Mixed finite element methods for linear Cosserat equations

TL;DR

This paper develops and analyzes mixed finite element methods for linear Cosserat equations, providing convergence proofs and stability analysis, with numerical verification demonstrating their effectiveness.

Contribution

It introduces two novel mixed finite element approaches for Cosserat materials, with rigorous convergence and stability analysis, including in the limit of vanishing characteristic length.

Findings

Proved convergence of both strongly and weakly coupled methods

Established stability as the micropolar length tends to zero

Numerical results confirm theoretical predictions

Abstract

We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weakly coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing characteristic length of the micropolar structure. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications.