Intrinsic mixed finite element methods for linear Cosserat elasticity

TL;DR

This paper introduces two structure-preserving mixed finite element methods for linear Cosserat elasticity that are robust across different material parameters and regimes, ensuring optimal convergence and addressing locking phenomena.

Contribution

The paper develops novel structure-preserving finite element schemes for Cosserat elasticity that remain stable and accurate regardless of the Cosserat coupling constant and material incompressibility.

Findings

Methods are robust in the Cosserat coupling constant $$c and nearly incompressible regimes.

Optimal convergence rates are achieved, independent of $$c.

Numerical benchmarks demonstrate the effectiveness of the proposed methods.

Abstract

We propose two parameter-robust mixed finite element methods for linear Cosserat elasticity. The Cosserat coupling constant $\mu_c$, connecting the displacement $u$ and rotation vector $\omega$, leads to possible locking phenomena in finite element methods. The formal limit of $\mu_c\to\infty$ enforces the constraint $\frac{1}{2}\operatorname{curl} u = \omega$ and leads to the fourth-order couple stress problem. Viewing the linear Cosserat model as the Hodge-Laplacian problem of a twisted de~Rham complex, we derive structure-preserving distributional finite element spaces, where the limit constraint is fulfilled in the discrete setting. Applying the mass conserving mixed stress (MCS) method for the rotations, the resulting scheme is robust in $\mu_c$. Combining it with the tangential-displacement normal-normal-stress (TDNNS) method for the displacement part, we obtain additional robustness in the nearly incompressible regime and for anisotropic structures. Using a post-processing scheme for the rotations, we prove optimal convergence rates independent of the Cosserat coupling constant $\mu_c$. We demonstrate the performance of the proposed methods in several numerical benchmark examples.