FEniCS: UMFPACK报告正在求解的矩阵是奇异的

Question

我在第一个子域中用stokes方程，在第二个子域中创建了一个混合泊松方程(darcy)。我使用UnitSquare，子域1应该是从0到0,5和子域2从0,5到1。

但是现在我得到了以下错误：

求解线性变分问题。与调用数值有关的UMFPACK问题*警告: UMFPACK报告正在求解的矩阵是奇异的。警告: UMFPACK报告正在求解的矩阵是奇异的。断言vmax>=vmin，“空范围，请指定vmin和/或vmax”断言错误:空范围，请指定vmin和/或vmax

有人能帮忙吗？谢谢!

以下是代码：

enter code here


#-*- coding: utf-8 -*-

from dolfin import *

import numpy as np

# Define mesh
mesh = UnitSquare(32,32)

#Subdomain 1
# Gitter übergeben
subdomains = CellFunction("uint", mesh)

# Klasse des Teilgebiets
class Domain_1(SubDomain):
   def inside(self, x, on_boundary):
       return between(x[0], (0, 0.5)) # Koordinatenangabe des Teilgebiets

# Objekt der Klasse erstellen
sub_domain1 = Domain_1()
sub_domain1.mark(subdomains,0)

# Definition Funktionenräume
U = FunctionSpace(mesh, "CG", 2)
V = FunctionSpace(mesh, "CG", 1)
W = U*V

# Definition Trial- und Testfunktion
(u, p) = TrialFunctions(W)
(v, q) = TestFunctions(W)

# boundary condition
p_in = 1
p_out = 0
noslip = DirichletBC(W.sub(0), (0),

                      "on_boundary && \

                      (x[1] <= DOLFIN_EPS | x[1] >= 0.5-DOLFIN_EPS)")

inflow = DirichletBC(W.sub(1), p_in, "x[0] <= 0.0 + DOLFIN_EPS*1000")
outflow = DirichletBC(W.sub(1), p_out, "x[0] >= 0.5 - DOLFIN_EPS*1000")

bcp = [noslip,inflow, outflow]

# Definition f
f = Expression("0")

# Variationsformulierung
a = inner(grad(u), grad(v))*dx + div(v)*p*dx(0) + q*div(u)*dx(0)
L = inner(f,v)*dx(0)

# Lösung berechnen
w = Function(W)
problem = LinearVariationalProblem(a, L, w, bcp)
solver = LinearVariationalSolver(problem)
solver.solve()
(u, p) = w.split()

# Subdomain 2
# Gitter übergeben
subdomains = CellFunction("uint", mesh)

# Klasse des Teilgebiets
class Domain_2(SubDomain):
   def inside(self,x,on_boundary):
       return between(x[0], (0.5,1.0)) # Koordinatenangabe des Teilgebiets

# Objekt der Klasse erstellen
sub_domain2 = Domain_2()
sub_domain2.mark(subdomains,1)

# Define function spaces and mixed (product) space
BDM = FunctionSpace(mesh, "BDM", 1)
DG = FunctionSpace(mesh, "DG", 0)
CG = FunctionSpace(mesh, "CG", 1)
W = MixedFunctionSpace([BDM, DG, CG])

# Define trial and test functions
(sigma, u, p) = TrialFunctions(W)
(tau, v, q) = TestFunctions(W)

#Define pressure boundary condition
p_in = 1
p_out = 0
noslip = DirichletBC(W.sub(1), (0),
                      "on_boundary && \
                      (x[1] <= 0.5 + DOLFIN_EPS | x[1] >= 1.0-DOLFIN_EPS)")

inflow = DirichletBC(W.sub(2), p_in, "x[0] <= 0.5 + DOLFIN_EPS*1000")
outflow = DirichletBC(W.sub(2), p_out, "x[0] >= 1.0 - DOLFIN_EPS*1000")
bcp = [noslip,inflow, outflow]

# Define f
#f = Expression("0")
f = Expression("10*exp(-(pow(x[0] - 0.5, 2) + pow(x[1] - 0.5, 2)) / 0.02)")

# Define variational form
a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx(1) + inner(p,q)*dx(1) + u*q*dx(1)
L = f*v*dx(1)

# Compute solution
w = Function(W)
problem = LinearVariationalProblem(a, L, w, bcp)
solver = LinearVariationalSolver(problem)
solver.solve()
(sigma, u, p) = w.split()

# plot

plot(u, axes = True, interactive=True, title = "u")
plot(p, axes = True, interactive=True, title = "p")

这学期我忘了dx(0)。但这不是问题所在。

在代码的第一部分(Stokes)中，我试图以以下方式编写无滑移条件：

# Randbedingungen

def top_bottom(x, on_boundary):
    return x[1] > 1.0 - DOLFIN_EPS or x[1] < DOLFIN_EPS

noslip = Constant((0.0,0.0))  

bc0 = DirichletBC(W.sub(0), noslip, top_bottom)

p_in = 1
p_out = 0

inflow = DirichletBC(W.sub(1), p_in, "x[0] <= 0.0 + DOLFIN_EPS*1000")
outflow = DirichletBC(W.sub(1), p_out, "x[0] >= 0.5 - DOLFIN_EPS*1000")

bcp = [bc0, inflow, outflow]

# Definition f

f = Expression("(0.0, 0.0)")

# Variationsformulierung Stokes
a = inner(grad(u), grad(v))*dx(0) + div(v)*p*dx(0) + q*div(u)*dx(0)
L = inner(f,v)*dx(0)

但是现在我得到了以下错误：

Shape mismatch: line 56, in <module> L = inner(f,v)*dx(0

有人能帮忙吗？谢谢!

Answer 1

我认为这里有几个错误。在代码的第一部分，我认为你是用泰勒-胡德元素来解斯托克斯方程。如果我们是这样的话，那你应该是：

U=VectorFunctionSpace(网格，"CG"，2)

同样在守则的这一部分：

A=内部(梯度(U)，梯度(V))*dx+ div(v)*p*dx(0) + q*div(u)*dx(0)

L=内(f，v)*dx(0)

我不知道你为什么在第一学期不使用dx(0)。我鼓励您查看演示：http://fenicsproject.org/documentation/dolfin/dev/python/demo/index.html，您可能会得到一些更多的提示。