On the numerical treatment of nonconvex energy problems: multilevel decomposition methods for hemivariational inequalities

Abstract

The variational formulation of mechanical problems involving nonmonotone, possibly multivalued, material or boundary laws leads to hemivariational inequalities. Since the underlying energy (super) potentials generally lack both convexity and smoothness, these problems cannot be treated by the classical nonlinear analysis methods.Here we propose a method for the solution of a wide family of hemivariational inequalities. It is based on a multilevel decomposition of the admissible solution space combined interactively with an appropriate structure decomposition, which gives rise to a finite number of variational inequalities involving convex (but nonsmooth) energy functionals. The latter can be solved easily using existing convex minimization algorithms.Concluding, numerical examples illustrate the applicability of the approach.