Modified element stacking method for multi-material topology optimization with anisotropic materials

  • Daozhong Li
  • Il Yong KimEmail author
Research Paper


This article presents a modified element stacking method for anisotropic multi-material topology optimization. This method can transform standard multi-material topology optimization formulations into a series of equivalent single-material topology optimization ones to overcome the various limitations inherent to conventional techniques. First, typical multi-material topology optimization methods utilize material interpolation schemes that restrict the study to the isotropic domain with a constant Poisson’s ratio, limiting practical applications and solution accuracy. Additionally, in attempts to further extend these classical multi-material topology optimization methods to anisotropic materials, preparing the necessary element information matrices becomes increasingly laborious or even impossible for complex models, preventing the application of sensitivity analysis for robust gradient-based optimization methods. To directly address these limitations, the element interpolation method replaces material interpolation schemes with an element interpolation framework, where each design cell property is determined using a weighted sum of various coincident single-material elements. This paper provides a thorough description of element interpolation and presents several numerical examples demonstrating successful implementation.


Topology optimization Multi-material Element interpolation Modified element stacking method SIMP SAMP 



Technical advice and direction were gratefully received from Balbir Sangha, Manish Pamwar, Derrick Chow, and Chandan Mozumder, at General Motors.

Funding information

This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) and General Motors of Canada.

Compliance with ethical Standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202. CrossRefGoogle Scholar
  2. 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
  3. Bendsøe M, Sigmund O (2003) Topology optimization: theory, methods and applications, 2nd edn. SpringerGoogle Scholar
  4. 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
  5. Cristello N, Kim IY (2007) Multidisciplinary design optimization of a zero-emission vehicle chassis considering crashworthiness and hydroformability. Proc Institut Mech Eng Part D: J Autom Eng 221(5):511–526. CrossRefGoogle Scholar
  6. 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
  7. Díaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10(1):40–45. CrossRefGoogle Scholar
  8. Eschenauer H, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–390. CrossRefGoogle Scholar
  9. 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., 1010.1724CrossRefzbMATHGoogle Scholar
  10. 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
  11. Hilchenbach CF, Ramm E (2015) Optimization of multiphase structures considering damage. Struct Multidiscip Optim 51(5):1083–1096. MathSciNetCrossRefGoogle Scholar
  12. 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
  13. 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
  14. Kim TS, Kim JE, Jeong JH, Kim YY (2004) Filtering technique to control member size in topology design optimization. KSME Int J 18(2):253–261CrossRefGoogle Scholar
  15. 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
  16. 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
  17. Li C, Kim IY (2015) Topology, size and shape optimization of an automotive cross car beam. Proc Institut Mech Eng Part D: J Autom Eng 229(10):1361–1378. CrossRefGoogle Scholar
  18. Li C, Kim IY (2017) Multi-material topology optimization for automotive design problems. Proc Institut Mech Eng Part D: J Autom Eng, 095440701773790.
  19. 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
  20. NAIR (2011) Hexcel 8552S AS4 plain weave fabric prepeg 193 gsm & 38% RC qualification material property data reportGoogle Scholar
  21. Sigmund O (2000) A new class of extremal composites. J Mech Phys Solids 48(2):397–428. MathSciNetCrossRefzbMATHGoogle Scholar
  22. Sigmund O (2001) Design of multiphysics actuators using topology optimization - Part II: two material structures. Comput Methods Appl Mech Eng 190(49–50):6605–6627CrossRefzbMATHGoogle Scholar
  23. Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055. MathSciNetCrossRefGoogle Scholar
  24. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75. CrossRefGoogle Scholar
  25. 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
  26. Stegmann J, Lund E (2005) Discrete material optimization of general composite shell structures. Int J Numer Methods Eng 62(14):2009–2027. CrossRefzbMATHGoogle Scholar
  27. 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
  28. Svanberg K (2002) A class of globally convergent pptimization methods based on conservative convex separable approximations. SIAM J Optim 12(2):555–573. MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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
  30. 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
  31. 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
  32. 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
  33. 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
  34. 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. MathSciNetCrossRefzbMATHGoogle Scholar
  35. 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
  36. Yoon G, Park YK, Kim Y (2007) Element stacking method for topology optimization with material-dependent boundary and loading conditions. J Mech Mater Struct 2(5):883–895. CrossRefGoogle Scholar
  37. Yun KS, Youn SK (2017) Multi-material topology optimization of viscoelastically damped structures under time-dependent loading. Finite Elem Anal Des 123(August 2016):9–18. MathSciNetCrossRefGoogle Scholar
  38. 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
  39. 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
  40. Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23(4):595–622. MathSciNetCrossRefzbMATHGoogle Scholar
  41. 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
  42. Zuo W, Saitou K (2017) Multi-material topology optimization using ordered SIMP interpolation. Struct Multidiscip Optim 55(2):477–491. MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations