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

- 1 Citations
- 767 Downloads

## 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.

## References

- 1.Ali, E.J. and Gao, D.Y. (2017). Improved canonical dual finite element method and algorithm for post buckling analysis of nonlinear gao beam,
*Canonical Duality-Triality: Unified Theory and Methodology for Multidisciplinary Study*, D.Y. Gao, N. Ruan and V. Latorre (Eds). Springer, New York, pp. 277–290.Google Scholar - 2.Bends
*ϕ*e, M. P. (1989.). Optimal shape design as a material distribution problem.*Structural Optimization, 1*, 193–202.CrossRefGoogle Scholar - 3.Bends
*ϕ*e, M. P. and Kikuchi, N. (1988). Generating optimal topologies in structural design using a homogenization method.*Computer Methods in Applied Mechanics and Engineering, 71(2)*, 197–224.MathSciNetCrossRefGoogle Scholar - 4.Bends
*ϕ*e, M. P. and Sigmund, O. (2004).*Topological optimization: theory, methods and applications.*Berlin: Springer-Verlag, 370.Google Scholar - 5.Ciarlet, P.G. (1988).
*Mathematical Elasticity*, Volume 1: Three Dimensional Elasticity. North-Holland, 449pp.Google Scholar - 6.Díaz, A. and Sigmund, O. (1995). Checkerboard patterns in layout optimization.
*Structural Optimization, 10(1)*, 40–45.CrossRefGoogle Scholar - 7.Gao, D.Y. (1986).
*On Complementary-Dual Principles in Elastoplastic Systems and Pan-Penalty Finite Element Method*, PhD Thesis, Tsinghua University.Google Scholar - 8.Gao, D.Y. (1988). Panpenalty finite element programming for limit analysis,
*Computers & Structures, 28*, 749–755.CrossRefGoogle Scholar - 9.Gao, D.Y. (1996). Complementary finite-element method for finite deformation nonsmooth mechanics,
*Journal of Engineering Mathematics, 30(3)*, 339–353.MathSciNetCrossRefGoogle Scholar - 10.Gao, D.Y. (1997). Dual extremum principles in finite deformation theory with applications to post-buckling analysis of extended nonlinear beam theory,
*Appl. Mech. Rev., 50(11),*S64-S71.CrossRefGoogle Scholar - 11.Gao, D.Y. (1999). Pure complementary energy principle and triality theory in finite elasticity,
*Mech. Res. Comm., 26(1)*, 31–37.Google Scholar - 12.Gao, D.Y. (2000).
*Duality Principles in Nonconvex Systems: Theory, Methods and Applications*, Springer, London/New York/Boston, xviii + 454pp.CrossRefGoogle Scholar - 13.Gao, D.Y. (2001). Complementarity, polarity and triality in nonsmooth, nonconvex and nonconservative Hamilton systems,
*Philosophical Transactions of the Royal Society: Mathematical, Physical and Engineering Sciences, 359*, 2347–2367.CrossRefGoogle Scholar - 14.Gao, D.Y. (2007). Solutions and optimality criteria to box constrained nonconvex minimization problems.
*Journal of Industrial & Management Optimization, 3(2)*293–304.MathSciNetCrossRefGoogle Scholar - 15.Gao, D.Y. (2009). Canonical duality theory: unified understanding and generalized solutions for global optimization.
*Comput. & Chem. Eng. 33,*1964–1972.CrossRefGoogle Scholar - 16.Gao, D.Y. (2016). On unified modeling, theory, and method for solving multi-scale global optimization problems, in
*Numerical Computations: Theory And Algorithms,*(Editors) Y. D. Sergeyev, D. E. Kvasov and M. S. Mukhametzhanov, AIP Conference Proceedings 1776, 020005.Google Scholar - 17.Gao, D.Y. (2016). On unified modeling, canonical duality-triality theory, challenges and breakthrough in optimization, https://arxiv.org/abs/1605.05534 .Google Scholar
- 18.Gao, D.Y. (2017). Canonical Duality Theory for Topology Optimization,
*Canonical Duality-Triality: Unified Theory and Methodology for Multidisciplinary Study*, D.Y. Gao, N. Ruan and V. Latorre (Eds). Springer, New York, pp.263–276.Google Scholar - 19.Gao, D.Y. (2017). Analytical solution to large deformation problems governed by generalized neo-Hookean model, in
*Canonical Duality Theory: Unified Methodology for Multidisciplinary Studies*, DY Gao, V. Latorre and N. Ruan (Eds). Springer, pp.49–68.Google Scholar - 20.Gao, D.Y. (2017). On Topology Optimization and Canonical Duality Solution. Plenary Lecture at
*Int. Conf. Mathematics, Trends and Development,*28–30 Dec. 2017, Cairo, Egypt, and Opening Address at*Int. Conf. on Modern Mathematical Methods and High Performance Computing in Science and Technology*, 4–6, January, 2018, New Delhi, India. Online first at https://arxiv.org/abs/1712.02919, to appear in*Computer Methods in Applied Mechanics and Engineering*. - 21.Gao, D.Y. (2018). Canonical duality-triality: Unified understanding modeling, problems, and NP-hardness in multi-scale optimization. In
*Emerging Trends in Applied Mathematics and High-Performance Computing*, V.K. Singh, D.Y. Gao and A. Fisher (eds), Springer, New York.Google Scholar - 22.Gao, DY and Hajilarov, E. (2016). On analytic solutions to 3-d finite deformation problems governed by St Venant-Kirchhoff material. in
*Canonical Duality Theory: Unified Methodology for Multidisciplinary Studies*, DY Gao, V. Latorre and N. Ruan (Eds). Springer, 69–88.Google Scholar - 23.Gao, D.Y., V. Latorre, and N. Ruan (2017).
*Canonical Duality Theory: Unified Methodology for Multidisciplinary Study*, Springer, New York, 377pp.CrossRefGoogle Scholar - 24.Gao, D.Y., Ogden, R.W. (2008). Multi-solutions to non-convex variational problems with implications for phase transitions and numerical computation.
*Q. J. Mech. Appl. Math.*61, 497–522.CrossRefGoogle Scholar - 25.Gao, D.Y. and Ruan, N. (2010). Solutions to quadratic minimization problems with box and integer constraints.
*J. Glob. Optim., 47*, 463–484.MathSciNetCrossRefGoogle Scholar - 26.Gao, D.Y. and Ruan, N. (2018). On canonical penalty-duality method for solving nonlinear constrained problems and a 66-line Matlable code for topology optimization. To appear.Google Scholar
- 27.Gao, D.Y. and Sherali, H.D. (2009). Canonical duality theory: Connection between nonconvex mechanics and global optimization, in
*Advances in Appl. Mathematics and Global Optimization*, 257–326, Springer.Google Scholar - 28.Gao, D.Y. and Strang, G.(1989). Geometric nonlinearity: Potential energy, complementary energy, and the gap function.
*Quart. Appl. Math., 47(3)*, 487–504.MathSciNetCrossRefGoogle Scholar - 29.Gao, D.Y., Yu, H.F. (2008). Multi-scale modelling and canonical dual finite element method in phase transitions of solids.
*Int. J. Solids Struct. 45,*3660–3673.CrossRefGoogle Scholar - 30.Huang, X. and Xie, Y.M. (2007). Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method.
*Finite Elements in Analysis and Design, 43(14)*1039–1049.CrossRefGoogle Scholar - 31.Huang, R. and Huang, X. (2011). Matlab implementation of 3D topology optimization using BESO.
*Incorporating Sustainable Practice in Mechanics of Structures and Materials*, 813–818.Google Scholar - 32.Isac, G.
*Complementarity Problems*. Springer, 1992.Google Scholar - 33.Karp, R. (1972). Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.)
*Complexity of Computer Computations*, Plenum Press, New York, 85–103.CrossRefGoogle Scholar - 34.Latorre, V. and Gao, D.Y. (2016). Canonical duality for solving general nonconvex constrained problems.
*Optimization Letters, 10(8)*, 1763–1779.MathSciNetCrossRefGoogle Scholar - 35.Li, S.F. and Gupta, A. (2006). On dual configuration forces,
*J. of Elasticity, 84*, 13–31.MathSciNetCrossRefGoogle Scholar - 36.Liu, K. and Tovar, A. (2014). An efficient 3D topology optimization code written in Matlab.
*Struct Multidisc Optim, 50*, 1175–1196.MathSciNetCrossRefGoogle Scholar - 37.Marsden, J.E. and Hughes, T.J.R.(1983).
*Mathematical Foundations of Elasticity,*Prentice-Hall.Google Scholar - 38.Osher, S. and Sethian, JA. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations.
*Journal of Computational Physics, 79(1)*, 12–49.MathSciNetCrossRefGoogle Scholar - 39.Querin, O. M., Steven, G.P. and Xie, Y.M. (2000). Evolutionary Structural optimization using an additive algorithm.
*Finite Element in Analysis and Design, 34(3–4)*, 291–308.CrossRefGoogle Scholar - 40.Querin, O.M., Young V., Steven, G.P. and Xie, Y.M. (2000). Computational Efficiency and validation of bi-directional evolutionary structural optimization.
*Comput Methods Applied Mechanical Engineering, 189(2)*, 559–573.CrossRefGoogle Scholar - 41.Rozvany, G.I.N. (2009). A critical review of established methods of structural topology optimization.
*Structural and Multidisciplinary Optimization, 37(3)*, 217–237.MathSciNetCrossRefGoogle Scholar - 42.Rozvany, G.I.N., Zhou, M. and Birker, T. (1992). Generalized shape optimization without homogenization.
*Structural Optimization, 4(3)*, 250–252.CrossRefGoogle Scholar - 43.Sethian, J.A. (1999). Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer version and material science.
*Cambridge, UK: Cambridge University Press*, 12–49.Google Scholar - 44.Sigmund, O. and Petersson, J. (1998). Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima.
*Structural Optimization, 16(1)*, 68–75.CrossRefGoogle Scholar - 45.Sigmund, O. and Maute, K. (2013). Topology optimization approaches: a comparative review.
*Structural and Multidisciplinary Optimization, 48(6)*, 1031–1055.MathSciNetCrossRefGoogle Scholar - 46.Sigmund, O. (2001). A 99 line topology optimization code written in matlab.
*Struct Multidiscip Optim, 21(2)*, 120–127.MathSciNetCrossRefGoogle Scholar - 47.Stolpe, M. and Bendsøe, M.P. (2011). Global optima for the Zhou–Rozvany problem,
*Struct Multidisc Optim, 43*, 151–164.MathSciNetCrossRefGoogle Scholar - 48.Strang, G. (1986).
*Introduction to Applied Mathematics*, Wellesley-Cambridge Press.Google Scholar - 49.Truesdell, C.A. and Noll, W. (1992).
*The Non-Linear Field Theories of Mechanics,*Second Edition, 591 pages. Springer-Verlag, Berlin-Heidelberg-New York.CrossRefGoogle Scholar - 50.Xie, Y.M. and Steven, G.P. (1993). A simple evolutionary procedure for structural optimization.
*Comput Struct, 49(5)*, 885–896.CrossRefGoogle Scholar - 51.Xie, Y.M. and Steven, G.P. (1997). Evolutionary structural optimization.
*London: Springer*.CrossRefGoogle Scholar - 52.Zuo, Z.H. and Xie, Y.M. (2015). A simple and compact Python code for complex 3D topology optimization.
*Advances in Engineering Software, 85*, 1–11.CrossRefGoogle Scholar - 53.Zhou, M. and Rozvany, G.I.N. (1991). The COC algorithm, Part II: Topological geometrical and generalized shape optimization.
*Computer Methods in Applied Mechanics and Engineering, 89(1)*, 309–336.CrossRefGoogle Scholar