def poisson_fem_2d_neuman_test(self, p=1):
pde = CosCosData()
mesh = pde.init_mesh(n=2)
for i in range(4):
space = LagrangeFiniteElementSpace(mesh, p=p)
A = space.stiff_matrix()
b = space.source_vector(pde.source)
uh = space.function()
bc = BoundaryCondition(space, neuman=pde.neuman)
bc.apply_neuman_bc(b)
c = space.integral_basis()
AD = bmat([[A, c.reshape(-1, 1)], [c, None]], format='csr')
bb = np.r_[b, 0]
x = spsolve(AD, bb)
uh[:] = x[:-1]
area = np.sum(space.integralalg.cellmeasure)
ubar = space.integralalg.integral(pde.solution,
barycenter=False) / area
def solution(p):
return pde.solution(p) - ubar
error = space.integralalg.L2_error(solution, uh)
print(error)
mesh.uniform_refine()
def test_poisson_fem_2d():
degree = 1
dim = 2
nrefine = 4
maxit = 4
from fealpy.pde.poisson_2d import CosCosData as PDE
pde = PDE()
mesh = pde.init_mesh(n=nrefine)
errorMatrix = np.zeros((2, maxit), dtype=np.float64)
NDof = np.zeros(maxit, dtype=np.int64)
for i in range(maxit):
space = LagrangeFiniteElementSpace(mesh, p=degree)
NDof[i] = space.number_of_global_dofs()
bc = DirichletBC(space, pde.dirichlet)
uh = space.function()
A = space.stiff_matrix()
F = space.source_vector(pde.source)
A, F = bc.apply(A, F, uh)
uh[:] = spsolve(A, F).reshape(-1)
errorMatrix[0, i] = space.integralalg.L2_error(pde.solution, uh)
errorMatrix[1, i] = space.integralalg.L2_error(pde.gradient, uh.grad_value)
if i < maxit-1:
mesh.uniform_refine()
def interpolation(self, n=0, p=0):
pde = CosCosData()
mesh = pde.init_mesh(n=n, meshtype='tri')
space = RaviartThomasFiniteElementSpace2d(mesh, p=p)
uI = space.interpolation(pde.flux)
print(space.basis.coordtype)
u = Function(space)
u[:] = uI
qf = mesh.integrator(3, 'cell')
bcs, ws = qf.get_quadrature_points_and_weights()
error = space.integralalg.L2_error(pde.source, u.div_value)
print('error:', error)
print(dir(u))
class QuadBilinearFiniteElementSpaceTest:
def __init__(self):
self.pde = CosCosData()
self.mesh = self.pde.init_mesh(meshtype='quad')
self.space = QuadBilinearFiniteElementSpace(self.mesh)
def test_intetpolation(self):
pass
def test_projection(self):
pass
def test_sovle_poisson_equation(self):
pass
def poisson_fem_2d_test(self, p=1):
pde = CosCosData()
mesh = pde.init_mesh(n=3)
for i in range(4):
space = LagrangeFiniteElementSpace(mesh, p=p)
A = space.stiff_matrix()
b = space.source_vector(pde.source)
uh = space.function()
bc = BoundaryCondition(space, dirichlet=pde.dirichlet)
A, b = bc.apply_dirichlet_bc(A, b, uh)
uh[:] = spsolve(A, b).reshape(-1)
error = space.integralalg.L2_error(pde.solution, uh)
print(error)
mesh.uniform_refine()
def interpolation(self, n=0, p=0, plot=True):
box = [-0.5, 1.5, -0.5, 1.5]
pde = CosCosData()
mesh = pde.init_mesh(n=n, meshtype='tri')
space = RaviartThomasFiniteElementSpace2d(mesh, p=p)
uI = space.interpolation(pde.flux)
error = space.integralalg.L2_error(pde.flux, uI)
print(error)
if plot:
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes, box=box)
plt.show()
def plane_pmesh():
pde = CosCosData()
mesh = pde.init_mesh(n=0)
node = mesh.entity('node')
cell = mesh.entity('cell')
NN = mesh.number_of_nodes()
pnode = np.zeros((2 * NN, 3), dtype=mesh.ftype)
pnode[:NN, 0:2] = node
pnode[NN:, 0:2] = node
pnode[NN:, 2] = 1
pcell = np.r_['1', cell, cell + NN]
pmesh = PrismMesh(pnode, pcell)
return pmesh
def project_test(self, p=2, m=0):
if m == 0:
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)
elif m == 1:
pde = CosCosData()
qtree = pde.init_mesh(n=4, meshtype='quadtree')
mesh = qtree.to_polygonmesh()
points = np.array([[(0.5, 0.5), (0.6, 0.6)]], dtype=np.float)
space = ScaledMonomialSpace2d(mesh, p)
gphi = space.grad_basis(points)
print(gphi)
def test_space_on_triangle(self, p=3):
pde = CosCosData()
mesh = pde.init_mesh(0)
space = LagrangeFiniteElementSpace(mesh, p=p)
ips = space.interpolation_points()
face2dof = space.dof.face_to_dof()
print(face2dof)
print(mesh.entity('cell'))
print('cell2dof:', space.cell_to_dof())
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes)
mesh.find_node(axes, node=ips, showindex=True)
mesh.find_edge(axes, showindex=True)
plt.show()
def print_method(self, n=0, p=0):
pde = CosCosData()
mesh = pde.init_mesh(n=n, meshtype='tri')
space = LagrangeFiniteElementSpace(mesh, p=p)
array = space.array(dim=2)
uI = Function(space, array=array)
qf = mesh.integrator(3, 'cell')
bcs, ws = qf.get_quadrature_points_and_weights()
print("__call__:", uI(bcs))
print("value:", uI.value(bcs))
print(uI[1:2])
print(uI.index(0))
print(uI.__dict__)
def sympy_compute(self, plot=True):
import sympy as sp
from sympy.abc import x, y, z
if plot:
pde = CosCosData()
mesh = pde.init_mesh(n=0, meshtype='tri')
n = mesh.edge_unit_normal()
print("n:\n", n)
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 test_poisson():
p = 1 # 有限元空间次数, 可以增大 p， 看输出结果的变化
n = 4 # 初始网格加密次数
maxit = 4 # 最大迭代次数
pde = PDE()
mesh = pde.init_mesh(n=n)
errorMatrix = np.zeros((2, maxit), dtype=np.float)
NDof = np.zeros(maxit, dtype=np.float)
for i in range(maxit):
space = LagrangeFiniteElementSpace(mesh, p=p) # 建立有限元空间
NDof[i] = space.number_of_global_dofs() # 有限元空间自由度的个数
bc = DirichletBC(space, pde.dirichlet) # DirichletBC 条件
uh = space.function() # 有限元函数
A = space.stiff_matrix() # 刚度矩阵
F = space.source_vector(pde.source) # 载荷向量
A, F = bc.apply(A, F, uh) # 处理边界条件
uh[:] = spsolve(A, F).reshape(-1) # 稀疏矩阵直接解法器
# ml = pyamg.ruge_stuben_solver(A) # 代数多重网格解法器
# uh[:] = ml.solve(F, tol=1e-12, accel='cg').reshape(-1)
errorMatrix[0, i] = space.integralalg.L2_error(
pde.solution, uh
) # 计算 L2 误差
errorMatrix[1, i] = space.integralalg.L2_error(
pde.gradient, uh.grad_value
) # 计算 H1 误差
if i < maxit - 1:
mesh.uniform_refine() # 一致加密网格
assert (errorMatrix < 1.0).all()
def simulation(queue=None):
pde = CosCosData()
mesh = pde.init_mesh(n=3, meshtype='tri')
if queue is not None:
queue.put({'mesh': mesh})
return mesh
import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D from fealpy.pde.poisson_2d import CosCosData as PDE from fealpy.functionspace import LagrangeFiniteElementSpace pde = PDE() mesh = pde.init_mesh(n=4, meshtype='tri') space = LagrangeFiniteElementSpace(mesh, 2) uI = space.interpolation(pde.solution) # 有限元空间的函数，同地它也是一个数组 bc = np.array([1 / 3, 1 / 3, 1 / 3], dtype=np.float64) val = uI(bc) # (NC, ) val = uI.value(bc) # (NC, ) gval = uI.grad_value(bc) #(NC, GD) L2error = space.integralalg.error(pde.solution, uI, q=5, power=2) H1error = space.integralalg.error(pde.gradient, uI.grad_value, q=5, power=2) print(H1error) fig = plt.figure() axes = fig.add_subplot(projection='3d') uI.add_plot(axes, cmap='rainbow') plt.show()
import numpy as np import matplotlib.pyplot as plt from fealpy.pde.poisson_2d import CosCosData from fealpy.functionspace import ConformingVirtualElementSpace2d, ScaledMonomialSpace2d p = 1 pde = CosCosData() quadtree = pde.init_mesh(n=5, meshtype='quadtree') options = quadtree.adaptive_options(method='numrefine', maxsize=1, HB=True) pmesh = quadtree.to_pmesh() space0 = ConformingVirtualElementSpace2d(pmesh, p=p) uh0 = space0.interpolation(pde.solution) sh0 = space0.project_to_smspace(uh0) error = space0.integralalg.L2_error(pde.solution, sh0.value) print(error) axes0 = plt.subplot(1, 2, 1) pmesh.add_plot(axes0) pmesh.find_cell(axes0, showindex=True) NC = pmesh.number_of_cells() eta = -1 * np.ones(NC, dtype=np.int) quadtree.adaptive(eta, options) pmesh = quadtree.to_pmesh() space1 = ConformingVirtualElementSpace2d(pmesh, p=p) uI = space1.interpolation(sh0, options['HB'])
def solve_poisson_2d(self):
pde = CosCosData()
mesh = pde.init_mesh(n=3, methtype='tri')
space = RaviartThomasFiniteElementSpace2d(mesh, p=0)
def solve_poisson_2d(self, n=3, p=0, plot=True):
pde = CosCosData()
mesh = pde.init_mesh(n=n, meshtype='tri')
space = RaviartThomasFiniteElementSpace2d(mesh, p=p)
udof = space.number_of_global_dofs()
pdof = space.smspace.number_of_global_dofs()
gdof = udof + pdof
uh = space.function()
ph = space.smspace.function()
A = space.stiff_matrix()
B = space.div_matrix()
F1 = space.source_vector(pde.source)
AA = bmat([[A, -B], [-B.T, None]], format='csr')
if True:
F0 = -space.set_neumann_bc(pde.dirichlet)
FF = np.r_['0', F0, F1]
x = spsolve(AA, FF).reshape(-1)
uh[:] = x[:udof]
ph[:] = x[udof:]
error0 = space.integralalg.L2_error(pde.flux, uh)
def f(bc):
xx = mesh.bc_to_point(bc)
return (pde.solution(xx) - ph(xx))**2
error1 = space.integralalg.integral(f)
print(error0, error1)
else:
isBdDof = -space.set_dirichlet_bc(uh, pde.neumann)
x = np.r_['0', uh, ph]
isBdDof = np.r_['0', isBdDof, np.zeros(pdof, dtype=np.bool_)]
FF = np.r_['0', np.zeros(udof, dtype=np.float64), F1]
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:]
error0 = space.integralalg.L2_error(pde.flux, uh)
def f(bc):
xx = mesh.bc_to_point(bc)
return (pde.solution(xx) - ph(xx))**2
error1 = space.integralalg.integral(f)
print(error0, error1)
if plot:
box = [-0.5, 1.5, -0.5, 1.5]
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes, box=box)
#mesh.find_node(axes, showindex=True)
#mesh.find_edge(axes, showindex=True)
#mesh.find_cell(axes, showindex=True)
node = ps.reshape(-1, 2)
uv = uh.reshape(-1, 2)
axes.quiver(node[:, 0], node[:, 1], uv[:, 0], uv[:, 1])
plt.show()
from fealpy.tools.show import showmultirate, show_error_table
from fealpy.pde.poisson_2d import CosCosData as PDE
n = 3
maxit = 4
pde = PDE() # 创建 pde 模型
# 误差类型与误差存储数组
errorType = ['$|| u - u_h||_0$', '$||\\nabla u - \\nabla u_h||_0$']
errorMatrix = np.zeros((len(errorType), maxit), dtype=np.float)
# 自由度数组
Ndof = np.zeros(maxit, dtype=np.int)
# 创建初始网格对象
mesh = pde.init_mesh(n, meshtype='quad')
for i in range(maxit):
fem = PoissonQBFEMModel(pde, mesh, q=3) # 创建 Poisson 有限元模型
ls = fem.solve() # 求解
Ndof[i] = fem.space.number_of_global_dofs() # 获得空间自由度个数
errorMatrix[0, i] = fem.L2_error() # 计算 L2 误差
errorMatrix[1, i] = fem.H1_semi_error() # 计算 H1 误差
if i < maxit - 1:
mesh.uniform_refine() # 一致加密网格
# 显示误差
show_error_table(Ndof, errorType, errorMatrix)
# 可视化误差收敛阶
showmultirate(plt, 0, Ndof, errorMatrix, errorType)
import numpy as np
import matplotlib.pyplot as plt
import sys
from fealpy.functionspace import ConformingVirtualElementSpace2d
from fealpy.pde.poisson_2d import CosCosData
n = int(sys.argv[1])
p = int(sys.argv[2])
pde = CosCosData()
mesh = pde.init_mesh(n=n, meshtype='quadtree')
pmesh = mesh.to_pmesh()
space = ConformingVirtualElementSpace2d(pmesh, p=p)
uI = space.interpolation(pde.solution)
S = space.project_to_smspace(uI)
error = space.integralalg.L2_error(pde.solution, S.value)
print(error)
uh = space.grad_recovery(uI)
S = space.project_to_smspace(uh)
def f(x, cellidx):
val = S.value(x, cellidx)
return np.sum(val**2, axis=-1)
a = space.integralalg.integral(f, celltype=True)
from fealpy.solver import MatlabSolver
from fealpy.pde.poisson_2d import CosCosData
from fealpy.functionspace import LagrangeFiniteElementSpace
from fealpy.boundarycondition import DirichletBC
p = int(sys.argv[1])
maxit = int(sys.argv[2])
start = timer()
solver = MatlabSolver()
end = timer()
print("The matalb start time:", end - start)
pde = CosCosData()
mesh = pde.init_mesh(4)
for i in range(maxit):
space = LagrangeFiniteElementSpace(mesh, p=1)
gdof = space.number_of_global_dofs()
print("The num of dofs:", gdof)
A = space.stiff_matrix()
b = space.source_vector(pde.source)
bc = DirichletBC(space, pde.dirichlet)
AD, b = bc.apply(A, b)
uh0 = solver.divide(AD, b)
start = timer()
uh1 = spsolve(AD, b)
end = timer()
print("The spsolver time:", end - start)
from fealpy.boundarycondition import NeumannBC
from fealpy.tools.show import showmultirate
p = 1
n = 2
maxit = 3
d = 4
if d == 2:
from fealpy.pde.poisson_2d import CosCosData as PDE
elif d == 3:
from fealpy.pde.poisson_3d import CosCosCosData as PDE
#pde = PDE()
pde = CosCosData()
mesh = pde.init_mesh(n=3)
errorType = [
'$|| u - u_h||_{\Omega,0}$', '$||\\nabla u - \\nabla u_h||_{\Omega, 0}$'
]
errorMatrix = np.zeros((2, maxit), dtype=np.float)
NDof = np.zeros(maxit, dtype=np.float)
for i in range(maxit):
space = LagrangeFiniteElementSpace(mesh, p=p)
NDof[i] = space.number_of_global_dofs()
uh = space.function()
A = space.stiff_matrix()
F = space.source_vector(pde.source)
help='默认网格加密求解的次数, 默认加密求解 4 次')
args = parser.parse_args()
degree = args.degree
dim = args.dim
nrefine = args.nrefine
maxit = args.maxit
if dim == 2:
from fealpy.pde.poisson_2d import CosCosData as PDE
elif dim == 3:
from fealpy.pde.poisson_3d import CosCosCosData as PDE
pde = PDE()
mesh = pde.init_mesh(n=nrefine)
errorType = ['$|| u - u_h||_{\Omega,0}$',
'$||\\nabla u - \\nabla u_h||_{\Omega, 0}$'
]
errorMatrix = np.zeros((2, maxit), dtype=np.float64)
NDof = np.zeros(maxit, dtype=np.float64)
for i in range(maxit):
space = LagrangeFiniteElementSpace(mesh, p=degree)
NDof[i] = space.number_of_global_dofs()
uh = space.function()
A = space.stiff_matrix()
F = space.source_vector(pde.source)