A FEniCS Tutorial - FEniCS Project

tol = 1E-14
def left_boundary(x, on_boundary):
return on_boundary and abs(x[0]) < tol
def right_boundary(x, on_boundary):
return on_boundary and abs(x[0]-1) < tol
Gamma_0 = DirichletBC(V, Constant(0.0), left_boundary)
Gamma_1 = DirichletBC(V, Constant(1.0), right_boundary)
bcs = [Gamma_0, Gamma_1]
# Define variational problem for initial guess (q(u)=1, i.e., m=0)
u = TrialFunction(V)
v = TestFunction(V)
a = inner(nabla_grad(u), nabla_grad(v))*dx
f = Constant(0.0)
L = f*v*dx
A, b = assemble_system(a, L, bcs)
u_k = Function(V)
U_k = u_k.vector()
solve(A, U_k, b)
Here, u_k denotes the solution function for the previous iteration, so that the
solution after each Newton iteration is u = u_k + omega*du. Initially, u_k is
the initial guess we call u 0 in the mathematics.
TheDirichletboundaryconditionsforδu, intheproblemtobesolvedineach
Newton iteration, are somewhat different than the conditions for u. Assuming
that u k fulfills the Dirichlet conditions for u, δu must be zero at the boundaries
where the Dirichlet conditions apply, in order for u k+1 = u k + ωδu to fulfill
the right boundary values. We therefore define an additional list of Dirichlet
boundary conditions objects for δu:
Gamma_0_du = DirichletBC(V, Constant(0), left_boundary)
Gamma_1_du = DirichletBC(V, Constant(0), right_boundary)
bcs_du = [Gamma_0_du, Gamma_1_du]
The nonlinear coefficient and its derivative must be defined before coding the
weak form of the Newton system:
def q(u):
return (1+u)**m
def Dq(u):
return m*(1+u)**(m-1)
du = TrialFunction(V) # u = u_k + omega*du
a = inner(q(u_k)*nabla_grad(du), nabla_grad(v))*dx + \
inner(Dq(u_k)*du*nabla_grad(u_k), nabla_grad(v))*dx
L = -inner(q(u_k)*nabla_grad(u_k), nabla_grad(v))*dx
The Newton iteration loop is very similar to the Picard iteration loop in
Section 2.1:
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