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This paper devises a novel lowest-order conforming virtual element method (VEM) for planar linear elasticity with the pure displacement/traction boundary condition. The main trick is to view a generic polygon $K$ as a new one…

Numerical Analysis · Mathematics 2022-01-13 Jianguo Huang , Sen Lin , Yue Yu

This work provides an efficient virtual element scheme for the modeling of nonlinear elastodynamics undergoing large deformations. The virtual element method (VEM) has been applied to various engineering problems such as elasto-plasticity,…

Numerical Analysis · Mathematics 2020-02-10 M. Cihan , F. Aldakheel , B. Hudobivnik , P. Wriggers

We consider the approximation of the 2D frictionless contact problem in elasticity using the Virtual Element Methods (VEMs). To overcome the volumetric locking phenomenon in the nearly incompressible case, we adopt a mixed…

Numerical Analysis · Mathematics 2026-01-07 C. Lovadina , L. Molinari

Within the framework of the displacement-based Virtual Element Method (VEM) for plane elasticity a significant problem is represented by an accurate evaluation of the stress field. In particular, in the classical VEM formulation, a suitable…

Numerical Analysis · Mathematics 2018-02-20 Edoardo Artioli , Stefano de Miranda , Carlo Lovadina , Luca Patruno

A virtual element method (VEM) with the first order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background…

Numerical Analysis · Mathematics 2022-02-17 Shuhao Cao , Long Chen , Ruchi Guo

In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. We adopt the Hellinger--Reissner variational principle to construct a weak equilibrium…

Numerical Analysis · Mathematics 2024-02-29 Alvin Chen , Joseph E. Bishop , N. Sukumar

Strain gradient plasticity theories are being widely used for fracture assessment, as they provide a richer description of crack tip fields by incorporating the influence of geometrically necessary dislocations. Characterizing the behavior…

Numerical Analysis · Mathematics 2017-11-29 Emilio Martínez-Pañeda , Sundar Natarajan , Stéphane Bordas

This paper presents the first time implementation of the eXtended Finite Element Method (XFEM) in the general purpose commercial software COMSOL Multiphysics. An enrichment strategy is proposed, consistent with the structure of the…

Computational Engineering, Finance, and Science · Computer Science 2021-09-08 Ahmad Jafari , Pooyan Broumand , Mohammad Vahab , Nasser Khalili

The numerical approximation of 2D elasticity problems is considered, in the framework of the small strain theory and in connection with the mixed Hellinger-Reissner variational formulation. A low-order Virtual Element Method (VEM) with…

Numerical Analysis · Mathematics 2017-10-11 E. Artioli , S. de Miranda , C. Lovadina , L. Patruno

A hybrid framework integrating the Virtual Element Method (VEM) with deep learning is presented as an initial step toward developing efficient and flexible numerical models for one-dimensional Euler-Bernoulli beams. The primary aim is to…

Machine Learning · Computer Science 2025-01-14 Paulo Akira F. Enabe , Rodrigo Provasi

The present paper is the second part of a twofold work, whose first part is reported in [3], concerning a newly developed Virtual Element Method (VEM) for 2D continuum problems. The first part of the work proposed a study for linear elastic…

Numerical Analysis · Mathematics 2018-10-24 Edoardo Artioli , Lourenco Beirão da Veiga , Carlo Lovadina , Elio Sacco

The numerical simulation of physical processes in the underground frequently entails challenges related to the geometry and/or data. The former are mainly due to the shape of sedimentary layers and the presence of fractures and faults,…

Numerical Analysis · Mathematics 2020-02-28 Alessio Fumagalli , Anna Scotti , Luca Formaggia

A robust $C^0$-continuous nonconforming virtual element method (VEM) is developed for a boundary value problem arising from strain gradient elasticity in two dimensions, with the family of polygonal meshes satisfying a very general…

Numerical Analysis · Mathematics 2024-12-24 Jianguo Huang , Yue Yu

This paper presents an initial exploration of stress-assisted diffusion problems in three dimensions within the framework of the virtual element method (VEM). Hilbert spaces enriched with parameter-weighted norms, the extended…

Numerical Analysis · Mathematics 2025-02-05 Andres E. Rubiano

We present and analyze a Virtual Element Method (VEM) of arbitrary polynomial order $k\in\mathbb{N}$ for the Laplace-Beltrami equation on a surface in $\mathbb{R}^3$. The method combines the Surface Finite Element Method (SFEM) [Dziuk,…

Numerical Analysis · Mathematics 2020-01-20 Massimo Frittelli , Ivonne Sgura

The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element…

Numerical Analysis · Mathematics 2021-12-28 Gianmarco Manzini , Annamaria Mazzia

This study presents a finite element and virtual element (FE-VE) coupled method for thermomechanical analysis in electronic packaging structures. The approach partitions computational domains strategically, employing FEM for regular…

Numerical Analysis · Mathematics 2026-01-14 Yanpeng Gong , Sishuai Li , Yue Mei , Bingbing Xu , Fei Qin , Xiaoying Zhuang , Timon Rabczuk

This work presents a practical finite element modeling strategy, the Crack Element Method (CEM), for simulating the dynamic crack propagation in two-dimensional structures. The method employs an element-splitting algorithm based on the…

Computational Engineering, Finance, and Science · Computer Science 2025-08-04 Yuxi Xie , Ethan J. Wu , Lu Xu , Jimmy Perez , Shaofan Li

The eXtended Finite Element Method (XFEM) is used to solve interface problems with an unfitted mesh. We present an implementation of the XFEM in the FEM-library deal.II. The main parts of the implementation are (i) the appropriate…

Numerical Analysis · Mathematics 2015-07-16 Thomas Carraro , Sven Wetterauer

This work presents a study on the computational homogenization of electro-magneto-mechanically coupled problems through the Virtual Element Method (VEM). VE-approaches have great potential for the homogenization of the physical properties…

Computational Engineering, Finance, and Science · Computer Science 2021-04-07 Christoph Böhm , Blaž Hudobivnik , Michele Marino , Peter Wriggers