Structural optimization

, Volume 1, Issue 4, pp 193–202 | Cite as

Optimal shape design as a material distribution problem

  • M. P. Bendsøe


Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.


Material Parameter Optimal Shape Boundary Variation Material Distribution Shape Design 
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  1. Avellaneda, M. 1987: Optimal bounds and microgeometries for elastic two-phase composites.SIAM J. Appl. Math. 47, 1216–1228Google Scholar
  2. Bendsøe, M.P. 1983: On obtaining a solution to optimization problems for solid, elastic plates by restriction of the design space.J. Struct. Mech. 11, 501–521Google Scholar
  3. Bendsøe, M.P. 1986: Generalized plate models and optimal design. In: Eriksen, J.L.; Kinderlehrer, D.; Kohn, R.; Lions, J.-L., (eds.)Homogenization and effective moduli of materials and media, the IMA volumes in mathematics and its applications, pp. 1–26. Berlin, Heidelberg, New York: SpringerGoogle Scholar
  4. Bendsøe, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenization method.Comput. Meth. Appl. Mech. Engrg. 71, 197–224Google Scholar
  5. Bendsøe, M.P.; Rodrigues, H.C. 1989: Integrated topology and boundary shape optimization of 2-D solids.MAT-Report No.1989–14, Math. Inst., Techn. Univ. of Denmark, DK-2800 LyngbyGoogle Scholar
  6. Bensousson, A.; Lions, J.-L.; Papanicolaou, G. 1978:Asymptotic analysis for periodic structures. Amsterdam: North-HollandGoogle Scholar
  7. Bourgat, J.F. 1977: Numerical experiments of the homogenization method for operators with periodic coefficients.Lecture Notes in Mathematics 704, pp. 330–356, Berlin, Heidelberg, New York: SpringerGoogle Scholar
  8. Cheng, K.T.; Olhoff, N. 1982: Regularized formulation for optimal design of axisymmetric plates.Int. J. Solids Struct. 18, 153–170Google Scholar
  9. Ding, Y. 1986: Shape optimization of structures: a literature survey.Comp. Struct. 24, 985–1004Google Scholar
  10. Goodman, J.; Kohn, R.V.; Reyna, L. 1986: Numerical study of a relaxed variational problem from optimal design.Comp. Meth. Appl. Mech. Engrg. 57, 107–127Google Scholar
  11. Haftka, R.T.; Gandhi, R.V. 1986: Structural shape optimization- a survey.Comp. Meth. Appl. Mech. Engrg. 57, 91–106Google Scholar
  12. Kohn, R.V.; Strang, G. 1986a: Optimal design and relaxation of variational problems.Comm. Pure Appl. Math. 39, 1–25 (Part I), 139–182 (Part II), and 353–377 (Part III)Google Scholar
  13. Kohn, R.V.; Strang, G. 1986b: Optimal design in elasticity and plasticity.Int. J. Numer. Meth. Eng. 22, 183–188Google Scholar
  14. Lurie, K.A.; Fedorov, A.V.; Cherkaev, A.V. 1982: Regularization of optimal design problems for bars and plates. Parts I and II.J. Optim. Theory Appl. 37, 499–521, 523–543Google Scholar
  15. Olhoff, N.; Lurie, K.A.; Cherkaev, A.V.; Fedorov, A.V. 1981: Sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates.Int. J. Solids Struct. 17, 931–948Google Scholar
  16. Olhoff, N.; Taylor, J.E. 1983: On structural optimization.J. Appl. Mech. 50, 1134–1151Google Scholar
  17. Pedersen, P. 1989a: On optimal orientation of orthotropic materials.Struct. Optim. 1, 101–106Google Scholar
  18. Pedersen, P. 1989b: Bounds on elastic energy in solids of orthotropic materials.DCAMM Report No.392, Techn. Univ. of Denamrk, DK-2800 LyngbyGoogle Scholar
  19. Rossow, M.P.; Taylor, J.E. 1973: A finite element method for the optimal design of variable thickness sheets.AIAA J. 11, 1566–1569Google Scholar
  20. Rozvany, G.I.N. 1984: Structural layout theory - the present state of knowledge. In: Atrek, E.; Gallagher, R.H.; Ragsdell, K.M.; Zienkiewicz, O.C. (eds.)Directions in optimum structural design, Chapter 7. Chichester: Wiley & SonsGoogle Scholar
  21. Rozvany, G.I.N.; Ong, T.G.; Szeto, W.T.; Olhoff, N.; Bendsøe, M.P. 1987: Least-weight design of perforated plates.Int. J. Solids and Struct. 23, 521–536 (Part I), 537–550 (Part II)Google Scholar
  22. Rozvany, G.I.N. 1989: Optimality criteria for continuous and segment-wise linear distribution of the cross-sectional parameters. In: Eschenauer, H.A.; Thierauf, G. (eds.)Discretization methods and structural optimization - procedures and applications, pp. 291–298. Berlin, Heidelberg, New York: SpringerGoogle Scholar
  23. Sanchez-Palencia, E. 1980: Non-homogeneous media and vibration theory.Lecture Notes in Physics 127, Berlin, Heidelberg, New York: SpringerGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • M. P. Bendsøe
    • 1
  1. 1.Mathematical InstituteThe Technical University of DenmarkLyngbyDenmark

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