On Topology Optimization and Canonical Duality Method

TL;DR

This paper introduces a novel canonical duality theory (CDT) approach to solve topology optimization problems, enabling analytical solutions for the linear knapsack problem and producing exact 0-1 density distributions without artificial techniques.

Contribution

The paper refines and applies the CDT method to general topology optimization, demonstrating its effectiveness and advantages over traditional methods like SIMP and BESO.

Findings

CDT provides analytical solutions for linear knapsack problems in topology optimization.

The method produces exact 0-1 optimal density distributions without checkerboard patterns.

Numerical tests confirm CDT's superior performance compared to existing approaches.

Abstract

Topology optimization for general materials is correctly formulated as a bi-level knapsack problem, which is considered to be NP-hard in global optimization and computer science. By using canonical duality theory (CDT) developed by the author, the linear knapsack problem can be solved analytically to obtain global optimal solution at each design iteration. Both uniqueness, existence, and NP-hardness are discussed. The novel CDT method for general topology optimization is refined and tested by both 2-D and 3-D benchmark problems. Numerical results show that without using filter and any other artificial technique, the CDT method can produce exactly 0-1 optimal density distribution with almost no checkerboard pattern. Its performance and novelty are compared with the popular SIMP and BESO approaches. Additionally, some mathematical and conceptual mistakes in literature are explicitly pointed out. A brief review on the canonical duality theory for solving a unified problem in multi-scale nonconvex/discrete systems is given in Appendix.