Nonlinear elasticity complex and a finite element diagram chase

TL;DR

This paper develops a nonlinear elasticity complex related to Riemannian geometry, reformulates classical theorems, and constructs finite element methods using algebraic and diagram chase techniques.

Contribution

It introduces a nonlinear elasticity complex encoding geometric properties and generalizes finite element construction for BGG complexes with degree of freedom reduction.

Findings

Reformulation of rigidity and Riemannian theorems as complex exactness

Construction of a 2D strain complex via diagram chase

Reduction of degrees of freedom in finite element methods

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.