Efficient Solvers for Mixed Finite Element Discretizations of Nonlinear Problems in Solid Mechanics

Abstract

A common goal of our projects in the three phases of GRK 615 was, among other issues, the development of efficient solvers for different mixed finite element approaches to nonlinear problems in solid mechanics. In the first phase, the PEERS (‘plane elasticity element with reduced symmetry’) was studied for elastoplastic deformationmodels. The nonlinear algebraic systems were solved with a fixed point iteration leading to a linear elasticity problem in each step which was treated by suitable constraint preconditioners.The treatment of elastoplastic deformations by least squaresmixed finite elementmethodswas the subject of the project in the second phase. In particular, appropriate regularizations for the non-smoothness of the nonlinear problems were investigated. In the third phase, the least squares finite element formulation of contact problems was studied. For the Signorini problem, the quadratic minimization problems under affine constraints were treated by an active set strategy. Preconditioned conjugate gradient iterations for a null space formulation were used for the systems arising in each step.