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Acta Mechanica Solida Sinica

, Volume 32, Issue 3, pp 310–325 | Cite as

Lightweight Topology Optimization with Buckling and Frequency Constraints Using the Independent Continuous Mapping Method

  • Weiwei Wang
  • Hongling YeEmail author
  • Yunkang Sui
Article
  • 101 Downloads

Abstract

This research focuses on the lightweight topology optimization method for structures under the premise of meeting the requirements of stability and vibration characteristics. A new topology optimization model with the constraints of natural frequencies and critical buckling loads and the objective of minimizing the structural volume is established and solved based on the independent continuous mapping method. The eigenvalue equations and composite exponential filter functions are applied to convert the optimization formulation into a continuous, solvable mathematical programming model. In the process of topology optimization, suitable initial values of the filter functions are chosen to avoid local modes, and the dynamic frequency gap constraints are added in the optimal model to prevent mode switches. Furthermore, for the optimal structures with grey elements obtained by the continuous optimization model, the bisection–inverse iteration is applied to search the optimal discrete structures. Finally, a detailed scheme is given for the buckling and frequency topology optimization problem. Numerical examples illustrate that the modelling method of minimizing the economic index with given performance requirements is practical and feasible for multi-performance topology optimization problems.

Keywords

Topology optimization Lightweight Buckling constraints Frequency constraints ICM method 

References

  1. 1.
    Luo Z, Du YX, Chen LP, Yang JZ, Karim AM. Continuum topology optimization for monolithic compliant mechanisms of micro-actuators. Acta Mech Solida Sin. 2006;19(1):58–68.CrossRefGoogle Scholar
  2. 2.
    Hassani B, Hinton E. A review of homogenization and topology optimization II–analytical and numerical solution of homogenization equations. Comput Struct. 1998;69(6):719–38.CrossRefzbMATHGoogle Scholar
  3. 3.
    Bendsøe MP, Sigmund O. Material interpolation schemes in topology optimization. Arch Appl Mech. 1999;69(9–10):635–54.zbMATHGoogle Scholar
  4. 4.
    Huang XD, Xie YM. A further review of ESO type methods for topology optimization. Struct Multidiscip Optim. 2010;41(5):671–83.CrossRefGoogle Scholar
  5. 5.
    Wang XJ, Zhang XA, Cheng KP. Computer program for directed structure topology optimization. Acta Mech Solida Sin. 2015;28(4):431–40.CrossRefGoogle Scholar
  6. 6.
    Wang MY, Wang XM, Guo DM. A level set method for structural topology optimization. Comput Methods Appl Mech Eng. 2003;192:227–46.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zhang WS, Zhou JH, Zhu YC, Guo X. Structural complexity control in topology optimization via moving morphable component (MMC) approach. Struct Multidiscip Optim. 2017;56(3):535–52.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guo X, Zhang WS, Zhang J, Yuan J. Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Comput Methods Appl Mech Eng. 2016;310:711–48.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Garcke H, Hecht C. Shape and topology optimization in stokes flow with a phase field approach. Appl Math Optim. 2016;73(1):23–70.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jakiela MJ, Chapman C, Duda J, Adewuya A, Saitou K. Continuum structural topology design with genetic algorithms. Comput Methods Appl Mech Eng. 2000;186(2–4):339–56.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sui YK, Ye HL. Continuum topology optimization methods ICM. 1st ed. Beijing: Science Press; 2013 (in Chinese).Google Scholar
  12. 12.
    Lund E. Buckling topology optimization of laminated multi-material composite shell structures. Compos Struct. 2009;91(2):158–67.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lindgaard E, Dahl J. On compliance and buckling objective functions in topology optimization of snap-through problems. Struct Multidiscip Optim. 2013;47(3):409–21.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Luo QT, Tong LY. Structural topology optimization for maximum linear buckling loads by using a moving iso-surface threshold method. Struct Multidiscip Optim. 2015;52(1):71–90.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Bochenek B, Tajs-Zielin’ska K. Minimal compliance topologies for maximal buckling load of columns. Struct Multidiscip Optim. 2015;51(5):1149–57.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cheng GD, Xu L. Two-scale topology design optimization of stiffened or porous plate subject to out-of-plane buckling constraint. Struct Multidiscip Optim. 2016;54(5):1283–96.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Browne PA, Budd C, Gould NIM, Kim HA, Scott JA. A fast method for binary programming using first-order derivatives, with application to topology optimization with buckling constraints. Int J Numer Methods Eng. 2012;92(12):1026–43.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gao XJ, Ma HT. Topology optimization of continuum structures under buckling constraints. Comput Struct. 2015;157:142–52.CrossRefGoogle Scholar
  19. 19.
    Dunning PD, Ovtchinnikov E, Scott J, Kim HA. Level-set topology optimization with many linear buckling constraints using an efficient and robust eigensolver. Int J Numer Methods Eng. 2016;107(12):1029–53.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ye HL, Wang WW, Chen N, Sui YK. Plate/shell topological optimization subjected to linear buckling constraints by adopting composite exponential filtering function. Acta Mech Sin. 2016;32(4):649–58.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ye HL, Wang WW, Chen N, Sui YK. Plate/shell structure topology optimization of orthotropic material for buckling problem based on independent continuous topological variables. Acta Mech Sin. 2017;33(5):899–911.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Deng SG, Suresh K. Topology optimization under thermo-elastic buckling. Struct Multidiscip Optim. 2017;55(5):1759–72.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tsai TD, Cheng CC. Structural design for desired eigenfrequencies and mode shapes using topology optimization. Struct Multidiscip Optim. 2013;47(5):673–86.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pedersen NL. Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim. 2000;20(1):2–11.CrossRefGoogle Scholar
  25. 25.
    Niu B, Yan J, Cheng GD. Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct Multidiscip Optim. 2009;39(2):115–32.CrossRefGoogle Scholar
  26. 26.
    Huang X, Zuo ZH, Xie YM. Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct. 2010;88(5–6):357–64.CrossRefGoogle Scholar
  27. 27.
    Xia Q, Shi TL, Wang MY. A level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration. Struct Multidiscip Optim. 2011;43(4):473–85.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zheng J, Long SY, Li GY. Topology optimization of free vibrating continuum structures based on the element free Galerkin method. Struct Multidiscip Optim. 2012;45(1):119–27.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Krog LA, Olhoff N. Optimum topology and reinforcement design of disk and plate structures with multiple stiffness and eigenfrequency objectives. Comput Struct. 1999;72(4–5):535–63.CrossRefzbMATHGoogle Scholar
  30. 30.
    Ma ZD, Cheng HC, Kikuchi N. Structural design for obtaining desired eigenfrequencies by using the topology and shape optimization method. Comput Syst Eng. 1994;5(1):77–89.CrossRefGoogle Scholar
  31. 31.
    Zhou PZ, Du JB, Lü ZH. Topology optimization of freely vibrating continuum structures based on nonsmooth optimization. Struct Multidiscip Optim. 2017;56(3):603–18.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Jensen JS, Pedersen NL. On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases. J Sound Vib. 2006;289(4):967–86.CrossRefGoogle Scholar
  33. 33.
    Du JB, Olhoff N. Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct Multidiscip Optim. 2007;34(2):91–110.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Munk DJ, Vio GA, Steven GP. A simple alternative formulation for structural optimization with dynamic and buckling objectives. Struct Multidiscip Optim. 2017;55(3):969–86.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sui YK, Peng XR. Modeling, solving and application for topology optimization of continuum structures ICM method based on step function. 1st ed. Beijing: Tsinghua University Press; 2018.Google Scholar
  36. 36.
    Sigmund O. Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim. 2007;33(4–5):401–24.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics 2019

Authors and Affiliations

  1. 1.College of Mechanical Engineering and Applied Electronics TechnologyBeijing University of TechnologyBeijingChina

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