Numerical Solution of a Variational Problem Arising in Stress Analysis : the Vector Case

In this article, we discuss the numerical solution of a constrained minimization problem arising from the stress analysis of elasto-plastic bodies. This minimization problem has the flavor of a generalized non-smooth eigenvalue problem, with the smallest eigenvalue corresponding to the load capacity ratio of the elastic body under consideration. An augmented Lagrangian method, together with finite element approximations, is proposed for the computation of the optimum of the non-smooth objective function, and the corresponding minimizer. The augmented Lagrangian approach allows the decoupling of some of the nonlinearities and of the differential operators. Similarly an appropriate Lagrangian functional, and associated Uzawa algorithm with projection, are introduced to treat non-smooth equality constraints. Numerical results validate the proposed methodology for various two-dimensional geometries.