Structural optimization

, Volume 11, Issue 1–2, pp 1–12 | Cite as

A new approach to variable-topology shape design using a constraint on perimeter

  • R. B. Haber
  • C. S. Jog
  • M. P. Bendsøe
Research Papers


This paper introduces a method for variable-topology shape optimization of elastic structures called theperimeter method. An upper-bound constraint on the perimeter of the solid part of the structure ensures a well-posed design problem. The perimeter constraint allows the designer to control the number of holes in the optimal design and to establish their characteristic length scale. Finite element implementations generate practical designs that are convergent with respect to grid refinement. Thus, an arbitrary level of geometric resolution can be achieved, so single-step procedures for topology design and detailed shape design are possible. The perimeter method eliminates the need for relaxation, thereby circumventing many of the complexities and restrictions of other approaches to topology design.


Optimal Design Design Problem Characteristic Length Shape Optimization Characteristic Length Scale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Allaire, G.; Francfort, G.A. 1993: A numerical algorithm for topology and shape optimization. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology design of structures, pp. 239–248. Dordrecht: KluwerGoogle Scholar
  2. Allaire, G.; Kohn, R.V. 1993: Topology design and optimal shape design using homogenization. In: Bendsøe, M.P.; Mota Soares, C. A. (eds.)Topology design of structures, pp. 207–218. Dordrecht: KluwerGoogle Scholar
  3. Ambrosio, L.; Buttazzo, G. 1993: An optimal design problem with perimeter penalization.Calculus of Variations and Partial Differential Equations 1, 55–69Google Scholar
  4. Bendsøe, M.P. 1989: Optimal shape design as a material distribution problem.Struct. Optim. 1, 193–202Google Scholar
  5. Bendsøe, M.P.; Diaz, A.; Kikuchi, N. 1993: Topology and generalized layout optimization of elastic structures. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology design of structures, pp. 159–205. Dordrecht: KluwerGoogle Scholar
  6. Bendsøe, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenization method.Comp. Meth. Appl. Mech. Eng. 71, 197–224Google Scholar
  7. Cheng, K.T.; Olhoff, N. 1981: An investigation concerning optimal design of solid elastic plates.Int. J. Solids & Struct. 17, 305–323Google Scholar
  8. Eschenauer, H.A.; Kobolev, V.; Schumacher, A. 1994: Bubble method of topology and shape optimization of structures.Struct. Optim. 8, 42–51Google Scholar
  9. Evans, L.C.; Gariepy, R.F. 1992:Measure theory and fine properties of functions. Boca Ratan: CRC PressGoogle Scholar
  10. Gibiansky, L.V.; Cherkaev, A.V. 1987: Microstructures of composites of extremal rigidity and exact estimates of the associated energy density.Ioffe Physicotechnical Institute, Preprint 1115, Leningrad (in Russian); also in: Kohn, R.V. (ed.) 1994:Topics in the mathematical modeling of composite materials. New York: BirkhauserGoogle Scholar
  11. Goodman, J.; Kohn, R.V.; Reyna, L. 1986: Numerical study of a relaxed variational problem for optimal design.Comp. Meth. Appl. Mech. Eng. 57, 107–127Google Scholar
  12. Haber, R.B.; Jog, C.S.; Bendsøe, M.P. 1994: Variable-topology shape optimization with a control on perimeter. In: Gilmore, B.J.; Hoeltzel, D.A.; Dutta, D.; Eschenauer, H.A. (eds.)Advances in design automation, pp. 261–272. Washington D.C.: AIAAGoogle Scholar
  13. Jog, C.S.; Haber, R.B. 1995: Stability of finite element models for distributed-parameter optimization and topology design.Comp. Meth. Appl. Mech. Eng. (to appear)Google Scholar
  14. Jog, C.S.; Haber, R.B.; Bendsøe, M.P. 1993: A displacement-based topology design method with self-adaptive layered materials. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology design of structures, pp. 219–238. Dordrecht: KluwerGoogle Scholar
  15. Jog, C.S.; Haber, R.B.; Bendsøe, M.P. 1994: Topology design with optimized, self-adaptive materials.Int. J. Num. Methods Engng. 37, 1323–1350Google Scholar
  16. Kohn, R.V. 1991: Composite materials and structural optimization.Proc. Workshop on Smart/Intelligent Materials and Systems. (held in Honolulu, 1990). Lancaster, Pa.: Technomic PressGoogle Scholar
  17. Kohn, R.V.; Strang, G. 1986: Optimal design and relaxation of variational problems.Comm. Pure Appl. Math. 39, 1–25 (Part I), 139–182 (Part II), 353–377 (Part III)Google Scholar
  18. Lurie, A.K.; Federov, A.V.; Cherkaev, A.V. 1982: Regularization of optimal design problems for bars and plates.J. Optimiz. Theory & Appl. 37, 499–521 (Part I), 523–543 (Part II)Google Scholar
  19. Mlejnek, H. 1992: Some aspects of the genesis of structures.Struct. Optim. 5, 64–69Google Scholar
  20. Murat, F. 1977: Contre-exemples pour divers problemes ou le controle intervient dans les coefficients.Ann. Mat. Pura et Appl. 112, 49–68Google Scholar
  21. Murat, F.; Tartar, L. 1985: Calcul des variations et homogeneisation. In:Les methodes de l'homogeneisation: theorie et applications en physique, pp. 319–370. Col. de la Dir. des Etudes et Recherches de Electricité de France, Eyrolles, ParisGoogle Scholar
  22. Rodrigues, H.; Fernandes, P. 1995: A material based model for topology optimization of thermoelastic structures.Int. J. Num. Methods Engng. 37, 1951–1965Google Scholar
  23. Rozvany, G.I.N. 1993: Layout theory for grid-type structures. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology design of structures, pp. 251–272. Dordrecht: KluwerGoogle Scholar
  24. Sigmund, O. 1995: Design of material structures using topology optimization.DCAMM Special Report No. S69, Technical University of DenmarkGoogle Scholar
  25. Sokolnikoff, I.S. 1987:Mathematical theory of elasticity. Florida: Robert KriegerGoogle Scholar
  26. Suzuki, K.; Kikuchi, N. 1991: Shape and topology optimization for generalized layout problems using the homogenization method.Comp. Meth. Appl. Mechs. Engng. 93, 291–318Google Scholar
  27. Tartar, L. 1977: Estimation de coefficents homogeneises.Lecture Notes in Mathematics 704, pp. 364–373. Berlin, Heidelberg, New York: SpringerGoogle Scholar
  28. Wheedon, R.L.; Zygmund, A. 1977:Measure and integral: an introduction to real analysis. Monographs in Pure and Applied Math.43. New York: Marcel DekkerGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • R. B. Haber
    • 1
  • C. S. Jog
    • 2
  • M. P. Bendsøe
    • 2
  1. 1.Department of Theoretical and Applied MechanicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Mathematical InstituteThe Technical University of DenmarkLyngbyDenmark

Personalised recommendations