A total Lagrangian Galerkin free element method for finite deformation in hyperelastic materials

authored by: Wei Long Fan, Xiao Wei Gao, Fan Peng, Bing Bing Xu
Abstract: In this research, a total Lagrangian Galerkin free element method (GFrEM) is proposed for the analysis of finite deformation in hyperelastic materials. This method derives the total Lagrangian formulation using the initial configuration as the reference. The mechanical behavior of hyperelastic materials is modeled by the non-Hookean strain energy function. Since Lagrangian isoparametric elements are freely formed in GFrEM by collocation nodes with their surrounding nodes, intrinsic boundary conditions can be imposed simply as in the finite elements method. In addition, the Galerkin method was used to ensure the stability of the results when constructing the equations for each collocation node. The validity and convergence of the proposed method are verified by several two- and three-dimensional numerical examples that include bending, compression, and torsion of hyperelastic materials. The example of nearly incompressible material shows that GFrEM remains highly accurate even with large deformations where the FEM cannot converge.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Dalian University of Technology
Chang'an University
Type: Article
Journal: Applied mathematical modelling
Volume: 137
No. of pages: 25
ISSN: 0307-904X
Publication date: 01.2025
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Modelling and Simulation, Applied Mathematics
Electronic version(s): https://doi.org/10.1016/j.apm.2024.115740 (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{aa759a3ab35844e185c99a5393ee8baa,
title = "A total Lagrangian Galerkin free element method for finite deformation in hyperelastic materials",
abstract = "In this research, a total Lagrangian Galerkin free element method (GFrEM) is proposed for the analysis of finite deformation in hyperelastic materials. This method derives the total Lagrangian formulation using the initial configuration as the reference. The mechanical behavior of hyperelastic materials is modeled by the non-Hookean strain energy function. Since Lagrangian isoparametric elements are freely formed in GFrEM by collocation nodes with their surrounding nodes, intrinsic boundary conditions can be imposed simply as in the finite elements method. In addition, the Galerkin method was used to ensure the stability of the results when constructing the equations for each collocation node. The validity and convergence of the proposed method are verified by several two- and three-dimensional numerical examples that include bending, compression, and torsion of hyperelastic materials. The example of nearly incompressible material shows that GFrEM remains highly accurate even with large deformations where the FEM cannot converge.",
keywords = "Galerkin free element method, Hyperelastic material, Large deformation, Meshless methods, Total Lagrangian",
author = "Fan, {Wei Long} and Gao, {Xiao Wei} and Fan Peng and Xu, {Bing Bing}",
note = "Publisher Copyright: {\textcopyright} 2024 The Author(s)",
year = "2025",
month = jan,
doi = "10.1016/j.apm.2024.115740",
language = "English",
volume = "137",
journal = "Applied mathematical modelling",
issn = "0307-904X",
publisher = "Elsevier Inc.",
}