Time Dependent Case

This chapter is devoted to finite element approximations of scalar time dependent hemivariational inequalities. We start with the parabolic case following closely Miettinen and Haslinger, 1998. At the end of this chapter we discuss, how the results can be extended to constrained problems. Our presentation will follow the structure used for the static case in Chapter 3. First, we introduce an abstract formulation of a class of parabolic hemivariational inequalities (see Miettinen, 1996, Miettinen and Panagiotopoulos, 1999).

Nonsmooth Optimization Methods

From the previous chapters we know that after the discretization, elliptic and parabolic hemivariational inequalities can be transformed into substationary point type problems for locally Lipschitz superpotentials and as such will be solved. There is a class of mathematical programming methods especially developed for this type of problems. The aim of this chapter is to give an overview of nonsmooth optimization techniques with special emphasis on the first and the second order bundle methods. We present their basic ideas in the convex case and necessary modifications for nonconvex optimization. We shall use them in the next chapter for the numerical realization of several model examples. L…

Approximation of Elliptic Hemivariational Inequalities

From the previous chapter we know that there exist many important problems in mechanics in which constitutive laws are expressed by means of nonmonotone, possibly multivalued relations (nonmonotone multivalued stress-strain or reaction-displacement relations,e.g). The resulting mathematical model leads to an inclusion type problem involving multivalued nonmonotone mappings or to a substationary type problem for a nonsmooth, nonconvex superpotential expressed in terms of calculus of variation. It is the aim of this chapter to give a detailed study of a discretization of such a type of problems including the convergence analysis. Here we follow closely Miettinen and Haslinger, 1995, Miettinen…

Nonsmooth Mechanics. Convex and Nonconvex Problems

Nonlinear, multivalued and possibly nonmonotone relations arise in several areas of mechanics. A multivalued or complete relation is a relation with complete vertical branches. Boundary laws of this kind connect boundary (or interface) quantities. A contact relation or a locking mechanism between boundary displacements and boundary tractions in elasticity is a representative example. Material constitutive relations with complete branches connect stress and strain tensors, or, in simplified theories, equivalent stress and strain quantities. A locking material or a perfectly plastic one is represented by such a relation. The question of nonmonotonicity is more complicated. One aspect concerns…

On parabolic hemivariational inequalities and applications