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

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Abstract

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.

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References

  1. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224

    Article  MathSciNet  Google Scholar 

  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):501–521

    Article  Google Scholar 

  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 16:68–75

    Article  Google Scholar 

  4. Allaire G, Jouve F, Maillot H (2004) Topology optimization and optimal shape design using homogenization. Struct Multidiscip Optim 28:87–98

    Article  MathSciNet  Google Scholar 

  5. Suzuki K, Kikuchi N (1991) A homogenization method for shape and topology optimization. Comput Methods Appl Mech Eng 93(3):291–318

    Article  Google Scholar 

  6. Zhou M, Rozvany GIN (1991) The coc algorithm, part II: Topological geometry and generalized shape optimization. Comput Meth Appl Mech Eng 89:197–224

    Article  Google Scholar 

  7. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 6(65):635–654

    MATH  Google Scholar 

  8. Bendsøe MP, Sigmund O (2003) Topology optimization–theory, methods, and applications. Springer, Germany

    MATH  Google Scholar 

  9. Gersborg-Hansen A, Bendsøe MP, Sigmund O (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 30(3):181–192

    Article  MathSciNet  Google Scholar 

  10. Yoon GH (2010) Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation. Int J. Numer Methods Eng 82(5):591–616

    Article  Google Scholar 

  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):1049–1075

    Article  MathSciNet  Google Scholar 

  12. Gersborg-Hansen A, Bendsøe MP, Sigmund O (2006) Topology optimization of heat conduction using the finite volume method. Struct Multidiscip Optim 31(4):251–259

    Article  MathSciNet  Google Scholar 

  13. Andreasen CS, Sigmund O (2013) Topology optimization of fluid-structure-interaction problems in poroelasticity. Comput Methods Appl Mech Eng 31(4):55–62

    Article  MathSciNet  Google Scholar 

  14. Sigmund O, Torquato S (1996) Composites with extremal thermal expansion coefficients. Appl Phys Lett 69(21):3203–3205

    Article  Google Scholar 

  15. Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM Control Optim Calc Var 9:19–48

    Article  MathSciNet  Google Scholar 

  16. Burger M, Stainko R (2006) Phase-field relaxation of topology optimization with local stress constraints. SIAM J Control Optim 45(4):1447–1466

    Article  MathSciNet  Google Scholar 

  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):2697–2718

    Article  MathSciNet  Google Scholar 

  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 229–258:2012

    MATH  Google Scholar 

  19. Dedè L, Borden MJ, Hughes TJR (2012) Isogeometric analysis for topology optimization with a phase field model. Arch Comput Methods Eng 19(3):427–465

    Article  MathSciNet  Google Scholar 

  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 231–246

    MATH  Google Scholar 

  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 115:403–412

    Article  Google Scholar 

  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 111:709–722

    Article  Google Scholar 

  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 104:28–38

    Article  Google Scholar 

  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 61:581–598

    Article  Google Scholar 

  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 27:540–551

    Article  Google Scholar 

  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 28:2313–2366

    Article  MathSciNet  Google Scholar 

  27. Brackett D, Ashcroft I, Hague R (2014) Topology optimization for additive manufacturing. In: solid freeform fabrication symposium (SFF), Austin

  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)

  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 19:81–94

    Article  Google Scholar 

  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 20:1025–1058

    Article  MathSciNet  Google Scholar 

  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 00016160

    Article  Google Scholar 

  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 progress

  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 344:334–359

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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.

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Authors and Affiliations

  1. Dipartimento di Ingegneria Civile ed Architettura (DICAr), Universitá degli Studi Pavia, via Ferrata 3, 27100, Pavia, Italy

    Massimo Carraturo, Alessandro Reali & Ferdinando Auricchio

  2. Chair for Computation in Engineering, Technical University of Munich, Arcisstr. 21, 80333, Munich, Germany

    Massimo Carraturo

  3. Dipartimento di Matematica, Universitá degli Studi Pavia, via Ferrata 5, 27100, Pavia, Italy

    Elisabetta Rocca

  4. IMATI-CNR, via Ferrata 1, 27100, Pavia, Italy

    Elisabetta Rocca & Elena Bonetti

  5. Dipartimento di Matematica “F.Enriques”, Universitá degli Studi di Milano, via Saldini 50, 20133, Milano, Italy

    Elena Bonetti

  6. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Dietmar Hömberg

  7. Department of Mathematical Sciences, NTNU, Alfred Getz vei 1, 7491, Trondheim, Norway

    Dietmar Hömberg

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  1. Massimo Carraturo
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  2. Elisabetta Rocca
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  3. Elena Bonetti
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  4. Dietmar Hömberg
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  5. Alessandro Reali
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  6. Ferdinando Auricchio
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Correspondence to Massimo Carraturo.

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Carraturo, M., Rocca, E., Bonetti, E. et al. Graded-material design based on phase-field and topology optimization. Comput Mech 64, 1589–1600 (2019). https://doi.org/10.1007/s00466-019-01736-w

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  • DOI: https://doi.org/10.1007/s00466-019-01736-w

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