Primal and mixed finite element formulations for the relaxed micromorphic model

Abstract

The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are $[H^1{]}^3$ for the displacement and $[H(\mathrm{curl}){]}^3$ for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart-Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length $L_{\mathrm{c}}$ and the correlation to the classical Cauchy continuum theory.

References in corpus (14)

• A unifying perspective: the relaxed linear micromorphic continuum
• High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows
• Orientation Embedded High Order Shape Functions for the Exact Sequence Elements of All Shapes
• Reflection and transmission of elastic waves at interfaces embedded in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model
• Complexes from complexes
• Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative
• Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements
• The Heterogeneous Multiscale Finite Element Method for the Homogenization of Linear Elastic Solids and a Comparison with the FE${}^2$ Method
• Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy
• Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures
• Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)
• Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields
• Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach
• $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative

Cited by in corpus (7)

• Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model
• Lagrange and $H(\mathrm{curl},\mathcal{B})$ based Finite Element formulations for the relaxed micromorphic model
• Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions
• A computational approach to identify the material parameters of the relaxed micromorphic model
• Green's functions for the isotropic planar relaxed micromorphic model -- concentrated force and concentrated couple
• A global higher regularity result for the static relaxed micromorphic model on smooth domains
• Yet another best approximation isotropic elasticity tensor in plane strain