Structural and Multidisciplinary Optimization

, Volume 58, Issue 3, pp 1081–1094 | Cite as

Multi-material topology optimization for practical lightweight design

  • Daozhong Li
  • Il Yong Kim


Topology optimization is one of the most effective tools for conducting lightweight design and has been implemented across multiple industries to enhance product development. The typical topology optimization problem statement is to minimize system compliance while constraining the design space to an assumed volume fraction. The traditional single-material compliance problem has been extended to include multiple materials, which allows increased design freedom for potentially better solutions. However, compliance minimization has the limitations for practical lightweight design because compliance lacks useful physical meanings and has never been a design criterion in industry. Additionally, the traditional compliance minimization problem statement requires volume fraction constraints to be selected a priori; however, designers do not know the optimized balance among materials. In this paper, a more practical method of multi-material topology optimization is presented to overcome the limitations. This method seeks the optimized balance among materials by minimizing the total weight while satisfying performance constraints. This paper also compares the weight minimization approach to compliance minimization. Several numerical examples prove the success of weight minimization and demonstrate its benefit over compliance minimization.


Topology optimization Multi-material Lightweight Weight minimization Stiffness constraints SIMP 



one design variable (i.e. nominal density) relevant to the j-th material in any element;

\({\rho }_{e}^{j}\):

one design variable (i.e. nominal density) relevant to the j-th material in the e-th element;


all design variables (i.e. a vector of nominal densities) relevant to the j-th material;

\((\rho ^{j})^{p}, ({\rho }_{e}^{j})^{p}\):

powers of ρj and \({\rho }_{e}^{j}\) with the component p;


original elastic modulus of the j-th material;


original weight of any element filled with the j-th material;

\({W_{e}^{j}}, W_{e}^{(j)}\):

original weight of the e-th element filled with the j-th material;

E(1,⋯ ,j),W(1,⋯ ,j):

interpolated elastic modulus and weight of the materials from the first to the j-th materials for any element.



This research was funded by Automotive Partnership Canada and General Motors of Canada. Technical advice and direction were gratefully received from Joe Moore, Balbir Sangha, Manish Pamwar, Derrick Chow, and Chandan Mozumder, at General Motors.


  1. Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393. MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Structural Optimization 1 (4):193–202. CrossRefGoogle Scholar
  3. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71 (2):197–224. MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bendsøe M, Sigmund O (2003) Topology optimization: theory, methods and applications, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  5. Bruyneel M (2011) SFP—a new parameterization based on shape functions for optimal material selection: application to conventional composite plies. Struct Multidiscip Optim 43(1):17–27. CrossRefGoogle Scholar
  6. Cai K, Cao J, Shi J, Liu L, Qin QH (2016) Optimal layout of multiple bi-modulus materials. Struct Multidiscip Optim 53(4):801–811. MathSciNetCrossRefGoogle Scholar
  7. Cristello N, Kim IY (2007) Multidisciplinary design optimization of a zero-emission vehicle chassis considering crashworthiness and hydroformability. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 221(5):511–526. Google Scholar
  8. Deaton J, Grandhi R (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38. MathSciNetCrossRefGoogle Scholar
  9. Eschenauer H, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–390. CrossRefGoogle Scholar
  10. Gao T, Zhang W, Duysinx P (2012) A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate. Int J Numer Methods Eng 91(1):98–114. arXiv:1010.1724 CrossRefzbMATHGoogle Scholar
  11. Guo X, Zhang W, Zhong W (2014) Stress-related topology optimization of continuum structures involving multi-phase materials. Comput Methods Appl Mech Eng 268:632–655. MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hilchenbach CF, Ramm E (2015) Optimization of multiphase structures considering damage. Struct Multidiscip Optim 51(5):1083–1096. MathSciNetCrossRefGoogle Scholar
  13. Hvejsel CF, Lund E (2011) Material interpolation schemes for unified topology and multi-material optimization. Struct Multidiscip Optim 43(6):811–825. CrossRefzbMATHGoogle Scholar
  14. Jung D, Gea HC (2006) Design of an energy-absorbing structure using topology optimization with a multimaterial model. Struct Multidiscip Optim 32(3):251–257. CrossRefGoogle Scholar
  15. Kim IY, Kwak BM (2002) Design space optimization using a numerical design continuation method. Int J Numer Methods Eng 53(8):1979–2002. CrossRefzbMATHGoogle Scholar
  16. Kim SY, Kim IY, Mechefske CK (2012) A new efficient convergence criterion for reducing computational expense in topology optimization: reducible design variable method. Int J Numer Methods Eng 90(6):752–783. CrossRefzbMATHGoogle Scholar
  17. Kim SY, Mechefske CK, Kim IY (2013) Optimal damping layout in a shell structure using topology optimization. J Sound Vib 332(12):2873–2883. CrossRefGoogle Scholar
  18. Li C, Kim IY (2015) Topology, size and shape optimization of an automotive cross car beam. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 229(10):1361–1378. Google Scholar
  19. Li C, Kim IY (2017) Multi-material topology optimization for automotive design problems. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering. 095440701773790.
  20. Li C, Kim IY, Jeswiet J (2015) Conceptual and detailed design of an automotive engine cradle by using topology, shape, and size optimization. Struct Multidiscip Optim 51(2):547–564. CrossRefGoogle Scholar
  21. Luo J, Luo Z, Chen L, Tong L, Wang MY (2008) A semi-implicit level set method for structural shape and topology optimization. J Comput Phys 227(11):5561–5581. MathSciNetCrossRefzbMATHGoogle Scholar
  22. Mirzendehdel AM, Suresh K (2015) A Pareto-Optimal approach to multimaterial topology optimization. J Mech Des 137(10): 101,701. CrossRefGoogle Scholar
  23. Ramani A (2010) A pseudo-sensitivity based discrete-variable approach to structural topology optimization with multiple materials. Struct Multidiscip Optim 41(6):913–934. CrossRefGoogle Scholar
  24. Ramani A (2011) Multi-material topology optimization with strength constraints. Struct Multidiscip Optim 43(5):597–615. CrossRefGoogle Scholar
  25. Sethian J, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489–528. arXiv:1011.1669v3 MathSciNetCrossRefzbMATHGoogle Scholar
  26. Sigmund O (2000) A new class of extremal composites. J Mech Phys Solids 48(2):397–428. MathSciNetCrossRefzbMATHGoogle Scholar
  27. Sigmund O (2001) Design of multiphysics actuators using topology optimization - Part II: Two-material structures. Comput Methods Appl Mech Eng 190(49-50):6605–6627. CrossRefzbMATHGoogle Scholar
  28. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055. MathSciNetCrossRefGoogle Scholar
  29. Sigmund O, 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
  30. Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067. MathSciNetCrossRefGoogle Scholar
  31. Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62(14):2009–2027. CrossRefzbMATHGoogle Scholar
  32. Stolpe M, Stegmann J (2008) A Newton method for solving continuous multiple material minimum compliance problems. Struct Multidiscip Optim 35(2):93–106. MathSciNetCrossRefzbMATHGoogle Scholar
  33. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373. MathSciNetCrossRefzbMATHGoogle Scholar
  34. Tavakoli R, Mohseni SM (2014) Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Struct Multidiscip Optim 49(4):621–642. MathSciNetCrossRefGoogle Scholar
  35. van Dijk NP, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48(3):437–472. MathSciNetCrossRefGoogle Scholar
  36. Vermaak N, Michailidis G, Parry G, Estevez R, Allaire G, Bréchet Y (2014) Material interface effects on the topology optimization of multi-phase structures using a level set method. Struct Multidiscip Optim 50 (4):623–644. MathSciNetCrossRefGoogle Scholar
  37. Wang MY, Wang X (2004) “Color” level sets: a multi-phase method for structural topology optimization with multiple materials. Comput Methods Appl Mech Eng 193(6–8):469–496. MathSciNetCrossRefzbMATHGoogle Scholar
  38. Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246. MathSciNetCrossRefzbMATHGoogle Scholar
  39. Wang X, Mei Y, Wang MY (2004) Level-set method for design of multi-phase elastic and thermoelastic materials. Int J Mech Mater Des 1(3):213–239. CrossRefGoogle Scholar
  40. Wang Y, Luo Z, Kang Z, Zhang N (2015) A multi-material level set-based topology and shape optimization method. Comput Methods Appl Mech Eng 283:1570–1586. MathSciNetCrossRefGoogle Scholar
  41. Yin L, Ananthasuresh G (2001) Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct Multidiscip Optim 23(1):49–62. CrossRefGoogle Scholar
  42. Zhou M, Rozvany G (1991) The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336. CrossRefGoogle Scholar
  43. Zhou S, Wang MY (2006) Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct Multidiscip Optim 33(2):89–111. MathSciNetCrossRefzbMATHGoogle Scholar
  44. Zhuang C, Xiong Z, Ding H (2010) Topology optimization of multi-material for the heat conduction problem based on the level set method. Eng Optim 42(9):811–831. MathSciNetCrossRefGoogle Scholar
  45. Zuo W, Saitou K (2016) Multi-material topology optimization using ordered SIMP interpolation. Struct Multidiscip Optim.

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Materials EngineeringQueen’s UniversityKingstonCanada

Personalised recommendations