def boundary conditions

variational formulation
1

Hi there,

The function boundary specifies which points that belong to the part of
the boundary where the boundary condition should be applied:

def boundary(x, on_boundary):
return on_boundary

FEniCS already knows whether the point belongs to the actual boundary
(the mathematical boundary of the domain) and kindly shares this
information with you in the variable on_boundary.
You may choose to use this information, or ignore it completely.

I’m trying to understand this thing, this def can be easily implemented,
in case of homogeneous, like u_D = 0, or u_D = expression, for all
points that belong to u_D.

What if the domain is rectangle where left boundary is the expression,
and all other boundaries are constant set to zero?

Thanks,
Evgeny.

2

See for instance: https://bitbucket.org/fenics-project/dolfin/raw/f0a90b8fffc958ed612484cd3465e59994b695f9/python/demo/documented/subdomains-poisson/demo_subdomains-poisson.py

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3

Jorgen, many thanks for your response. Let me see the boundary formulation in my case then, should I declare the class for left boundary? like

class(left)
def boundary(x, on_boundary):
return near(x[0], 'expression')
class(boundaries)
def boundary(x, on_boundary):
return on_boundary

And above formulation thanks to FEniCS would compile my boundary value problem - boundary conditions (mentioned above)?

Or instead of declaring class, should I have provided unique label right after def … e.g., def left(…) & def walls(…)?

Many thanks,
Evgeny.

4

As shown in the link above, you should do:


# Create classes for defining parts of the boundaries and the interior
# of the domain
class Left(SubDomain):
    def inside(self, x, on_boundary):
        return near(x[0], 0.0)

class AllBoundaries(SubDomain):
    def inside(self, x, on_boundary):
        return on_boundary
# Initialize sub-domain instances
left = Left()
allbndry = AllBoundaries()
# Define mesh
mesh = UnitSquareMesh(64, 64)

# Initialize mesh function for boundary domains
boundaries = MeshFunction("size_t", mesh, mesh.topology().dim()-1)
boundaries.set_all(0)
allbndry.set(boundaries,1)
left.mark(boundaries, 2)

This will create a mesh function (a value for each facet) which is 0 in the interior, 1 on all of the boundary, except at the left boundary,where it is 2

5

Jorgen, many thanks, it seems to be a little bit different than in FEniCS tutorial & its some demo python files, are you referring to somewhat old version of FEniCS like Dolfin or similar ?

Many thanks,
Evgeny.

6

I am referring to the latest version of dolfin, located at bitbucket.
If you are referring to dolfinx, located at github, there is indeed a different syntax.

7

I was referring to FEniCS, that can be imported as python module, and it seems like this Fenics env needs somewhat different syntax, and that’s why I confused on the link you have provided kindly to me. Basically, this syntax is also somewhat relative to Fenics for sure;

8

The fenics Python module, and the dolfin python module is the same module, i.e.
from dolfin import * and from fenics import * is equivalent.

9

Then I can’t understand this syntax and FEniCS tutorial especially boundary conditions formulation, it seems like they use different syntax, and for my ordinary case, I should have used another way & syntax to formulate; thought the tutorial should provide the basic thing that i can do it myself further

10

As you have provided no links to the tutorials you are using, i cannot help you any further.

There are many different ways of applying boundary conditions, depending on the PDE and complexity of the problem.
In the bitbucket repository i referred to in the previous post, there is a large variety of documented demos illustrating usage

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