“Two-Scale Topology Optimization with Microstructures” by Zhu, Skouras, Chen and Matusik

  • ©Bo Zhu, Melina Skouras, Desai Chen, and Wojciech Matusik




    Two-Scale Topology Optimization with Microstructures


Session Title: Fabricating Look & Feel



    In this article, we present a novel two-scale framework to optimize the structure and the material distribution of an object given its functional specifications. Our approach utilizes multi-material microstructures as low-level building blocks of the object. We start by precomputing the material property gamut—the set of bulk material properties that can be achieved with all material microstructures of a given size. We represent the boundary of this material property gamut using a level set field. Next, we propose an efficient and general topology optimization algorithm that simultaneously computes an optimal object topology and spatially varying material properties constrained by the precomputed gamut. Finally, we map the optimal spatially varying material properties onto the microstructures with the corresponding properties to generate a high-resolution printable structure. We demonstrate the efficacy of our framework by designing, optimizing, and fabricating objects in different material property spaces on the level of a trillion voxels, that is, several orders of magnitude higher than what can be achieved with current systems.


    1. Joe Alexandersen and Boyan S. Lazarov. 2015. Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner. Comput. Methods Appl. Mech. Eng. 290 (2015), 156–182.Google ScholarCross Ref
    2. Grégoire Allaire. 2012. Shape Optimization by the Homogenization Method. Vol. 146. Springer Science 8 Business Media.Google Scholar
    3. G. Allaire and R.V. Kohn. 1993. Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials. Quaterly of Applied Mathematics 51, 4 (1993), 643–674.Google ScholarCross Ref
    4. Ryoichi Ando, Nils Thürey, and Chris Wojtan. 2013. Highly adaptive liquid simulations on tetrahedral meshes. ACM Trans. Graph. 32, 4 (2013), 103:1–103:10. Google ScholarDigital Library
    5. Erik Andreassen, Boyan S. Lazarov, and Ole Sigmund. 2014. Design of manufacturable 3D extremal elastic microstructure. Mech. Mater. 69, 1 (2014), 1–10.Google ScholarCross Ref
    6. Sahab Babaee, Jongmin Shim, James C. Weaver, Elizabeth R. Chen, Nikita Patel, and Katia Bertoldi. 2013. 3D soft metamaterials with negative poisson’s ratio. Adv. Mater. 25, 36 (2013), 5044–5049.Google ScholarCross Ref
    7. Martin P. Bendsøe. 1989. Optimal shape design as a material distribution problem. Struct. Optimiz. 1, 4 (1989), 193–202.Google ScholarCross Ref
    8. Martin P. Bendsøe and Ole Sigmund. 1999. Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69, 9–10 (1999), 635–654.Google Scholar
    9. Martin Philip Bendsøe and Ole Sigmund. 2004. Topology Optimization: Theory, Methods, and Applications. Springer Science 8 Business Media.Google ScholarCross Ref
    10. Haimasree Bhatacharya, Yue Gao, and Adam Bargteil. 2011. A level-set method for skinning animated particle data. In Proceedings of the 2011 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. ACM, 17–24. Google ScholarDigital Library
    11. Bernd Bickel, Moritz Bächer, Miguel A. Otaduy, Hyunho Richard Lee, Hanspeter Pfister, Markus Gross, and Wojciech Matusik. 2010. Design and fabrication of materials with desired deformation behavior. ACM Trans. Graph. 29, 4 (2010), 63:1–63:10. Google ScholarDigital Library
    12. J. Bonet and R. D. Wood. 1997. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press.Google Scholar
    13. Joseph E. Cadman, Shiwei Zhou, Yuhang Chen, and Qing Li. 2013. On design of multi-functional microstructural materials. J. Mater. Sci. 48, 1 (2013), 51–66.Google ScholarCross Ref
    14. Vivien J. Challis and James K. Guest. 2009. Level set topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Eng. 79, 10 (2009), 1284–1308.Google ScholarCross Ref
    15. Desai Chen, David I. W. Levin, Piotr Didyk, Pitchaya Sitthi-Amorn, and Wojciech Matusik. 2013. Spec2Fab: A reducer-tuner model for translating specifications to 3D prints. ACM Trans. Graph. 32, 4 (2013), 135:1–135:10. Google ScholarDigital Library
    16. Xiang Chen, Changxi Zheng, Weiwei Xu, and Kun Zhou. 2014. An asymptotic numerical method for inverse elastic shape design. ACM Trans. Graph. 33, 4 (Aug. 2014), 95:1–95:11. Google ScholarDigital Library
    17. P. G. Coelho, P. R. Fernandes, J. M. Guedes, and H. C. Rodrigues. 2008. A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct. Multidiscipl. Optimiz. 35, 2 (2008), 107–115.Google ScholarCross Ref
    18. Christian Dick, Joachim Georgii, and Rüdiger Westermann. 2011. A real-time multigrid finite hexahedra method for elasticity simulation using CUDA. Simul. Model. Pract. Theory 19, 2 (2011), 801–816.Google ScholarCross Ref
    19. Yue Dong, Jiaping Wang, Fabio Pellacini, Xin Tong, and Baining Guo. 2010. Fabricating spatially-varying subsurface scattering. ACM Trans. Graph. 29, 4 (2010), 62:1–62:10. Google ScholarDigital Library
    20. Randal Douc. 2005. Comparison of resampling schemes for particle filtering. In 4th International Symposium on Image and Signal Processing and Analysis (ISPA’05). 64–69.Google ScholarCross Ref
    21. R. B. Haber, P. Pedersen, and J. E. Taylor. 1994. An analytical model to predict optimal material properties in the context of optimal structural design. Urbana 51 (1994), 61801.Google Scholar
    22. Miloš Hašan, Martin Fuchs, Wojciech Matusik, Hanspeter Pfister, and Szymon Rusinkiewicz. 2010. Physical reproduction of materials with specified subsurface scattering. ACM Trans. Graph. 29, 4 (2010), 61:1–61:10. Google ScholarDigital Library
    23. Steven G. Johnson. 2014. The NLopt nonlinear-optimization package. Retrieved from http://ab-initio.mit.edu/nlopt.Google Scholar
    24. Lily Kharevych, Patrick Mullen, Houman Owhadi, and Mathieu Desbrun. 2009. Numerical coarsening of inhomogeneous elastic materials. ACM Trans. Graph. 28, 3 (2009), 51:1–51:8. Google ScholarDigital Library
    25. Roderic Lakes. 1987. Foam structures with a negative poisson’s ratio. Science 235, 4792 (1987), 1038–1040.Google Scholar
    26. Robert Lipton. 1994. Optimal bounds on effective elastic tensors for orthotropic composites. Proc. Roy. Soc. A 444, 1921 (1994), 399–410.Google ScholarCross Ref
    27. Jonàs Martínez, Jérémie Dumas, Sylvain Lefebvre, and Li-Yi Wei. 2015. Structure and appearance optimization for controllable shape design. ACM Trans. Graph. 34, 6, Article 229 (Oct. 2015), 11 pages. Google ScholarDigital Library
    28. Graeme W. Milton and Andrej V. Cherkaev. 1995. Which elasticity tensors are realizable? J. Eng. Mater. Technol. 117 (1995), 483.Google ScholarCross Ref
    29. P. B. Nakshatrala, D. A. Tortorelli, and K. B. Nakshatrala. 2013. Nonlinear structural design using multiscale topology optimization. Comput. Methods Appl. Mech. Eng. 261 (2013), 167–176.Google ScholarCross Ref
    30. Stanley Osher and Ronald Fedkiw. 2006. Level Set Methods and Dynamic Implicit Surfaces. Vol. 153. Springer Science 8 Business Media.Google Scholar
    31. Julian Panetta, Qingnan Zhou, Luigi Malomo, Nico Pietroni, Paolo Cignoni, and Denis Zorin. 2015. Elastic textures for additive fabrication. ACM Trans. Graph. 34, 4, Article 135 (July 2015), 12 pages. Google ScholarDigital Library
    32. U. T. Ringertz. 1993. On finding the optimal distribution of material properties. Struct. Multidiscipl. Optimiz. 5, 4 (1993), 265–267.Google ScholarCross Ref
    33. Daniel Ritchie, Ben Mildenhall, Noah D. Goodman, and Pat Hanrahan. 2015. Controlling procedural modeling programs with stochastically-ordered sequential monte carlo. ACM Trans. Graph. 34, 4 (2015), 105:1–105:11. Google ScholarDigital Library
    34. H. Rodrigues, Jose M. Guedes, and M. P. Bendsøe. 2002. Hierarchical optimization of material and structure. Struct. Multidiscipl. Optimiz. 24, 1 (2002), 1–10.Google ScholarCross Ref
    35. Christian Schumacher, Bernd Bickel, Jan Rys, Steve Marschner, Chiara Daraio, and Markus Gross. 2015. Microstructures to control elasticity in 3D printing. ACM Trans. Graph. 34, 4 (2015), 136. Google ScholarDigital Library
    36. Eftychios Sifakis and Jernej Barbic. 2012. FEM simulation of 3D deformable solids: A practitioner’s guide to theory, discretization and model reduction. In ACM SIGGRAPH 2012 Courses. Google ScholarDigital Library
    37. Ole Sigmund. 1997. On the design of compliant mechanisms using topology optimization. Mech. Struct. Mach. 25, 4 (1997), 493–524.Google ScholarCross Ref
    38. Ole Sigmund. 2007. Morphology-based black and white filters for topology optimization. Struct. Multidiscipl. Optimiz. 33 (2007), 401–424.Google ScholarCross Ref
    39. Ole Sigmund and Kurt Maute. 2013. Topology optimization approaches. Struct. Multidiscipl. Optimiz. 48, 6 (2013), 1031–1055.Google ScholarCross Ref
    40. Ole Sigmund and Salvatore Torquato. 1996. Composites with extremal thermal expansion coefficients. Appl. Phys. Lett. 69, 21 (1996), 3203–3205.Google ScholarCross Ref
    41. Mélina Skouras, Bernhard Thomaszewski, Stelian Coros, Bernd Bickel, and Markus Gross. 2013. Computational design of actuated deformable characters. ACM Trans. Graph. 32, 4 (2013), 82:1–82:10. Google ScholarDigital Library
    42. Ondrej Stava, Juraj Vanek, Bedrich Benes, Nathan Carr, and Radomír Měch. 2012. Stress relief: Improving structural strength of 3D printable objects. ACM Trans. Graph. 31, 4 (2012), 48:1–48:11. Google ScholarDigital Library
    43. Krister Svanberg. 1987. The method of moving asymptotes—a new method for structural optimization. Int. J. Numer. Methods Eng. 24, 2 (1987), 359–373.Google ScholarCross Ref
    44. T. C. T. Ting and Tungyang Chen. 2005. Poisson’s ratio for anisotropic elastic materials can have no bounds. Quart. J. Mech. Appl. Math. 58, 1 (2005), 73–82.Google ScholarCross Ref
    45. Kiril Vidimče, Szu-Po Wang, Jonathan Ragan-Kelley, and Wojciech Matusik. 2013. OpenFab: A programmable pipeline for multi-material fabrication. ACM Trans. Graph. 32, 4 (2013), 136:1–136:12. Google ScholarDigital Library
    46. A. Wächter and L. T. Biegler. 2006. On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106, 1 (2006), 25–57. Google ScholarDigital Library
    47. F. Wang, O. Sigmund, and J. S. Jensen. 2014. Design of materials with prescribed nonlinear properties. J. Mech. Phys. Solids 69 (2014), 156–174.Google ScholarCross Ref
    48. Jun Wu, Christian Dick, and Rüdiger Westermann. 2016. A system for high-resolution topology optimization. IEEE Trans. Vis. Comput. Graph. 22, 3 (2016), 1195–1208. Google ScholarDigital Library
    49. Liang Xia and Piotr Breitkopf. 2014. Concurrent topology optimization design of material and structure within nonlinear multiscale analysis framework. Comput. Methods Appl. Mech. Eng. 278 (2014), 524–542.Google ScholarCross Ref
    50. Liang Xia and Piotr Breitkopf. 2015a. Design of materials using topology optimization and energy-based homogenization approach in matlab. Struct. Multidiscipl. Optimiz. 52, 6 (2015), 1229–1241. Google ScholarDigital Library
    51. Liang Xia and Piotr Breitkopf. 2015b. Multiscale structural topology optimization with an approximate constitutive model for local material microstructure. Comput. Methods Appl. Mech. Eng. 286 (2015), 147–167.Google ScholarCross Ref
    52. Hongyi Xu, Yijing Li, Yong Chen, and Jernej Barbivč. 2015. Interactive material design using model reduction. ACM Trans. Graph. 34, 2 (2015), 18:1–18:14. Google ScholarDigital Library
    53. Xiaolei Yan, Xiaodong Huang, Guangyong Sun, and Yi Min Xie. 2015. Two-scale optimal design of structures with thermal insulation materials. Comp. Struct. 120 (2015), 358–365.Google ScholarCross Ref
    54. X. Yan, X. Huang, Y. Zha, and Y. M. Xie. 2014. Concurrent topology optimization of structures and their composite microstructures. Comput. Struct. 133 (2014), 103–110. Google ScholarDigital Library
    55. Qingnan Zhou, Julian Panetta, and Denis Zorin. 2013. Worst-case structural analysis. ACM Trans. Graph. 32, 4 (2013), 137:1–137:12. Google ScholarDigital Library
    56. Yongning Zhu and Robert Bridson. 2005. Animating sand as a fluid. ACM Trans. Graph. 24, 3 (2005), 965–972. Google ScholarDigital Library

ACM Digital Library Publication: