Stability and continuation of solutions to obstacle problems

AbstractIn this paper we will give a summary of some of our results which we have obtained recently. We mainly consider the question whether solutions to variational inequalities with an eigenvalue parameter are stable in the sense defined in Section 1. More precisely, we ask whether a solution to the variational inequality yields a strict local minimum of an associated energy functional defined on a closed convex subset of a real Hilbert space. This nonlinearity of the space of admissible vectors implies a new and interesting stability behavior of the solutions which is not present in the case of equations.Moreover, it is noteworthy that optimal regularity properties of the solutions to the variational inequality are needed for the stability criterion which we will describe in Section 2. Applications to the beam and plate are considered in Sections 4 and 5. In the case of a plate, numerical computations are crucial because it is impossible to find an analytical expression for a branch of solutions to the variational inequality which is not also a solution to the free problem. Closely connected to the question of stability of a given solution to a variational inequality is the question of the continuation of this solution, which we will discuss in Section 3.In Section 6 a survey will be given on the methods used for the computation of stability bounds. This includes in particular a short introduction to continuation algorithms for both equations and variational inequalities.Frequent references will be made to the literature of direct relevance to the material presented. A few additional related research papers or monographs have been included in the bibliography (Courant and Hilbert (1962/1968), Fichera (1972), Funk (1962), Glowinski et al. (1981), Kikuchi and Oden (1988), Landau and Lifschitz (1970), Lions (1971) and Lions and Stampacchia (1967))

Analytical and numerical aspects of time-dependent models with internal variables

In this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented