Structural and Multidisciplinary Optimization

, Volume 57, Issue 2, pp 665–688 | Cite as

Multiobjective and multi-physics topology optimization using an updated smart normal constraint bi-directional evolutionary structural optimization method

  • David J. Munk
  • Timoleon Kipouros
  • Gareth A. Vio
  • Geoffrey T. Parks
  • Grant P. Steven


To date the design of structures using topology optimization methods has mainly focused on single-objective problems. Since real-world design problems typically involve several different objectives, most of which counteract each other, it is desirable to present the designer with a set of Pareto optimal solutions that capture the trade-off between these objectives, known as a smart Pareto set. Thus far only the weighted sums and global criterion methods have been incorporated into topology optimization problems. Such methods are unable to produce evenly distributed smart Pareto sets. However, recently the smart normal constraint method has been shown to be capable of directly generating smart Pareto sets. Therefore, in the present work, an updated smart Normal Constraint Method is combined with a Bi-directional Evolutionary Structural Optimization (SNC-BESO) algorithm to produce smart Pareto sets for multiobjective topology optimization problems. Two examples are presented, showing that the Pareto solutions found by the SNC-BESO method make up a smart Pareto set. The first example, taken from the literature, shows the benefits of the SNC-BESO method. The second example is an industrial design problem for a micro fluidic mixer. Thus, the problem is multi-physics as well as multiobjective, highlighting the applicability of such methods to real-world problems. The results indicate that the method is capable of producing smart Pareto sets to industrial problems in an effective and efficient manner.


Multiobjective optimization Multi-physics optimization Normal constraint method BESO Pareto 



D.J. Munk thanks the Australian government for their financial support through the Endeavour Fellowship scheme.

The authors would like to thank Dr Tiziano Ghisu for producing the data displayed in Fig. 14.


  1. Abrahamson S, Lonnes S (1995) Uncertainty in calculating vorticity from 2D velocity fields using circulation and least-squares approach. Exp Fluids 20:10–20CrossRefGoogle Scholar
  2. Athan TW, Papalambros PY (1996) A note on weighted criteria methods for compromise solutions in multi-objective optimization. Eng Optim 27:155–176CrossRefGoogle Scholar
  3. Barber CB, Dobkin DP, Huhdanpaa HT (1996) The quickhull algorithm for convex hulls. ACM Trans Math Softw 22:469–483MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bendsoe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202CrossRefGoogle Scholar
  5. Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Method Appl Mech Eng 71:197–224MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bendsoe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654CrossRefzbMATHGoogle Scholar
  7. Bendsoe MP, Sigmund O (2004) Topology optimization: theory methods and applications, 2nd edn. Springer, Berlin, Heidelberg, New YorkCrossRefzbMATHGoogle Scholar
  8. Boyce NO, Mattson CA (2008) Reducing computational time of the normal constraints method by eliminating redundant optimization runs. In: 12th AIAA/ISSMO multidisciplinary analysis and optimization conference. AIAAGoogle Scholar
  9. Chen W, Wiecek MM, Zhang J (1999) Quality utility - a compromise programming approach to robust design. J Mech Des 121:179–187CrossRefGoogle Scholar
  10. Chen W, Sahai A, Messac A, Sundararaj G (2000) Exploration of the effectiveness of physical programming in robust design. J Mech Des 122:155–163CrossRefGoogle Scholar
  11. Chu D, Xie Y M, Hira A, Steven GP (1996) Evolutionary structural optimization for problems with stiffness constraints. Finite Elem Anal Des 21:239–251CrossRefzbMATHGoogle Scholar
  12. Coello CAC, Lamont GB, Veldhuizen DAV (2007) Evolutionary algorithms for solving multi-objective problems, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  13. Das I (1999) An improved technique for choosing parameters for Pareto surface generation using normal-boundary intersection. University of Buffalo, Center for Advanced Design, BuffaloGoogle Scholar
  14. Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Optim 14:63–69CrossRefGoogle Scholar
  15. Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8:631–657MathSciNetCrossRefzbMATHGoogle Scholar
  16. Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49:1–38MathSciNetCrossRefGoogle Scholar
  17. Deb K (2009) Multi-objective optimization using evolutionary algorithms, 1st edn. Wiley, New YorkzbMATHGoogle Scholar
  18. Du J, Olhoff N (2007) Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct Multidiscip Optim 34:91–110MathSciNetCrossRefzbMATHGoogle Scholar
  19. Haddock ND, Mattson CA, Knight DC (2008) Exploring direct generation of smart Pareto sets. In: 12th AIAA/ISSMO multidisciplinary analysis and optimization conference. AIAAGoogle Scholar
  20. Hancock BJ, Mattson CA (2013) The smart normal constraints method for directly generating a smart Pareto set. Struct Multidiscip Optim 48:763–775CrossRefGoogle Scholar
  21. Huang HZ, Gu YK, Du X (2006) An interactive fuzzy multi-objective optimization method for engineering design. Eng Appl Artif Intel 19:451–460CrossRefGoogle Scholar
  22. Huang X, Xie YM (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43:1039–1049CrossRefGoogle Scholar
  23. Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43:393–401MathSciNetCrossRefzbMATHGoogle Scholar
  24. Huang X, Xie YM (2010) Evolutionary topology optimization of continuum structures, 1st edn. Wiley, ChichesterCrossRefzbMATHGoogle Scholar
  25. Huang X, Zuo ZH, Xie YM (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88:357–364CrossRefGoogle Scholar
  26. Ismail-Yahaya A, Messac A (2002) Effective generation of the Pareto frontier using the normal constraint method. In: 40th AIAA aerospace sciences meeting & exhibit. AIAAGoogle Scholar
  27. Jaeggi DM, Parks GT, Kipouros T, Clarkson PJ (2008) The development of a multi-objective Tabu search algorithm for continuous optimisation problems. Eur J Oper Res 185:1192–1212MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kasumba H, Kunisch K (2012) Vortex control in channel flows using translation invariant cost functionals. Comput Optim Appl 52:691–717MathSciNetCrossRefzbMATHGoogle Scholar
  29. Kim WY, Grandhi RV, Haney M (2006) Multiobjective evolutionary structural optimization using combined static/dynamic control parameters. AIAA J 44:794–802CrossRefGoogle Scholar
  30. Koski J (1985) Defectiveness of weighting method in multicriterion optimization of structures. Commun Appl Numer Methods 1:333–337CrossRefzbMATHGoogle Scholar
  31. Kunakote T, Bureerat S (2011) Multi-objective topology optimization using evolutionary algorithms. Eng Optim 43:541–557MathSciNetCrossRefGoogle Scholar
  32. Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26:369– 395MathSciNetCrossRefzbMATHGoogle Scholar
  33. Marler RT, Arora JS (2010) The weighted sum method for multi-objective optimization: new insights. Struct Multidiscip Optim 41:853–862MathSciNetCrossRefzbMATHGoogle Scholar
  34. Martinez JSM, Blasco X, Salceo JV (2009a) A new perspective on multiobjective optimization by enhanced normalized normal constraint method. Struct Multidiscip Optim 36:537–546Google Scholar
  35. Martinez M, Sanchis J, Blasco X (2007) Global and well-distributed Pareto frontier by modifying normal normalized constraint methods for bicriterion problems. Struct Multidiscip Optim 34:197–209MathSciNetCrossRefzbMATHGoogle Scholar
  36. Martinez M, Garcia-Nieto S, Sanchis J, Blasco X (2009b) Genetic algorithms optimization for normalized normal constraint method under Pareto construction. Adv Eng Softw 40:260–267Google Scholar
  37. Martinez MP, Messac A, Rais-Rohani M (2001) Manufacturability-based optimization of aircraft structures using physical programming. AIAA J 39:517–525CrossRefGoogle Scholar
  38. Mattson CA, Mullur AA, Messac A (2004) Smart Pareto filter: Obtaining a minimal representation of multiobjective design space. Eng Optim 36:721–740MathSciNetCrossRefGoogle Scholar
  39. Messac A (2000) From dubious construction of objective functions to the application of physical programming. AIAA J 38:155–163CrossRefGoogle Scholar
  40. Messac A, Hattis P (1996) Physical programming design optimization for high speed civil transport (HSCT). J Aircraft 33:446–449CrossRefGoogle Scholar
  41. Messac A, Ismail-Yahaya A (2001) Required relationship between objective function and Pareto frontier orders: practical implications. AIAA J 11:2168–2174CrossRefGoogle Scholar
  42. Messac A, Mattson CA (2002) Generating well-distributed sets of Pareto points for engineering design using physical programming. Optim Eng 3:431–450CrossRefzbMATHGoogle Scholar
  43. Messac A, Mattson CA (2004) Normal constraints method with guarantee of even representation of complete Pareto frontier. AIAA J 42:2101–2111CrossRefGoogle Scholar
  44. Messac A, Sukam CP, Melachrinoudis E (2000a) Aggregate objective functions and Pareto frontiers: required relationships and practical implications. Optim Eng 1:171–188Google Scholar
  45. Messac A, Sundararaj GJ, Tappeta RV, Renaud JE (2000b) Ability of objective functions to generate points on nonconvex Pareto frontiers. AIAA J 38:1084–1091Google Scholar
  46. Messac A, Puemi-Sukam C, Melachrinoudis E (2001) Mathematical and pragmatic perspectives of physical programming. AIAA J 39:885–893CrossRefzbMATHGoogle Scholar
  47. Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized normal constraint method for generating the Pareto frontier. Struct Multidiscip Optim 25:86–98MathSciNetCrossRefzbMATHGoogle Scholar
  48. Messac A, Dessel SV, Mullur S, Maria AA (2004) Optimization of large scale rigidified inflatable structures for housing using physical programming. Struct Multidiscip Optim 26:139–151CrossRefGoogle Scholar
  49. Michell AGM (1904) The limits of economy of material in frame structures. Philos Mag 8:589–597CrossRefzbMATHGoogle Scholar
  50. Moghtaderi B, Shames I, Djenidi L (2006) Microfluidic characteristics of a multi-holed baffle plate micro-reactor. Int J Heat Fluid Fl 127:1069–1077CrossRefGoogle Scholar
  51. Motta RS, Afonso SMB, Lyra PRM (2012) A modified NBI and NC method for the solution of n-multiobjective optimization problems. Struct Multidiscip Optim 46:239–259MathSciNetCrossRefzbMATHGoogle Scholar
  52. Munk D, Vio G, Kipouros T, Parks G (2016a) Computational design for micro fluidic devices using a tightly coupled Lattice Boltzmann and level set-based optimization algorithm. In: Proceedings of the 11th ASMO UK/ISSMO conference on engineering design optimization. ASMO, UKGoogle Scholar
  53. Munk DJ, Vio GA, Steven GP (2015) Topology and shape optimization methods using evolutionary algorithms: a review. Struct Multidiscip Optim 52(3):613–631MathSciNetCrossRefGoogle Scholar
  54. Munk DJ, Kipouros T, Vio GA, Parks GT, Steven GP (2016b) Topology optimization of micro fluidic mixers considering fluid-structure interactions with a coupled Lattice Boltzmann method. J Comput Phys Under reviewGoogle Scholar
  55. Munk DJ, Vio GA, Steven GP (2017) A bi-directional evolutionary structural optimization algorithm with an added connectivity constraint. Finite Elem Anal Des 131:25–42CrossRefGoogle Scholar
  56. Pareto V (1964) Cour deconomie politique (the first edition in 1896) edn. Librarie Droz-Geneve, GenevaGoogle Scholar
  57. Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim 20:2–11CrossRefGoogle Scholar
  58. Prager W, Rozvany GIN (1977) Optimization of the structural geometry. In: Bednarek A, Cesari L (eds) Dynamical systems. Academic Press, pp 265–293Google Scholar
  59. Proos KA, Steven GP, Querin OM, Xie YM (2001a) Multicriterion evolutionary structural optimization using the weighting and the global criterion methods. AIAA J 39(10):2006–2012Google Scholar
  60. Proos KA, Steven GP, Querin OM, Xie YM (2001b) Stiffness and inertia multicriteria evolutionary structural optimization. Eng Comput 18:1031–1054Google Scholar
  61. Ray T, Tai K, Seow KC (2001) An evolutionary algorithm for multiobjective optimization. Eng Optim 33:399–424CrossRefGoogle Scholar
  62. Rozvany GIN (2009) A critical review of established methods of structural topology optimization. Struct Multidiscip Optim 37:217–237MathSciNetCrossRefzbMATHGoogle Scholar
  63. Rozvany GIN, Lewinski T (eds) (2013) Topology optimization in structural and continuum mechanics, 1st edn. Springer, DordrechtGoogle Scholar
  64. Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4:250–254CrossRefGoogle Scholar
  65. Ruzika S, Wiecek MM (2005) Approximation methods in multiobjective programming. J Optimiz Theory App 126:473–501MathSciNetCrossRefzbMATHGoogle Scholar
  66. Rynne B (2007) Linear functional analysis, 1st edn. Springer, New YorkGoogle Scholar
  67. Sigmund O (2001) Design of multiphysics actuators using topology optimization – Part I: One-material structures. Comput Method Appl Mech Eng 190:6577–6604CrossRefzbMATHGoogle Scholar
  68. Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43:589–596MathSciNetCrossRefzbMATHGoogle Scholar
  69. Sigmund O, Maute K (2013) Topology optimization approaches: a comparative review. Struct Multidiscip Optim 48:1031–1055MathSciNetCrossRefGoogle Scholar
  70. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75CrossRefGoogle Scholar
  71. Stadler W (1995) Caveats and boons of multicriteria optimization. Comput-Aided Civ Infrastruct Eng 10:291–299CrossRefGoogle Scholar
  72. Steven GP, Li Q, Xie YM (2000) Evolutionary topology and shape design for generating physical field problems. Comput Mech 26(2):129–139CrossRefzbMATHGoogle Scholar
  73. Tanaka M, Watanabe H, Furukawa Y, Tanino T (1995) GA-based decision support system for multicriteria optimization. In: 1995 IEEE international conference on systems, man, and cybernetics, vol 2. IEEE, pp 1556–1561Google Scholar
  74. Tsotskas C, Kipouros T, Savill M (2015) Fast multi-objective optimisation of a micro-fluidic device by using graphics accelerators. Proc Comput Sci 51:2237–2246CrossRefGoogle Scholar
  75. Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896CrossRefGoogle Scholar
  76. Yang XY, Xie YM, Steven GP, Querin OM (1999) Bidirectional evolutionary method for stiffness optimization. AIAA J 37:1483–1488CrossRefGoogle Scholar
  77. Zadeh LA (1963) Optimality and non-scalar-valued performance criteria. IEEE Trans Automat Contr 8:59–60CrossRefGoogle Scholar
  78. Zhou A, Qu BO, Li H, Zhoa SZ, Suganthan PN, Zhang Q (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol Comput 1:32–49CrossRefGoogle Scholar
  79. Zuo Z, Xie YM, Huang X (2010) An improved bi-directional evolutionary topology optimization method for frequencies. Int J Struct Stab Dynam 10:55–75MathSciNetCrossRefzbMATHGoogle Scholar
  80. Zuo Z, Xie YM, Huang X (2012) Evolutionary topology optimization of structures with multiple displacements and frequency constraints. Adv Struct Eng 15:385–398CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • David J. Munk
    • 1
  • Timoleon Kipouros
    • 2
  • Gareth A. Vio
    • 1
  • Geoffrey T. Parks
    • 2
  • Grant P. Steven
    • 1
  1. 1.The University of SydneySydneyAustralia
  2. 2.University of CambridgeCambridgeUK

Personalised recommendations