Topological Optimization with Interfaces

  • N. VermaakEmail author
  • G. Michailidis
  • A. Faure
  • G. Parry
  • R. Estevez
  • F. Jouve
  • G. Allaire
  • Y. Bréchet
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 282)


Design of architectured materials and structures, whether in nature or in engineering, often relies on forms of optimization. In nature, controlling architecture or spatial heterogeneity is usually adaptive and incremental. Naturally occuring architectured materials exploit heterogeneity with typically graded interfaces, smoothly transitioning across properties and scales in the pursuit of performance and longevity. This chapter explores an engineering tool, topology optimization, that is at the frontier of designing architectured materials and structures. Topology optimization offers a mathematical framework to determine the most efficient material layout for prescribed constraints and loading conditions. In engineering, topology optimization is identifying designs with interfaces, materials, manufacturing methods, and functionalities unavailable to the natural world. The particular focus is on the variety of roles that interfaces may play in advancing architectured materials and structures with topology optimization.



This material is, in part, based upon work supported by the National Science Foundation under Grant No. 1538125.


  1. 1.
    M. Ashby, Designing architectured materials. Scr. Mater. 68(1), 4–7 (2013)CrossRefGoogle Scholar
  2. 2.
    M. Ashby, Y. Brechet, Designing hybrid materials. Acta Mater. 51(19), 5801–5821 (2003)CrossRefGoogle Scholar
  3. 3.
    S. Torquato, Optimal design of heterogeneous materials. Ann. Rev. Mater. Res. 40, 101–129 (2010)CrossRefGoogle Scholar
  4. 4.
    M. Bendsoe, O. Sigmund, in Topology Optimization: Theory, Methods and Applications (Springer, 2004)Google Scholar
  5. 5.
    D. Wolf, J. Jaszczak, in Materials Interfaces: Atomic-level Structure and Properties (1992)Google Scholar
  6. 6.
    G. Allaire, Conception optimale de structures, vol. 58, in Mathématiques & Applications (Springer-Verlag, Berlin, 2007)Google Scholar
  7. 7.
    P. Christensen, A. Klarbring, in An Introduction to Structural Optimization, vol. 153. (Springer, 2009)Google Scholar
  8. 8.
    J.D. Deaton, R.V. Grandhi, A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multi. Optim. 49(1), 1–38 (2014)CrossRefGoogle Scholar
  9. 9.
    G. Allaire, E. Bonnetier, G. Francfort, F. Jouve, Shape optimization by the homogenization method. Numer. Math. 76(1), 27–68 (1997)CrossRefGoogle Scholar
  10. 10.
    M. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988)CrossRefGoogle Scholar
  11. 11.
    L. Gibiansky and A. Cherkaev, Design of composite plates of extremal rigidity, in Topics in the Mathematical Modelling of Composite Materials (Springer, 1997), pp. 95–137Google Scholar
  12. 12.
    R. Kohn, G. Strang, Optimal design and relaxation of variational problems, I. Commun. Pure Appl. Math. 39(1), 113–137 (1986a)CrossRefGoogle Scholar
  13. 13.
    R. Kohn, G. Strang, Optimal design and relaxation of variational problems, II. Commun. Pure Appl. Math. 39(2), 139–182 (1986b)CrossRefGoogle Scholar
  14. 14.
    R. Kohn, G. Strang, Optimal design and relaxation of variational problems, III. Commun. Pure Appl. Math. 39(3), 353–377 (1986c)CrossRefGoogle Scholar
  15. 15.
    F. Murat, L. Tartar, Calcul des variations et homogénéisation. Les méthodes de lhomogénéisation: théorie et applications en physique 57, 319–369 (1985)Google Scholar
  16. 16.
    S. Osher, J. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)CrossRefGoogle Scholar
  17. 17.
    J. Sethian, A. Wiegmann, Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163(2), 489–528 (2000)CrossRefGoogle Scholar
  18. 18.
    G. Allaire, F. Jouve, A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194(1), 363–393 (2004)CrossRefGoogle Scholar
  19. 19.
    M. Wang, X. Wang, D. Guo, A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192(1), 227–246 (2003)CrossRefGoogle Scholar
  20. 20.
    A. Christiansen, M. Nobel-Jørgensen, N. Aage, O. Sigmund, J. Bærentzen, Topology optimization using an explicit interface representation, in Structural and Multidisciplinary Optimization (2013), pp. 1–13Google Scholar
  21. 21.
    L. Blank, M. Farshbaf-Shaker, H. Garcke, C. Rupprecht, V. Styles, Multi-material phase field approach to structural topology optimization, in Trends in PDE Constrained Optimization (Springer, 2014), pp. 231–246Google Scholar
  22. 22.
    S. Zhou, M. Wang, Multimaterial structural topology optimization with a generalized cahn-hilliard model of multiphase transition. Struct. Multi. Optim. 33(2), 89–111 (2007)CrossRefGoogle Scholar
  23. 23.
    O. Querin, G. Steven, Y. Xie, Evolutionary structural optimisation (eso) using a bidirectional algorithm. Eng. Comput. 15(8), 1031–1048 (1998)CrossRefGoogle Scholar
  24. 24.
    A. Baumgartner, L. Harzheim, C. Mattheck, Sko (soft kill option): the biological way to find an optimum structure topology. Int. J. Fatigue 14(6), 387–393 (1992)CrossRefGoogle Scholar
  25. 25.
    C. Mattheck, Design and growth rules for biological structures and their application to engineering. Fatigue Fract. Eng. Mater. Struct. 13(5), 535–550 (1990)CrossRefGoogle Scholar
  26. 26.
    O. Sigmund, On the usefulness of non-gradient approaches in topology optimization. Struct. Multi. Optim. 43(5), 589–596 (2011)CrossRefGoogle Scholar
  27. 27.
    J. Guest, Topology optimization with multiple phase projection. Comput. Methods Appl. Mech. Eng. 199(1), 123–135 (2009)CrossRefGoogle Scholar
  28. 28.
    A. Clausen, N. Aage, O. Sigmund, Topology optimization of coated structures and material interface problems. Comput. Methods Appl. Mech. Eng. 290, 524–541 (2015)CrossRefGoogle Scholar
  29. 29.
    T. Abballe, M. Albertelli, G. Allaire, A. Caron, P. Conraux, L. Dall’Olio, C. Dapogny, C. Dobrzynski, B. Jeannin, F. Jouve, et al., Rodin Project, Topology Optimization 2.0? (2015). HAL preprint:
  30. 30.
    F. Murat, J. Simon, Etude de problèmes d’optimal design. Optim. Tech. Model. Optim. Serv. Man Part 2, 54–62 (1976)CrossRefGoogle Scholar
  31. 31.
    J. Simon, F. Murat, in Sur le contrôle par un domaine géométrique. Publication 76015 du Laboratoire d’Analyse Numérique de l’Université Paris VI, (76015):222 pages (1976)Google Scholar
  32. 32.
    B. Merriman, J.K. Bence, S.J. Osher, Motion of multiple junctions: a level set approach. J. Comput. Phys. 112(2), 334–363 (1994)CrossRefGoogle Scholar
  33. 33.
    M. Wang, X. Wang, Color level sets: a multi-phase method for structural topology optimization with multiple materials. Comput. Methods Appl. Mech. Eng. 193(6), 469–496 (2004)CrossRefGoogle Scholar
  34. 34.
    G. Allaire, C. Dapogny, G. Delgado, G. Michailidis, Mutli-phase structural optimization via a level-set method. ESAIM Control Optim. Calc. Var. 20(2), 576–611 (2014). Scholar
  35. 35.
    O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer Series in Computational Physics. (Springer-Verlag, New York, 1984)Google Scholar
  36. 36.
    J. Sokołowski and J.-P. Zolésio. Introduction to Shape Optimization, vol. 16, Springer Series in Computational Mathematics. (Springer-Verlag, Berlin, 1992). Shape sensitivity analysisGoogle Scholar
  37. 37.
    S. Osher, R. Fedkiw, Level set methods and dynamic implicit surfaces, in Applied Mathematical Sciences,vol. 153 (Springer-Verlag, New York, 2003)Google Scholar
  38. 38.
    J. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (Cambridge University Press, Cambridge, 1999)Google Scholar
  39. 39.
    G. Allaire, C. Dapogny, P. Frey, A mesh evolution algorithm based on the level set method for geometry and topology optimization. Struct. Multi. Optim. 48(4), 711–715 (2013)CrossRefGoogle Scholar
  40. 40.
    Q. Xia, T. Shi, S. Liu, M. Wang, A level set solution to the stress-based structural shape and topology optimization. Comput. Struct. 9091, 55–64 (2012)CrossRefGoogle Scholar
  41. 41.
    L. Ambrosio, G. Buttazzo, An optimal design problem with perimeter penalization. Calc. Var. Partial. Differ. Equ. 1(1), 55–69 (1993)CrossRefGoogle Scholar
  42. 42.
    C. Dapogny, Optimisation de formes, méthode des lignes de niveaux sur maillages non structurés et évolution de maillages. Ph.D. thesis, Université Pierre et Marie Curie-Paris VI, 2013. Available at
  43. 43.
    G. Allaire, F. Jouve, G. Michailidis, Thickness control in structural optimization via a level set method. Struct. Multi. Optim. 53(6), 1349–1382 (2016)CrossRefGoogle Scholar
  44. 44.
    F. Feppon, Design and Optimization for Wear of Bi-material Composite Surfaces. Master’s thesis, Ecole Polytechnique, Promotion X2012, 2015Google Scholar
  45. 45.
    F. Feppon, G. Michailidis, M. Sidebottom, G. Allaire, B. Krick, N. Vermaak, Introducing a level-set based shape and topology optimization method for the wear of composite materials with geometric constraints. Struct. Multi. Optim. 55(2), 547–568 (2017). Scholar
  46. 46.
    N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire, Y. Bréchet, Material interface effects on the topology optimizationof multi-phase structures using a level set method. Struct. Multi. Optim. 50(4), 623–644 (2014)CrossRefGoogle Scholar
  47. 47.
    O. Sigmund, Tailoring materials with prescribed elastic properties. Mech. Mater. 20(4), 351–368 (1995)CrossRefGoogle Scholar
  48. 48.
    A. Faure, G. Michailidis, G. Parry, N. Vermaak, R. Estevez, Design of thermoelastic multi-material structures with graded interfaces using topology optimization. Struct. Multi. Optim. 56(4), 823–837 (2017).
  49. 49.
    A. Clausen, N. Aage, O. Sigmund, Exploiting additive manufacturing infill in topology optimization for improved buckling load. Engineering 2(2), 250–257 (2016)CrossRefGoogle Scholar
  50. 50.
    P. Zhang, J. Liu, A. To, Role of anisotropic properties on topology optimization of additive manufactured load bearing structures. Scripta Mater. 135, 148–152 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • N. Vermaak
    • 1
    Email author
  • G. Michailidis
    • 2
  • A. Faure
    • 2
  • G. Parry
    • 2
  • R. Estevez
    • 2
  • F. Jouve
    • 3
  • G. Allaire
    • 4
  • Y. Bréchet
    • 2
  1. 1.Department of Mechanical Engineering and Mechanics, Lehigh UniversityBethlehemUSA
  2. 2.SIMaP, CNRSUniversité GrenobleGrenobleFrance
  3. 3.Laboratoire J.L. Lions (UMR 7598)University Paris DiderotParisFrance
  4. 4.CMAP, Ecole PolytechniquePalaiseauFrance

Personalised recommendations