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Abstract

Advanced manufacturing processes provide a tremendous opportunity to fabricate materials with precisely defined architectures. To fully leverage these capabilities, however, materials architectures must be optimally designed according to the target application, base material used, and specifics of the fabrication process. Computational topology optimization offers a systematic, mathematically driven framework for navigating this new design challenge. The design problem is posed and solved formally as an optimization problem with unit cell and upscaling mechanics embedded within this formulation. This article briefly reviews the key requirements to apply topology optimization to materials architecture design and discusses several fundamental findings related to optimization of elastic, thermal, and fluidic properties in periodic materials. Emerging areas related to topology optimization for manufacturability and manufacturing variations, nonlinear mechanics, and multiscale design are also discussed.

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2016-07-01
2024-06-23
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Literature Cited

  1. Billington DP.1.  1983. The Tower and the Bridge Princeton, NJ: Princeton Univ. Press [Google Scholar]
  2. Sigmund O.2.  1997. On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach. 25:493–524 [Google Scholar]
  3. Sigmund O.3.  2001. Design of multiphysics actuators using topology optimization. Part II. Two-material structures. Comput. Methods Appl. Mech. Eng. 190:6605–27 [Google Scholar]
  4. Evgrafov A, Maute K, Yang RG, Dunn ML. 4.  2009. Topology optimization for nano-scale heat transfer. Int. J. Numer. Methods Eng. 77:285–300 [Google Scholar]
  5. Otomori M, Yamada T, Izui K, Nishiwaki S, Kogiso N. 5.  2014. Level set–based topology optimization for the design of light-trapping structures. IEEE Trans. Magn. 50:729–32 [Google Scholar]
  6. Lin S, Zhou L, Guest JK, Weihs T, Liu Z. 6.  2015. Topology optimization of a passive fluid diode. ASME J. Mech. Des. 137:081402 [Google Scholar]
  7. Fleck NA, Deshpande VS, Ashby MF. 7.  2011. Micro-architectured materials: past, present and future. Proc. R. Soc. A 466:2495–516 [Google Scholar]
  8. Wadley HNG.8.  2006. Multifunctional periodic cellular metals. Philos. Trans. R. Soc. A 364:31–68 [Google Scholar]
  9. Schaedler TA, Jacobsen AJ, Torrents A, Sorensen AE, Lian J. 9.  et al. 2011. Ultralight metallic microlattices. Science 334:962–65 [Google Scholar]
  10. Torrents A, Schaedler TA, Jacobsen AJ, Carter WB, Valdevit L. 10.  2012. Characterization of nickel-based microlattice materials with structural hierarchy from the nanometer to the millimeter scale. Acta Mater. 60:3511–23 [Google Scholar]
  11. Sigmund O.11.  1994. Design of material structures using topology optimization PhD Thesis, Dep. Solid Mech., Tech. Univ. Den. [Google Scholar]
  12. Sigmund O.12.  1994. Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31:2313–29 [Google Scholar]
  13. Sigmund O.13.  1995. Tailoring materials with prescribed elastic properties. Mech. Mater. 20:351–68 [Google Scholar]
  14. Bendsøe MP, Sigmund O. 14.  2003. Topology Optimization: Theory, Methods, and Applications. Berlin: Springer [Google Scholar]
  15. Cadman J, Zhou S, Chen Y, Li Q. 15.  2013. On design of multi-functional microstructural materials. J. Mater. Sci. 48:51–66 [Google Scholar]
  16. Bendsøe MP, Kikuchi N. 16.  1988. Generating optimal topologies in structural design using homogenization method. Comput. Methods Appl. Mech. Eng. 71:197–224 [Google Scholar]
  17. Wang MY, Wang X, Guo D. 17.  2003. A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192:227–46 [Google Scholar]
  18. Allaire G, Jouve F, Toader AM. 18.  2004. Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194:363–93 [Google Scholar]
  19. Amstutz S, Andra H. 19.  2006. A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216:573–88 [Google Scholar]
  20. Villanueva CH, Maute K. 20.  2014. Density and level set-XFEM schemes for topology optimization of 3-D structures. Comp. Mech. 52:133–50 [Google Scholar]
  21. Sigmund O, Maute K. 21.  2013. Topology optimization approaches: a comparative review. Struct. Multidiscip. Optim. 48:1031–55 [Google Scholar]
  22. Bensoussan A, Lions J, Papanicolaou G. 22.  1978. Asymptotic Analysis for Periodic Structures Amsterdam: North-Holland [Google Scholar]
  23. Sanchez-Palencia E.23.  1980. Non-homogeneous media and vibration theory. Lecture Notes in Physics 127 Berlin: Springer [Google Scholar]
  24. Guedes JM, Kikuchi N. 24.  1990. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput. Methods Appl. Mech. Eng. 83:143–98 [Google Scholar]
  25. Guest JK, Prévost JH. 25.  2007. Design of maximum permeability material structures. Comput. Methods Appl. Mech. Eng. 196:1006–17 [Google Scholar]
  26. Kim I, de Weck O. 26.  2005. Adaptive weighted-sum method for bi-objective optimization: Pareto front generation. Struct. Multidiscip. Optim. 29:149–58 [Google Scholar]
  27. Challis VJ, Guest JK, Grotowski JF, Roberts AP. 27.  2012. Computationally generated cross-property bounds for stiffness and fluid permeability using topology optimization. Int. J. Solids Struct. 49:3397–408 [Google Scholar]
  28. Bendsøe MP.28.  1989. Optimal shape design as a material distribution problem. Struct. Optim. 1:193–202 [Google Scholar]
  29. Zhou M, Rozvany GIN. 29.  1991. The COC algorithm. Part II. Topological, geometry and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89:309–36 [Google Scholar]
  30. Stolpe M, Svanberg K. 30.  2001. An alternative interpolation scheme for minimum compliance topology optimization. Struct. Multidiscip. Optim. 22:2116–24 [Google Scholar]
  31. Michaleris P, Tortorelli DA, Vidal CA. 31.  1994. Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity. Int. J. Numer. Methods Eng. 37:2471–99 [Google Scholar]
  32. Jameson A.32.  2001. A perspective on computational algorithms for aerodynamic analysis and design. Progr. Aerosp. Sci. 37:197–243 [Google Scholar]
  33. Tsay JJ, Arora JS. 33.  1990. Nonlinear structural design sensitivity analysis for path dependent problems. Part I. General theory. Comp. Methods Appl. Mech. Eng. 81:183–208 [Google Scholar]
  34. Deaton JD, Grandhi RV. 34.  2014. A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49:1–38 [Google Scholar]
  35. Hashin Z, Shtrikman S. 35.  1962. A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 35:3125–31 [Google Scholar]
  36. Hashin Z, Shtrikman S. 36.  1963. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11:127–40 [Google Scholar]
  37. Bendsøe MP, Sigmund O. 37.  1999. Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69:9–10635–54 [Google Scholar]
  38. Borrvall T, Petersson J. 38.  2003. Topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Fluids 41:77–107 [Google Scholar]
  39. Guest JK, Pr évost JH. 39.  2006. Topology optimization of creeping fluid flows using a Darcy-Stokes finite element. Int. J. Numer. Methods Eng. 66:461–84 [Google Scholar]
  40. Guest JK, Prévost JH. 40.  2006. Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int. J. Solids Struct. 43:7028–47 [Google Scholar]
  41. Svanberg K.41.  1987. The method of moving asymptotes—a new method for structural optimization. Int. J. Numer. Methods Eng. 24:359–73 [Google Scholar]
  42. Svanberg K.42.  2002. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Opt. 12:555–73 [Google Scholar]
  43. Xie YM, Steven GP. 43.  1993. A simple evolutionary procedure for structural optimization. Comp. Struct. 49:885–96 [Google Scholar]
  44. Young V, Querin OM, Steven GP, Xie YM. 44.  1999. 3D and multiple load case bi-directional evolutionary structural optimization (BESO). Struct. Opt. 18:183–92 [Google Scholar]
  45. Huang X, Xie YM, Jia B, Li Q, Zhou SW. 45.  2012. Evolutionary topology optimization of periodic composites for extremal magnetic permeability and electrical permittivity. Struct. Multidiscip. Optim. 46:3385–98 [Google Scholar]
  46. Sigmund O, Petersson J. 46.  1998. Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Optim. 16:68–75 [Google Scholar]
  47. Diaz A, Sigmund O. 47.  1995. Checkerboard patterns in layout optimization. Struct. Optim. 10:40–45 [Google Scholar]
  48. Jog CS, Haber RB. 48.  1996. Stability of finite element models for distributed—parameter optimization and topology design. Comput. Methods Appl. Mech. Eng. 130:203–26 [Google Scholar]
  49. Poulsen TA.49.  2003. A new scheme for imposing minimum length scale in topology optimization. Int. J. Numer. Methods Eng. 57:741–60 [Google Scholar]
  50. Sigmund O.50.  2007. Morphology-based black and white filters for topology optimization. Struct. Multidiscip. Optim. 33:401–24 [Google Scholar]
  51. Guest JK, Prévost JH, Belytschko T. 51.  2004. Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int. J. Numer. Methods Eng. 61:238–54 [Google Scholar]
  52. Diaz A, Sigmund O. 52.  2010. A topology optimization method for design of negative permeability metamaterials. Struct. Multidiscip. Optim. 41:163–77 [Google Scholar]
  53. Zhou S, Li W, Chen Y, Sun G, Li Q. 53.  2011. Topology optimization for negative permeability metamaterials using level-set algorithm. Acta Mater. 59:2624–36 [Google Scholar]
  54. Sigmund O, Torquato S, Aksay IA. 54.  1998. On the design of 1-3 piezocomposites using topology optimization. J. Mater. Res. 13:1038–48 [Google Scholar]
  55. Nelli Silva EC, Ono Fonesca J, Kikuchi N. 55.  1997. Optimal design of piezoelectric microstructures. Comput. Mech. 19:5397–410 [Google Scholar]
  56. Sigmund O, Jensen JS. 56.  2003. Systematic design of phononic band-gap materials and structures by topology optimization. Philos. Trans. R. Soc. A 361:1001–19 [Google Scholar]
  57. Rupp CJ, Evgrafov A, Maute K, Dunn ML. 57.  2007. Design of phononic materials/structures for surface wave devices using topology optimization. Struct. Multidiscip. Optim. 34:111–22 [Google Scholar]
  58. Jensen JS, Sigmund O. 58.  2011. Topology optimization of nano-photonics. Laser Photonics Rev. 5:2308–12 [Google Scholar]
  59. Prasad J, Diaz AR. 59.  2009. Viscoelastic material design with negative stiffness components using topology optimization. Struct. Multidiscip. Optim. 38:6583–97 [Google Scholar]
  60. Andreassen E, Jensen JS. 60.  2014. Topology optimization of periodic microstructures for enhanced dynamic properties of viscoelastic composite materials. Struct. Multidiscip. Optim. 49:5695–705 [Google Scholar]
  61. Larsen UD, Sigmund O, Bouwstra S. 61.  1997. Design and fabrication of compliant mechanisms and material structures with negative Poisson's ratio. J. MicroElectroMech. Syst. 6:99–106 [Google Scholar]
  62. Andreassen E, Lazarov BS, Sigmund O. 62.  2014. Design of manufacturable 3D extremal elastic microstructure. Mech. Mater. 69:1–10 [Google Scholar]
  63. Lakes R.63.  1987. Foam structures with a negative Poisson's ratio. Science 235:47921038–40 [Google Scholar]
  64. Sigmund O.64.  1999. On the optimality of bone microstructure. Synthesis in Bio Solid Mechanics P Pedersen, MP Bendsøe 221–34 Dordrecht/Boston: Kluwer [Google Scholar]
  65. Neves MM, Rodrigues H, Guedes JM. 65.  2000. Optimal design of periodic linear elastic microstructures. Eng. Comput. Struct. 76:1421–29 [Google Scholar]
  66. Challis VJ, Roberts AP, Wilkins AH. 66.  2008. Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. Int. J. Solids Struct. 45:4130–46 [Google Scholar]
  67. Ashby MF, Evans A, Fleck NA, Gibson LJ, Hutchinson JW, Wadley HNG. 67.  2000. Metal Foams: A Design Guide. Burlington, MA: Elsevier Sci [Google Scholar]
  68. Aage N, Andreassen E, Lazarov BS. 68.  2015. Topology optimization using PETSc: An easy-to-use, fully parallel, open source topology optimization framework. Struct. Multidiscip. Optim. 51:3565–72 [Google Scholar]
  69. Sigmund O, Aage N, Andreassen E. 69.  2016. On the (non-)optimality of Michell structures. Struct. Multidiscip. Optim.In press [Google Scholar]
  70. Sigmund O.70.  2000. A new class of extremal composites. J. Mech. Phys. Solids 48:397–428 [Google Scholar]
  71. Gibiansky LV, Sigmund O. 71.  2000. Multiphase composites with extremal bulk modulus. J. Mech. Phys. Solids 48:461–98 [Google Scholar]
  72. Vigdergauz SB.72.  1989. Regular structures with extremal elastic properties. Mech. Solids 24:357–63 [Google Scholar]
  73. Vigdergauz SB.73.  1994. Three-dimensional grained composites of extreme thermal properties. J. Mech. Phys. Solids 42:5729–40 [Google Scholar]
  74. Andreasen CS, Andreassen E, Søndergaard Jensen J, Sigmund O. 74.  2014. On the realization of the bulk modulus bounds for two-phase viscoelastic composites. J. Mech. Phys. Solids 63:1228–41 [Google Scholar]
  75. Paulino GH, Silva ECN, Le CH. 75.  2009. Optimal design of periodic functionally graded composites with prescribed properties. Struct. Multidiscip. Optim. 38:469–89 [Google Scholar]
  76. Radman A, Huang X, Xie Y. 76.  2013. Topology optimization of functionally graded cellular materials. J. Mater. Sci. 48:41503–10 [Google Scholar]
  77. Despois JF, Mortensen A. 77.  2005. Permeability of open-pore microcellular materials. Acta Mater. 53:1381–88 [Google Scholar]
  78. Sigmund O, Torquato S. 78.  1996. Composites with extremal thermal expansion coefficients. Appl. Phys. Lett. 69:213203–5 [Google Scholar]
  79. Sigmund O, Torquato S. 79.  1997. Design of materials with extreme thermal expansion using a three-phase topology optimization method. J. Mech. Phys. Solids 45:61037–67 [Google Scholar]
  80. Schapery RA.80.  1968. Thermal expansion coefficients of composite materials based on energy principles. J. Compos. Mater. 2:380–404 [Google Scholar]
  81. Rosen BW, Hashin Z. 81.  1970. Effective thermal expansion and specific heat of composite materials. Int. J. Eng. Sci. 8:157–73 [Google Scholar]
  82. Gibiansky LV, Torquato S. 82.  1997. Thermal expansion of isotropic multi-phase composites and polycrystals. J. Mech. Phys. Solids 45:1223–52 [Google Scholar]
  83. Andreassen E.83.  2015. Optimal design of porous materials. PhD Thesis, Dep. Mech. Eng., Tech. Univ. Den 176 [Google Scholar]
  84. Andreassen E, Jensen JS, Sigmund O, Thomsen JJ. 84.  2015. Optimal design of porous materials DCAMM Special Rep. S172, Dep. Mech. Eng., Tech. Univ. Den. [Google Scholar]
  85. Torquato S, Hyun S, Donev A. 85.  2002. Multifunctional composites: optimizing microstructures for simultaneous transport of heat and electricity. Phys. Rev. Lett. 89:266601–1266601-4 [Google Scholar]
  86. de Kruijf N, Zhou S, Li Q, Mai YW. 86.  2007. Topological design of structures and composite materials with multiobjectives. Int. J. Solids Struct. 44:7092–109 [Google Scholar]
  87. Gibiansky L, Torquato S. 87.  1996. Connection between the conductivity and bulk modulus of isotropic composite materials. Proc. R. Soc. A 452:253–83 [Google Scholar]
  88. Hollister SJ.88.  2009. Scaffold design and manufacturing: from concept to clinic. Adv. Eng. Mater. 21:3330–42 [Google Scholar]
  89. Challis VJ, Roberts AP, Grotowski JF, Zhang LC, Sercombe TB. 89.  2010. Prototypes for bone implant scaffolds designed via topology optimization and manufactured by solid freeform fabrication. Adv. Eng. Mater. 12:1106–10 [Google Scholar]
  90. Chen Y, Zhou S, Li Q. 90.  2011. Microstructure design of biodegradable scaffold and its effect on tissue regeneration. Biomaterials 32:5003–14 [Google Scholar]
  91. Zhao L, Ryan SM, Ortega JK, Ha S, Sharp KW. 91.  et al. 2016. Experimental investigation of 3D woven Cu lattices for heat exchanger applications. Int. J. Heat Mass Transf. 96:296–311 [Google Scholar]
  92. Alexandersen J, Sigmund O, Aage N. 92.  2015. Topology optimisation of passive coolers for light-emitting diode lamps. Proc. World Cong. Struct. Multidiscip. Optim., Sydney1–5 [Google Scholar]
  93. Andreasen C, Sigmund O. 93.  2011. Saturated poroelastic actuators generated by topology optimization. Struct. Multidiscip. Optim. 43:5693–706 [Google Scholar]
  94. Jung Y, Torquato S. 94.  2005. Fluid permeabilities of triply periodic minimal surfaces. Phys. Rev. E 72:056319 [Google Scholar]
  95. Chen Y, Schellekens M, Zhou S, Cadman J, Li W. 95.  et al. 2011. Design optimization of scaffold microstructures using wall shear stress criterion towards regulated flow-induced erosion. J. Biomech. Eng. 133:8081008 [Google Scholar]
  96. Zhao L, Ha S, Sharp KW, Geltmacher AB, Fonda RW. 96.  et al. 2014. Permeability measurements and modeling of topology-optimized metallic 3D woven lattices. Acta Mater. 81:326–36 [Google Scholar]
  97. Zhang Y, Ha S, Sharp K, Guest JK, Weihs TP, Hemker KJ. 97.  2015. Fabrication and mechanical characterization of 3D woven Cu lattice materials. Mater. Des. 85:743–51 [Google Scholar]
  98. Guest JK.98.  2009. Imposing maximum length scale in topology optimization. Struct. Multidiscip. Optim. 37:463–73 [Google Scholar]
  99. Guest JK.99.  2009. Topology optimization with multiple phase projection. Comput. Methods Appl. Mech. Eng. 199:123–35 [Google Scholar]
  100. Guest JK, Zhu M. 100.  2012. Casting and milling restrictions in topology optimization via projection-based algorithms. Proc. ASME Des. Eng. Tech. Conf. 3:A–B913–20 [Google Scholar]
  101. Guest JK.101.  2015. Optimizing discrete object layouts in structures and materials: a projection-based topology optimization approach. Comput. Methods Appl. Mech. Eng. 283:330–51 [Google Scholar]
  102. Ha S, Guest JK. 102.  2014. Optimizing inclusion shapes and patterns in periodic materials using Discrete Feature Projection. Struct. Multidiscip. Optim. 50:65–80 [Google Scholar]
  103. Asadpoure A, Guest JK, Valdevit L. 103.  2015. Incorporating fabrication cost into topology optimization of discrete structures and lattices. Struct. Multidiscip. Optim. 51:2385–96 [Google Scholar]
  104. Asadpoure A, Valdevit L. 104.  2015. Topology optimization of lightweight periodic lattices under simultaneous compressive and shear stiffness constraints. Int. J. Solids Struct. 60–61:1–16 [Google Scholar]
  105. Gaynor AT, Guest JK. 105.  2014. Topology optimization for additive manufacturing considering maximum overhang constraint. Proc. AIAA/ISSMO Multidiscip. Anal. Optim. Conf. 15th, Atlanta1–8 [Google Scholar]
  106. Guest JK, Igusa T. 106.  2008. Structural optimization under uncertain loads and nodal locations. Comput. Methods Appl. Mech. Eng. 198:1116–24 [Google Scholar]
  107. Jalalpour M, Igusa T, Guest JK. 107.  2011. Optimal design of trusses with geometric imperfections—accounting for global instability. Int. J. Solids Struct. 48:213011–19 [Google Scholar]
  108. Jalalpour M, Guest JK, Igusa T. 108.  2013. Reliability-based topology optimization of trusses with stochastic stiffness matrix. J. Struct. Saf. 43:41–49 [Google Scholar]
  109. Asadpoure A, Tootkaboni M, Guest JK. 109.  2011. Robust topology optimization of structures with uncertainties in stiffness—application to truss structures. Comput. Struct. 89:11–121131–41 [Google Scholar]
  110. Tootkaboni M, Asadpoure A, Guest JK. 110.  2012. Topology optimization of continuum structures under uncertainty—a polynomial chaos approach. Comput. Methods Appl. Mech. Eng. 201–204:263–75 [Google Scholar]
  111. Sigmund O.111.  2009. Manufacturing tolerant topology optimization. Acta Mech. Sin. 25:227–39 [Google Scholar]
  112. Schevenels M, Lazarov BS, Sigmund O. 112.  2011. Robust topology optimization accounting for spatially varying manufacturing errors. Comput. Methods Appl. Mech. Eng. 200:3613–27 [Google Scholar]
  113. Jansen M, Lombaert G, Schevenels M. 113.  2015. Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis. Comput. Methods Appl. Mech. Eng. 285:452–67 [Google Scholar]
  114. Wang F, Jensen JS, Sigmund O. 114.  2011. Robust topology optimization of photonic crystal waveguides with tailored dispersion properties. J. Opt. Soc. Am. B 28:3387–97 [Google Scholar]
  115. Wang F, Lazarov BS, Sigmund O. 115.  2011. On projection methods, convergence and robust formulations in topology optimization. Struct. Multidiscip. Optim. 43:767–84 [Google Scholar]
  116. Chen S, Chen W, Lee S. 116.  2010. Level set based robust shape and topology optimization under random field uncertainties,. Struct. Multidiscip. Optim. 41:507–24 [Google Scholar]
  117. Maute K, Schwarz S, Ramm E. 117.  1998. Adaptive topology optimization of elastoplastic structures. Struct. Optim. 2:81–91 [Google Scholar]
  118. Swan CC, Kosaka O. 118.  1997. Voigt-Reuss topology optimization for structures with nonlinear elastic material behaviors. Int. J. Numer. Methods Eng. 40:3785–814 [Google Scholar]
  119. Buhl T, Pedersen CBW, Sigmund O. 119.  2000. Stiffness design of geometrically nonlinear structures using topology optimization. Struct. Multidiscip. Optim. 19:93–104 [Google Scholar]
  120. Bruns TE, Tortorelli DA. 120.  2001. Topology optimization of non-linear elastic structures and compliant mechanisms. Comput. Methods Appl. Mech. Eng. 190:3443–59 [Google Scholar]
  121. Gea HC, Luo J. 121.  2001. Topology optimization of structures with geometrical nonlinearities. Comput. Struct. 79:20–211977–85 [Google Scholar]
  122. Bruns T, Tortorelli D. 122.  2003. An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. Int. J. Numer. Methods Eng. 57:101413–30 [Google Scholar]
  123. Wang F, Lazarov BS, Sigmund O, Jensen JS. 123.  2014. Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems. Comput. Methods Appl. Mech. Eng. 276:453–72 [Google Scholar]
  124. Xie Y, Zuo Z, Huang X, Rong J. 124.  2012. Convergence of topological patterns of optimal periodic structures under multiple scales. Struct. Multidiscip. Optim. 46:141–50 [Google Scholar]
  125. Zuo ZH, Huang X, Yang X, Rong JH, Xie YM. 125.  2013. Comparing optimal material microstructures with optimal periodic structures. Comput. Mater. Sci. 69:137–47 [Google Scholar]
  126. Carstensen JV, Lotfi R, Chen W, Schroers J, Guest JK. 126.  2015. Topology optimization of cellular materials with maximized energy absorption. Proc. ASME Des. Eng. Tech. Conf.1–12 Boston: ASME [Google Scholar]
  127. Shim J, Shan S, Košmrlj A, Kang SH, Chen ER. 127.  et al. 2013. Harnessing instabilities for design of soft reconfigurable auxetic/chiral materials. Soft Matter 9:8198–202 [Google Scholar]
  128. Rodrigues H, Guedes J, Bendsøe M. 128.  2002. Hierarchical optimization of material and structure. Struct. Multidiscip. Optim. 24:11–10 [Google Scholar]
  129. Nakshatrala PB, Tortorelli DA, Nakshatrala KB. 129.  2013. Nonlinear structural design using multiscale topology optimization. Part I. Static formulation. Comput. Methods Appl. Mech. Eng. 261–263:167–76 [Google Scholar]
  130. Xia L, Breitkopf P. 130.  2014. Concurrent topology optimization design of material and structure within nonlinear multiscale analysis framework. Comput. Methods Appl. Mech. Eng. 278:524–42 [Google Scholar]
  131. Schury F, Stingl M, Wein F. 131.  2012. Efficient two-scale optimization of manufacturable graded structures. SIAM J. Sci. Comput. 34:6B711–33 [Google Scholar]
  132. Zhang W, Sun S. 132.  2006. Scale-related topology optimization of cellular materials and structures. Int. J. Numer. Methods Eng. 68:9993–1011 [Google Scholar]
  133. Alexandersen J, Lazarov BS. 133.  2015. Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner. Comput. Methods Appl. Mech. Eng. 290:156–82 [Google Scholar]
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