Numerical solution of a variational–hemivariational inequality modelling simplified adhesion of an elastic body

The paper is devoted to the Galerkin method and Finite Element Method for a stationary variational–hemivariational inequality modelling unilateral adhesive and frictionless contact of an elastic body with a foundation. Adhesion is modelled by a simplified Winkler-type law. An abstract theorem on the convergence of the Galerkin method for a class of nonlinear and pseudomonotone elasticity operators is proved. The theorem generalizes the result of Haslinger et al. (1999, Finite Element Method for Hemivariational Inequalities, Theory, Methods and Applications. Boston: Kluwer Academic Publishers). The problem is solved numerically on a mesh of linear triangles, by minimization of an associated energy functional for the linear, coercive and symmetric case. For nonsmooth optimization the Proximal Bundle Method (PBM) is used. For the benchmark problem of 2D linear elasticity, the numerical assessment of the method convergence rate is done. Moreover, tests are performed to establish the speed of PBM convergence.