Acta Mechanica Sinica

, Volume 34, Issue 2, pp 315–326 | Cite as

Concurrent topology optimization for minimization of total mass considering load-carrying capabilities and thermal insulation simultaneously

  • Kai Long
  • Xuan Wang
  • Xianguang Gu
Research Paper


The present work introduces a novel concurrent optimization formulation to meet the requirements of lightweight design and various constraints simultaneously. Nodal displacement of macrostructure and effective thermal conductivity of microstructure are regarded as the constraint functions, which means taking into account both the load-carrying capabilities and the thermal insulation properties. The effective properties of porous material derived from numerical homogenization are used for macrostructural analysis. Meanwhile, displacement vectors of macrostructures from original and adjoint load cases are used for sensitivity analysis of the microstructure. Design variables in the form of reciprocal functions of relative densities are introduced and used for linearization of the constraint function. The objective function of total mass is approximately expressed by the second order Taylor series expansion. Then, the proposed concurrent optimization problem is solved using a sequential quadratic programming algorithm, by splitting into a series of sub-problems in the form of the quadratic program. Finally, several numerical examples are presented to validate the effectiveness of the proposed optimization method. The various effects including initial designs, prescribed limits of nodal displacement, and effective thermal conductivity on optimized designs are also investigated. An amount of optimized macrostructures and their corresponding microstructures are achieved.


Concurrent design Topology optimization Homogenization Thermal insulation Nodal displacement Independent continuous mapping method 



The project was supported by the National Natural Science Foundation of China (Grants 11202078, 51405123) and the Fundamental Research Funds for the Central Universities (Grant 2017MS077). We are thankful for Professor Krister Svanberg for MMA program made freely available for research purposes.


  1. 1.
    Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bendsøe, M.P.: Optimal shape design as a material distribution problem. Struct. Optim. 1, 193–202 (1989)CrossRefGoogle Scholar
  3. 3.
    Zhou, M., Rozvany, G.I.N.: The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89, 309–336 (1991)CrossRefGoogle Scholar
  4. 4.
    Xie, Y.M., Steven, G.P.: A simple evolutionary procedure for structural optimization. Comput. Struct. 49, 885–896 (1993)CrossRefGoogle Scholar
  5. 5.
    Huang, X., Xie, Y.M.: Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem. Anal. Des. 43, 1039–1049 (2007)CrossRefGoogle Scholar
  6. 6.
    Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192, 227–246 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sethian, J.A., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163, 489–528 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Allaire, G., Jouve, F., Toader, A.M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhou, S., Wang, M.Y.: Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition. Struct. Multidiscip. Optim. 33, 89 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guo, X., Zhang, W., Zhong, W.: Doing topology optimization explicitly and geometricallyła new moving morphable components based framework. J. Appl. Mech. 81, 081009 (2014)CrossRefGoogle Scholar
  11. 11.
    Guo, X., Zhang, W., Zhang, J., et al.: Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Comput. Methods Appl. Mech. Eng. 310, 711–748 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zhang, W., Zhang, J., Guo, X.: Lagrangian description based topology optimization—a revival of shape optimization. J. Appl. Mech. 83, 041010 (2016)CrossRefGoogle Scholar
  13. 13.
    Zhang, W., Yang, W., Zhou, J., et al.: Structural topology optimization through explicit boundary evolution. J. Appl. Mech. 84, 011011 (2016)CrossRefGoogle Scholar
  14. 14.
    Zhang, W., Chen, J., Zhu, X., et al.: Explicit three dimensional topology optimization via moving morphable void (MMV) approach. Comput. Methods Appl. Mech. Eng. 322, 590–614 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Guo, X., Zhou, J., Zhang, W., et al.: Self-supporting structure design in additive manufacturing through explicit topology optimization. Comput. Methods Appl. Mech. Eng. 323, 27–63 (2017)Google Scholar
  16. 16.
    Eschenauer, H.A., Olhoff, N.: Topology optimization of continuum structures: a review. J. Appl. Mech. Appl. Mech. Rev. 54, 331–390 (2001)CrossRefGoogle Scholar
  17. 17.
    Rozvany, G.I.N.: A critical review of established methods of structural topology optimization. Struct. Multidiscip. Optim. 37, 217–237 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sigmund, O., Maute, K.: Topology optimization approaches. Struct. Multidiscip. Optim. 48, 1031–1055 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31, 2313–2329 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sigmund, O.: Tailoring materials with prescribed elastic properties. Mech. Mater. 20, 351–368 (1995)CrossRefGoogle Scholar
  21. 21.
    Clausen, A., Wang, F., Jensen, J.S., et al.: Topology optimized architectures with programmable Poisson’s ratio over large deformations. Adv. Mater. 27, 5523–5527 (2015)CrossRefGoogle Scholar
  22. 22.
    Xie, Y.M., Yang, X., Shen, J., et al.: Designing orthotropic materials for negative or zero compressibility. Int. J. Solids Struct. 51, 4038–4051 (2014)CrossRefGoogle Scholar
  23. 23.
    Wang, X., Xu, S., Zhou, S., et al.: Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: a review. Biomaterials 83, 127–141 (2016)CrossRefGoogle Scholar
  24. 24.
    Rodrigues, H., Guedes, J.M., Bendsoe, M.P.: Hierarchical optimization of material and structure. Struct. Multidiscip. Optim. 24, 1–10 (2002)CrossRefGoogle Scholar
  25. 25.
    Coelho, P.G., Fernandes, P.R., Guedes, J.M., et al.: A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct. Multidiscip. Optim. 35, 107–115 (2008)CrossRefGoogle Scholar
  26. 26.
    Liu, L., Yan, J., Cheng, G.: Optimum structure with homogeneous optimum truss-like material. Comput. Struct. 86, 1417–1425 (2008)CrossRefGoogle Scholar
  27. 27.
    Niu, B., Yan, J., Cheng, G.: Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct. Multidiscip. Optim. 39, 115–132 (2009)CrossRefGoogle Scholar
  28. 28.
    Deng, J., Yan, J., Cheng, G.: Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material. Struct. Multidiscip. Optim. 47, 583–597 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Guo, X., Zhao, X., Zhang, W., et al.: Multi-scale robust design and optimization considering load uncertainties. Comput. Methods Appl. Mech. Eng. 283, 994–1009 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Huang, X., Zhou, S.W., Xie, Y.M.: Topology optimization of microstructures of cellular materials and composites for macrostructures. Comput. Mater. Sci. 67, 397–407 (2013)CrossRefGoogle Scholar
  31. 31.
    Yan, X., Huang, X., Sun, G., et al.: Two-scale optimal design of structures with thermal insulation materials. Compos. Struct. 120, 358–365 (2015)CrossRefGoogle Scholar
  32. 32.
    Liu, Q., Chan, R., Huang, X.: Concurrent topology optimization of macrostructures and material microstructures for natural frequency. Mater. Des. 106, 380–390 (2016)CrossRefGoogle Scholar
  33. 33.
    Xu, B., Jiang, J.S., Xie, Y.M.: Concurrent design of composite macrostructure and multi-phase material microstructure for minimum dynamic compliance. Compos. Struct. 128, 221–233 (2015)CrossRefGoogle Scholar
  34. 34.
    Xu, B., Xie, Y.M.: Concurrent design of composite macrostructure and cellular microstructure under random excitations. Compos. Struct. 123, 65–77 (2015)CrossRefGoogle Scholar
  35. 35.
    Xu, B., Huang, X., Xie, Y.M.: Two-scale dynamic optimal design of composite structures in the time domain using equivalent static loads. Compos. Struct. 142, 335–345 (2016)CrossRefGoogle Scholar
  36. 36.
    Zhang, W., Sun, S.: Scale-related topology optimization of cellular materials and structures. Int. J. Numer. Methods Eng. 68, 993–1011 (2006)CrossRefzbMATHGoogle Scholar
  37. 37.
    Xia, L., Breitkopf, P.: Concurrent topology optimization design of material and structure within FE\(_2\) nonlinear multiscale analysis framework. Comput. Methods Appl. Mech. Eng. 278, 524–542 (2014)CrossRefGoogle Scholar
  38. 38.
    Xia, L., Breitkopf, P.: Recent advances on topology optimization of multiscale nonlinear structures. Arch. Comput. Methods Eng. 24, 227–249 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Jia, J., Cheng, W., Long, K., et al.: Hierarchical design of structures and multiphase material cells. Comput. Struct. 165, 136–144 (2016)CrossRefGoogle Scholar
  40. 40.
    Long, K., Han, D., Gu, X.: Concurrent topology optimization of composite macrostructure and microstructure constructed by constituent phases of distinct Poisson’s ratios for maximum frequency. Comput. Mater. Sci. 129, 194–201 (2017)CrossRefGoogle Scholar
  41. 41.
    Chen, W., Tong, L., Liu, S.: Concurrent topology design of structure and material using a two-scale topology optimization. Comput. Struct. 178, 119–128 (2017)CrossRefGoogle Scholar
  42. 42.
    Sui, Y., Peng, X.: The ICM method with objective function transformed by variable discrete condition for continuum structure. Acta Mech. Sin. 22, 68–75 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Sui, Y., Yang, D.: A new method for structural topological optimization based on the concept of independent continuous variables and smooth model. Acta Mech. Sin. 14, 179–185 (1998)CrossRefGoogle Scholar
  44. 44.
    Sui, Y.: Modelling, Transformation and Optimizationł New Developments of Structural Synthesis Method. Dalian University of Technology Press, Dalian (1996)Google Scholar
  45. 45.
    Andreassen, E., Andreasen, C.S.: How to determine composite material properties using numerical homogenization. Comput. Mater. Sci. 83, 488–495 (2014)CrossRefGoogle Scholar
  46. 46.
    Zuo, Z.H., Xie, Y.M.: Evolutionary topology optimization of continuum structures with a global displacement control. Comput. Aided Des. 56, 58–67 (2014)CrossRefGoogle Scholar
  47. 47.
    Lazarov, B.S., Sigmund, O.: Filters in topology optimization based on Helmholtz-type differential equations. Int. J. Numer. Methods Eng. 86, 765–781 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Amstutz, S., Giusti, S.M., Novotny, A.A., et al.: Topological derivative for multi-scale linear elasticity models applied to the synthesis of microstructures. Int. J. Numer. Methods Eng. 84, 733–756 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Svanberg, K.: The method of moving asymptotes-a new method for structural optimization. Int. J. Numer. Methods Eng. 24, 359–373 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Beijing Key Laboratory of Energy Safety and Clean UtilizationNorth China Electric Power UniversityBeijingChina
  2. 2.State Key Laboratory for Alternate Electrical Power System with Renewable Energy SourcesNorth China Electric Power UniversityBeijingChina
  3. 3.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering MechanicsDalian University of TechnologyDalianChina
  4. 4.School of Automobile and Traffic EngineeringHefei University of TechnologyHefeiChina

Personalised recommendations