DOI： 10.1016/j.finel.2026.104586

Abstract：This work presents a novel Virtual Element Method (VEM) based on a divergence-free projection for the numerical simulation of two-dimensional frictionless contact problems. The proposed formulation belongs to a new class of stabilization-free VEMs, where the stiffness matrix is constructed without introducing any stabilization terms. Unlike conventional stabilization-free VEMs (Berrone et al., 2023), the projection operator is defined in a divergence-free space, leading to a formulation that completely avoids stabilization. Since no stabilization terms are
required, contact constraints can be imposed in a more natural and consistent manner. To the best of the author’s knowledge, there has been no previous work applying a stabilization-free VEM-type method to the field of contact mechanics. In addition, fracture phenomena induced
by contact interactions are also investigated by coupling the proposed VEM with phase field models. The intrinsic flexibility of polygonal meshes in VEM is fully exploited to develop an adaptive mesh refinement strategy, which is applied in the vicinity of the contact interfaces and crack surfaces. This adaptive strategy significantly enhances computational efficiency and accuracy in capturing localized deformation, stress concentration, and crack evolution. Several numerical examples involving frictionless contact and contact-induced fracture are presented to demonstrate the accuracy and robustness of the proposed method.