Stress‐hybrid virtual element method on quadrilateral meshes for compressible and nearly‐incompressible linear elasticity

Abstract In this article, we propose a robust low‐order stabilization‐free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress‐hybrid principle. We refer to this approach as the stress‐hybrid virtual element method (SH‐VEM). In this method, the Hellinger–Reissner variational principle is adopted, wherein both the equilibrium equations and the strain‐displacement relations are variationally enforced. We consider small‐strain deformations of linear elastic solids in the compressible and near‐incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are virtual . The stress field, similar to the stress‐hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5‐parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain‐displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH‐VEM is free of volumetric and shear locking, and it converges optimally in the norm and energy seminorm of the displacement field, and in the norm of the hydrostatic stress.