Journal of Mechanical Science and Technology

The Korean Society of Mechanical Engineers (대한기계학회)
권호 표지

The Role of S-Shape Mapping Functions in the SIMP Approach for Topology Optimization

  • Yoon, Gil-Ho (School of Mechanical and Aerospace Engineering, and National Creative Research Initiatives Center for Multiscale Design Seoul National University) ;
  • Kim, Yoon-Young (School of Mechanical and Aerospace Engineering, and National Creative Research Initiatives Center for Multiscale Design Seoul National University)
  • Published : 2003.10.01
Download PDF
⟨ Previous Next ⟩

Abstract

The SIMP (solid isotropic material with penalization) approach is perhaps the most popular density variable relaxation method in topology optimization. This method has been very successful in many applications, but the optimization solution convergence can be improved when new variables, not the direct density variables, are used as the design variables. In this work, we newly propose S-shape functions mapping the original density variables nonlinearly to new design variables. The main role of S-shape function is to push intermediate densities to either lower or upper bounds. In particular, this method works well with nonlinear mathematical programming methods. A method of feasible directions is chosen as a nonlinear mathematical programming method in order to show the effects of the S-shape scaling function on the solution convergence.

Keywords

References

  1. Arora, J. S., 1989, Introduction to Optimum design, McGRAW-HILL
  2. Bendsoe, M. P. and Kikuchi, N., 1988, 'Generating Optimal Topologies in Structural Design Using a Homogenization Method,' Comp. Meth. Appl. Mechs. Engng., Vol. 20, pp. 197-224 https://doi.org/10.1016/0045-7825(88)90086-2
  3. Bendsoe. M. P., 1995, Shape and Material,Optimization of Structural Topology, Springer, New York
  4. Bendsoe. M. P. and Sigmund, O., 1999, 'Material Interpolation Schemes in Topology Optimization,' Archive of Applied Mechanics, Vol. 69, pp. 635-654 https://doi.org/10.1007/s004190050248
  5. Bendsoe. M. P. and Sigmund, O., 2003, Topology Optimization Theory,Methods and Applications, Springer-Verlag, New York
  6. Bendsoe. M. P., 1989, 'Optimal Shape Design as a Material Distribution Problem,' Struct. Optim, Vol. 1, pp. 193-202 https://doi.org/10.1007/BF01650949
  7. Duysinx, P., 1997, 'Layout Optimization : A Mathematical Programming Approach,' DCAMM, Vol. 540, pp. 1-30
  8. Fujii, D. and Kikuchi, N., 2000, 'Improvement of Numerical Instabilities in Topology Optimization Using the SLP Method,' Struct. Multidisc. Optim., Vol. 19, pp. 113-121 https://doi.org/10.1007/s001580050091
  9. Hassani, B. and Hinton, E., 1998a, 'A Review of Homogenization and Topology Optimization I-Homogenization Theory for Media with Periodic Structure,' Computers & Structures, Vol. 69, pp. 707-717 https://doi.org/10.1016/S0045-7949(98)00131-X
  10. Hassani, B. and Hinton, E., 1998b, 'A Review of Homogenization and Topology Optimization II-Analytical and Numerical Solution of Homogenization Equations,' Computers & Structures, Vol. 69, pp. 719-738 https://doi.org/10.1016/S0045-7949(98)00132-1
  11. Hassani, B. and Hinton, E., 1998c, 'A Review of Homogenization and Topology Optimization III-Topology Optimization Using Optimality Criteria,' Computers & Structures, Vol. 69, pp. 739-756 https://doi.org/10.1016/S0045-7949(98)00133-3
  12. Hasin, Z. and Shtrikman, S., 1963, 'A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials,' Journal of the Mechanics and Physics of Solids, Vol. 11, pp. 127-140 https://doi.org/10.1016/0022-5096(63)90060-7
  13. Kim. Y. Y. and Yoon, G. H., 2000, 'Multi-Resolution Multi-Scale Topology Optimization A New Paradigm,' International Journal of Solids and Structures, Vol. 37, pp. 5529-5559 https://doi.org/10.1016/S0020-7683(99)00251-6
  14. Nishiwaki, S., Frecker, M. I., Min, S. J. and Kikuchi, N., 1998, 'Topology Optimization of Compliant Mechanism Using the Homogenization Method,' Int. J. Numer. Meth. Engrg. Vol. 42, pp. 535-560 https://doi.org/10.1002/(SICI)1097-0207(19980615)42:3<535::AID-NME372>3.0.CO;2-J
  15. Sigmund, O., 1997, 'On the Design of Compliant Mechanisms Using Topology Optimization,' Mechanics of Structures and Machines, Vol. 25, No. 4, pp. 495-526 https://doi.org/10.1080/08905459708945415
  16. 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, Vol. 16, No. 1, pp. 68-75 https://doi.org/10.1007/BF01214002
  17. Poulsen, T. A., 2002, 'Topology Optimization in Wavelet Space,' International Journal for Numerical Methods in Engineering, Vol. 53, No. 3, pp. 567-582 https://doi.org/10.1002/nme.285
  18. Vanderplaats, G. N., 1984, 'ADS-A Fortran Program for Automated Design Synthesis,' NASA CR 172460
  19. Vanderplaats, G. N., 1999, Numerical Optimization Techniques for Engineering Design, Colorado Springs
  20. Yang, R. J. and Chuang, C. H., 1994, 'Optimal Topology Design Using Linear Programming,' Computers & Structures, Vol. 52, pp. 256-275 https://doi.org/10.1016/0045-7949(94)90279-8
  21. Zhou, M. and Rozvany, G. I. N., 1991, 'The COC Algorithm, Part II : Topological, Geometry and Generalized Shape Optimization,' Comp. Meth. Appl. Mechs. Engng., Vol. 89, pp. 197-224 https://doi.org/10.1016/0045-7825(91)90046-9
  22. Mechanics of Structures and Machines v.25 no.4 On the Design of Compliant Mechanisms Using Topology Optimization Sigmund,O.