Combining Finite Elements with Discrete Differential Geometry for Curvature Approximation and Nonlinear Shell Analysis
Date:
The slides can be found here.
Abstract:
Curvature quantities are central in nonlinear shell models, geometric flows, and differential geometry. Discrete differential geometry provides natural curvature approximations on discretized surfaces and manifolds, while finite element methods offer a framework for stability, convergence analysis, and implementation. In this talk, we combine these two viewpoints, with a primary focus on approximating the extrinsic curvature and its application to nonlinear shells.
The extrinsic curvature of a surface is represented by the shape operator, which enters directly into the bending energy of shell models. On discrete surfaces, however, classical derivatives are not available. We therefore introduce a generalized shape operator defined in a distributional sense. The construction combines the dihedral-angle idea from discrete differential geometry with Hellan-Herrmann-Johnson finite elements [Comodi, Mathematics of Computation 52, 1989]. Based on an integral error representation, we obtain convergence in negative Sobolev norms under mild assumptions and optimal L2 convergence under stronger assumptions [Gopalakrishnan and Neunteufel, in preparation].
We then discuss how this curvature approximation leads to a mixed finite element discretization of nonlinear shells, in which the bending moment tensor appears as an additional unknown. This avoids C1-conforming surface elements and yields a consistent bending discretization even on nonsmooth geometries. A second key issue is membrane locking: to obtain robust shell elements, membrane strains are projected into Regge finite element spaces, whose tangential-tangential continuity is natural for metric and strain quantities [Neunteufel and Schöberl, Computers & Structures 305, 2024]. We present several numerical examples implemented in NGSolve to illustrate the performance of the resulting discretization methods.