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 schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i~otropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

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Generating optimal topologies in structural design using a homogenization method

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.

Optimal shape design as a material distribution problem

A level set method for structural topology optimization

In topology optimization of structures, materials and mechanisms, parametrization of geometry is often performed by a grey-scale density-like interpolation function. In this paper we analyze and compare the various approaches to this concept in the light of variational bounds on effective properties of composite materials. This allows us to derive simple necessary conditions for the possible realization of grey-scale via composites, leading to a physical interpretation of all feasible designs as well as the optimal design. Thus it is shown that the so-called artificial interpolation model in many circumstances actually falls within the framework of microstructurally based models. Single material and multi-material structural design in elasticity as well as in multi-physics problems is discussed.

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Material interpolation schemes in topology optimization

A simple evolutionary procedure for structural optimization

The COC algorithm, Part II: Topological, geometrical and generalized shape optimization

Two types of solutions may be considered in generalized shape optimization. Absolute minimum weight solutions, which are rather unpractical, consist of solid, empty and porous regions. In more practical solutions of shape optimization, porous regions are suppressed and only solid and empty regions remain. This note discusses this second class of problems and shows that a solid, isotropic microstructure with an adjustable penalty for intermediate densities is efficient in generating optimal topologies.

Generalized shape optimization without homogenization

The aim of this article is to evaluate and compare established numerical methods of structural topology optimization that have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive debate on this important topic.

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A critical review of established methods of structural topology optimization

Layout Optimization of Structures

Topology optimization of structures and composite continua has two main subfields: Layout Optimization (LO) deals with grid-like structures having very low volume fractions and Generalized Shape Optimization (GSO) is concerned with higher volume fractions, optimizing simultaneously the topology and shape of internal boundaries of porous or composite continua. The solutions for both problem classes can be exact/analytical or discretized/FE-based.

This review article discusses FE-based generalized shape optimization, which can be classified with respect to the types of topologies involved, namely Isotropic-Solid/Empty (ISE), Anisotropic-Solid/Empty (ASE), and Isotropic-Solid/Empty/Porous (ISEP) topologies.

Considering in detail the most important class of (i.e. ISE) topologies, the computational efficiency of various solution strategies, such as SIMP (Solid Isotropic Microstructure with Penalization), OMP (Optimal Microstructure with Penalization) and NOM (NonOptimal Microstructures) are compared.

The SIMP method was proposed under the terms "direct approach" or "artificial density approach" by Bendsoe over a decade ago; it was derived independently, used extensively and promoted by the author's research group since 1990. The term "SIMP" was introducted by the author in 1992. After being out of favour with most other research schools until recently, SIMP is becoming generally accepted in topology optimization as a technique of considerable advantages. It seems, therefore, useful to review in greater detail the origins, theoretical background, history, range of validity and major advantages of this method.

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George I. N. Rozvany

Papers

The COC algorithm, Part II: Topological, geometrical and generalized shape optimization

Generalized shape optimization without homogenization

A critical review of established methods of structural topology optimization

Layout Optimization of Structures

Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics