On a Branch and Bound Algorithm for the Solutionof a Family of Softening Material Problems of Mechanics withApplications to the Analysis of Metallic Structures

Abstract

The variational formulation of mechanical problems involving nonmonotone, possibly multivalued, material or boundary laws leads to hemivariational inequalities. The solutions of the hemivariational inequalities constitute substationarity points of the related energy (super)potentials. For their computation convex and global optimization algorithms have been proposed instead of the earlier nonlinear optimization methods, due to the lack of smoothness and convexity of the potential. In earlier works one of us has proposed an approach based on the decomposition of the solutions space into convex parts resulting in a sequence of convex optimization subproblems with different feasible sets. In this case nonconvexity of the potential was attributed to (generalized) gradient jumps. In order to treat ’softening‘ material effects, in the present paper this method is extended to cover also energy functionals where nonconvexity is caused by the existence of concave sections. The nonconvex minimization problem is formulated as d.c. (difference convex) minimization and an algorithm of the branch and bound type based on simplex partitions is adapted for its treatment. The partitioning scheme employed here is adapted to the large dimension of the problem and the approximation steps are equivalent to convex minimization subproblems of the same structure as the ones arising in unilateral problems of mechanics. The paper concludes with a numerical example and a discussion of the properties and the applicability of the method.