Extended virtual element method for 2D elasticity problems

The virtual element method (VEM) [2] is a stabilized Galerkin formulation on arbitrary polygonal meshes, in which the explicit knowledge of the basis functions within the problem domain is not required – i.e. such functions are virtual. Suitable projection operators are used to decompose the bilinear form into a consistent part, that reproduces the polynomial space, and a correction term ensuring stability. Taking inspiration from the well-established extended finite element (X-FEM) technique [3], we have formerly devised in [1] an extended virtual element method (X-VEM) for the Laplace problem with discontinuous and singular solutions. In this contribution, we formulate the extended virtual element method (X-VEM) for two dimensional elasticity problems in the presence of cracks. X-VEM's concept is to augment the standard virtual element space with the product of standard virtual nodal basis functions and enrichment functions. In particular, we first formulate an extended projector that maps functions lying in the extended virtual element space onto a set spanned by linear polynomials and the enrichment function. For cracks, the enrichment function is a classic crack tip function [3], while crack discontinuities are dealt with by means of Hansbo & Hansbo FE method [4]. Once the element projection matrix has been computed, we get the element stabilization matrix as in the standard VEM. Numerical experiments are shown for two dimensional elastic domains with mode I crack opening, to highlight the accuracy of the proposed method in both cases of quadrilateral and polygonal meshes.