Journal of Central South University

, Volume 21, Issue 2, pp 558–566

A topology optimization method based on element independent nodal density

  • Ji-jun Yi (易继军)
  • Tao Zeng (曾韬)
  • Jian-hua Rong (荣见华)
  • Yan-mei Li (李艳梅)
Article
  • 193 Downloads

Abstract

A methodology for topology optimization based on element independent nodal density (EIND) is developed. Nodal densities are implemented as the design variables and interpolated onto element space to determine the density of any point with Shepard interpolation function. The influence of the diameter of interpolation is discussed which shows good robustness. The new approach is demonstrated on the minimum volume problem subjected to a displacement constraint. The rational approximation for material properties (RAMP) method and a dual programming optimization algorithm are used to penalize the intermediate density point to achieve nearly 0-1 solutions. Solutions are shown to meet stability, mesh dependence or non-checkerboard patterns of topology optimization without additional constraints. Finally, the computational efficiency is greatly improved by multithread parallel computing with OpenMP.

Key words

topology optimization element independent nodal density Shepard interpolation parallel computation 

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References

  1. [1]
    BENDSØE E M P, KIKUCHI N. Generating optimal topologies in structural design using a homogenization method [J]. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224.CrossRefMathSciNetGoogle Scholar
  2. [2]
    BENDSOE M P. Optimal shape design as a material distribution problem [J]. Structural and Multidisciplinary Optimization, 1989, 1(4): 193–202.CrossRefGoogle Scholar
  3. [3]
    ROZVANY G N. Aims scope methods history and unified terminology of computer-aided topology optimization in structural mechanics [J]. Structural and Multidisciplinary Optimization, 2001, 21(2): 90–108.CrossRefGoogle Scholar
  4. [4]
    ZHAO H, LONG K, MA Z D. Homogenization topology optimization method based on continuous field [J]. Advances in Mechanical Engineering, 2010: 1–7.Google Scholar
  5. [5]
    ROZVANY G, ZHOU M, BIRKER T. Generalized shape optimization without homogenization [J]. Structural and Multidisciplinary Optimization, 1992, 4(3): 250–252.CrossRefGoogle Scholar
  6. [6]
    L G J. Simp type topology optimization procedure considering uncertain load position [J]. Civil Engineering, 2012, 56(2): 213–219.Google Scholar
  7. [7]
    AMSTUTZ S. Connections between topological sensitivity analysis and material interpolation schemes in topology optimization [J]. Structural and Multidisciplinar Optimization, 2011, 43(6): 755–765.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    XIE Y M, STEVEN G P. Evolutionary structural optimization [M]. Berlin: Springer, 1997: 37–51.Google Scholar
  9. [9]
    XIE Y M, STEVEN G P. A simple evolutionary procedure for structural optimization [J]. Computers & Structures, 1993, 49(5): 885–896.CrossRefGoogle Scholar
  10. [10]
    HUANG X, ZUO Z, XIE Y. Evolutionary topological optimization of vibrating continuum structures for natural frequencies [J]. Computers & Structures, 2010, 88(5): 357–364.CrossRefGoogle Scholar
  11. [11]
    HUANG X, XIE Y M. Evolutionary topology optimization of continuum structures: Methods and applications [M]. Chichester: John Wiley & Sons, 2010.CrossRefGoogle Scholar
  12. [12]
    SETHIAN J A, WIEGMANN A. Structural boundary design via level set and immersed interface methods [J]. Journal of Computational Physics, 2000, 163(2): 489–528.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    RONG J H, LIANG Q Q. A level set method for topology optimization of continuum structures with bounded design domains [J]. Computer Methods in Applied Mechanics and Engineering, 2008, 197(17/18): 1447–1465.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    CHEN S, CHEN W, LEE S. Level set based robust shape and topology optimization under random field uncertainties [J]. Structural and Multidisciplinary Optimization, 2010, 41(4): 507–524.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    GUO X, ZHANG W S, WANG M Y, WEI P. Stress-related topology optimization via level set approach [J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47): 3439–3452.CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    SIGMUND O, PETERSSON J. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima [J]. Struct Optim, 1998, 16(1): 68–75.CrossRefGoogle Scholar
  17. [17]
    CARBONARI R C, SILVA E C N, NISHIWAKI S. Topology optimization applied to the design of multi-actuated piezoelectric micro-tools [C]// Proceedings of SPIE 5383. Smart Structures and Materials 2004: Modeling, Signal Processing, and Control. San Diego: SPIE, 2004: 277–288.CrossRefGoogle Scholar
  18. [18]
    MATSUI K, TERADA K. Continuous approximation of material distribution for topology optimization [J]. International Journal for Numerical Methods in Engineering, 2004, 59(14): 1925–1944.CrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    RAHMATALLA S F, SWAN C C. A q4/q4 continuum structural topology optimization implementation [J]. Structural and Multidisciplinary Optimization, 2004, 27(1): 130–135.CrossRefGoogle Scholar
  20. [20]
    PAULINO G H, LE C H. A modified q4/q4 element for topology optimization [J]. Structural and Multidisciplinary Optimization, 2009, 37(3): 255–264.CrossRefMathSciNetGoogle Scholar
  21. [21]
    POULSEN T A. Topology optimization in wavelet space [J]. International Journal for Numerical Methods in Engineering, 2002, 53(3): 567–582.CrossRefMATHMathSciNetGoogle Scholar
  22. [22]
    JOG C S, HABER R B. Stability of finite elements models for distributed-parameter optimization and topology design [J]. Computer Methods in Applied Meehanies and Engineering, 1996, 130(3/4): 203–226.CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    KANG Z, WANG Y Q. Structural topology optimization based on non-local shepard interpolation of density field [J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(49): 3515–3525.CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    WANG Y, LUO Z, ZHANG N. Topological optimization of structures using a multilevel nodal density-based approximant [J]. Computer Modeling in Engineering and Sciences, 2012, 84(3): 229–236.MathSciNetGoogle Scholar
  25. [25]
    WANG Y, KANG Z, HE Q. An adaptive refinement approach for topology optimization based on separated density field description [J]. Computers & Structures, 2013, 117: 10–22.CrossRefGoogle Scholar
  26. [26]
    GUEST J K, PREVOST J H, BELYTSCHKO T. Achieving minimum length scale in topology optimization using nodal design variables and projection functions [J]. International Journal for Numerical Methods in Engineering, 2004, 61(2): 238–254.CrossRefMATHMathSciNetGoogle Scholar
  27. [27]
    DIAZ A, SIGMUND O. Checkerboard patterns in layout optimization [J]. Structural and Multidisciplinary Optimization, 1995, 10(1): 40–45.CrossRefGoogle Scholar
  28. [28]
    SHEPARD D. A two-dimensional interpolation function for irregularly-spaced data [C]// Proceedings of the 1968 3rd ACM National Conference. New York: ACM, 1968: 517–524.CrossRefGoogle Scholar
  29. [29]
    BRODLIE K W, ASIM M R, UNSWORTH K. Constrained visualization using the shepard interpolation family [J]. Computer Graphics Forum, 2005, 24(4): 809–820.CrossRefGoogle Scholar
  30. [30]
    YANG R J, CHUANG C H. Optimal topology design using linear programming [J]. Computers & Structures, 1994, 52(2): 265–275.CrossRefMATHGoogle Scholar
  31. [31]
    STOLPE M, SVANBERG K. An alternative interpolation scheme for minimum compliance topology optimization [J]. Structural and Multidisciplinary Optimization, 2001, 22(2): 116–124.CrossRefGoogle Scholar
  32. [32]
    BENDSOE M P, SIGMUND O. Topology optimization: Theory, methods, and applications [M]. New York, USA: Springer-Verlag, 2003.Google Scholar
  33. [33]
    BECKERS M. Topology optimization using a dual method with discrete variables [J]. Structural and Multidisciplinary Optimization, 1999, 17(1): 14–24.CrossRefGoogle Scholar
  34. [34]
    RONG J H, LI W X, FENG B. A structural topological optimization method based on varying displacement limits and design space adjustments [J]. Advanced Materials Research, 2010, 97-101: 3609–3616.CrossRefGoogle Scholar
  35. [35]
    CHAPMAN B, JOST G, PAS R V D. Using openmp: Portable shared memory parallel programming [M]. Cambridge, Massachusetts: The MIT Press, 2007: 35–50.Google Scholar

Copyright information

© Central South University Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ji-jun Yi (易继军)
    • 1
    • 2
  • Tao Zeng (曾韬)
    • 1
  • Jian-hua Rong (荣见华)
    • 2
  • Yan-mei Li (李艳梅)
    • 3
  1. 1.School of Mechanical and Electrical EngineeringCentral South UniversityChangshaChina
  2. 2.School of Automobile and Mechanical EngineeringChangsha University of Science and TechnologyChangshaChina
  3. 3.Hunan Technical College of Water Resources and Hydro PowerChangshaChina

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