Canonical Duality Theory for Topology Optimization

This paper presents a canonical duality approach for solving a general topology optimization problem of nonlinear elastic structures. Based on the principle of minimum total potential energy, this most challenging problem can be formulated as a bi-level mixed integer nonlinear programming problem (MINLP), i.e., for a given deformation, the first-level optimization is a typical linear constrained 0–1 programming problem, while for a given structure, the second-level optimization is a general nonlinear continuous minimization problem in computational nonlinear elasticity. It is discovered that for linear elastic structures, first-level optimization is a typical Knapsack problem, which is considered to be NP-complete in computer science. However, by using canonical duality theory, this well-known problem can be solved analytically to obtain exact integer solution. A perturbed canonical dual algorithm (CDT) is proposed and illustrated by benchmark problems in topology optimization. Numerical results show that the proposed CDT method produces desired optimal structure without any gray elements. The checkerboard issue in traditional methods is much reduced. Additionally, an open problem on NP-hardness of the Knapsack problem is proposed.