A Primal-Dual Finite Element Approximation for a Nonlocal Model in Plasticity

We study the numerical approximation of a static infinitesimal plasticity model of kinematic hardening with a nonlocal extension. Here, the free energy to be minimized is a combination of the elastic energy and an additional term depending on the curl of the plastic variable. First, we introduce the stress as dual variable and provide an equivalent primal-dual formulation resulting in a local flow rule. The discretization is based on curl-conforming Nédélec elements. To obtain optimal a priori estimates, the finite element spaces have to satisfy a uniform inf-sup condition. This can be guaranteed by adding locally defined face and element bubbles. Second, the discrete variational inequality system is reformulated as a nonlinear equality. We show that the classical radial return algorithm applied to the mixed inequality formulation is equivalent to a semismooth Newton method for the nonlinear system of equations. Numerical results illustrate the convergence of the applied discretization and the solver.