Lagrange and \(H(\mathrm{curl}, \mathcal{B})\) based finite element formulations for the relaxed micromorphic model. (English) Zbl 1509.74048
Summary: 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 significant aspects related to the relaxed micromorphic model realization with the finite element method (FEM). The variational problem is derived and different FEM-formulations for the two-dimensional case are presented. These are a nodal standard formulation \(H^1(\mathcal{B})\times H^1(\mathcal{B})\) and a nodal-edge formulation \(H^1(\mathcal{B}) \times H(\mathrm{curl}, \mathcal{B})\), where the latter employs the Nédélec space. In this framework, the implementation of higher-order Nédélec elements is not trivial and requires some technicalities which are demonstrated. We discuss the computational 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 parameter.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74E30 | Composite and mixture properties |
| 74M25 | Micromechanics of solids |
Keywords:
Nédélec element; mechanical metamaterial; variational problem; consistent coupling boundary condition; size effect; patch testReferences:
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