Download this article
 Download this article For screen
For printing
Recent Issues
Volume 13, Issue 3
Volume 13, Issue 2
Volume 13, Issue 1
Volume 12, Issue 4
Volume 12, Issue 3
Volume 12, Issue 2
Volume 12, Issue 1
Volume 11, Issue 4
Volume 11, Issue 3
Volume 11, Issue 2
Volume 11, Issue 1
Volume 10, Issue 4
Volume 10, Issue 3
Volume 10, Issue 2
Volume 10, Issue 1
Volume 9, Issue 4
Volume 9, Issue 3
Volume 9, Issue 2
Volume 9, Issue 1
Volume 8, Issue 4
Volume 8, Issue 3
Volume 8, Issue 2
Volume 8, Issue 1
Volume 7, Issue 4
Volume 7, Issue 3
Volume 7, Issue 2
Volume 7, Issue 1
Volume 6, Issue 4
Volume 6, Issue 3
Volume 6, Issue 2
Volume 6, Issue 1
Volume 5, Issue 3-4
Volume 5, Issue 2
Volume 5, Issue 1
Volume 4, Issue 3-4
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 4
Volume 3, Issue 3
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN 2325-3444 (online)
ISSN 2326-7186 (print)
 
Author index
To appear
 
Other MSP journals
A comparative analysis for different finite element types in strain-gradient elasticity simulations performed on Firedrake and FEniCS

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
Mathematical Subject Classification
Primary: 00A71, 35Q74, 65N30, 70-10, 74S05
Milestones
Received: 20 November 2024
Revised: 25 April 2025
Accepted: 4 June 2025
Published: 6 September 2025

Communicated by Emilio Barchiesi
Authors
B. Cagri Sarar
Università degli Studi dell’Aquila
67100 L’Aquila
Italy
M. Erden Yildizdag
Istanbul Technical University
34467 Istanbul
Turkey
Francesco Fabbrocino
Telematic University Pegaso
80143 Napoli
Italy
B. Emek Abali
Uppsala University
75310 Uppsala
Sweden