A hybrid H1xH(curl) finite element formulation for a relaxed micromorphic continuum model of antiplane shear

TL;DR

This paper develops a hybrid finite element method combining H1 and H(curl) functions to effectively simulate a relaxed micromorphic continuum model for antiplane shear, addressing convergence issues in standard approaches.

Contribution

It introduces a novel hybrid H1xH(curl) finite element formulation tailored for relaxed micromorphic models, improving solution accuracy for complex metamaterial simulations.

Findings

Hybrid formulation demonstrates better convergence rates.

Mixed formulations ensure existence and uniqueness of solutions.

Simplified 2D model captures essential 3D behavior.

Abstract

One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to H1, such that standard nodal H1-finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces H1 and H(curl), demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates.