Lagrange and $H(\operatorname{curl},{\cal B})$ based Finite Element formulations for the relaxed micromorphic model

TL;DR

This paper develops finite element formulations for the relaxed micromorphic model, comparing standard and Nédélec-based approaches, and analyzes their convergence and size-effect properties for modeling metamaterials.

Contribution

It introduces and compares $H^1$ and $H( ext{curl})$ finite element formulations for the relaxed micromorphic model, including implementation details and convergence analysis.

Findings

Nédélec-based formulation effectively captures size effects.

Convergence behavior differs between Lagrange and Nédélec elements.

Higher-order Nédélec elements require specialized implementation.

Abstract

Modeling the unusual mechanical properties of metamaterials is a challenging topic for the mechanics community and enriched continuum theories are promising computational tools for such materials. The so-called relaxed micromorphic model has shown many advantages in this field. In this contribution, we present the significant aspects related to the relaxed micromorphic model realization with the finite element method. The variational problem is derived and different FEM-formulations for the two-dimensional case are presented. These are a nodal standard formulation $H^1({\cal B}) \times H^1({\cal B})$ and a nodal-edge formulation $H^1({\cal B}) \times H(\operatorname{curl}, {\cal B})$, where the latter employs the N\'ed\'elec space. However, the implementation of higher-order N\'ed\'elec elements is not trivial and requires some technicalities which are demonstrated. We discuss the convergence behavior of Lagrange-type and tangential-conforming finite element discretizations. Moreover, we analyze the characteristic length effect on the different components of the model and reveal how the size-effect property is captured via this characteristic length.