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.
Ⓒ Springer International Publishing, AG 2018 V.K. Singh, D.Y. Gao, A. Fisher (eds). Emerging Trends in Applied Mathematics and High-Performance Computing
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Notes
- 1.
The linear inequality constraint A ρ ≤b in [36] is ignored in this paper.
- 2.
Due to this conceptual mistake, the general problem for topology optimization was originally formulated as a double-min optimization \(({\mathcal {P}}_{bl})\) in [18]. Although this model is equivalent to a knapsack problem for linear elastic structures under the condition f = K(ρ)u, it contradicts the popular theory in topology optimization.
- 3.
This algorithm was called the CDT algorithm in [18]. Since a new CDT algorithm without β perturbation has been developed, this algorithm based on the canonical penalty-duality method should be called CPD algorithm.
- 4.
According to Professor Y.M. Xie at RMIT, this BESO code was poorly implemented and has never been used for any of their further research simply because it was extremely slow compared to their other BESO codes. Therefore, the comparison for computing time between CPD and BESO provided in this section may not show the reality if the other commercial BESO codes are used.
- 5.
The so-called compliance in this section is actually a doubled strain energy, i.e., c = 2C(ρ, u) as used in [36].
- 6.
Indeed, since the first author was told that the strain energy is also called the compliance in topology optimization and (P c) is a correct model for topology optimization, the general problem \(({\mathcal {P}}_{bl})\) was originally formulated as a minimum total potential energy so that using \(\mathbf {f} = \mathbf {K}(\boldsymbol {\rho }) \bar {\mathbf {u}} \), \(\min \{ \Pi _h(\bar {\mathbf {u}}, \boldsymbol {\rho })|\;\boldsymbol {\rho } \in {\mathcal {Z}}_a\} = \min \{ - \frac {1}{2} \mathbf {c}(\mathbf {u}) \boldsymbol {\rho }^T | \;\;\boldsymbol {\rho } \in {\mathcal {Z}}_a \} \) is a knapsack problem [18].
- 7.
- 8.
- 9.
- 10.
This terminology is used mainly in the English literature. The function f(x) is correctly called the target function in Chinese and Japanese literature.
- 11.
The celebrated textbook Introduction to Applied Mathematics by Gil Strang is a required course for all engineering graduate students at MIT. Also, the well-known MIT online teaching program was started from this course.
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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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Gao, D., Ali, E.J. (2019). A Novel Canonical Duality Theory for Solving 3-D Topology Optimization Problems. In: Singh, V., Gao, D., Fischer, A. (eds) Advances in Mathematical Methods and High Performance Computing. Advances in Mechanics and Mathematics, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-030-02487-1_13
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