A Unified Finite Strain Theory for Membranes and Ropes

TL;DR

This paper introduces a unified finite strain theory based on Tangential Differential Calculus that applies to membranes, ropes, and 3D continua, compatible with various numerical methods and geometries, achieving higher-order convergence.

Contribution

It reformulates finite strain theory using TDC, unifying different structures and numerical approaches without relying on parametrizations or specific coordinate systems.

Findings

Higher-order convergence rates achieved

Applicable to parametrized and implicit geometries

Compatible with Surface FEM and Trace FEM

Abstract

The finite strain theory is reformulated in the frame of the Tangential Differential Calculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, membranes and three-dimensional continua are treated with one set of governing equations. Secondly, the reformulated boundary value problem applies to parametrized and implicit geometries. Therefore, the formulation is more general than classical ones as it does not rely on parametrizations implying curvilinear coordinate systems and the concept of co- and contravariant base vectors. This leads to the third unification: TDC-based models are applicable to two fundamentally different numerical approaches. On the one hand, one may use the classical Surface FEM where the geometry is discretized by curved one-dimensional elements for ropes and two-dimensional surface elements for membranes. On the other hand, it also applies to recent Trace FEM approaches where the geometry is immersed in a higher-dimensional background mesh. Then, the shape functions of the background mesh are evaluated on the trace of the immersed geometry and used for the approximation. As such, the Trace FEM is a fictitious domain method for partial differential equations on manifolds. The numerical results show that the proposed finite strain theory yields higher-order convergence rates independent of the numerical methodology, the dimension of the manifold, and the geometric representation type.