Concurrent optimization of structural topology and infill properties with a CBF-based level set method

  • Long Jiang
  • Yang Guo
  • Shikui ChenEmail author
  • Peng Wei
  • Na Lei
  • Xianfeng David Gu
Open Access
Research Article
Part of the following topical collections:
  1. Structural Topology Optimization


In this paper, a parametric level-set-based topology optimization framework is proposed to concurrently optimize the structural topology at the macroscale and the effective infill properties at the micro/meso scale. The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming. Within the proposed framework, both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming, which is more advantageous compared with the conventional partial differential equation-driven level set approach. Moreover, the proposed approach will be more efficient in handling nonlinear problems with multiple constraints. Instead of using radial basis functions (RBF), in this paper, we propose to construct a new type of cardinal basis functions (CBF) for the level set function parameterization. The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables. This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method, where the lower and upper bounds of the design variables oftentimes have to be set by trial and error. A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distanceregularized shape. With the distance information embedded in the level set model, the wrapping boundary layer and the interior infill region can be naturally defined. The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method, which leads to a hierarchical physical structure with optimized overall topology and effective infill properties. The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design.


concurrent topology optimization parametric level set method cardinal basis function shell-infill structure design conformal mapping 



The authors acknowledge the support from the National Science Foundation of the United States (Grant Nos. CMMI1462270 and CMMI1762287), Ford University Research Program (URP), and the start-up fund from the State University of New York at Stony Brook.


  1. 1.
    Gibson L J, Ashby M F. Cellular Solids: Structure and Properties. 2nd ed. Cambridge: Cambridge University Press, 1999zbMATHGoogle Scholar
  2. 2.
    Christensen R M. Mechanics of cellular and other low-density materials. International Journal of Solids and Structures, 2000, 37 (1–2): 93–104MathSciNetzbMATHGoogle Scholar
  3. 3.
    Zheng X, Lee H, Weisgraber T H, et al. Ultralight, ultrastiff mechanical metamaterials. Science, 2014, 344(6190): 1373–1377Google Scholar
  4. 4.
    Valdevit L, Jacobsen A J, Greer J R, et al. Protocols for the optimal design of multi-functional cellular structures: From hypersonics to micro-architected materials. Journal of the American Ceramic Society, 2011, 94(s1): s15–s34Google Scholar
  5. 5.
    Clausen A, Wang F, Jensen J S, et al. Topology optimized architectures with programmable Poisson’s ratio over large deformations. Advanced Materials, 2015, 27(37): 5523–5527Google Scholar
  6. 6.
    Schwerdtfeger J, Wein F, Leugering G, et al. Design of auxetic structures via mathematical optimization. Advanced Materials, 2011, 23(22–23): 2650–2654Google Scholar
  7. 7.
    Murr L E, Gaytan S M, Ramirez D A, et al. Metal fabrication by additive manufacturing using laser and electron beam melting technologies. Journal of Materials Science and Technology, 2012, 28(1): 1–14Google Scholar
  8. 8.
    Han S C, Lee J W, Kang K. A new type of low density material: Shellular. Advanced Materials, 2015, 27(37): 5506–5511Google Scholar
  9. 9.
    Xia Q, Wang M Y. Simultaneous optimization of the material properties and the topology of functionally graded structures. Computer Aided Design, 2008, 40(6): 660–675Google Scholar
  10. 10.
    Xia L, Breitkopf P. Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework. Computer Methods in Applied Mechanics and Engineering, 2014, 278: 524–542MathSciNetzbMATHGoogle Scholar
  11. 11.
    Sigmund O. Design of material structures using topology optimization. Dissertation for the Doctoral Degree. Lyngby: Technical University of Denmark, 1994Google Scholar
  12. 12.
    Bendsøe M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fujii D, Chen B, Kikuchi N. Composite material design of twodimensional structures using the homogenization design method. International Journal for Numerical Methods in Engineering, 2001, 50(9): 2031–2051MathSciNetzbMATHGoogle Scholar
  14. 14.
    Neves M M, Sigmund O, Bendsøe M P. Topology optimization of periodic microstructures with a penalization of highly localized buckling modes. International Journal for Numerical Methods in Engineering, 2002, 54(6): 809–834MathSciNetzbMATHGoogle Scholar
  15. 15.
    Bendsøe M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202Google Scholar
  16. 16.
    Rozvany G I, Zhou M, Birker T. Generalized shape optimization without homogenization. Structural Optimization, 1992, 4(3–4): 250–252Google Scholar
  17. 17.
    Sethian J A, Wiegmann A. Structural boundary design via level set and immersed interface methods. Journal of Computational Physics, 2000, 163(2): 489–528MathSciNetzbMATHGoogle Scholar
  18. 18.
    Wang M Y, Wang X, Guo D. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1–2): 227–246MathSciNetzbMATHGoogle Scholar
  19. 19.
    Allaire G, Jouve F, Toader A M. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 2004, 194(1): 363–393MathSciNetzbMATHGoogle Scholar
  20. 20.
    Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896Google Scholar
  21. 21.
    Sigmund O. Materials with prescribed constitutive parameters: An inverse homogenization problem. International Journal of Solids and Structures, 1994, 31(17): 2313–2329MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sigmund O. Tailoring materials with prescribed elastic properties. Mechanics of Materials, 1995, 20(4): 351–368Google Scholar
  23. 23.
    Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids, 1997, 45(6): 1037–1067MathSciNetGoogle Scholar
  24. 24.
    Mei Y, Wang X. A level set method for structural topology optimization and its applications. Advances in Engineering Software, 2004, 35(7): 415–441zbMATHGoogle Scholar
  25. 25.
    Vogiatzis P, Chen S, Wang X, et al. Topology optimization of multimaterial negative Poisson’s ratio metamaterials using a reconciled level set method. Computer Aided Design, 2017, 83: 15–32MathSciNetGoogle Scholar
  26. 26.
    Wang Y, Luo Z, Zhang N, et al. Topological shape optimization of microstructural metamaterials using a level set method. Computational Materials Science, 2014, 87: 178–186Google Scholar
  27. 27.
    Huang X, Radman A, Xie Y. Topological design of microstructures of cellular materials for maximum bulk or shear modulus. Computational Materials Science, 2011, 50(6): 1861–1870Google Scholar
  28. 28.
    Huang X, Xie Y M, Jia B, et al. Evolutionary topology optimization of periodic composites for extremal magnetic permeability and electrical permittivity. Structural and Multidisciplinary Optimization, 2012, 46(3): 385–398MathSciNetzbMATHGoogle Scholar
  29. 29.
    Huang X, Zhou S, Xie Y, et al. Topology optimization of microstructures of cellular materials and composites for macrostructures. Computational Materials Science, 2013, 67: 397–407Google Scholar
  30. 30.
    Xia L, Xia Q, Huang X, et al. Bi-directional evolutionary structural optimization on advanced structures and materials: A comprehensive review. Archives of Computational Methods in Engineering, 2018, 25(2): 437–478MathSciNetzbMATHGoogle Scholar
  31. 31.
    Xia L, Breitkopf P. Recent advances on topology optimization of multiscale nonlinear structures. Archives of Computational Methods in Engineering, 2017, 24(2): 227–249MathSciNetzbMATHGoogle Scholar
  32. 32.
    Rodrigues H, Guedes J M, Bendsøe M P. Hierarchical optimization of material and structure. Structural and Multidisciplinary Optimization, 2002, 24(1): 1–10Google Scholar
  33. 33.
    Coelho P, Fernandes P, Guedes J, et al. A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Structural and Multidisciplinary Optimization, 2008, 35(2): 107–115Google Scholar
  34. 34.
    Liu L, Yan J, Cheng G. Optimum structure with homogeneous optimum truss-like material. Computers & Structures, 2008, 86(13–14): 1417–1425Google Scholar
  35. 35.
    Deng J, Chen W. Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty. Structural and Multidisciplinary Optimization, 2017, 56(1): 1–19MathSciNetGoogle Scholar
  36. 36.
    Wu J, Clausen A, Sigmund O. Minimum compliance topology optimization of shell-infill composites for additive manufacturing. Computer Methods in Applied Mechanics and Engineering, 2017, 326: 358–375MathSciNetGoogle Scholar
  37. 37.
    Wu J, Aage N, Westermann R, et al. Infill optimization for additive manufacturing approaching bone-like porous structures. IEEE Transactions on Visualization and Computer Graphics, 2018, 24(2): 1127–1140Google Scholar
  38. 38.
    Deng J, Yan J, Cheng G. Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material. Structural and Multidisciplinary Optimization, 2013, 47(4): 583–597MathSciNetzbMATHGoogle Scholar
  39. 39.
    Sivapuram R, Dunning P D, Kim H A. Simultaneous material and structural optimization by multiscale topology optimization. Structural and Multidisciplinary Optimization, 2016, 54(5): 1267–1281MathSciNetGoogle Scholar
  40. 40.
    Wang Y, Chen F, Wang M Y. Concurrent design with connectable graded microstructures. Computer Methods in Applied Mechanics and Engineering, 2017, 317: 84–101MathSciNetGoogle Scholar
  41. 41.
    Li H, Luo Z, Zhang N, et al. Integrated design of cellular composites using a level-set topology optimization method. Computer Methods in Applied Mechanics and Engineering, 2016, 309: 453–475MathSciNetGoogle Scholar
  42. 42.
    Li H, Luo Z, Gao L, et al. Topology optimization for concurrent design of structures with multi-patch microstructures by level sets. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 536–561MathSciNetGoogle Scholar
  43. 43.
    Yan X, Huang X, Zha Y, et al. Concurrent topology optimization of structures and their composite microstructures. Computers & Structures, 2014, 133: 103–110Google Scholar
  44. 44.
    Yan X, Huang X, Sun G, et al. Two-scale optimal design of structures with thermal insulation materials. Composite Structures, 2015, 120: 358–365Google Scholar
  45. 45.
    Da D, Cui X, Long K, et al. Concurrent topological design of composite structures and the underlying multi-phase materials. Computers & Structures, 2017, 179: 1–14Google Scholar
  46. 46.
    Vicente W, Zuo Z, Pavanello R, et al. Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures. Computer Methods in Applied Mechanics and Engineering, 2016, 301: 116–136MathSciNetGoogle Scholar
  47. 47.
    Osher S, Sethian J A. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 1988, 79(1): 12–49MathSciNetzbMATHGoogle Scholar
  48. 48.
    Osher S J, Santosa F. Level set methods for optimization problems involving geometry and constraints: I. Frequencies of a two-density inhomogeneous drum. Journal of Computational Physics, 2001, 171(1): 272–288MathSciNetzbMATHGoogle Scholar
  49. 49.
    Allaire G, Jouve F, Toader A M. A level-set method for shape optimization. Comptes Rendus Mathematique, 2002, 334(12): 1125–1130MathSciNetzbMATHGoogle Scholar
  50. 50.
    Luo Z, Tong L, Wang M Y, et al. Shape and topology optimization of compliant mechanisms using a parameterization level set method. Journal of Computational Physics, 2007, 227(1): 680–705MathSciNetzbMATHGoogle Scholar
  51. 51.
    Luo Z, Wang M Y, Wang S, et al. A level set-based parameterization method for structural shape and topology optimization. International Journal for Numerical Methods in Engineering, 2008, 76(1): 1–26MathSciNetzbMATHGoogle Scholar
  52. 52.
    Jiang L, Chen S. Parametric structural shape & topology optimization with a variational distance-regularized level set method. Computer Methods in Applied Mechanics and Engineering, 2017, 321: 316–336MathSciNetGoogle Scholar
  53. 53.
    Wang S, Wang M Y. Radial basis functions and level set method for structural topology optimization. International Journal for Numerical Methods in Engineering, 2006, 65(12): 2060–2090MathSciNetzbMATHGoogle Scholar
  54. 54.
    Buhmann M D. Radial Basis Functions: Theory and Implementations. Vol. 12. Cambridge: Cambridge University Press, 2003zbMATHGoogle Scholar
  55. 55.
    Svanberg K. MMA and GCMM A-two methods for nonlinear optimization. 2007. Retrieved from mmagcmma.pdf.Google Scholar
  56. 56.
    Chen J, Shapiro V, Suresh K, et al. Shape optimization with topological changes and parametric control. International Journal for Numerical Methods in Engineering, 2007, 71(3): 313–346MathSciNetzbMATHGoogle Scholar
  57. 57.
    Qian X. Full analytical sensitivities in NURBS based isogeometric shape optimization. Computer Methods in Applied Mechanics and Engineering, 2010, 199(29–32): 2059–2071MathSciNetzbMATHGoogle Scholar
  58. 58.
    Wang Y, Benson D J. Isogeometric analysis for parameterized LSMbased structural topology optimization. Computational Mechanics, 2016, 57(1): 19–35MathSciNetzbMATHGoogle Scholar
  59. 59.
    Safdari-Vaighani A, Heryudono A, Larsson E. A radial basis function partition of unity collocation method for convectiondiffusion equations arising in financial applications. Journal of Scientific Computing, 2015, 64(2): 341–367MathSciNetzbMATHGoogle Scholar
  60. 60.
    Li C, Xu C, Gui C, et al. Distance regularized level set evolution and its application to image segmentation. IEEE Transactions on Image Processing, 2010, 19(12): 3243–3254MathSciNetzbMATHGoogle Scholar
  61. 61.
    Zhu B, Wang R, Li H, et al. A level set method with a bounded diffusion for structural topology optimization. Journal of Mechanical Design, 2018, 140(7): 071402Google Scholar
  62. 62.
    Jiang L, Chen S, Jiao X. Parametric shape and topology optimization: A new level set approach based on cardinal basis functions. International Journal for Numerical Methods in Engineering, 2018, 114(1): 66–87MathSciNetGoogle Scholar
  63. 63.
    Jiang L, Ye H, Zhou C, et al. Parametric topology optimization toward rational design and efficient prefabrication for additive manufacturing. In: Proceedings of ASME 2017 12th International Manufacturing Science and Engineering Conference Collocated with the JSME/ASME 2017 6th International Conference on Materials and Processing. Los Angeles: ASME, 2017, V004T05A006Google Scholar
  64. 64.
    Wang Y, Kang Z. A level set method for shape and topology optimization of coated structures. Computer Methods in Applied Mechanics and Engineering, 2018, 329: 553–574MathSciNetGoogle Scholar
  65. 65.
    Fu J, Li H, Xiao M, et al. Topology optimization of shell-infill structures using a distance regularized parametric level-set method. Structural and Multidisciplinary Optimization, 2019, 59(1): 249–262MathSciNetGoogle Scholar
  66. 66.
    Tushinsky L, Kovensky I, Plokhov A, et al. Coated Metal: Structure and Properties of Metal-Coating Compositions. Berlin: Springer, 2013Google Scholar
  67. 67.
    Challis V, Roberts A, Wilkins A. Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. International Journal of Solids and Structures, 2008, 45(14–15): 4130–4146zbMATHGoogle Scholar
  68. 68.
    Neves M, Rodrigues H, Guedes J M. Optimal design of periodic linear elastic microstructures. Computers & Structures, 2000, 76(1–3): 421–429Google Scholar
  69. 69.
    Radman A, Huang X, Xie Y. Topological optimization for the design of microstructures of isotropic cellular materials. Engineering Optimization, 2013, 45(11): 1331–1348MathSciNetGoogle Scholar
  70. 70.
    Guth D, Luersen M, Muñoz-Rojas P. Optimization of threedimensional truss-like periodic materials considering isotropy constraints. Structural and Multidisciplinary Optimization, 2015, 52(5): 889–901MathSciNetGoogle Scholar
  71. 71.
    Vogiatzis P, Ma M, Chen S, et al. Computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping. Computer Methods in Applied Mechanics and Engineering, 2018, 328: 477–497MathSciNetGoogle Scholar
  72. 72.
    Sethian J A. Theory, algorithms, and applications of level set methods for propagating interfaces. Acta Numerica, 1996, 5: 309–395MathSciNetzbMATHGoogle Scholar
  73. 73.
    Tumbleston J R, Shirvanyants D, Ermoshkin N, et al. Continuous liquid interface production of 3D objects. Science, 2015, 347(6228): 1349–1352Google Scholar
  74. 74.
    Thomsen C R, Wang F, Sigmund O. Buckling strength topology optimization of 2D periodic materials based on linearized bifurcation analysis, Computer Methods in Applied Mechanics and Engineering, 2018, 339: 115–136Google Scholar
  75. 75.
    Ahlfors L V. Conformal Invariants: Topics in Geometric Function Theory. Vol. 371. Providence: American Mathematical Society, 2010zbMATHGoogle Scholar
  76. 76.
    Jin M, Kim J, Luo F, et al. Discrete surface Ricci flow. IEEE Transactions on Visualization and Computer Graphics, 2008, 14(5): 1030–1043Google Scholar
  77. 77.
    Chow B, Luo F. Combinatorial Ricci flows on surfaces. Journal of Differential Geometry, 2003, 63(1): 97–129MathSciNetzbMATHGoogle Scholar
  78. 78.
    Zeng W, Gu X D. Ricci Flow for Shape Analysis and Surface Registration: Theories, Algorithms and Applications. New York: Springer, 2013zbMATHGoogle Scholar
  79. 79.
    Gu X, He Y, Jin M, et al. Manifold splines with a single extraordinary point. Computer Aided Design, 2008, 40(6): 676–690zbMATHGoogle Scholar
  80. 80.
    Jin M, Luo F, Gu X. Computing surface hyperbolic structure and real projective structure. In: Proceedings of the 2006 ACM Symposium on Solid and Physical Modeling. New York: ACM, 2006, 105–116Google Scholar
  81. 81.
    Gu X D, Zeng W, Luo F, et al. Numerical computation of surface conformal mappings. Computational Methods and Function Theory, 2012, 11(2): 747–787MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder

To view a copy of this licence, visit by/4.0/.

Authors and Affiliations

  • Long Jiang
    • 1
  • Yang Guo
    • 2
  • Shikui Chen
    • 1
    Email author
  • Peng Wei
    • 3
  • Na Lei
    • 4
  • Xianfeng David Gu
    • 2
  1. 1.Department of Mechanical EngineeringState University of New York at Stony BrookStony BrookUSA
  2. 2.Department of Computer ScienceState University of New York at Stony BrookStony BrookUSA
  3. 3.State Key Laboratory of Subtropical Building Science, School of Civil Engineering and TransportationSouth China University of TechnologyGuangzhouChina
  4. 4.DUT-RU International School of Information Science & EngineeringDalian University of TechnologyDalianChina

Personalised recommendations