A FEniCS Tutorial - FEniCS Project

FunctionSpace, etc.). Many use instance and object as interchangeable terms.
In other computer programming languages one may also use the term variable
for the same thing. We mostly use the well-known term object in this text.
A subclass of SubDomain with an inside method offers functionality for
marking parts of the domain or the boundary. Now we need to define one class
for the subdomain Ω 0 where y ≤ 1/2 and another for the subdomain Ω 1 where
y ≥ 1/2:
class Omega0(SubDomain):
def inside(self, x, on_boundary):
return True if x[1] = 0.5 else False
Notice the use of = in both tests. For a cell to belong to, e.g., Ω 1 ,
the inside method must return True for all the vertices x of the cell. So to
make the cells at the internal boundary y = 1/2 belong to Ω 1 , we need the test
x[1] >= 0.5.
The next task is to use a MeshFunction to mark all cells in Ω 0 with the
subdomain number 0 and all cells in Ω 1 with the subdomain number 1. Our
convention is to number subdomains as 0,1,2,....
A MeshFunction is a discrete function that can be evaluated at a set of socalled
mesh entities. Examples of mesh entities are cells, facets, and vertices. A
MeshFunction over cells is suitable to represent subdomains (materials), while
a MeshFunction over facets is used to represent pieces of external or internal
boundaries. Mesh functions over vertices can be used to describe continuous
fields.
Since we need to define subdomains of Ω in the present example, we must
make use of a MeshFunction over cells. The MeshFunction constructor is fed
with three arguments: 1) the type of value: ’int’ for integers, ’uint’ for
positive (unsigned) integers, ’double’ for real numbers, and ’bool’ for logical
values; 2) a Mesh object, and 3) the topological dimension of the mesh entity in
question: cells have topological dimension equal to the number of space dimensions
in the PDE problem, and facets have one dimension lower. Alternatively,
the constructor can take just a filename and initialize the MeshFunction from
data in a file.
We start with creating a MeshFunction whose values are non-negative integers
(’uint’) for numbering the subdomains. The mesh entities of interest are
the cells, which have dimension 2 in a two-dimensional problem (1 in 1D, 3 in
3D). The appropriate code for defining the MeshFunction for two subdomains
then reads
subdomains = MeshFunction(’uint’, mesh, 2)
# Mark subdomains with numbers 0 and 1
subdomain0 = Omega0()
subdomain0.mark(subdomains, 0)
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