A hybrid H1xH(curl) finite element formulation for a planar relaxed micromorphic continuum

09/28/2020
by   Adam Chejanovsky, et al.
0

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 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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

12/01/2021

Lagrange and H(curl, B) based Finite Element formulations for the relaxed micromorphic model

Modeling the unusual mechanical properties of metamaterials is a challen...
02/17/2022

Primal and mixed finite element formulations for the relaxed micromorphic model

The classical Cauchy continuum theory is suitable to model highly homoge...
11/20/2019

The Hodge Laplacian on Axisymmetric Domains

We study the mixed formulation of the abstract Hodge Laplacian on axisym...
12/17/2020

A priori error analysis of high-order LL* (FOSLL*) finite element methods

A number of non-standard finite element methods have been proposed in re...
02/17/2022

On H1, H(curl) and H(sym Curl) finite elements for matrix-valued Curl problems

In this work we test the numerical behaviour of matrix-valued fields app...
10/20/2019

A Conformal Three-Field Formulation for Nonlinear Elasticity: From Differential Complexes to Mixed Finite Element Methods

We introduce a new class of mixed finite element methods for 2D and 3D c...
08/21/2019

Adaptive Morley FEM for the von Kármán equations with optimal convergence rates

The adaptive nonconforming Morley finite element method (FEM) approximat...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.