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

Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 41)

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.

Notes

Acknowledgements

This research is supported by the US Air Force Office for Scientific Research (AFOSR) under the grants FA2386-16-1-4082 and FA9550-17-1-0151. The authors would like to express their sincere gratitude to Professor Y.M. Xie at RMIT for providing his BESO3D code in Python and for his important comments and suggestions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Science and TechnologyFederation University AustraliaMt HelenAustralia

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