Mixed finite element and TPSA finite volume methods for linearized elasticity and Cosserat materials

TL;DR

This paper introduces a mixed finite element method and a TPSA finite volume method for solving Cosserat elasticity equations, offering computational efficiency and robustness in various physical limits.

Contribution

It develops a new finite volume approach with minimal stress variables, improving computational efficiency over existing mixed finite element methods.

Findings

Both methods are robust in the incompressible limit.

The TPSA finite volume method reduces computational complexity.

Numerical comparisons show competitive performance against existing methods.

Abstract

Cosserat theory of elasticity is a generalization of classical elasticity that allows for asymmetry in the stress tensor by taking into account micropolar rotations in the medium. The equations involve a rotation field and associated "couple stress" as variables, in addition to the conventional displacement and Cauchy stress fields. In recent work, we derived a mixed finite element method (MFEM) for the linear Cosserat equations that converges optimally in these four variables. The drawback of this method is that it retains the stresses as unknowns, and therefore leads to relatively large saddle point system that are computationally demanding to solve. As an alternative, we developed a finite volume method in which the stress variables are approximated using a minimal, two-point stencil (TPSA). The system consists of the displacement and rotation variables, with an additional "solid pressure" unknown. Both the MFEM and TPSA methods are robust in the incompressible limit and in the Cauchy limit, for which the Cosserat equations degenerate to classical linearized elasticity. We report on the construction of the methods, their a priori properties, and compare their numerical performance against an MPSA finite volume method.