Recent Advances on Topology Optimization of Multiscale Nonlinear Structures

Original Paper

Abstract

Research on topology optimization mainly deals with the design of monoscale structures, which are usually made of homogeneous materials. Recent advances of multiscale structural modeling enables the consideration of microscale material heterogeneities and constituent nonlinearities when assessing the macroscale structural performance. However, due to the modeling complexity and the expensive computing requirement of multiscale modeling, there has been very limited research on topology optimization of multiscale nonlinear structures. This paper reviews firstly recent advances made by the authors on topology optimization of multiscale nonlinear structures, in particular techniques regarding to nonlinear topology optimization and computational homogenization (also known as FE2) are summarized. Then the conventional concurrent material and structure topology optimization design approaches are reviewed and compared with a recently proposed FE2-based design approach, which treats the microscale topology optimization process integrally as a generalized nonlinear constitutive behavior. In addition, discussions on the use of model reduction techniques is provided in regard to the prohibitive computational cost.

Keywords

Topology optimization Multiscale analysis Microstructure Homogenization Model reduction 

Notes

Acknowledgments

This work was carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program “Investments for the future” managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).

References

  1. 1.
    Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Andreassen E, Jensen J (2014) Topology optimization of periodic microstructures for enhanced dynamic properties of viscoelastic composite materials. Struct Multidiscip Optim 49(5):695–705MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andreassen E, Lazarov B, Sigmund O (2014) Design of manufacturable 3d extremal elastic microstructure. Mech Mater 69:1–10CrossRefGoogle Scholar
  4. 4.
    Bendsøe M, Guedes J, Plaxton S, Taylor J (1996) Optimization of structure and material properties for solids composed of softening material. Int J Solids Struct 33(12):1799–1813MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optimiz 1(4):193–202CrossRefGoogle Scholar
  6. 6.
    Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654MATHCrossRefGoogle Scholar
  8. 8.
    Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, BerlinMATHGoogle Scholar
  9. 9.
    Bendsøe MP, Diaz AR, Lipton R, Taylor JE (1995) Optimal design of material properties and material distribution for multiple loading conditions. Int J Numer Meth Eng 38(7):1149–1170MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bruggi M, Duysinx P (2012) Topology optimization for minimum weight with compliance and stress constraints. Struct Multidiscip Optimiz 46(3):369–384MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bruns T, Tortorelli D (2003) An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. Int J Numer Meth Eng 57(10):1413–1430MATHCrossRefGoogle Scholar
  12. 12.
    Buhl T, Pedersen C, Sigmund O (2000) Stiffness design of geometrically nonlinear structures using topology optimization. Struct Multidiscip Optimiz 19(2):93–104CrossRefGoogle Scholar
  13. 13.
    Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Cadman J, Zhou S, Chen Y, Li Q (2013) On design of multi-functional microstructural materials. J Mater Sci 48(1):51–66CrossRefGoogle Scholar
  15. 15.
    Cai S, Zhang W (2015) Stress constrained topology optimization with free-form design domains. Comput Methods Appl Mech Eng 289:267–290MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cai S, Zhang W, Zhu J, Gao T (2014) Stress constrained shape and topology optimization with fixed mesh: a b-spline finite cell method combined with level set function. Comput Methods Appl Mech Eng 278:361–387MathSciNetCrossRefGoogle Scholar
  17. 17.
    Challis VJ, Roberts AP, Wilkins AH (2008) Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. Int J Solids Struct 45(14–15):4130–4146MATHCrossRefGoogle Scholar
  18. 18.
    Challis VJ, Guest JK, Grotowski JF, Roberts AP (2012) Computationally generated cross-property bounds for stiffness and fluid permeability using topology optimization. Int J Solids Struct 49(23–24):3397–3408CrossRefGoogle Scholar
  19. 19.
    Chen W, Liu S (2014) Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus. Struct Multidiscip Optimiz 50(2):287–296MathSciNetCrossRefGoogle Scholar
  20. 20.
    Clément A, Soize C, Yvonnet J (2012) Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis. Int J Numer Meth Eng 91(8):799–824CrossRefGoogle Scholar
  21. 21.
    Clément A, Soize C, Yvonnet J (2013) Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials. Comput Methods Appl Mech Eng 254:61–82MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Coelho PG, Fernandes PR, Guedes JM, Rodrigues HC (2008) A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidiscip Optimiz 35(2):107–115CrossRefGoogle Scholar
  23. 23.
    Coenen EWC, Kouznetsova VG, Geers MGD (2012) Multi-scale continuous–discontinuous framework for computational- homogenization-localization. J Mech Phys Solids 60(8):1486–1507MathSciNetCrossRefGoogle Scholar
  24. 24.
    Cremonesi M, Néron D, Guidault PA, Ladevèze P (2013) A PGD-based homogenization technique for the resolution of nonlinear multiscale problems. Comput Methods Appl Mech Eng 267:275–292Google Scholar
  25. 25.
    Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optimiz 49(1):1–38MathSciNetCrossRefGoogle Scholar
  26. 26.
    Deng J, Yan J, Cheng G (2013) Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material. Struct Multidiscip Optimiz 47(4):583–597MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Duva J, Hutchinson J (1984) Constitutive potentials for dilutely voided nonlinear materials. Mech Mater 3(1):41–54CrossRefGoogle Scholar
  28. 28.
    Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Meth Eng 43(8):1453–1478MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    El Halabi F, González D, Chico A, Doblaré M (2013) Fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition. Comput Methods Appl Mech Eng 257:183–202MATHCrossRefGoogle Scholar
  30. 30.
    Feyel F, Chaboche J (2000) \(\text{FE }^{2}\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials. Comput Methods Appl Mech Eng 183(3–4):309–330MATHCrossRefGoogle Scholar
  31. 31.
    Filomeno Coelho R, Breitkopf P, Knopf-Lenoir C (2008) Model reduction for multidisciplinary optimization—application to a 2D wing. Struct Multidiscip Optimiz 37(1):29–48MATHCrossRefGoogle Scholar
  32. 32.
    Filomeno Coelho R, Breitkopf P, Knopf-Lenoir C, Villon P (2009) Bi-level model reduction for coupled problems. Struct Multidiscip Optimiz 39(4):401–418MATHCrossRefGoogle Scholar
  33. 33.
    Forrester A, Keane A (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1–3):50–79CrossRefGoogle Scholar
  34. 34.
    Fritzen F, Böhlke T (2011) Nonuniform transformation field analysis of materials with morphological anisotropy. Compos Sci Technol 71:433–442CrossRefGoogle Scholar
  35. 35.
    Fritzen F, Böhlke T (2013) Reduced basis homogenization of viscoelastic composites. Compos Sci Technol 76:84–91CrossRefGoogle Scholar
  36. 36.
    Fritzen F, Leuschner M (2013) Reduced basis hybrid computational homogenization based on a mixed incremental formulation. Comput Methods Appl Mech Eng 260:143–154MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Fritzen F, Hodapp M, Leuschner M (2014) Gpu accelerated computational homogenization based on a variational approach in a reduced basis framework. Comput Methods Appl Mech Eng 278:186–217MathSciNetCrossRefGoogle Scholar
  38. 38.
    Fujii D, Chen BC, Kikuchi N (2001) Composite material design of two-dimensional structures using the homogenization design method. Int J Numer Meth Eng 50(9):2031–2051MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Ganapathysubramanian B, Zabaras N (2007) Modeling diffusion in random heterogeneous media: data-driven models, stochastic collocation and the variational multiscale method. J Comput Phys 226(1):326–353MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    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 Meth Eng 91(1):98–114MATHCrossRefGoogle Scholar
  41. 41.
    Gao T, Zhang WH, Duysinx P (2013) Simultaneous design of structural layout and discrete fiber orientation using bi-value coding parameterization and volume constraint. Struct Multidiscip Optimiz 48(6):1075–1088MathSciNetCrossRefGoogle Scholar
  42. 42.
    Gea H, Luo J (2001) Topology optimization of structures with geometrical nonlinearities. Comput Struct 79(20–21):1977–1985CrossRefGoogle Scholar
  43. 43.
    Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175–2182MATHCrossRefGoogle Scholar
  44. 44.
    Ghosh S, Lee K, Raghavan P (2001) A multi-level computational model for multi-scale damage analysis in composite and porous materials. Int J Solids Struct 38(14):2335–2385MATHCrossRefGoogle Scholar
  45. 45.
    Gibiansky L, Sigmund O (2000) Multiphase composites with extremal bulk modulus. J Mech Phys Solids 48(3):461–498MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Gu X, Zhu J, Zhang W (2012) The lattice structure configuration design for stereolithography investment casting pattern using topology optimization. Rapid Prototyping J 18(5):353–361CrossRefGoogle Scholar
  47. 47.
    Guedes J, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 83(2):143–198MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Guessasma S, Babin P, Della Valle G, Dendieve R (2008) Relating cellular structure of open solid food foams to their young’s modulus: finite element calculation. Int J Solids Struct 45(10):2881–2896MATHCrossRefGoogle Scholar
  49. 49.
    Guest JK, Prévost JH (2006) Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int J Solids Struct 43(22–23):7028–7047MATHCrossRefGoogle Scholar
  50. 50.
    Guest JK, Prévost JH (2007) Design of maximum permeability material structures. Comput Methods Appl Mech Eng 196(4–6):1006–1017MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Guo X, Zhang W, Wang M, Wei P (2011) Stress-related topology optimization via level set approach. Comput Methods Appl Mech Eng 200(47–48):3439–3452MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Guo X, Zhao X, Zhang W, Yan J, Sun G (2015) Multi-scale robust design and optimization considering load uncertainties. Comput Methods Appl Mech Eng 283:994–1009MathSciNetCrossRefGoogle Scholar
  53. 53.
    Hashin Z (1983) Analysis of composite materials—a survey. J Appl Mech Trans ASME 50(3):481–505MATHCrossRefGoogle Scholar
  54. 54.
    Hassani B, Hinton E (1998a) A review of homogenization and topology optimization. I. Homogenization theory for media with periodic structure. Comput Struct 69(6):707–717MATHCrossRefGoogle Scholar
  55. 55.
    Hassani B, Hinton E (1998b) A review of homogenization and topology opimization. II. Analytical and numerical solution of homogenization equations. Comput Struct 69(6):719–738CrossRefGoogle Scholar
  56. 56.
    Hernandez J, Oliver J, Huespe A, Caicedo M, Cante J (2014) High-performance model reduction techniques in computational multiscale homogenization. Comput Methods Appl Mech Eng 276:149–189MathSciNetCrossRefGoogle Scholar
  57. 57.
    Huang X, Xie Y (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43(14):1039–1049CrossRefGoogle Scholar
  58. 58.
    Huang X, Xie Y (2008) Topology optimization of nonlinear structures under displacement loading. Eng Struct 30(7):2057–2068CrossRefGoogle Scholar
  59. 59.
    Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43(3):393–401MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Huang X, Xie YM (2010) Topology optimization of continuum structures: methods and applications. Wiley, ChichesterMATHCrossRefGoogle Scholar
  61. 61.
    Huang X, Xie Y, Lu G (2007) Topology optimization of energy-absorbing structures. Int J Crashworthiness 12(6):663–675CrossRefGoogle Scholar
  62. 62.
    Huang X, Radman A, Xie YM (2011) Topological design of microstructures of cellular materials for maximum bulk or shear modulus. Comput Mater Sci 50(6):1861–1870CrossRefGoogle Scholar
  63. 63.
    Huang X, Xie YM, Jia B, Li Q, Zhou SW (2012) Evolutionary topology optimization of periodic composites for extremal magnetic permeability and electrical permittivity. Struct Multidiscipl Optimiz 46(3):385–398MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Huang X, Zhou SW, Xie YM, Li Q (2013) Topology optimization of microstructures of cellular materials and composites for macrostructures. Comput Mater Sci 67:397–407CrossRefGoogle Scholar
  65. 65.
    Huang X, Zhou S, Sun G, Li G, Xie Y (2015) Topology optimization for microstructures of viscoelastic composite materials. Comput Methods Appl Mech Eng 283:503–516CrossRefGoogle Scholar
  66. 66.
    Ibrahimbegovic A, Markovic D (2003) Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures. Comput Methods Appl Mech Eng 192(28–30):3089–3107MATHCrossRefGoogle Scholar
  67. 67.
    Ibrahimbegovic A, Papadrakakis M (2010) Multi-scale models and mathematical aspects in solid and fluid mechanics. Comput Methods Appl Mech Eng 199(21–22):1241MathSciNetCrossRefGoogle Scholar
  68. 68.
    Jung D, Gea H (2004) Topology optimization of nonlinear structures. Finite Elem Anal Des 40(11):1417–1427CrossRefGoogle Scholar
  69. 69.
    Kato J, Yachi D, Terada K, Kyoya T (2014) Topology optimization of micro-structure for composites applying a decoupling multi-scale analysis. Struct Multidiscipl Optimiz 49(4):595–608MathSciNetCrossRefGoogle Scholar
  70. 70.
    Kouznetsova V, Brekelmans WAM, Baaijens FPT (2001) An approach to micro-macro modeling of heterogeneous materials. Comput Mech 27(1):37–48MATHCrossRefGoogle Scholar
  71. 71.
    Lamari H, Ammar A, Cartraud P, Legrain G, Chinesta F, Jacquemin F (2010) Routes for efficient computational homogenization of nonlinear materials using the proper generalized decompositions. Arch Comput Methods Eng 17(4):373–391MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Le B, Yvonnet J, He QC (2015) Computational homogenization of nonlinear elastic materials using neural networks. Int J Numer Meth Eng. doi: 10.1002/nme.4953 MathSciNetMATHGoogle Scholar
  73. 73.
    Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscipl Optimiz 41(4):605–620CrossRefGoogle Scholar
  74. 74.
    Li Q, Steven G, Xie Y (2001) A simple checkerboard suppression algorithm for evolutionary structural optimization. Struct Multidiscipl Optimiz 22(3):230–239CrossRefGoogle Scholar
  75. 75.
    Liu S, Hou Y, Sun X, Zhang Y (2012) A two-step optimization scheme for maximum stiffness design of laminated plates based on lamination parameters. Compos Struct 94(12):3529–3537CrossRefGoogle Scholar
  76. 76.
    Luo Y, Wang M, Kang Z (2015) Topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique. Comput Methods Appl Mech Eng 286:422–441MathSciNetCrossRefGoogle Scholar
  77. 77.
    Lv J, Zhang H, Chen B (2014) Shape and topology optimization for closed liquid cell materials using extended multiscale finite element method. Struct Multidiscipl Optimiz 49(3):367–385MathSciNetCrossRefGoogle Scholar
  78. 78.
    Maute K, Schwarz S, Ramm E (1998) Adaptive topology optimization of elastoplastic structures. Struct Optimiz 15(2):81–91CrossRefGoogle Scholar
  79. 79.
    Michel JC, Suquet P (2003) Nonuniform transformation field analysis. Int J Solids Struct 40:6937–6955MathSciNetMATHCrossRefGoogle Scholar
  80. 80.
    Michel JC, Suquet P (2004) Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis. Comput Methods Appl Mech Eng 193:5477–5502MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    Michel JC, Moulinec H, Suquet P (1999) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 172(1–4):109–143MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Miehe C (2002) Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int J Numer Meth Eng 55:1285–1322MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171(3–4):387–418MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    Miled B, Ryckelynck D, Cantournet S (2013) A priori hyper-reduction method for coupled viscoelastic-viscoplastic composites. Comput Struct 119:95–103CrossRefGoogle Scholar
  85. 85.
    Mosby M, Matous K (2014) Hierarchically parallel coupled finite strain multiscale solver for modeling heterogeneous layers. Int J Numer Meth Eng 102(3–4):748–765MathSciNetMATHGoogle Scholar
  86. 86.
    Nakshatrala PB, Tortorelli DA, Nakshatrala KB (2013) Nonlinear structural design using multiscale topology optimization. part I: static formulation. Comput Methods Appl Mech Eng 261–262:167–176MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Neves MM, Rodrigues H, Guedes JM (2000) Optimal design of periodic linear elastic microstructures. Comput Struct 76(1):421–429CrossRefGoogle Scholar
  88. 88.
    Neves MM, Sigmund O, Bendsøe MP (2002) Topology optimization of periodic microstructures with a penalization of highly localized buckling modes. Int J Numer Meth Eng 54(6):809–834MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Niu B, Yan J, Cheng G (2009) Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct Multidiscipl Optimiz 39(2):115–132CrossRefGoogle Scholar
  90. 90.
    Nix W, Gao H (1998) Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 46(3):411–425MATHCrossRefGoogle Scholar
  91. 91.
    Oskay C, Fish J (2007) Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials. Comput Methods Appl Mech Eng 196(7):1216–1243MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    Pedersen C, Buhl T, Sigmund O (2001) Topology synthesis of large-displacement compliant mechanisms. Int J Numer Meth Eng 50(12):2683–2705MATHCrossRefGoogle Scholar
  93. 93.
    Queipo NB, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28CrossRefGoogle Scholar
  94. 94.
    Raghavan B, Xia L, Breitkopf P, Rassineux A, Villon P (2013) Towards simultaneous reduction of both input and output spaces for interactive simulation-based structural design. Comput Methods Appl Mech Eng 265(1):174–185MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    Rodrigues H, Guedes JM, Bendsøe MP (2002) Hierarchical optimization of material and structure. Struct Multidiscipl Optimiz 24(1):1–10CrossRefGoogle Scholar
  96. 96.
    Schwarz S, Maute K, Ramm E (2001) Topology and shape optimization for elastoplastic structural response. Comput Methods Appl Mech Eng 190(15–17):2135–2155MATHCrossRefGoogle Scholar
  97. 97.
    Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489–528MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Setoodeh S, Abdalla MM, Gürdal Z (2005) Combined topology and fiber path design of composite layers using cellular automata. Struct Multidiscipl Optimiz 30(6):413–421CrossRefGoogle Scholar
  99. 99.
    Setoodeh S, Abdalla M, Gürdal Z (2006) Design of variable-stiffness laminates using lamination parameters. Compos Part B Eng 37(4–5):301–309CrossRefGoogle Scholar
  100. 100.
    Sigmund O (1994) Materials with prescribed constitutive parameters: an inverse homogenization problem. Int J Solids Struct 31(17):2313–2329MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    Sigmund O (2000) New class of extremal composites. J Mech Phys Solids 48(2):397–428MathSciNetMATHCrossRefGoogle Scholar
  102. 102.
    Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscipl Optimiz 21(2):120–127MathSciNetCrossRefGoogle Scholar
  103. 103.
    Sigmund O, Maute K (2013) Topology optimization approaches—a comparative review. Struct Multidiscipl Optimiz 48(6):1031–1055CrossRefGoogle Scholar
  104. 104.
    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):1037–1067MathSciNetCrossRefGoogle Scholar
  105. 105.
    Smit RJM, Brekelmans WAM, Meijer HEH (1998) Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput Methods Appl Mech Eng 155(1–2):181–192MATHCrossRefGoogle Scholar
  106. 106.
    Su W, Liu S (2010) Size-dependent optimal microstructure design based on couple-stress theory. Struct Multidiscipl Optimiz 42(2):243–254CrossRefGoogle Scholar
  107. 107.
    Temizer I, Wriggers P (2007) An adaptive method for homogenization in orthotropic nonlinear elasticity. Comput Methods Appl Mech Eng 196(35–36):3409–3423MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    Temizer I, Zohdi T (2007) A numerical method for homogenization in non-linear elasticity. Comput Mech 40(2):281–298MATHCrossRefGoogle Scholar
  109. 109.
    Theocaris PS, Stavroulaki GE (1999) Optimal material design in composites: an iterative approach based on homogenized cells. Comput Methods Appl Mech Eng 169(1–2):31–42MATHCrossRefGoogle Scholar
  110. 110.
    Tran A, Yvonnet J, He QC, Toulemonde C, Sanahuja J (2011) A simple computational homogenization method for structures made of linear heterogeneous viscoelastic materials. Comput Methods Appl Mech Eng 200(45–46):2956–2970MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Wang F, Lazarov B, Sigmund O, Jensen J (2014a) Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems. Comput Methods Appl Mech Eng 276:453–472MathSciNetCrossRefGoogle Scholar
  112. 112.
    Wang F, Sigmund O, Jensen J (2014b) Design of materials with prescribed nonlinear properties. J Mech Phys Solids 69(1):156–174MathSciNetCrossRefGoogle Scholar
  113. 113.
    Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    Xia L, Breitkopf P (2014a) Concurrent topology optimization design of material and structure within Fe\(^{2}\) nonlinear multiscale analysis framework. Comput Methods Appl Mech Eng 278:524–542MathSciNetCrossRefGoogle Scholar
  115. 115.
    Xia L, Breitkopf P (2014b) A reduced multiscale model for nonlinear structural topology optimization. Comput Methods Appl Mech Eng 280:117–134MathSciNetCrossRefGoogle Scholar
  116. 116.
    Xia L, Breitkopf P (2015a) Multiscale structural topology optimization with an approximate constitutive model for local material microstructure. Comput Methods Appl Mech Eng 286:147–167MathSciNetCrossRefGoogle Scholar
  117. 117.
    Xia L, Breitkopf P (2015b) Design of of materials using topology optimization and energy-based homogenization approach in matlab. Struct Multidiscipl Optimiz 52(6):1229–1241MathSciNetCrossRefGoogle Scholar
  118. 118.
    Xia L, Raghavan B, Breitkopf P, Zhang W (2013) Numerical material representation using proper orthogonal decomposition and diffuse approximation. Appl Math Comput 224:450–462MathSciNetMATHGoogle Scholar
  119. 119.
    Xia Z, Zhang Y, Ellyin F (2003) A unified periodical boundary conditions for representative volume elements of composites and applications. Int J Solids Struct 40(8):1907–1921MATHCrossRefGoogle Scholar
  120. 120.
    Xia Z, Zhou C, Yong Q, Wang X (2006) On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites. Int J Solids Struct 43(2):266–278MathSciNetMATHCrossRefGoogle Scholar
  121. 121.
    Xiao M, Breitkopf P, Filomeno Coelho R, Knopf-Lenoir C, Sidorkiewicz M, Villon P (2010) Model reduction by cpod and kriging: application to the shape optimization of an intake port. Struct Multidiscipl Optimiz 41(4):555–574MathSciNetMATHCrossRefGoogle Scholar
  122. 122.
    Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896CrossRefGoogle Scholar
  123. 123.
    Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, LondonMATHCrossRefGoogle Scholar
  124. 124.
    Xu B, Xie Y (2015) Concurrent design of composite macrostructure and cellular microstructure under random excitations. Compos Struct 123:65–77CrossRefGoogle Scholar
  125. 125.
    Xu B, Jiang J, Xie Y (2015b) Concurrent design of composite macrostructure and multi-phase material microstructure for minimum dynamic compliance. Compos Struct 128:221–233CrossRefGoogle Scholar
  126. 126.
    Xu Y, Zhang W (2011) Numerical modelling of oxidized microstructure and degraded properties of 2d c/sic composites in air oxidizing environments below 800 °C. Mater Sci Eng A 528(27):7974–7982CrossRefGoogle Scholar
  127. 127.
    Xu Y, Zhang W (2012) A strain energy model for the prediction of the effective coefficient of thermal expansion of composite materials. Comput Mater Sci 53(1):241–250CrossRefGoogle Scholar
  128. 128.
    Yan X, Huang X, Zha Y, Xie YM (2014) Concurrent topology optimization of structures and their composite microstructures. Comput Struct 133:103–110CrossRefGoogle Scholar
  129. 129.
    Yi YM, Park SH, Youn SK (2000) Design of microstructures of viscoelastic composites for optimal damping characteristics. Int J Solids Struct 37(35):4791–4810MATHCrossRefGoogle Scholar
  130. 130.
    Yoon G, Kim Y (2005) Element connectivity parameterization for topology optimization of geometrically nonlinear structures. Int J Solids Struct 42(7):1983–2009MathSciNetMATHCrossRefGoogle Scholar
  131. 131.
    Yoon G, Kim Y (2007) Topology optimization of material-nonlinear continuum structures by the element connectivity parameterization. Int J Numer Meth Eng 69(10):2196–2218MathSciNetMATHCrossRefGoogle Scholar
  132. 132.
    Yuan Z, Fish J (2009) Multiple scale eigendeformation-based reduced order homogenization. Comput Methods Appl Mech Eng 198(21–26):2016–2038MATHCrossRefGoogle Scholar
  133. 133.
    Yuge K, Kikuchi N (1995) Optimization of a frame structure subjected to a plastic deformation. Struct Optimiz 10(3–4):197–208CrossRefGoogle Scholar
  134. 134.
    Yuge K, Iwai N, Kikuchi N (1999) Optimization of 2-d structures subjected to nonlinear deformations using the homogenization method. Struct Optimiz 17(4):286–299CrossRefGoogle Scholar
  135. 135.
    Yvonnet J, He QC (2007) The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys 223(1):341–368MathSciNetMATHCrossRefGoogle Scholar
  136. 136.
    Yvonnet J, Gonzalez D, He QC (2009) Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials. Comput Methods Appl Mech Eng 198(33–36):2723–2737MATHCrossRefGoogle Scholar
  137. 137.
    Yvonnet J, Monteiro E, He QC (2013) Computational homogenization method and reduced database model for hyperelastic heterogeneous structures. Int J Multiscale Comput Eng 11(3):201–225CrossRefGoogle Scholar
  138. 138.
    Zhang W, Sun S (2006) Scale-related topology optimization of cellular materials and structures. Int J Numer Meth Eng 68(9):993–1011MATHCrossRefGoogle Scholar
  139. 139.
    Zhang W, Dai G, Wang F, Sun S, Bassir H (2007) Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures. Acta Mech Sinica/Lixue Xuebao 23(1):77–89MathSciNetMATHCrossRefGoogle Scholar
  140. 140.
    Zhang W, Guo X, Wang M, Wei P (2013) Optimal topology design of continuum structures with stress concentration alleviation via level set method. Int J Numer Meth Eng 93(9):942–959MathSciNetMATHCrossRefGoogle Scholar
  141. 141.
    Zhou M, Rozvany GIN (1991) The COC algorithm, part II: Topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336CrossRefGoogle Scholar
  142. 142.
    Zhu J, Zhang W, Qiu K (2007) Bi-directional evolutionary topology optimization using element replaceable method. Comput Mech 40(1):97–109MATHCrossRefGoogle Scholar
  143. 143.
    Zhu J, Zhang W, Xia L (2015) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng. doi: 10.1007/s11831-015-9151-2
  144. 144.
    Zuo Z, Huang X, Rong J, Xie Y (2013) Multi-scale design of composite materials and structures for maximum natural frequencies. Mater Des 51:1023–1034CrossRefGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2016

Authors and Affiliations

  1. 1.CNRS, UMR 7337 Roberval, Centre de Recherches de RoyallieuSorbonne Universités, Université de Technologie de CompiègneCompiègne CedexFrance

Personalised recommendations