A comparative analysis for different finite element types in strain-gradient elasticity simulations performed on Firedrake and FEniCS

TL;DR

This paper compares various finite element methods, including Lagrange, Argyris, Hermite, mixed formulations, and NURBS-based isogeometric analysis, for strain-gradient elasticity problems using Firedrake and FEniCS, providing open-source implementations.

Contribution

It offers a comprehensive comparison of finite element formulations for strain-gradient elasticity within open-source FEM packages, including implementation details and open access codes.

Findings

Different finite element types show varying accuracy and computational efficiency.

NURBS-based isogeometric analysis provides smooth solutions suitable for higher-gradient models.

Open-source codes facilitate further research in generalized continuum mechanics.

Abstract

The layer-upon-layer approach in additive manufacturing, open or closed cells in polymeric or metallic foams involve an intrinsic microstructure tailored to the underlying applications. Homogenization of such architectured materials creates metamaterials modeled by higher-gradient models, specifically when the microstructure's characteristic length is comparable to the length scale of the structure. In this study, we conduct a comparative analysis of various finite elements methods for solving problems in strain-gradient elasticity. We employ open-source packages from Firedrake and FEniCS. Different finite element formulations are tested: we implement Lagrange, Argyris, Hermite elements, a Hu--Washizu type (mixed) formulation, as well as isogeometric analysis with Non-Uniform Rational B-Splines (NURBS). For the numerical study, we investigate one- and two-dimensional problems discussed in the literature of strain-gradient modeling. All developed codes are open-access to encourage research in Finite Element Method (FEM) based computation of generalized continua.