Canonical Duality Theory for Topology Optimization

TL;DR

This paper introduces a canonical duality approach and a perturbed algorithm to solve complex topology optimization problems in nonlinear elastic structures, achieving exact solutions and reducing common issues like gray elements and checkerboarding.

Contribution

It develops a novel canonical duality framework and a perturbed algorithm to analytically solve the NP-complete Knapsack problem within topology optimization, improving solution accuracy.

Findings

Exact solutions for topology optimization problems without gray elements.

Reduced checkerboard artifacts in optimized structures.

Effective handling of NP-complete subproblems using canonical duality.

Abstract

This paper presents a canonical duality approach for solving a general topology optimization problem of nonlinear elastic structures. By using finite element method, this most challenging problem can be formulated as a 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.