Exploiting Lagrange duality for topology optimizationwith frictionless unilateral contact

Abstract

This paper presents tractable reformulations of topology optimization problems of structures subject to frictionless unilateral contact conditions. Specifically, we consider stiffness maximization problems of trusses and continua. Based on the Lagrange duality theory, we derive formulations that do not involve complementarity constraints. It is often that a structural optimization problem with contact conditions is formulated as a mathematical programming problem with complementarity constraints (MPCC problem). However, MPCC usually requires special treatment for numerical solution, because it does not satisfy standard constraint qualifications. In contrast, to the formulation presented in this paper, we can apply standard optimization approaches. Numerical experiments of trusses and continua are performed to examine efficiency of the proposed approach.

Robust optimization of structures subjected to frictionless unilateral contact with uncertain initial gaps

“…In this section, we recall a problem formulation for structural optimization with frictionless unilateral contacts (Kanno, 2020a). Uncertainty is not taken into account here.…”

“…where q e is the axial force of member e, r j is the normal contact reaction at node j, and w e is a subsidiary variable corresponding to the twice of the complementary strain energy stored in member e. Theorem 1 in Kanno (2020a) shows…”

“…Therefore, a design optimization problem of structure subjected to frictionless unilateral contacts is often formulated as a mathematical programming problem with complementarity constraints (an MPCC problem). Such formulations can be found in, e.g., Tin-Loi (1999), Strömberg and Klarbring (2010), and the references complied in Kanno (2020a). Since any feasible solution of an MPCC problem does not satisfy any standard constraint qualification (Luo et al, 1996), conventional nonlinear programming (NLP) techniques are likely to fail to find a local optimal solution.…”

“…This brief paper is a sequel to Kanno (2020a). Specifically, we consider a robust counterpart, in which the initial gaps of contact conditions are supposed to be uncertain.…”

“…Specifically, we consider a robust counterpart, in which the initial gaps of contact conditions are supposed to be uncertain. As a natural extension of Kanno (2020a), we see that this robust optimization problem can also be reduced to an NLP problem, which does not involve complementarity constraints.…”