A FEniCS Tutorial - FEniCS Project

The code for the y = 1 boundary is similar and is seen in dnr_p2D.py.
The Dirichlet boundaries are marked similarly, using subdomain number 2
for Γ 0 and 3 for Γ 1 :
class LeftBoundary(SubDomain):
def inside(self, x, on_boundary):
tol = 1E-14 # tolerance for coordinate comparisons
return on_boundary and abs(x[0]) < tol
Gamma_0 = LeftBoundary()
Gamma_0.mark(boundary_parts, 2)
class RightBoundary(SubDomain):
def inside(self, x, on_boundary):
tol = 1E-14 # tolerance for coordinate comparisons
return on_boundary and abs(x[0] - 1) < tol
Gamma_1 = RightBoundary()
Gamma_1.mark(boundary_parts, 3)
Specifying the DirichletBC objects may now make use of the mesh function
(instead of a SubDomain subclass object) and an indicator for which subdomain
each condition should be applied to:
u_L = Expression(’1 + 2*x[1]*x[1]’)
u_R = Expression(’2 + 2*x[1]*x[1]’)
bcs = [DirichletBC(V, u_L, boundary_parts, 2),
DirichletBC(V, u_R, boundary_parts, 3)]
Some functions need to be defined before we can go on with the a and L of
the variational problem:
g = Expression(’-4*x[1]’)
q = Expression(’1 + x[0]*x[0] + 2*x[1]*x[1]’)
p = Constant(100) # arbitrary function can go here
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(-6.0)
The new aspect of the variational problem is the two distinct boundary integrals.
Having a mesh function over exterior cell facets (our boundary_parts
object), where subdomains (boundary parts) are numbered as 0,1,2,..., the
special symbol ds(0) implies integration over subdomain (part) 0, ds(1) denotes
integration over subdomain (part) 1, and so on. The idea of multiple
‘ds‘-type objects generalizes to volume integrals too: dx(0), dx(1), etc., are
used to integrate over subdomain 0, 1, etc., inside Ω.
The variational problem can be defined as
a = inner(nabla_grad(u), nabla_grad(v))*dx + p*u*v*ds(0)
L = f*v*dx - g*v*ds(1) + p*q*v*ds(0)
For the ds(0) and ds(1) symbols to work we must obviously connect them (or
a and L) to the mesh function marking parts of the boundary. This is done by
a certain keyword argument to the assemble function:
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