Regularization Methods and Finite Element Approximation of Hemivariational Inequalities with Applications to Nonmonotone Contact Problems

In this thesis, we consider mechanical problems with nonmonotone contact, like adhesive problems, delamination problems, bilateral contact problems with nonmonotone friction law, nonmonotone unilateral contact, etc. In all of them the contact phenomena are described by nonmonotone and multivalued laws, which can be expressed by means of the Clarke subdifferential of a locally Lipschitz function called a nonconvex, nonsmooth superpotential. Problems involving such laws give rise to hemivariational inequalities introduced for the first time by the engineer Panagiotopoulos in the eighties.

In this work, we combine the regularization techniques with the finite element method to approximate a special class of hemivariational inequalities with maximum (resp. minimum) superpotential. Using some classes of smoothing approximations for nonsmooth functions based on convolution, we provide a regularization procedure to smooth the nonsmooth superpotential. The non-differentiable functional is approximated by a family of differentiable ones. Convergence of the solution based on the regularized problem to the solution of the original problem is shown. Then, the finite element approach for the regularized problem is analysed and convergence results are given. As an application we consider some model examples from continuum mechanics with nonmonotone contact and present some numerical results.