Structural optimization

, Volume 1, Issue 4, pp 193-202

First online:

Optimal shape design as a material distribution problem

  • M. P. BendsøeAffiliated withMathematical Institute, The Technical University of Denmark

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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.