Multi-objective Topology Optimization Design of Micro-structures

  • Sebastián Miguel GiustiEmail author
  • Antonio André Novotny
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 49)


This contribution proposes a methodology for the multi-objective synthesis and/or topology optimization of microstructures based on the topological derivative concept. The macroscopic properties are estimated by a standard multi-scale constitutive theory where the macroscopic responses are volume averages of their microscopic counterparts over a Representative Volume Element (RVE). We introduce a macroscopic cost functional that combines the mechanical and thermal effects in a single expression, allowing to design an RVE satisfying a specific thermo-mechanical macroscopic behavior. The algorithm is of simple computational implementation and relies in a level-set domain representation method. The effectiveness of the proposed methodology is illustrated by a set of finite element-based numerical examples.


  1. 1.
    Allaire, G., Jouve, F., Van Goethem, N.: Damage and fracture evolution in brittle materials by shape optimization methods. J. Comput. Phys. 230(12), 5010–5044 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Almgreen, R.F.: An isotropic three-dimensional structure with Poisson’s ratio–1. J. Elast. 15(4), 427–430 (1985)CrossRefGoogle Scholar
  3. 3.
    Amstutz, S., Andrä, H.: A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216(2), 573–588 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Amstutz, S., Giusti, S.M., Novotny, A.A., de Souza Neto, E.A.: Topological derivative for multi-scale linear elasticity models applied to the synthesis of microstructures. Int. J. Numer. Methods Eng. 84, 733–756 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amstutz, S., Novotny, A.A., de Souza Neto, E.A.: Topological derivative-based topology optimization of structures subject to Drucker-Prager stress constraints. Comput. Methods Appl. Mech. Eng. 233–236, 123–136 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Auriault, J.L.: Effective macroscopic description for heat conduction in periodic composites. Int. J. Heat Mass Transf. 26(6), 861–869 (1983)CrossRefzbMATHGoogle Scholar
  7. 7.
    Auriault, J.L., Royer, P.: Double conductivity media: a comparison between phenomenological and homogenization approaches. Int. J. Heat Mass Transf. 36(10), 2613–2621 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bensoussan, A., Lions, J.L., Papanicolau, G.: Asymptotic Analysis for Periodic Microstructures. North Holland, Amsterdam (1978)Google Scholar
  9. 9.
    Celentano, D.J., Dardati, P.M., Godoy, L.A., Boeri, R.E.: Computational simulation of microstructure evolution during solidification of ductile cast iron. Int. J. Cast Met. Res. 21(6), 416–426 (2008)CrossRefGoogle Scholar
  10. 10.
    de Souza Neto, E.A., Amstutz, S., Giusti, S.M., Novotny, A.A.: Topology optimization design of micro-structures considering different multi-scale models. Comput. Model. Eng. Sci. 62(1), 23–56 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    de Souza Neto, E.A., Feijóo, R.A.: On the equivalence between spatial and material volume averaging of stress in large strain multi-scale solid constitutive models. Mech. Mater. 40(10), 803–811 (2008)CrossRefGoogle Scholar
  12. 12.
    de Souza Neto, E.A., Feijóo, R.A.: Variational foundations of large strain multiscale solid constitutive models: kinematical formulation. In: Advanced Computational Materials Modeling: From Classical to Multi-Scale Techniques. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany (2011)Google Scholar
  13. 13.
    Germain, P., Nguyen, Q.S., Suquet, P.: Continuum thermodynamics. J. Appl. Mech. Trans. ASME 50(4), 1010–1020 (1983)CrossRefzbMATHGoogle Scholar
  14. 14.
    Giusti, S.M., Blanco, P.J., de Souza Neto, E.A., Feijóo, R.A.: An assessment of the Gurson yield criterion by a computational multi-scale approach. Eng. Comput. 26(3), 281–301 (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Giusti, S.M., Novotny, A.A., de Souza Neto, E.A.: Sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions. In: Proceeding of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 466, pp. 1703–1723 (2010)Google Scholar
  16. 16.
    Giusti, S.M., Novotny, A.A., de Souza Neto, E.A., Feijóo, R.A.: Sensitivity of the macroscopic elasticity tensor to topological microstructural changes. J. Mech. Phys. Solids 57(3), 555–570 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Giusti, S.M., Novotny, A.A., de Souza Neto, E.A., Feijóo, R.A.: Sensitivity of the macroscopic thermal conductivity tensor to topological microstructural changes. Comput. Methods Appl. Mech. Eng. 198(5–8), 727–739 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gurson, A.L.: Continuum theory of ductile rupture by void nucleation and growth: part I—yield criteria and flow rule for porous ductile media. J. Eng. Mater. Technol. Trans. ASME 99(1), 2–15 (1977)CrossRefGoogle Scholar
  19. 19.
    Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2), 127–140 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13(4), 213–222 (1965)CrossRefGoogle Scholar
  21. 21.
    Hintermüller, M., Laurain, A.: Multiphase image segmentation and modulation recovery based on shape and topological sensitivity. J. Math. Imaging Vis. 35, 1–22 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hintermüller, M., Laurain, A., Novotny, A.A.: Second-order topological expansion for electrical impedance tomography. Adv. Comput. Math. 36(2), 235–265 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lakes, R.: Foam structures with negative Poisson’s ratio. Sci. AAAS 235(4792), 1038–1040 (1987)Google Scholar
  24. 24.
    Mandel, J.: Plasticité Classique et Viscoplasticité. CISM Lecture Notes. Springer, Udine (1971)zbMATHGoogle Scholar
  25. 25.
    Michel, J.C., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172(1–4), 109–143 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Miehe, C., Schotte, J., Schröder, J.: Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput. Mater. Sci. 16(1–4), 372–382 (1999)CrossRefGoogle Scholar
  27. 27.
    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, Amsterdam (1993)zbMATHGoogle Scholar
  28. 28.
    Novotny, A.A.: Topological derivative for multi-scale linear elasticity models in three spatial dimensions. Optimization of Structures and Components. Advanced Structured Materials, vol. 43. Springer, Switzerland (2013)CrossRefGoogle Scholar
  29. 29.
    Novotny, A.A., Sokołowski, J.: Topological Derivatives in Shape Optimization. Interaction of Mechanics and Mathematics. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  30. 30.
    Ostoja-Starzewski, M., Schulte, J.: Bounding of effective thermal conductivities of multiscale materials by essential and natural boundary conditions. Phys. Rev. B 54(1), 278–285 (1996)CrossRefGoogle Scholar
  31. 31.
    Oyen, M.L., Ferguson, V.L., Bembey, A.K., Bushby, A.J., Boyde, A.: Composite bounds on the elastic modulus of bone. J. Biomech. 41(11), 2585–2588 (2008)CrossRefGoogle Scholar
  32. 32.
    Sanchez-Palencia, E.: Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin (1980)zbMATHGoogle Scholar
  33. 33.
    Sokołowski, J., Żochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Speirs, D.C.D., de Souza Neto, E.A., Perić, D.: An approach to the mechanical constitutive modelling of arterial tissue based on homogenization and optimization. J. Biomech. 41(12), 2673–2680 (2008)CrossRefGoogle Scholar
  35. 35.
    Van Goethem, N., Novotny, A.A.: Crack nucleation sensitivity analysis. Math. Methods Appl. Sci. 33(16), 197–1994 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Sebastián Miguel Giusti
    • 1
    Email author
  • Antonio André Novotny
    • 2
  1. 1.Departamento de Ingeniería Civil, Facultad Regional Córdoba UTN/FRC - CONICETUniversidad Tecnológica NacionalCórdobaArgentina
  2. 2.Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e ComputacionalPetrópolisBrazil

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