B. Cagri Sarar, M. Erden Yildizdag, Francesco Fabbrocino
and B. Emek Abali
Vol. 13 (2025), No. 3, 237–252
DOI: 10.2140/memocs.2025.13.237
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 FEM) 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. Among the examined formulations, Argyris and
mixed FEM demonstrate superior accuracy, whereas Hermite and IGA lack of
convergence behavior. Displacements predicted by Hermite elements also differ
noticeably in the 1-D case. All developed codes are open-access to encourage
research in finite element method (FEM) based computation of generalized
continua.
Keywords
strain-gradient elasticity, finite element method,
variational method, higher-gradient modeling