Graded-material design based on phase-field and topology optimization

  • Massimo CarraturoEmail author
  • Elisabetta Rocca
  • Elena Bonetti
  • Dietmar Hömberg
  • Alessandro Reali
  • Ferdinando Auricchio
Original Paper


In the present work we introduce a novel graded-material design based on phase-field and topology optimization. The main novelty of this work comes from the introduction of an additional phase-field variable in the classical single-material phase-field topology optimization algorithm. This new variable is used to grade the material properties in a continuous fashion. Two different numerical examples are discussed, in both of them, we perform sensitivity studies to asses the effects of different model parameters onto the resulting structure. From the presented results we can observe that the proposed algorithm adds additional freedom in the design, exploiting the higher flexibility coming from additive manufacturing technology.


Phase-field Functionally graded material Multi-material design Topology optimization Additive manufacturing 



This work was partially supported by Regione Lombardia through the Project “TPro.SL - Tech Profiles for Smart Living” (No. 379384) within the Smart Living program, and through the project “MADE4LO - Metal ADditivE for LOmbardy” (No. 240963) within the POR FESR 2014-2020 program. MC and AR have been partially supported by Fondazione Cariplo - Regione Lombardia through the project “Verso nuovi strumenti di simulazione super veloci ed accurati basati sull’analisi isogeometrica”, within the program RST - rafforzamento. This research has been performed in the framework of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica d’Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l’ATtRattività dell’ateneo pavese”. The present paper also benefits from the support of the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) for ER. A grateful acknowledgment goes to Dr. Ing. Gianluca Alaimo for his support and precious suggestions on additive manufacturing technology.


  1. 1.
    Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bendsøe MP (1983) On obtaining a solution to optimization problems for solid, elastic plates by restriction of the design space. J Struct Mech 11(4):501–521CrossRefGoogle Scholar
  3. 3.
    Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with cheackboards, mesh-dependencies and local minima. Struct Optim 16:68–75CrossRefGoogle Scholar
  4. 4.
    Allaire G, Jouve F, Maillot H (2004) Topology optimization and optimal shape design using homogenization. Struct Multidiscip Optim 28:87–98MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Suzuki K, Kikuchi N (1991) A homogenization method for shape and topology optimization. Comput Methods Appl Mech Eng 93(3):291–318CrossRefzbMATHGoogle Scholar
  6. 6.
    Zhou M, Rozvany GIN (1991) The coc algorithm, part II: Topological geometry and generalized shape optimization. Comput Meth Appl Mech Eng 89:197–224CrossRefGoogle Scholar
  7. 7.
    Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 6(65):635–654zbMATHGoogle Scholar
  8. 8.
    Bendsøe MP, Sigmund O (2003) Topology optimization–theory, methods, and applications. Springer, GermanyzbMATHGoogle Scholar
  9. 9.
    Gersborg-Hansen A, Bendsøe MP, Sigmund O (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 30(3):181–192MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yoon GH (2010) Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation. Int J. Numer Methods Eng 82(5):591–616CrossRefzbMATHGoogle Scholar
  11. 11.
    Yoon GH, Jensen J, Sigmund O (2007) Topology optimization of acoustic-structure interaction problems using a mixed finite element formulation. Int J. Numer Methods Eng 70(9):1049–1075MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gersborg-Hansen A, Bendsøe MP, Sigmund O (2006) Topology optimization of heat conduction using the finite volume method. Struct Multidiscip Optim 31(4):251–259MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Andreasen CS, Sigmund O (2013) Topology optimization of fluid-structure-interaction problems in poroelasticity. Comput Methods Appl Mech Eng 31(4):55–62MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sigmund O, Torquato S (1996) Composites with extremal thermal expansion coefficients. Appl Phys Lett 69(21):3203–3205CrossRefGoogle Scholar
  15. 15.
    Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM Control Optim Calc Var 9:19–48MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Burger M, Stainko R (2006) Phase-field relaxation of topology optimization with local stress constraints. SIAM J Control Optim 45(4):1447–1466MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229(7):2697–2718MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Penzler P, Rumpf M, Wirth B (2012) A phase-field model and minimal compliance shape optimization in nonlinear elasticity. ESAIM Control Optim Calc Var 229–258:2012zbMATHGoogle Scholar
  19. 19.
    Dedè L, Borden MJ, Hughes TJR (2012) Isogeometric analysis for topology optimization with a phase field model. Arch Comput Methods Eng 19(3):427–465MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Blank L, Farshbaf-Shaker MH, Garcke H, Rupprecht C, Styles V (2014) Multi-material phase field approach to structural topology optimization. In: Leugering G, Benner P, Engell S, Griewank A, Harbrecht H, Hinze M, Rannacher R, Ulbrich S (eds) Trends in PDE constrained optimization, vol 165. Springer, Cham, pp 231–246Google Scholar
  21. 21.
    Yan W, Ge W, Smith J, Lin S, Kafka OL, Lin F, Liu WK (2016) Multi-scale modeling of electron beam melting of functionally graded materials. Acta Mater 115:403–412CrossRefGoogle Scholar
  22. 22.
    Gan Z, Liu H, Li S, He X, Gang Y (2017a) Modeling of thermal behavior and mass transport in multi-layer laser additive manufacturing of Ni-based alloy on cast iron. Int J Heat Mass Transfer 111:709–722CrossRefGoogle Scholar
  23. 23.
    Gan Z, Gang Y, He X, Li S (2017b) Numerical simulation of thermal behavior and multicomponent mass transfer in direct laser deposition of Co-base alloy on steel. Int J Heat Mass Transfer 104:28–38CrossRefGoogle Scholar
  24. 24.
    Yang KK, Zhu JH, Wang C, Jia DS, Song LL, Zhang WH (2018) Experimental validation of 3D printed material behaviors and their influence on the structural topology design. Comput Mech 61:581–598CrossRefzbMATHGoogle Scholar
  25. 25.
    Wolff SJ, Gan Z, Lin S, Bennett JL, Yan W, Hyatt G, Ehmann KF, Wagner GJ, Liu WK, Cao J (2019) Experimentally validated predictions of thermal history and microhardness in laser-deposited Inconel 718 on carbon steel. Addit Manuf 27:540–551CrossRefGoogle Scholar
  26. 26.
    Allaire G, Jakabcin L (2018) Taking into account thermal residual stresses in topology optimization of structures built by additive manufacturing. Math Models Methods Appl Sci 28:2313–2366MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Brackett D, Ashcroft I, Hague R (2014) Topology optimization for additive manufacturing. In: solid freeform fabrication symposium (SFF), AustinGoogle Scholar
  28. 28.
    Cheng L, Zhang P, Biyikli E, Bai J, Pilz S, To AC (2015) Integration of topology optimization with efficient design of additive manufactured cellular structures. In: Solid freeform fabrication symposium (SFF), Austin (2015)Google Scholar
  29. 29.
    Hickman D, Panesar A, Abdi M, Ashcroft I (2018) Strategies for functionally graded lattice structures derived using topology optimisation for additive manufacturing. Addit Manuf 19:81–94CrossRefGoogle Scholar
  30. 30.
    Blank L, Garcke H, Farshbaf-Shaker MH, Styles V (2014) Relating phase field and sharp interface approaches to structural topology optimization. ESAIM Control Optim Calc Var 20:1025–1058MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Allen SM, Cahn JW (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall 27(6):1085–1095 ISSN 00016160Google Scholar
  32. 32.
    Auricchio F, Bonetti E, Carraturo M, Hömberg D, Reali A, Rocca E (2018) Structural multiscale topology optimization with stress constraint for additive manufacturing. Work in progressGoogle Scholar
  33. 33.
    Cheng L, Bai J, Albert CT (2019) Functionally graded lattice structure topology optimization for the design of additive manufactured components with stress constraints. Comput Methods Appl Mech Eng 344:334–359MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Civile ed Architettura (DICAr)Universitá degli Studi PaviaPaviaItaly
  2. 2.Chair for Computation in EngineeringTechnical University of MunichMunichGermany
  3. 3.Dipartimento di MatematicaUniversitá degli Studi PaviaPaviaItaly
  4. 4.IMATI-CNRPaviaItaly
  5. 5.Dipartimento di Matematica “F.Enriques”Universitá degli Studi di MilanoMilanoItaly
  6. 6.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  7. 7.Department of Mathematical SciencesNTNUTrondheimNorway

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