Multigrid Methods for Problems in Solid Mechanics 32

2.1 INTRODUCTION We consider the eecient simulationof the quasi-static deformation process of materials with memory, where the history of the deformation is described by internal variables. Here, we focus on techniques which are required for large scale computations in three space dimensions on very ne grids with more than 10 6 unknowns. This involves the combination of parallel multigrid methods and stable discretizations in space and time. We discuss the construction of eecient solver from the point of view of scientiic computing, which includes many diierent aspects. For the application to plasticity we comment on the following topics. The simulation of inelastic problems requires an appropriate physical model. The basic models for the application to computational engineering are recently summarized in SH98]; we use this reference as an algorithmic background for our method. The algorithmic formulation is investigated in detail by the means of numerical analysis. For the Prandtl-Reuu model with linear hardening an comprehensive analysis is presented in HR99]; we comment on the extension of these results to more general cases. The analysis requires an analytical theory for obtaining existence and uniqueness results in appropriate function spaces. For the classical plasticity models the theory can be derived in the framework of convex analysis, cf. DL76,Joh76,Tem85]. Here, we follow a more general modern presentation in Alb98], which includes a large variety of diierent models for materials with memory.