# Optimal shape design as a material distribution problem

@article{Bendse1989OptimalSD, title={Optimal shape design as a material distribution problem}, author={Martin P. Bends{\o}e}, journal={Structural optimization}, year={1989}, volume={1}, pages={193-202} }

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… Expand

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#### References

SHOWING 1-10 OF 32 REFERENCES

Generating optimal topologies in structural design using a homogenization method

- Mathematics
- 1988

Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational… Expand

Integrated topology and boundary shape optimization of 2-D solids

- Mathematics
- 1991

Abstract This study is concerned with the development of an integrated procedure for the computation of the optimal topology as well as the optimal boundary shape of a two-dimensional, linear elastic… Expand

Regularization of optimal design problems for bars and plates, part 1

- Mathematics
- 1982

In this paper, we consider a number of optimal design problems for elastic bars and plates. The material characteristics of rigidity of an elastic nonhomogeneous medium are taken as the control… Expand

On Obtaining a Solution to Optimization Problems for Solid, Elastic Plates by Restriction of the Design Space*

- Mathematics
- 1983

ABSTRACT The problem of minimizing the static compliance of a solid elastic plate, for given total plate volume, is considered. Nonexistence of solutions in the case of a design space consisting of… Expand

Shape optimization of structures: a literature survey

- Mathematics
- 1986

Abstract Numerical and analytical methods for shape optimization of structures are reviewed. Several steps in the shape optimization process, such as model description, selection of the objective… Expand

Numerical study of a relaxed variational problem from optimal design

- Mathematics
- 1986

Abstract We revisit a well-known problem of optimal design, the placement of two elastic materials in the cross-section of a rod for maximum torsional rigidity. Another interpretation is the… Expand

Structural shape optimization — a survey

- Mathematics
- 1986

Abstract This paper is a survey of structural shape optimization with an emphasis on techniques dealing with shape optimization of the boundaries of two- and three-dimensional bodies. Attention is… Expand

Sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates

- Mathematics
- 1981

Abstract This paper deals with optimal design of solid, elastic, axisymmetric plates performing free, transverse vibrations. It is the objective to determine the plate thickness distribution from the… Expand

Regularized formulation for optimal design of axisymmetric plates

- Mathematics
- 1982

Abstract This paper presents a regularized mathematical formulation, necessary conditions of optimality and solutions of problems of optimal design of solid, elastic, axisymmetric plates of… Expand

Least-weight design of perforated elastic plates. II

- Mathematics
- 1987

Abstract The paper investigates implications of some recent mathematical developments in the fields of shape optimization and relaxation of variational problems. Considering the least-weight design… Expand