Applying a period mesh in a non-regular geometry

General
1

Hello,
Please find a part of my code where I apply periodic boundary condition.

coord = mesh.coordinates()
xmax = max(coord[:,0]); xmin = min(coord[:,0])
ymax = max(coord[:,1]); ymin = min(coord[:,1])

def inside_2D(x, on_boundary):
        return bool(on_boundary and (dolfin.near(x[0], xmin, self.tol) or dolfin.near(x[1], ymin, self.tol)) and not (dolfin.near(x[0], xmax, self.tol) or dolfin.near(x[1], ymax, self.tol)))

def map_2D(self, x, y):
        # Vertex
        if  (dolfin.near(x[0], xmax, self.tol) \
        and  dolfin.near(x[1], ymax, self.tol))\
        or  (dolfin.near(x[0], xmax, self.tol) \
        and  dolfin.near(x[1], ymin, self.tol))\
        or  (dolfin.near(x[0], xmin, self.tol) \
        and  dolfin.near(x[1], ymax, self.tol)):
            y[0] = xmin
            y[1] = ymin
        # Edges
        elif dolfin.near(x[0], xmax, self.tol):
            y[0] = self.xmin
            y[1] = x[1]
        elif dolfin.near(x[1], ymax, self.tol):
            y[0] = x[0]
            y[1] = ymin
        else:
            y[0] = -1.
            y[1] = -1.

But, the problem is that I can’t apply this to a geometry like the following (as the left and the right boundaries are not along the y-axis), unless I can find the left and right boundary vertices. Well, I already know the vertices as I have created the mesh, but how can I find them directly in Dolfin?

Screenshot 2022-12-22 at 11.05.40
Screenshot 2022-12-22 at 11.05.401318×878 54.8 KB

Thanks in advance!

2

In DOLFIN, you would need to use the parametric expression of your boundary to map from one boundary to the other. i.e. if the line can be described as y=Ax+b you would have to check

    def inside_2D(x, on_boundary):
        return bool(on_boundary and dolfin.near(x[1], A*x[0] + b))

and then create the maps based on the location of each boundary. (i.e. if on a single line do something, if on a curve map with something else).

If you want better support for such geometries, I would suggest using my library, dolfinx_mpc, where you can send in marked facets from GMSH as input to be a certain periodic surface, and give it a corresponding geometrical mapping function.

3

Thanks a lot! It seems a good solution. But if I don’t use dolfinx_mpc, is there any way to find the parameters if the line y=Ax+b from DOLFIN?

4

You have generated the mesh, so it should be easy to compute if you know the two corners.

5

Of course, but what if I’m given a geometry which I don’t know the corners? I want to make my code more general.
Thanks

6

Thrn you would need to use dolfinx and mesh information. Both DOLFIN and DOLFINx will require a mapping function map(x) that maps the coordinates of one boundary to the other. This function will be specific to each mesh (up to some constants, as described above)

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