Finite Element Methods on Surfaces

In this post, we return to one of the roots of Nested Tori, namely, visualization of problems solved using finite element methods. There's quite a few projects that I worked on for that PhD, whose scattered results have not managed to make it on here today, concerning curved domains. For some topics we have covered, see posts with the tag finite element methods. These include, e.g., basic diffusion in planar domains, wave equations in planar domains using symplectic methods, and electromagnetic wave simulation using vector finite elements. These posts will be a useful preview of today's post.

Our first two examples concern heat diffusion on evolving surfaces with two interesting topologies: the sphere and the torus. The evolution is fairly simple: rotation about an axis. The heat source remains fixed in the surrounding 3D space, and focuses on a fixed (in the ambient space) spot on the sphere; the rotation carries the heat away and it spreads throughout the sphere (or torus).

Next, we consider a case where the solution can't be effected by an obvious or simple change of coordinates (i.e. in the terminology of fluids and continuum mechanics, switching from Eulerian to Lagarangian coordinates): heat on a bouncing sphere that stretches and shrinks vertically. Here the heat source builds with time and cuts off midway to just let the heat spread:

The implementation of all of these cases are via surface finite elements in the Eulerian description: the surface is considered a material that moves through space, but the spatial coordinates are fixed.

Our last example is a wave equation on a sphere, using surface finite elements and symplectic methods. The oscillating quantity $u$ is depicted as displacement along the normal direction of a sphere (so that, for example, if $u$ has some small positive value at one point and decreases rapidly to 0 as you move away, it looks like a small lump over that point). The initial condition is a ruffly structure (with speed 0), and we just let it go.

Enjoy!