A new class of multilevel decomposition algorithms for non monotone problems based on the quasidifferentiability concept

Abstract

A convex, multilevel decomposition approach is proposed for the solution of static analysis problems involving non-monotone, possibly multivalued laws. The theory is developed here for model problems of structures having non-monotone interface or boundary conditions. First we decompose appropriately the non-monotone laws writing them as a difference of monotone constituents. In the general case, this is related to the quasidifferentiability concept. This permits us to obtain a system of convex variational inequalities, the solution s) of which describe the position s) of static equilibrium of the considered structure. Then the problems are formulated as min-min problems for appropriately defined Lagrangian functions. Convex optimization algorithms of various complexity are used in a multilevel scheme for the numerical solution of the considered structural analysis problem. Numerical results concerning the calculation of an elastic contact problem and an elastic stamp problem illustrate the theory.