A Novel Canonical Duality Theory for Solving 3-D Topology Optimization Problems

TL;DR

This paper introduces a new canonical duality theory-based method for efficiently solving 3D topology optimization problems, providing exact solutions and outperforming traditional methods like BESO and SIMP.

Contribution

It presents a deterministic polynomial-time solution to NP-hard topology optimization problems using canonical duality theory, linking it with Gao's energy principle and demonstrating its effectiveness on benchmark problems.

Findings

The CPD method yields precise 0-1 global optimal solutions.

It outperforms BESO and SIMP in computational efficiency and solution quality.

The approach addresses and corrects modeling and methodological issues in existing topology optimization techniques.

Abstract

This paper demonstrates a mathematically correct and computationally powerful method for solving 3D topology optimization problems. This method is based on canonical duality theory (CDT) developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard knapsack problem in topology optimization can be solved deterministically in polynomial time via a canonical penalty-duality (CPD) method to obtain precise 0-1 global optimal solution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Additionally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly addressed.