Stabilization-free virtual element method for 3D hyperelastic problems

authored by: Bing Bing Xu, Fan Peng, Peter Wriggers
Abstract: In this work, we present a first-order stabilization-free virtual element method (SFVEM) for three-dimensional hyperelastic problems. Different from the conventional virtual element method, which necessitates additional stabilization terms in the bilinear formulation, the method developed in this work operates without the need for any stabilization. Consequently, it proves highly suitable for the computation of nonlinear problems. The stabilization-free virtual element method has been applied in two-dimensional hyperelasticity and three-dimensional elasticity problems. In this work, the format will be applied to three-dimensional hyperelasticity problems for the first time. Similar to the techniques used in the two-dimensional stabilization-free virtual element method, the new virtual element space is modified to allow the computation of the higher-order L₂ projection of the gradient. This paper reviews the calculation process of the traditional H₁ projection operator; and describes in detail how to calculate the high-order L₂ projection operator for three-dimensional problems. Based on this high-order L₂ projection operator, this paper extends the method to more complex three-dimensional nonlinear problems. Some benchmark problems illustrate the capability of the stabilization-free VEM for three-dimensional hyperelastic problems.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Chang'an University
Type: Article
Journal: Computational mechanics
ISSN: 0178-7675
Publication date: 27.05.2024
Publication status: E-pub ahead of print
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Ocean Engineering, Mechanical Engineering, Computational Theory and Mathematics, Computational Mathematics, Applied Mathematics
Electronic version(s): https://doi.org/10.1007/s00466-024-02501-4 (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{2352793119574d12a2ade887b557b66d,
title = "Stabilization-free virtual element method for 3D hyperelastic problems",
abstract = "In this work, we present a first-order stabilization-free virtual element method (SFVEM) for three-dimensional hyperelastic problems. Different from the conventional virtual element method, which necessitates additional stabilization terms in the bilinear formulation, the method developed in this work operates without the need for any stabilization. Consequently, it proves highly suitable for the computation of nonlinear problems. The stabilization-free virtual element method has been applied in two-dimensional hyperelasticity and three-dimensional elasticity problems. In this work, the format will be applied to three-dimensional hyperelasticity problems for the first time. Similar to the techniques used in the two-dimensional stabilization-free virtual element method, the new virtual element space is modified to allow the computation of the higher-order L2 projection of the gradient. This paper reviews the calculation process of the traditional H1 projection operator; and describes in detail how to calculate the high-order L2 projection operator for three-dimensional problems. Based on this high-order L2 projection operator, this paper extends the method to more complex three-dimensional nonlinear problems. Some benchmark problems illustrate the capability of the stabilization-free VEM for three-dimensional hyperelastic problems.",
keywords = "Elastoplasticity, Nonlinear problems, Stabilization free, Virtual element method",
author = "Xu, {Bing Bing} and Fan Peng and Peter Wriggers",
note = "Publisher Copyright: {\textcopyright} The Author(s) 2024.",
year = "2024",
month = may,
day = "27",
doi = "10.1007/s00466-024-02501-4",
language = "English",
journal = "Computational mechanics",
issn = "0178-7675",
publisher = "Springer Verlag",
}