Elsevier

Computers & Structures

Volume 69, Issue 6, December 1998, Pages 707-717
Computers & Structures

A review of homogenization and topology optimization I—homogenization theory for media with periodic structure

https://doi.org/10.1016/S0045-7949(98)00131-XGet rights and content

Abstract

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.

Introduction

Advances in technology in recent years have been paralleled by the increased use of composite materials in industry. Since materials have different properties, it seems sensible to make use of the good properties of each single ingredient by using them in a proper combination. For example, a simple mixture of clay, sand and straw produced a composite building material which was used by the oldest known civilizations. The further development of non-metallic materials and composites has attracted the attention of scientists and engineers in various fields, for example, aerospace, transportation, and other branches of civil and mechanical engineering. Apart from the considerably low ratio of weight to strength, some composites benefit from other desirable properties, such as corrosion and thermal resistance, toughness and lower cost. Usually, composite materials comprise of a matrix which could be metal, polymeric (like plastics) or ceramic, and a reinforcement or inclusion, which could be particles or fibres of steel, aluminum, silicon etc.

Composite materials may be defined as a man-made material with different dissimilar constituents, which occupy different regions with distinct interfaces between them[1]. The properties of a composite are different from its individual constituents. A cellular body can be considered as a simple case of a composite, comprising solids and voids. This is the case which is used in the structural topology optimization.

In this study, composites with a regular or nearly regular structure are considered. Having sufficiently regular heterogenities enables us to assume a periodic structure for the composite. It should be emphasized that in comparison with the dimensions of the body the size of these non-homogeneities should be very small. Owing to this, these types of material are sometimes called composites with periodic microstructures.

Even with the help of high-speed modern computers, the analysis of the boundary value problems consisting of such media with a large number of heterogenities, is extremely difficult. A natural way to overcome this difficulty is to replace the composite with a kind of equivalent material model. This procedure is usually called homogenization. One way of finding the properties of such composites is by carrying out experimental tests. It is quite evident that because of the volume and cost of the required tests for all possible reinforcement types, experimental measurements are often impracticable.

The mathematical theory of homogenization, which has developed since the 1970 s is used as an alternative approach to find the effective properties of the equivalent homogenized material2, 3, 4. This theory can be applied in many areas of physics and engineering having finely heterogeneous continuous media, like heat transfer or fluid flow in porous media or, for example, electromagnetism in composites. In fact, the basic assumption of continuous media in mechanics and physics can be thought of as sort of homogenization, as the materials are composed of atoms or molecules.

From a mathematical point of view, the theory of homogenization is a limit theory which uses the asymptotic expansion and the assumption of periodicity to substitute the differential equations with rapidly oscillating coefficients, with differential equations whose coefficients are constant or slowly varying in such a way that the solutions are close to the initial equations[5].

This method makes it possible to predict both the overall and local properties of processes in composites. In the first step, the appropriate local problem on the unit cell of the material is solved and the effective material properties are obtained. In the second step, the boundary value problem for a homogenized material is solved.

Section snippets

Periodicity and Asymptotic Expansion

A heterogeneous medium is said to have a regular periodicity if the functions denoting some physical quantity of the the medium—either geometrical or some other characteristics—have the following property: F(x+NY)=F(x).x=(x1, x2, x3) is the position vector of the point, N is a 3×3 diagonal matrix: N=n1000n2000n3,where n1, n2 and n3 are arbitrary integer numbers, and Y=〈Y1 Y2 Y3T is a constant vector which determines the period of the structure; F can be a scalar or vectorial or even tensorial

One-dimensional Elasticity Problem

To clarify the homogenization method, the simple case of calculation of deformation of an inhomogeneous bar in the longitudinal direction is considered. Here, we attempt to derive the modulus of elasticity without recourse to advanced mathematics.

According to the assumptions of the theory, the medium has a periodic composite microstructure (Fig. 5).

The governing equations, in the form of Hooke's law of linear elasticity and the Cauchy's first law of motion (equilibrium equation), are: σϵ=Eϵ∂uϵ∂x

Problem of Heat Conduction

The 1D heat conduction is very similar to the 1D elasticity problem. The governing equations, Fourier's law of heat conduction and the equation of heat balance, are: qϵ(x)=KϵdTϵ(x)/dx∂qϵ/∂x+f=0.qϵ is the heat flux, Tϵ is the temperature, and Kϵ(x) is the conductivity coefficient. Following a very similar procedure to the 1D elasticity problem, the homogenized coefficient of heat conduction can be obtained as: KH=1/1Y0YdηK(η),which as is expected, is the same as Eq. (30).

Similarly, starting

General Boundary Value Problem

Many physical systems which do not change with time—sometimes called steady-state problems—can be modelled by elliptic equations. As a general problem, the divergent elliptic equation in a non-homogeneous medium with regular structure is now explained.

Let Ω⊂R3 be an unbounded medium tissued by parallelepiped unit cells Y, whose material properties are determined by a symmetric matrix aij(x, y)=aij(y), where y=x/ϵ and x=(x1, x2, x3) and the functions aij are periodic in the spatial variables y=(y

General Elasticity Problem

So far, the application of the homogenization theory in 1D elasticity, heat conduction, and as a more general problem in elliptic partial differential equations, has been discussed. For the sake of completeness the homogenization method for cellular media in weak form, which is suitable for the derivation of the finite element formulation, using the procedure and notation used by Guedes and Kikuchi in Ref.[9], is briefly explained. This is the case applied in topological structural optimization

Conclusion and Final Remarks

In this first part of a three paper review we have focused on the theory of the homogenization method for the computation of effective constitutive parameters of complex materials with a periodic microstructure. In the second part of this review we will consider the motives for using the homogenization theory for topological structural optimization. In particular, the finite element formulation will be explained for the material model based on a microstructure consisting of an isotropic

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