreeHDof, True]
u = np.zeros(len(isFreeDof))
## 求解特征值
A = AA[isFreeDof, :][:, isFreeDof].tocsr()
M = MM[isFreeDof, :][:, isFreeDof].tocsr()
if self.multieigs is True:
self.A = A
self.M = M
self.ml = pyamg.ruge_stuben_solver(self.A)
self.eigs()
else:
if self.matlab is False:
u[isFreeDof], d = self.eig(A, M)
else:
u[isFreeDof], d = self.meigs(A, M)
print("smallest eigns:", d)
uh *= u[-1]
uh += I @ u[:-1]
uh /= np.max(np.abs(uh))
uh = space.function(array=uh)
return uh
end = timer()
print("with time: ", end - start)
def alg_3_3(self, maxit=None):
"""
1. 最粗网格上求解最小特征特征值问题，得到最小特征值 d_H 和特征向量 u_H
2. 自适应求解 - \Delta u_h = u_H
* u_H 插值到下一层网格上做为新 u_H
3. 在最细网格上求解一次最小特征值问题
自适应 maxit, picard:2
"""
print("算法 3.3")
if maxit is None:
maxit = self.maxit
start = timer()
if self.step == 0:
idx = []
else:
idx = list(range(0, self.maxit, self.step))
mesh = self.pde.init_mesh(n=self.numrefine)
# 1. 粗网格上求解最小特征值问题
space = LagrangeFiniteElementSpace(mesh, 1)
AH = self.get_stiff_matrix(space)
MH = self.get_mass_matrix(space)
isFreeHDof = ~(space.boundary_dof())
gdof = space.number_of_global_dofs()
uH = np.zeros(gdof, dtype=mesh.ftype)
print("initial mesh :", gdof)
A = AH[isFreeHDof, :][:, isFreeHDof].tocsr()
M = MH[isFreeHDof, :][:, isFreeHDof].tocsr()
if self.matlab is False:
uH[isFreeHDof], d = self.eig(A, M)
else:
uH[isFreeHDof], d = self.meigs(A, M)
uh = space.function()
uh[:] = uH
GD = mesh.geo_dimension()
if (self.step > 0) and (0 in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_3_0_' + str(NN) + '.pdf')
plt.close()
self.savemesh(mesh,
self.resultdir + 'mesh_3_3_0_' + str(NN) + '.mat')
# 2. 以 u_H 为右端项自适应求解 -\Deta u = u_H
I = eye(gdof)
final = 0
for i in range(maxit - 1):
eta = self.residual_estimate(uh)
markedCell = mark(eta, self.theta)
IM = mesh.bisect(markedCell, returnim=True)
NN = mesh.number_of_nodes()
print(i + 1, "refine : ", NN)
if (self.step > 0) and (i in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_3_' + str(i + 1) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_3_' + str(i + 1) + '_' +
str(NN) + '.mat')
final = i + 1
if NN > self.maxdof:
break
I = IM @ I
uH = IM @ uH
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
b = M @ uH
uh = space.function()
if self.matlab is False:
uh[isFreeDof] = self.psolve(
A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof],
M[isFreeDof, :][:, isFreeDof].tocsr())
else:
uh[isFreeDof] = self.msolve(
A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof])
# 3. 在最细网格上求解一次最小特征值问题
if self.step > 0:
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_3_' + str(final) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_3_' + str(final) + '_' +
str(NN) + '.mat')
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
uh = space.function(array=uh)
A = A[isFreeDof, :][:, isFreeDof].tocsr()
M = M[isFreeDof, :][:, isFreeDof].tocsr()
uh = space.function()
if self.multieigs is True:
self.A = A
self.M = M
self.ml = pyamg.ruge_stuben_solver(self.A)
self.eigs()
else:
if self.matlab is False:
uh[isFreeDof], d = self.eig(A, M)
else:
uh[isFreeDof], d = self.meigs(A, M)
print("smallest eigns:", d)
return uh
end = timer()
print("with time: ", end - start)
def alg_3_1(self, maxit=None):
"""
1. 自适应在每层网格上求解最小特征值问题
refine: maxit, picard： maxit + 1
"""
print("算法 3.1")
if maxit is None:
maxit = self.maxit
start = timer()
if self.step == 0:
idx = []
else:
idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1]
mesh = self.pde.init_mesh(n=self.numrefine)
GD = mesh.geo_dimension()
if (self.step > 0) and (0 in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_1_0_' + str(NN) + '.pdf')
plt.close()
self.savemesh(mesh,
self.resultdir + 'mesh_3_1_0_' + str(NN) + '.mat')
space = LagrangeFiniteElementSpace(mesh, 1)
isFreeDof = ~(space.boundary_dof())
gdof = space.number_of_global_dofs()
uh = np.ones(gdof, dtype=mesh.ftype)
uh[~isFreeDof] = 0
IM = eye(gdof)
for i in range(maxit + 1):
area = mesh.entity_measure('cell')
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
uh = IM @ uh
A = A[isFreeDof, :][:, isFreeDof].tocsr()
M = M[isFreeDof, :][:, isFreeDof].tocsr()
if self.matlab is False:
uh[isFreeDof], d = self.eig(A, M)
else:
uh[isFreeDof], d = self.meigs(A, M)
if i < maxit:
uh = space.function(array=uh)
eta = self.residual_estimate(uh)
markedCell = mark(eta, self.theta)
IM = mesh.bisect(markedCell, returnim=True)
print(i + 1, "refine: ", mesh.number_of_nodes())
if (self.step > 0) and (i in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_1_' + str(i + 1) +
'_' + str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_1_' + str(i + 1) + '_' +
str(NN) + '.mat')
space = LagrangeFiniteElementSpace(mesh, 1)
isFreeDof = ~(space.boundary_dof())
gdof = space.number_of_global_dofs()
if gdof > self.maxdof:
break
if self.step > 0:
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_1_' + str(i + 1) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_1_' + str(i + 1) + '_' +
str(NN) + '.mat')
if self.multieigs is True:
self.A = A
self.M = M
self.ml = pyamg.ruge_stuben_solver(self.A)
self.eigs()
end = timer()
print("smallest eigns:", d, "with time: ", end - start)
uh = space.function(array=uh)
return uh
def alg_3_2(self, maxit=None):
"""
1. 自适应求解 -\Delta u = 1。
1. 在最细网格上求最小特征值和特征向量。
refine maxit， picard: 1
"""
print("算法 3.2")
if maxit is None:
maxit = self.maxit
start = timer()
if self.step == 0:
idx = []
else:
idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1]
mesh = self.pde.init_mesh(n=self.numrefine)
GD = mesh.geo_dimension()
if (self.step > 0) and (0 in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_2_0_' + str(NN) + '.pdf')
plt.close()
self.savemesh(mesh,
self.resultdir + 'mesh_3_2_0_' + str(NN) + '.mat')
final = 0
integrator = mesh.integrator(self.q)
for i in range(maxit):
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
b = M @ np.ones(gdof)
isFreeDof = ~(space.boundary_dof())
A = A[isFreeDof, :][:, isFreeDof].tocsr()
M = M[isFreeDof, :][:, isFreeDof].tocsr()
uh = space.function()
if self.matlab is False:
uh[isFreeDof] = self.psolve(A, b[isFreeDof], M)
else:
uh[isFreeDof] = self.msolve(A, b[isFreeDof])
eta = self.residual_estimate(uh)
markedCell = mark(eta, self.theta)
IM = mesh.bisect(markedCell, returnim=True)
NN = mesh.number_of_nodes()
print(i + 1, "refine :", NN)
if (self.step > 0) and (i in idx):
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_2_' + str(i + 1) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_2_' + str(i + 1) + '_' +
str(NN) + '.mat')
final = i + 1
if NN > self.maxdof:
break
if self.step > 0:
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_2_' + str(final) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_2_' + str(final) + '_' +
str(NN) + '.mat')
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
A = A[isFreeDof, :][:, isFreeDof].tocsr()
M = M[isFreeDof, :][:, isFreeDof].tocsr()
if self.multieigs is True:
self.A = A
self.M = M
self.ml = pyamg.ruge_stuben_solver(self.A)
self.eigs()
else:
uh = IM @ uh
if self.matlab is False:
uh[isFreeDof], d = self.eig(A, M)
else:
uh[isFreeDof], d = self.meigs(A, M)
print("smallest eigns:", d)
return space.function(array=uh)
end = timer()
print("with time: ", end - start)
def alg_0(self, maxit=None):
"""
1. 最粗网格上求解最小特征特征值问题，得到最小特征值 d_H 和特征向量 u_H
2. 自适应求解 - \Delta u_h = d_H*u_H
* 每层网格上求出的 u_h，插值到下一层网格上做为 u_H
* 并更新 d_H = u_h@A@u_h/u_h@M@u_h， 其中 A 是当前网格层上的刚度矩
阵，M 为当前网格层的质量矩阵。
自适应 maxit， picard： 1
"""
print("算法 0")
if maxit is None:
maxit = self.maxit
start = timer()
if self.step == 0:
idx = []
else:
idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1]
mesh = self.pde.init_mesh(n=self.numrefine)
# 1. 粗网格上求解最小特征值问题
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
print("initial mesh:", gdof)
uh = np.zeros(gdof, dtype=np.float)
AH = self.get_stiff_matrix(space)
MH = self.get_mass_matrix(space)
isFreeHDof = ~(space.boundary_dof())
A = AH[isFreeHDof, :][:, isFreeHDof].tocsr()
M = MH[isFreeHDof, :][:, isFreeHDof].tocsr()
if self.picard is True:
uh[isFreeHDof], d = picard(A,
M,
np.ones(sum(isFreeHDof)),
sigma=self.sigma)
else:
uh[isFreeHDof], d = self.eig(A, M)
GD = mesh.geo_dimension()
if (self.step > 0) and (0 in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_0_0_' + str(NN) + '.pdf')
plt.close()
self.savemesh(mesh,
self.resultdir + 'mesh_0_0_' + str(NN) + '.mat')
# 2. 以 u_h 为右端项自适应求解 -\Deta u = d*u_h
I = eye(gdof)
for i in range(maxit):
uh = space.function(array=uh)
eta = self.residual_estimate(uh)
markedCell = mark(eta, self.theta)
IM = mesh.bisect(markedCell, returnim=True)
print(i + 1, "refine: ", mesh.number_of_nodes())
if (self.step > 0) and (i in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_0_' + str(i + 1) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_0_' + str(i + 1) + '_' +
str(NN) + '.mat')
I = IM @ I
uh = IM @ uh
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
b = d * M @ uh
if self.sigma is None:
ml = pyamg.ruge_stuben_solver(
A[isFreeDof, :][:, isFreeDof].tocsr())
uh[isFreeDof] = ml.solve(b[isFreeDof],
x0=uh[isFreeDof],
tol=1e-12,
accel='cg').reshape((-1, ))
else:
K = A[isFreeDof, :][:, isFreeDof].tocsr(
) + self.sigma * M[isFreeDof, :][:, isFreeDof].tocsr()
b += self.sigma * M @ uh
ml = pyamg.ruge_stuben_solver(K)
uh[isFreeDof] = ml.solve(b[isFreeDof],
x0=uh[isFreeDof],
tol=1e-12,
accel='cg').reshape(-1)
# uh[isFreeDof] = spsolve(A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof])
d = uh @ A @ uh / (uh @ M @ uh)
if gdof > self.maxdof:
break
if self.multieigs is True:
self.A = A[isFreeDof, :][:, isFreeDof].tocsr()
self.M = M[isFreeDof, :][:, isFreeDof].tocsr()
self.ml = pyamg.ruge_stuben_solver(self.A)
self.eigs()
end = timer()
print("smallest eigns:", d, "with time: ", end - start)
uh = space.function(array=uh)
return uh
