Convex multilevel decomposition algorithms for non‐monotone problems

Abstract

AbstractA convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non‐monotone, possibly multivalued laws. The theory is developed here for a model structure with non‐monotone interface or boundary conditions. First the non‐monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non‐linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non‐convex min‐min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,1 is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non‐monotone stick‐slip frictional effects illustrate the theory.