FEM meets Differential Geometry: Curvature approximation with distributional finite elements (upcoming)
Date:
The slides are WIP and will be shared when finished.
Abstract:
In several applications, such as shells, general relativity, and geometric flows, the curvature of surfaces and Riemannian manifolds plays a fundamental role. Discrete differential geometry (DDG) aims to approximate curvature quantities of discretized domains. By embedding DDG into a FEM context, rigorous numerical analysis tools become available to investigate and extend those algorithms. So-called distributional finite elements are essential in the process. They entail weak regularity such that differential operators have to be understood in the sense of distributions.
In this talk, we approximate the curvature of surfaces, where the surface is described by Lagrange finite elements, and the intrinsic curvature of a Riemannian manifold, where symmetric and solely tangential-tangential continuous Regge elements discretize the metric tensor. Then, we discuss their application to nonlinear shells. We present numerical examples using the finite element library NGSolve (www.ngsolve.org).