Adaptive Least Squares Finite Element Methods in Elasto-Plasticity

In computational mechanics applications, one is often interested not only in accurate approximations for the displacements but also for the stress tensor. Least squares finite element methods are perfectly suited for such problems since they approximate both process variables simultaneously in suitable finite element spaces. We consider an H-1 x H(div) least squares formulation for the incremental formulation of elasto-plasticity using a plastic flow rule of von Mises type. The non-linear least squares functional constitutes an a posteriori error estimator on which an adaptive refinement strategy may be based. The variational formulation under plane strain and plane stress conditions is investigated in detail. Standard conforming elements are used for the displacement approximation while the stress is represented by Raviart-Thomas elements. The algebraic least squares problems arising from the finite element discretization are nonlinear and nonsmooth and may be solved by generalized Newton methods.