Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design

  • Seth WattsEmail author
  • William Arrighi
  • Jun Kudo
  • Daniel A. Tortorelli
  • Daniel A. White
Research Paper


Elastic meta-materials are those whose unique properties come from their micro-architecture, rather than, e.g., from their chemistry. The introduction of such architecture, which is increasingly able to be fabricated due to advances in additive manufacturing, expands the design domain and enables improved design, from the most complex multi-physics design problems to the simple compliance design problem that is our focus. Unfortunately, concurrent design of both the micro-scale and the macroscale is computationally very expensive when the former can vary spatially, particularly in three dimensions. Instead, we provide simple, accurate surrogate models of the homogenized linear elastic response of the isotruss, the octet truss, and the ORC truss based on high-fidelity continuum finite element analyses. These surrogate models are relatively accurate over the full range of relative densities, in contrast to analytical models in the literature, which we show lose accuracy as relative density increases. The surrogate models are also simple to implement, which we demonstrate by modifying Sigmund’s 99-line code to solve a three-dimensional, multiscale compliance design problem with spatially varying relative density. We use this code to generate examples in both two and three dimensions that illustrate the advantage of elastic meta-materials over structures with a single length scale, i.e., those without micro-architectures.


Topology design Elasticity Finite element methods Surrogate model Multiscale 



This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Funding information

This work received funding from LDRD number 17-SI-005. LLNL-JRNL-758077.

Compliance with ethical standards

Conflict of interests

The authors declare that they have no conflict of interest.


  1. Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550(7674):84CrossRefGoogle Scholar
  2. Allaire G (2002) Shape optimization by the homogenization method. Springer, New YorkCrossRefzbMATHGoogle Scholar
  3. Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 164(1):363–393MathSciNetCrossRefzbMATHGoogle Scholar
  4. Alzahrani M, Choi SK, Rosen DW (2015) Design of truss-like cellular structures using relative density mapping method. Mater Des 85:349–360CrossRefGoogle Scholar
  5. Bendsøe M, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bendsøe M, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, BerlinzbMATHGoogle Scholar
  7. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Structural Optimization 1 (4):193–202CrossRefGoogle Scholar
  8. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9-10):635– 654CrossRefzbMATHGoogle Scholar
  9. Berger JB, Wadley HNG, McMeeking RM (2017) Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness. Nature 543:233–537CrossRefGoogle Scholar
  10. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bower AF (2009) Applied mechanics of solids, 1st edn. CRC Press, Boca RatonCrossRefGoogle Scholar
  12. Bruns T, Tortorelli D (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26):3443–3459CrossRefzbMATHGoogle Scholar
  13. Chang PS, Rosen DW, Chang PS, Rosen DW (2011) An improved size, matching, and scaling method for the design of deterministic mesoscale truss structures. In: ASME 2011 International design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers, pp 697–707Google Scholar
  14. Coelho PG, Fernandes PR, Guedes JM, Rodrigues HC (2008) A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidiscip Optim 35(2):107–115CrossRefGoogle Scholar
  15. Cowin S, Mehrabadi M (1995) Anisotropic symmetries of linear elasticity. Appl Mech Rev 48(5):247–285CrossRefzbMATHGoogle Scholar
  16. Deshpande VS, Fleck NA, Ashby MF (2001) Effective properties of the octet-truss lattice material. J Mech Phys Solids 49(8):1747–1769CrossRefzbMATHGoogle Scholar
  17. Feppon F, Michailidis G, Sidebottom MA, Allaire G, Krick BA, Vermaak N (2017) Introducing a level-set based shape and topology optimization method for the wear of composite materials with geometric constraints. Struct Multidiscip Optim 55(2):547–568MathSciNetCrossRefGoogle Scholar
  18. Francfort GA, Murat F (1986) Homogenization and optimal bounds in linear elasticity. Arch Ration Mech Anal 94(4):307–334MathSciNetCrossRefzbMATHGoogle Scholar
  19. Fuller RB (1961) Synergetic building construction. U.S. Patent No. 2,986, 241Google Scholar
  20. Gaynor AT, Guest JK, Gaynor AT, Guest JK (2016) Topology optimization considering overhang constraints: eliminating sacrificial support material in additive manufacturing through design. Struct Multidiscip Optim 54(5):1157–1172MathSciNetCrossRefGoogle Scholar
  21. Graf GC, Chu J, Engelbrecht S, Rosen DW (2009) Synthesis methods for lightweight lattice structures. In: ASME 2009 International design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers, pp 579–589Google Scholar
  22. Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(2):127–140MathSciNetCrossRefzbMATHGoogle Scholar
  23. Horn TJ, Harrysson OLA (2012) Overview of current additive manufacturing technologies and selected applications. Sci Prog 95(3):255–282CrossRefGoogle Scholar
  24. Li H, Luo Z, Zhang N, Gao L, Brown T (2016) Integrated design of cellular composites using a level-set topology optimization method. Comput Methods Appl Mech Eng 309:453–475MathSciNetCrossRefGoogle Scholar
  25. Liu J, Cheng L, To AC (2017) Arbitrary void feature control in level set topology optimization. Comput Methods Appl Mech Eng 324:595–618MathSciNetCrossRefGoogle Scholar
  26. Messner MC (2016) Optimal lattice-structured materials. J Mech Phys Solids 96:162–183MathSciNetCrossRefGoogle Scholar
  27. Messner MC, Barham MI, Kumar M, Barton NR (2015) Wave propagation in equivalent continuums representing truss lattice materials. Int J Solids Struct 73-74:55–66CrossRefGoogle Scholar
  28. MFEM (2018) Modular finite element methods library,
  29. Mirzendehdel AM, Suresh K (2016) Support structure constrained topology optimization for additive manufacturing. Comput Aided Des 81:1–13CrossRefGoogle Scholar
  30. Nakshatrala P, Tortorelli D, Nakshatrala K (2013) Nonlinear structural design using multiscale topology optimization. Part I: static formulation. Comput Methods Appl Mech Eng 261-262:167– 176MathSciNetCrossRefzbMATHGoogle Scholar
  31. Rupp CJ, Evgrafov A, Maute K, Dunn ML (2009) Design of piezoelectric energy harvesting systems: a topology optimization approach based on multilayer plates and shells. J Intell Mater Syst Struct 20(16):1923–1939CrossRefGoogle Scholar
  32. Sigmund O (1997) On the design of compliant mechanisms using topology optimization. J Struct Mech 25 (4):493–524Google Scholar
  33. Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21 (2):120–127CrossRefGoogle Scholar
  34. Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43(5):589–596MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sigmund O, Petersson J (1998) Numerical instabilities in toplogy optimization: a survey on procedures dealing with checkerboards, mesh-dependencies, and local minima. Struct Multidiscip Optim 16(1):68–75CrossRefGoogle Scholar
  36. Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067MathSciNetCrossRefGoogle Scholar
  37. Sivapuram R, Dunning PD, Kim HA (2016) Simultaneous material and structural optimization by multiscale topology optimization. Struct Multidiscip Optim 54(5):1267–1281MathSciNetCrossRefGoogle Scholar
  38. Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124CrossRefGoogle Scholar
  39. Tancogne-Dejean T, Diamantopoulou M, Gorji MB, Bonatti C, Mohr D (2018) 3d platelattices: an emerging class of lowdensity metamaterial exhibiting optimal isotropic stiffness. Adv Mater 30(45):1803334CrossRefGoogle Scholar
  40. Wang M, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1):227–246MathSciNetCrossRefzbMATHGoogle Scholar
  41. 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):469–496MathSciNetCrossRefzbMATHGoogle Scholar
  42. Watts S, Tortorelli DA (2016) An n-material thresholding method for improving integerness of solutions in topology optimization. Int J Numer Methods Eng 108(12):1498–1524MathSciNetCrossRefGoogle Scholar
  43. Watts S, Tortorelli DA (2017) A geometric projection method for designing three-dimensional open lattices with inverse homogenization. Int J Numer Methods Eng 112(11):1564– 1588MathSciNetCrossRefGoogle Scholar
  44. White DA, Arrighi WJ, Kudo J, Watts SE (2019) Multiscale topology optimization using neural network surrogate models. Comput Meth Appl Mech Eng 346:1118–1135MathSciNetCrossRefGoogle Scholar
  45. Xia L, Breitkopf P (2014) Concurrent topology optimization design of material and structure within nonlinear multiscale analysis framework. Comput Methods Appl Mech Eng 278:524–542MathSciNetCrossRefzbMATHGoogle Scholar
  46. Zener C (1948) Elasticity and anelasticity of metals. University of Chicago Press, ChicagozbMATHGoogle Scholar
  47. Zheng X, Lee H, Weisgraber TH, Shusteff M, DeOtte J, Duoss EB, Kuntz JD, Biener MM, Ge Q, Jackson JA et al (2014) Ultralight, ultrastiff mechanical metamaterials. Science 344(6190):1373–1377CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Computational Engineering DivisionLawrence Livermore National LaboratoryLivermoreUSA
  2. 2.Applications, Simulations, and Quality DivisionLawrence Livermore National LaboratoryLivermoreUSA
  3. 3.Center for Design and OptimizationLawrence Livermore National LaboratoryLivermoreUSA
  4. 4.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations