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Hierarchical Infills for Additive Manufacturing Through a Multiscale Approach

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Abstract

A numerical method is presented to generate hierarchical infills for additive manufacturing, using homogenization and optimization. Given the shape and the allowed stages of grading, the macroscopic properties of each level of the hierarchical infill are computed through numerical homogenization. Then, a multi-material optimization problem is formulated to find the distribution of the prescribed discrete set of candidates that maximizes the structural stiffness of the object to be printed for a limited volume fraction. The formulation is endowed with an additional overturning constraint to achieve objects that resist gravity in a stable configuration. Numerical simulations, addressing the design of a self-supporting orthotropic rhombic infill and a stiff isotropic triangular one, are shown.

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References

  1. Chu, C., Graf, G., Rosen, D.W.: Design for additive manufacturing of cellular structures. Comput. Aided Design Appl. 5(5), 686–696 (2008)

    Google Scholar 

  2. Allaire, G., Bogosel, B.: Optimizing supports for additive manufacturing. Struct. Multidiscip. Optim. 58(6), 2493–2515 (2018)

    MathSciNet  Google Scholar 

  3. Bruggi, M., Parolini, N., Regazzoni, F., Verani, M.: Topology optimization with a time-integral cost functional. Finite Elem. Anal. Design 140, 11–22 (2018)

    MathSciNet  Google Scholar 

  4. Groen, J.P., Sigmund, O.: Homogenization-based topology optimization for high-resolution manufacturable microstructures. Int. J. Numer. Methods Eng. 113(8), 1148–1163 (2018)

    MathSciNet  Google Scholar 

  5. Allaire, G., Geoffroy-Donders, P., Pantz, O.: Topology optimization of modulated and oriented periodic microstructures by the homogenization method. Comput. Math. Appl. 78(7), 2197–2229 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Lewiński, T., Sokół, T., Graczykowski, C.: Michell structures. Springer, Berlin (2018)

    Google Scholar 

  7. Eschenauer, H.A., Olhoff, N.: Topology optimization of continuum structures: a review. Appl. Mech. Rev. 54(4), 331–390 (2001)

    Google Scholar 

  8. Liu, J., Gaynor, A.T., Chen, S., Kang, Z., Suresh, K., Takezawa, A., Li, L., Kato, J., Tang, J., Wang, C.C.L., Cheng, L., Liang, X., To, A.C.: Current and future trends in topology optimization for additive manufacturing. Struct. Multidiscip. Optim. 57(6), 2457–2483 (2018)

    Google Scholar 

  9. Meng, L., Zhang, W., Quan, D., Shi, G., Tang, L., Hou, Y., Breitkopf, P., Zhu, J., Gao, T.: From topology optimization design to additive manufacturing: Today’s success and tomorrow’s roadmap. Arch. Comput. Methods Eng. (2019). https://doi.org/10.1007/s11831-019-09331-1

    Article  Google Scholar 

  10. Plocher, J., Panesar, A.: Review on design and structural optimisation in additive manufacturing: towards next-generation lightweight structures. Mater. Design (2019). https://doi.org/10.1007/s11831-019-09331-1

    Article  Google Scholar 

  11. Wu, J., Wang, C.C.L., Zhang, X., Westermann, R.: Self-supporting rhombic infill structures for additive manufacturing. CAD Comput. Aided Design 80, 32–42 (2016)

    Google Scholar 

  12. Wu, J., Aage, N., Westermann, R., Sigmund, O.: Infill optimization for additive manufacturing-approaching bone-like porous structures. IEEE Trans. Vis. Comput. Graph. 24(2), 1127–1140 (2018)

    Google Scholar 

  13. Wu, Z., Xia, L., Wang, S., Shi, T.: Topology optimization of hierarchical lattice structures with substructuring. Comput. Methods Appl. Mech. Eng. 345, 602–617 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Alzahrani, M., Choi, S., Rosen, D.W.: Design of truss-like cellular structures using relative density mapping method. Mater. Design 85, 349–360 (2015)

    Google Scholar 

  15. Cheng, L., Bai, J., To, A.C.: Functionally graded lattice structure topology optimization for the design of additive manufactured components with stress constraints. Comput. Methods Appl. Mech. Eng. 344, 334–359 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Han, Y., Lu, W.F.: A novel design method for nonuniform lattice structures based on topology optimization. J. Mech. Design. Trans. ASME (2018). https://doi.org/10.1115/1.4040546

    Article  Google Scholar 

  17. Wang, Y., Zhang, L., Daynes, S., Zhang, H., Feih, S., Wang, M.Y.: Design of graded lattice structure with optimized mesostructures for additive manufacturing. Mater. Design 142, 114–123 (2018)

    Google Scholar 

  18. Panesar, A., Abdi, M., Hickman, D., Ashcroft, I.: Strategies for functionally graded lattice structures derived using topology optimisation for additive manufacturing. Additive Manuf. 19, 81–94 (2018)

    Google Scholar 

  19. Sigmund, O., Aage, N., Andreassen, E.: On the (non-)optimality of michell structures. Struct. Multidiscip. Optim. 54(2), 361–373 (2016)

    MathSciNet  Google Scholar 

  20. Gibiansky, L.V., Sigmund, O.: Multiphase composites with extremal bulk modulus. J. Mech. Phys. Solids 48(3), 461–498 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Stegmann, J., Lund, E.: Discrete material optimization of general composite shell structures. Int. J. Numer. Methods Eng. 62(14), 2009–2027 (2005)

    MATH  Google Scholar 

  22. Svanberg, K.: The method of moving asymptotes-a new method for structural optimization. Int. J. Numer. Methods Eng. 24(2), 359–373 (1987). https://doi.org/10.1002/nme.1620240207

    Article  MathSciNet  MATH  Google Scholar 

  23. Arabnejad, S., Pasini, D.: Mechanical properties of lattice materials via asymptotic homogenization and comparison with alternative homogenization methods. Int. J. Mech. Sci. 77, 249–262 (2013)

    Google Scholar 

  24. Noor, A.K.: Continuum modeling for repetitive lattice structures. Appl. Mech. Rev. 41(7), 285–297 (1988)

    Google Scholar 

  25. Kumar, R.S., McDowell, D.L.: Generalized continuum modeling of 2-D periodic cellular solids. Int. J. Solids Struct. 41(26), 7399–7422 (2004)

    MATH  Google Scholar 

  26. Hassani, B., Hinton, E.: A review of homogenization and topology optimization I—Homogenization theory for media with periodic structure. Comput. Struct. 69(6), 707–717 (1998)

    MATH  Google Scholar 

  27. Theerakittayakorn, K., Nanakorn, P., Sam, P., Suttakul, P.: Exact forms of effective elastic properties of frame-like periodic cellular solids. Arch. Appl. Mech. 86(8), 1465–1482 (2016)

    Google Scholar 

  28. Vigliotti, A., Pasini, D.: Linear multiscale analysis and finite element validation of stretching and bending dominated lattice materials. Mech. Mater. 46, 57–68 (2012)

    Google Scholar 

  29. Andreassen, E., Andreasen, C.S.: How to determine composite material properties using numerical homogenization. Comput. Mater. Sci. 83, 488–495 (2014)

    Google Scholar 

  30. Timoshenko, S.P., Goodier, J.N.: Theory of elasticity, 3rd edn. McGraw-Hill, Singapore (1970)

    MATH  Google Scholar 

  31. Bendsøe, M.P., Sigmund, O.: Topology optimization. Theory, methods and applications. Springer, Berlin (2004)

    MATH  Google Scholar 

  32. Kaw, K.: Mechanics of composite materials. CRC Press, Boca Raton (2006)

    MATH  Google Scholar 

  33. Vannucci, P.: Anisotropic elasticity. Springer, Berlin (2018)

    MATH  Google Scholar 

  34. Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2), 197–224 (1988)

    MathSciNet  MATH  Google Scholar 

  35. Bruggi, M.: Conceptual design of diagrids and hexagrids by distribution of lattice structures. Front. Built Environ. 6, 80 (2020). https://doi.org/10.3389/fbuil.2020.00080

  36. Wang, M.Y., Wang, X.: “color” level sets: a multi-phase method for structural topology optimization with multiple materials. Comput. Methods Appl. Mech. Eng. 193(6–8), 469–496 (2004)

    MathSciNet  MATH  Google Scholar 

  37. Sanders, E.D., Aguiló, M.A., Paulino, G.H.: Multi-material continuum topology optimization with arbitrary volume and mass constraints. Comput. Methods Appl. Mech. Eng. 340, 798–823 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Bruggi, M., Duysinx, P.: A stress-based approach to the optimal design of structures with unilateral behavior of material or supports. Struct. Multidiscip. Optim. 48(2), 311–326 (2013)

    MathSciNet  Google Scholar 

  39. Christensen, P.W., Klarbring, A.: An introduction to structural optimization. Springer, Berlin (2009)

    MATH  Google Scholar 

  40. Sigmund, O., Maute, K.: Topology optimization approaches. Struct. Multidiscip. Optim. 48, 1031–1055 (2013)

    MathSciNet  Google Scholar 

  41. Beckers, M.: Topology optimization using a dual method with discrete variables. Struct. Optim. 17, 14–24 (1999)

    Google Scholar 

  42. Stolpe, M., Bendsøe, M.: Global optima for the Zhou–Rozvany problem. Struct. Multidiscip. Optim. 43, 151–165 (2011)

    MathSciNet  MATH  Google Scholar 

  43. Aage, N., Andreassen, E., Lazarov, B.: Topology optimization using PETSC: an easy-to-use, fully parallel, open source topology optimization framework. Struct. Multidiscip. Optim. 51, 565–572 (2013)

    MathSciNet  Google Scholar 

  44. Bruggi, M.: Topology optimization with mixed finite elements on regular grids. Comput. Methods Appl. Mech. Eng. 305, 133–153 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Błachowski, B., Gutkowski, W.: Discrete structural optimization by removing redundant material. Eng. Optim. 40(7), 685–694 (2008)

    Google Scholar 

  46. Bourdin, B.: Filters in topology optimization. Int. J. Numer. Methods Eng. 50(9), 2143–2158 (2001)

    MathSciNet  MATH  Google Scholar 

  47. Bruyneel, M., Duysinx, P.: Note on topology optimization of continuum structures including self-weight. Struct. Multidiscip. Optim. 29(4), 245–256 (2005)

    Google Scholar 

  48. Divakara Shetty, S., Shetty, N.: Investigation of mechanical properties and applications of polylactic acids—a review. Mater. Res. Exp. 6(11), 112002 (2019)

    Google Scholar 

  49. Wang, W., Wang, T.Y., Yang, Z., Liu, L., Tong, X., Tong, W., Deng, J., Chen, F., Liu, X.: Cost-effective printing of 3D objects with skin-frame structures. ACM Trans. Graph. 32(6), 1–10 (2013)

    Google Scholar 

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Acknowledgements

The financial support of Italian Ministry of Education, University and Research through PRIN Grant Number 2015JW9NJT is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Politecnico di Milano, Milan, Italy

    Matteo Bruggi & Alberto Taliercio

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  1. Matteo Bruggi
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  2. Alberto Taliercio
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Correspondence to Matteo Bruggi.

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Communicated by Zenon Mróz.

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Bruggi, M., Taliercio, A. Hierarchical Infills for Additive Manufacturing Through a Multiscale Approach. J Optim Theory Appl 187, 654–682 (2020). https://doi.org/10.1007/s10957-020-01685-y

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