solve Anisotropy equation


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4 months ago by
kevin  
Hello everyone!
∇.(σ∇u)=0
In this problem, σ Is a tensor (σ is as_matrix([[1, 2],[2, 3]]))
How should I write this variational equation and how to solve it?
thank you!
Community: FEniCS Project

1 Answer


7
4 months ago by
You can modify the Poisson demo here

http://fenics.readthedocs.io/projects/dolfin/en/stable/demos/poisson/python/demo_poisson.py.html

by changing the bilinear form to

a = inner(sigma*grad(u), grad(v))*dx​

However, note that \(\sigma\) should be positive definite for the problem to be stable.  (Physically: "heat needs to flow from hot to cold".)  The suggested matrix has a negative eigenvalue, and will produce a very strange-looking discrete solution.
sorry, i try to use the method, but running error
the demo is following:

from dolfin import *
import numpy as np
import matplotlib.pyplot as plt
mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, "Lagrange", 1)
def boundary(x):
return x[0] < DOLFIN_EPS or x[0] > 1.0 - DOLFIN_EPS
u0 = Constant(0.0)
bc = DirichletBC(V, u0, boundary)
u = TrialFunction(V)
v = TestFunction(V)
f = Expression("10*exp(-(pow(x[0] - 0.5, 2) + pow(x[1] - 0.5, 2)) / 0.02)", degree=2)
g = Expression("sin(5*x[0])", degree=2)
tao = np.array([[1,2],[3,4]])
a = inner(inner(tao,grad(u)), grad(v))*dx
L = f*v*dx + g*v*ds
u = Function(V)
solve(a == L, u, bc)
plot(u)
plt.show()
interactive()


how can i solve the equation?
written 4 months ago by kevin  
1
You did not incorporate what David Kamensky wrote. The expression inside your bilinear form:
inner(tao,grad(u))​
does not make sense. There is no inner product between tensors and vectors. Use
inner(tao*grad(u), grad(v))*dx​
written 4 months ago by klunkean  
By the modification, the following form:
inner(tao*grad(u), grad(v))*dx​
L = f*v*dx + g*v*ds

but  the running is error
written 4 months ago by kevin  
What error is the running? Maybe paste the message?
written 4 months ago by klunkean  
Use tao = as_tensor([[1,2],[3,4]])
written 4 months ago by Adam Janecka  
thanks a lot! successful
written 4 months ago by kevin  
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