Nonlinear elasticity complex and a finite element diagram chase
Abstract
In this paper, we present a nonlinear version of the linear elasticity (Calabi, Kr\"oner, Riemannian deformation) complex which encodes isometric embedding, metric, curvature and the Bianchi identity. We reformulate the rigidity theorem and a fundamental theorem of Riemannian geometry as the exactness of this complex. Then we generalize an algebraic approach for constructing finite elements for the Bernstein-Gelfand-Gelfand (BGG) complexes. In particular, we discuss the reduction of degrees of freedom with injective connecting maps in the BGG diagrams. We derive a strain complex in two space dimensions with a diagram chase.
Keywords
Cite
@article{arxiv.2302.02442,
title = {Nonlinear elasticity complex and a finite element diagram chase},
author = {Kaibo Hu},
journal= {arXiv preprint arXiv:2302.02442},
year = {2023}
}
Comments
Manuscript prepared for proceedings of the INdAM conference "Approximation Theory and Numerical Analysis meet Algebra, Geometry, Topology'', which was held in September 2022 at Cortona, Italy