FEM meets Discrete Differential Geometry: Extrinsic & intrinsic curvature approximation
Date:
The slides can be found here.
Abstract:
The curvature of surfaces and Riemannian manifolds is essential in several fields such as nonlinear shell analysis, general relativity, and geometric flows. Discrete differential geometry (DDG) seeks to approximate curvature quantities on discretized surfaces and manifolds. Incorporating DDG into a finite element method (FEM) framework provides robust tools for analysis and development of algorithms. Distributional finite elements are crucial here. However, due to their weak regularity, (nonlinear) differential operators have to be interpreted in a sense of distributions.
In this talk, we present how the DDG algorithms of dihedral angles for extrinsic curvature and angle defects for intrinsic curvature approximation can be incorporated into FEM. To this end, we approximate the surface using Lagrange finite elements and the Riemannian metric tensor by symmetric, tangential-tangential continuous Regge elements. We discuss convergence results by utilizing an integral representation of the distributional curvatures involving covariant differential operators. We present the application of distributional extrinsic curvature in nonlinear shell models. Numerical examples are demonstrated with the finite element library NGSolve (www.ngsolve.org) and NGSDiffGeo, an add-on package for differential geometry support.