Structural Engineering and Mechanics

Techno-Press (테크노프레스)
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A topology optimization method of multiple load cases and constraints based on element independent nodal density

  • Yi, Jijun (School of Mechanical and Electrical Engineering, Central South University) ;
  • Rong, Jianhua (School of Mechanical and Electrical Engineering, Changsha University of Science and Technology) ;
  • Zeng, Tao (School of Mechanical and Electrical Engineering, Central South University) ;
  • Huang, X. (School of Civil, Environmental and Chemical Engineering, RMIT University)
  • Received : 2012.06.28
  • Accepted : 2013.02.19
  • Published : 2013.03.25
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Abstract

In this paper, a topology optimization method based on the element independent nodal density (EIND) is developed for continuum solids with multiple load cases and multiple constraints. The optimization problem is formulated ad minimizing the volume subject to displacement constraints. Nodal densities of the finite element mesh are used a the design variable. The nodal densities are interpolated into any point in the design domain by the Shepard interpolation scheme and the Heaviside function. Without using additional constraints (such ad the filtering technique), mesh-independent, checkerboard-free, distinct optimal topology can be obtained. Adopting the rational approximation for material properties (RAMP), the topology optimization procedure is implemented using a solid isotropic material with penalization (SIMP) method and a dual programming optimization algorithm. The computational efficiency is greatly improved by multithread parallel computing with OpenMP to run parallel programs for the shared-memory model of parallel computation. Finally, several examples are presented to demonstrate the effectiveness of the developed techniques.

Keywords

References

  1. Beckers, M. (1999), "Topology optimization using a dual method with discrete variables", Struct. Optim., 17(1), 14-24. https://doi.org/10.1007/BF01197709
  2. Bendsoe, M.P. and Kikuchi, N. (1988), "Generating optimal topologies in structural design using a homogenization method", Comput. Meth. Appl. Mech. Eng., 71(2), 197-224. https://doi.org/10.1016/0045-7825(88)90086-2
  3. Bendsoe, M.P. (1989), "Optimal shape design as a material distribution problem", Struct. Multidisc. Optim., 1(4), 193-202. https://doi.org/10.1007/BF01650949
  4. Bendsoe, M.P. and Sigmund, O. (2003), Topology Optimization: Theory, Methods, and Applications. Springer-Verlag, New York, USA
  5. Brodlie, K.W., Asim, M.R. and Unsworth, K. (2005), "Constrained visualization using the shepard interpolation family", Comput. Graphics Forum, 24(4), 809-820. https://doi.org/10.1111/j.1467-8659.2005.00903.x
  6. Carbonari, R.C., Silva, E.C.N. and Nishiwaki, S. (2004), "Topology optimization applied to the design of multi-actuated piezoelectric micro-tools", Smart Structures and Materials 2004: Modeling, Signal Processing, and Control (Proceedings of SPIE), San Diego, USA, March.
  7. Chapman, B., Jost, G. and Pas, R.V.D. (2007), Using OpenMP: Portable Shared Memory Parallel Programming, The MIT Press, Cambridge, Massachusetts, USA.
  8. Colominas, I., Parıs, J., Navarrina, F. and Casteleiro, M. (2009), "High performance parallel computing in structural topology optimization", Proceedings of the 12th International Conference on Civil, Structural and Environmental Engineering Computing, Funchal, Portugal, September.
  9. Diaz, A. and Sigmund, O. (1995), "Checkerboard patterns in layout optimization", Struct. Optim., 10(1), 40-45. https://doi.org/10.1007/BF01743693
  10. Guest, J.K., Prevost, J.H. and Belytschko, T. (2004), "Achieving minimum length scale in topology optimization using nodal design variables and projection functions", Int. J. Numer. Meth. Eng., 61(2), 238-254. https://doi.org/10.1002/nme.1064
  11. Huang, X. and Xie, Y. (2007), "Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method", FINITE ELEM. ANAL. DES., 43(14), 1039-1049. https://doi.org/10.1016/j.finel.2007.06.006
  12. Huang, X. and Xie, Y.M. (2010), Evolutionary Topology Optimization of Continuum Structures: Methods and Applications, John Wiley & Sons, Chichester, West Sussex, UK.
  13. Jog, C.S. and Haber, R.B. (1996), "Stability of finite elements models for distributed-parameter optimization and topology design", Comput. Meth. Appl. Mech. Eng., 130(3-4), 203-226. https://doi.org/10.1016/0045-7825(95)00928-0
  14. Kang, Z. and Wang, Y.Q. (2011), "Structural topology optimization based on non-local Shepard interpolation of density field", Comput. Meth. Appl. Mech. Eng., 200(49-52), 3515-3525. https://doi.org/10.1016/j.cma.2011.09.001
  15. Lee, D.K. (2007), "Combined topology and shape optimization of structures using nodal density as design parameter ", J. ASIAN. ARCHIT. BUILD., 6(1), 159-166. https://doi.org/10.3130/jaabe.6.159
  16. Lee, D.K., Kim, J.H., Starossek, U. and Shin, S.M. (2012), "Evaluation of structural outrigger belt truss layouts for tall buildings by using topology optimization", Struct. Eng. Mech., 43(6), 711-724. https://doi.org/10.12989/sem.2012.43.6.711
  17. Lee, E.H. and Park, J. (2011), "Structural design using topology and shape optimization", Struct. Eng. Mech., 38(4), 517-527. https://doi.org/10.12989/sem.2011.38.4.517
  18. Matsui, K. and Terada, K. (2004), "Continuous approximation of material distribution for topology optimization", Int. J. Numer. Meth. Eng., 59(14), 1925-1944. https://doi.org/10.1002/nme.945
  19. Paulino, G.H. and Le, C.H. (2009), "A modified Q4/Q4 element for topology optimization", Struct. Multidisc. Optim., 37(3), 255-264. https://doi.org/10.1007/s00158-008-0228-5
  20. Poulsen, T.A. (2002), "Topology optimization in wavelet space", Int. J. Numer. Meth. Eng., 53(3), 567-582. https://doi.org/10.1002/nme.285
  21. Rahmatalla, S.F. and Swan, C.C. (2004), "A Q4/Q4 continuum structural topology optimization implementation", Struct. Multidisc. Optim., 27(1), 130-135. https://doi.org/10.1007/s00158-003-0365-9
  22. Rong, J.H., Li, W.X. and Feng, B. (2010), "A structural topological optimization method based on varying displacement limits and design space adjustments", Adv. Mater. Res., 97-101, 3609. https://doi.org/10.4028/www.scientific.net/AMR.97-101.3609
  23. Rong, J.H. and Liang, Q.Q. (2008), "A level set method for topology optimization of continuum structures with bounded design domains", Comput. Meth. Appl. Mech. Eng., 197(17-18), 1447-1465. https://doi.org/10.1016/j.cma.2007.11.026
  24. Rozvany, G.I.N. (2001), "Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics", Struct. Multidisc. Optim., 21(2), 90-108. https://doi.org/10.1007/s001580050174
  25. Rozvany, G.I.N., Zhou, M. and Birker, T. (1992), "Generalized shape optimization without homogenization", Struct. Multidisc. Optim., 4(3-4), 250-254. https://doi.org/10.1007/BF01742754
  26. Sethian, A. and Wiegmann, A. (2000), "Structural boundary design via level set and immersed interface methods", J. Comput. Phys., 163(2), 489-528. https://doi.org/10.1006/jcph.2000.6581
  27. Shepard, D. (1968), "A two-dimensional interpolation function for irregularly-spaced data", Proceedings of the 1968 23rd ACM National Conference, New York, January
  28. Sigmund, O. and Petersson, J. (1998), "Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima", Struct. Optim., 16(1), 68-75. https://doi.org/10.1007/BF01214002
  29. Stolpe, M. and Svanberg, K. (2001), "An alternative interpolation scheme for minimum compliance topology optimization", Struct. Multidisc. Optim., 22(2), 116-124. https://doi.org/10.1007/s001580100129
  30. Swan, C.C. and Kosaka, I. (1997), "Voigt-Reuss topology optimization for structures with linear elastic material behaviours", Int. J. Numer. Meth. Eng., 40(16), 3033-3057. https://doi.org/10.1002/(SICI)1097-0207(19970830)40:16<3033::AID-NME196>3.0.CO;2-Z
  31. Wang, M.Y., Wang, X.M. and Guo, D.M. (2003), "A level set method for structural topology optimization", Comput. Meth. Appl. Mech. Eng., 192(1), 227-246. https://doi.org/10.1016/S0045-7825(02)00559-5
  32. Xie, Y.M. and Steven, G.P. (1993), "A simple evolutionary procedure for structural optimization", Comput. Struct., 49(5), 885-896. https://doi.org/10.1016/0045-7949(93)90035-C
  33. Xie, Y.M. and Steven, G.P. (1997), Evolutionary Structural Optimization. Springer, Berlin, Germany.
  34. Yang, R.J. and Chuang, C.H. (1994), "Optimal topology design using linear programming", Comput. Struct., 52(2), 265-275. https://doi.org/10.1016/0045-7949(94)90279-8

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