Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Optimal shape design as a material distribution problem


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.

This is a preview of subscription content, log in to check access.


  1. Avellaneda, M. 1987: Optimal bounds and microgeometries for elastic two-phase composites.SIAM J. Appl. Math. 47, 1216–1228

  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, 501–521

  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: Springer

  4. Bendsøe, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenization method.Comput. Meth. Appl. Mech. Engrg. 71, 197–224

  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 Lyngby

  6. Bensousson, A.; Lions, J.-L.; Papanicolaou, G. 1978:Asymptotic analysis for periodic structures. Amsterdam: North-Holland

  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: Springer

  8. Cheng, K.T.; Olhoff, N. 1982: Regularized formulation for optimal design of axisymmetric plates.Int. J. Solids Struct. 18, 153–170

  9. Ding, Y. 1986: Shape optimization of structures: a literature survey.Comp. Struct. 24, 985–1004

  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, 107–127

  11. Haftka, R.T.; Gandhi, R.V. 1986: Structural shape optimization- a survey.Comp. Meth. Appl. Mech. Engrg. 57, 91–106

  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)

  13. Kohn, R.V.; Strang, G. 1986b: Optimal design in elasticity and plasticity.Int. J. Numer. Meth. Eng. 22, 183–188

  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, 523–543

  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, 931–948

  16. Olhoff, N.; Taylor, J.E. 1983: On structural optimization.J. Appl. Mech. 50, 1134–1151

  17. Pedersen, P. 1989a: On optimal orientation of orthotropic materials.Struct. Optim. 1, 101–106

  18. Pedersen, P. 1989b: Bounds on elastic energy in solids of orthotropic materials.DCAMM Report No.392, Techn. Univ. of Denamrk, DK-2800 Lyngby

  19. Rossow, M.P.; Taylor, J.E. 1973: A finite element method for the optimal design of variable thickness sheets.AIAA J. 11, 1566–1569

  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 & Sons

  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)

  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: Springer

  23. Sanchez-Palencia, E. 1980: Non-homogeneous media and vibration theory.Lecture Notes in Physics 127, Berlin, Heidelberg, New York: Springer

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bendsøe, M.P. Optimal shape design as a material distribution problem. Structural Optimization 1, 193–202 (1989).

Download citation


  • Material Parameter
  • Optimal Shape
  • Boundary Variation
  • Material Distribution
  • Shape Design