Virtual element method for modeling the deformation of multiphase composites

TL;DR

This paper applies the virtual element method (VEM) to simulate deformation in multiphase composites, highlighting its flexibility with complex meshes and demonstrating its accuracy through various elastic problems.

Contribution

It introduces the use of VEM for multiphase composites, enabling simplified mesh generation with arbitrarily-shaped elements and validating its effectiveness through comparative simulations.

Findings

VEM accurately models deformation in multiphase composites.

VEM handles complex polygonal and polyhedral meshes effectively.

Simulations confirm the method's stability and precision.

Abstract

In this paper, we study applications of the virtual element method (VEM) for simulating the deformation of multiphase composites. The VEM is a Galerkin approach that is applicable to meshes that consist of arbitrarily-shaped polygonal and polyhedral (simple and nonsimple) elements. In the VEM, the basis functions are defined as the solution of a local elliptic partial differential equation, and are never explicitly computed in the implementation of the method. The stifness matrix of each element is built by using the elliptic projection operator of the internal virtual work (bilinear form) and it consists of two terms: a consistency term that is exactly computed (linear patch test is satisfied) and a correction term (ensures stability) that is orthogonal to affine displacement fields and has the right scaling. The VEM simplifies mesh generation for a multiphase composite: a stiff inclusion can be modeled using a single polygonal or polyhedral element. Attributes of the virtual element approach are highlighted through comparisons with Voronoi-cell lattice models, which provide discrete representations of material structure. The comparisons involve a suite of two-dimensional linear elastic problems: patch test, axisymmetric circular inclusion problem, and the deformation of a three-phase composite. The simulations demonstrate the accuracy and flexibility of the virtual element method.