def matrix_test(self, p=2):
"""
node = np.array([
(-1.0, -1.0),
( 0.0, -1.0),
( 1.0, -1.0),
(-1.0, 0.0),
( 1.0, 0.0),
(-1.0, 1.0),
( 0.0, 1.0),
( 1.0, 1.0)], dtype=np.float)
cell = np.array([0, 1, 2, 4, 7, 6, 5, 3], dtype=np.int)
cellLocation = np.array([0, 8], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
"""
node = np.array([(0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)],
dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
space = DivFreeNonConformingVirtualElementSpace2d(mesh, p)
print("G:", space.G)
print("B:", space.B)
print("R:", space.R)
print("J:", space.J)
print("Q:", space.Q)
print("L:", space.L)
print("D:", space.D)
def stokes_equation_test(self, p=2, maxit=4):
from scipy.sparse import bmat
from fealpy.pde.Stokes_Model_2d import CosSinData
from fealpy.mesh.simple_mesh_generator import triangle
h = 0.4
pde = CosSinData()
error = np.zeros((maxit, ), dtype=np.float)
for i in range(maxit):
mesh = pde.init_mesh(n=i + 2, meshtype='poly')
if 0:
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes)
plt.show()
uspace = DivFreeNonConformingVirtualElementSpace2d(mesh, p)
pspace = ScaledMonomialSpace2d(mesh, p - 1)
isBdDof = uspace.boundary_dof()
udof = uspace.number_of_global_dofs()
pdof = pspace.number_of_global_dofs()
uh = uspace.function()
ph = pspace.function()
uspace.set_dirichlet_bc(uh, pde.dirichlet)
A = uspace.matrix_A()
P = uspace.matrix_P()
F = uspace.source_vector(pde.source)
AA = bmat([[A, P.T], [P, None]], format='csr')
FF = np.block([F, np.zeros(pdof, dtype=uspace.ftype)])
x = np.block([uh.T.flat, ph])
isBdDof = np.block(
[isBdDof, isBdDof,
np.zeros(pdof, dtype=np.bool)])
gdof = 2 * udof + pdof
FF -= AA @ x
bdIdx = np.zeros(gdof, dtype=np.int)
bdIdx[isBdDof] = 1
Tbd = spdiags(bdIdx, 0, gdof, gdof)
T = spdiags(1 - bdIdx, 0, gdof, gdof)
AA = T @ AA @ T + Tbd
FF[isBdDof] = x[isBdDof]
x[:] = spsolve(AA, FF)
uh[:, 0] = x[:udof]
uh[:, 1] = x[udof:2 * udof]
ph[:] = x[2 * udof:]
up = uspace.project_to_smspace(uh)
integralalg = uspace.integralalg
error[i] = integralalg.L2_error(pde.velocity, up)
h /= 2
print(error)
print(error[0:-1] / error[1:])
def one_cell_test_0(self, p=2):
from fealpy.pde.stokes_model_2d import StokesModelData_7
pde = StokesModelData_7()
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)], dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
uspace = DivFreeNonConformingVirtualElementSpace2d(mesh, p)
pspace = ScaledMonomialSpace2d(mesh, p - 1)
ldof = pspace.number_of_local_dofs()
isBdDof = uspace.boundary_dof()
udof = uspace.number_of_global_dofs()
pdof = pspace.number_of_global_dofs()
uh = uspace.function()
ph = pspace.function()
uspace.set_dirichlet_bc(uh, pde.dirichlet)
A = uspace.matrix_A()
P = uspace.matrix_P()
C = uspace.CM[:, 0, :ldof].reshape(-1)
F = uspace.source_vector(pde.source)
AA = bmat([[A, P.T, None], [P, None, C[:, None]], [None, C, None]],
format='csr')
FF = np.block([F, np.zeros(pdof + 1, dtype=uspace.ftype)])
x = np.block([uh, ph, np.zeros((1, ), dtype=uspace.ftype)])
isBdDof = np.r_['0', isBdDof, np.zeros(pdof + 1, dtype=np.bool)]
gdof = udof + pdof + 1
FF -= AA @ x
bdIdx = np.zeros(gdof, dtype=np.int)
bdIdx[isBdDof] = 1
Tbd = spdiags(bdIdx, 0, gdof, gdof)
T = spdiags(1 - bdIdx, 0, gdof, gdof)
AA = T @ AA @ T + Tbd
FF[isBdDof] = x[isBdDof]
x[:] = spsolve(AA, FF)
uh[:] = x[:udof]
ph[:] = x[udof:-1]
print('ph:', ph)
print('uh:', uh)
up = uspace.project_to_smspace(uh)
print('up:', up)
integralalg = uspace.integralalg
error = integralalg.L2_error(pde.velocity, up)
print(error)
uv = uspace.project(pde.velocity)
print('uproject:', uv)
up = uspace.project_to_smspace(uv)
print('up', up)
error = integralalg.L2_error(pde.velocity, up)
print(error)
def matrix_A_test(self, p=2):
node = np.array([(0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)],
dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
space = DivFreeNonConformingVirtualElementSpace2d(mesh, p)
A = space.matrix_A()
print(A)
def index2_test(self, p=2):
node = np.array([(0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)],
dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
space = DivFreeNonConformingVirtualElementSpace2d(mesh, 3)
idx = space.index2(p=p)
print("p=", p, idx)
def project_test(self, u, p=2, mtype=0, plot=True):
from fealpy.mesh.simple_mesh_generator import triangle
if mtype == 0:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
elif mtype == 1:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([0, 1, 2, 3, 0, 2], dtype=np.int)
cellLocation = np.array([0, 3, 6], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
elif mtype == 2:
h = 0.1
mesh = triangle([-1, 1, -1, 1], h, meshtype='polygon')
elif mtype == 3:
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)],
dtype=np.float)
cell = np.array([[0, 1, 2, 3]], dtype=np.int)
mesh = QuadrangleMesh(node, cell)
mesh.uniform_refine()
mesh = PolygonMesh.from_mesh(mesh)
cell, cellLocation = mesh.entity('cell')
edge = mesh.entity('edge')
cell2edge = mesh.ds.cell_to_edge()
uspace = DivFreeNonConformingVirtualElementSpace2d(mesh, p)
up = uspace.project(u)
print(up)
up = uspace.project_to_smspace(up)
print(up)
integralalg = uspace.integralalg
error = integralalg.L2_error(u, up)
print(error)
if plot:
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes)
mesh.find_node(axes, showindex=True)
mesh.find_edge(axes, showindex=True)
mesh.find_cell(axes, showindex=True)
plt.show()
def matrix_G_test(self, p=2):
node = np.array([(-1, -1), (1, -1), (1, 1), (-1, 1)], dtype=np.float)
cell = np.array([0, 1, 2, 3], dtype=np.int)
cellLocation = np.array([0, 4], dtype=np.int)
mesh = PolygonMesh(node, cell, cellLocation)
space = DivFreeNonConformingVirtualElementSpace2d(mesh, p)
errorMatrix = np.zeros((5, maxit), dtype=np.float)
dof = np.zeros(maxit, dtype=np.float)
for i in range(maxit):
#mesh = pde.init_mesh(n=i+2, meshtype='poly')
mesh = pde.init_mesh(n=i+2, meshtype='quad')
mesh = PolygonMesh.from_mesh(mesh)
NE = mesh.number_of_edges()
NC = mesh.number_of_cells()
idof = (p-2)*(p-1)//2
dof[i] = NC
# 完全非协调向量虚单元空间
uspace = DivFreeNonConformingVirtualElementSpace2d(mesh, p, q=6)
pspace = ScaledMonomialSpace2d(mesh, p-1)
uh = uspace.function()
ph = pspace.function()
# 缩减非协调向量虚单元空间
tuspace = ReducedDivFreeNonConformingVirtualElementSpace2d(mesh, p, q=6)
# 分片常数压力空间
tpspace = ScaledMonomialSpace2d(mesh, 0)
isBdDof = tuspace.boundary_dof()
tudof = tuspace.number_of_global_dofs()
tpdof = tpspace.number_of_global_dofs()
tuh = tuspace.function()