Sensitivity of the macroscopic elasticity tensor to topological microstructural changes

ARTICLE IN PRESS Journal of the Mechanics and Physics of Solids 57 (2009) 555–570

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Sensitivity of the macroscopic elasticity tensor to topological microstructural changes S.M. Giusti a,, A.A. Novotny a, E.A. de Souza Neto b, R.A. Feijo´o a a b

´rio Nacional de Computac- ˜ ´ lio Vargas 333, 25651-075 Petro ´polis, RJ, Brasil Laborato ao Cientı´fica LNCC/MCT, Av. Getu Civil and Computational Engineering Centre, School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK

a r t i c l e in fo

abstract

Article history: Received 3 April 2008 Received in revised form 16 September 2008 Accepted 15 November 2008

A remarkably simple analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological microstructural changes of the underlying material is proposed. The derivation of the proposed formula relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. The derived sensitivity—a symmetric fourth order tensor field over the RVE domain—measures how the estimated two-dimensional macroscopic elasticity tensor changes when a small circular hole is introduced at the microscale level. This information has potential use in the design and optimisation of microstructures. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Homogenized Elasticity tensor Topological derivative Sensitivity analysis Multi-scale modelling Synthesis of microstructures

1. Introduction Composite materials have become one of the most important classes of engineering materials. Their macroscopic mechanical behaviour is of paramount importance in the design of load bearing components for a vast number of applications in civil, mechanical, aerospace, biomedical and nuclear industries. In a broad sense, one can argue that much of material science is about improving macroscopic material properties by means of topological and shape changes at a microstructural level. For example, changes in shape of graphite inclusions in a cast iron matrix may produce dramatic changes in the corresponding macroscopic properties of this material. In this context, the ability to accurately predict the macroscopic mechanical behaviour from the corresponding microscopic properties as well as its sensitivity to changes in microstructure becomes essential in the analysis and potential purpose-design and optimisation of heterogeneous media. Such concepts have been successfully used, for instance, in Almgreen (1985) and Lakes (1987) by means of a relaxationbased technique in the design of microstructural topologies that produce negative macroscopic Poisson’s ratio. This type of approach relies on the use of a fictitious material density field and mimics, in a regularised sense, the introduction of localised topological microstructural changes (voids) wherever the artificial density is sufficiently close to zero (refer, for instance, to the fundamental papers Bendsoe and Kikuchi, 1988; Zochowski, 1988). In contrast to the regularised approach, this paper proposes a general exact analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological changes of the microstructure of the underlying material. The macroscopic linear elastic response is estimated by means of a well-established homogenisation-based multi-scale

 Corresponding author. Tel.: +55 24 2233 6169; fax: +55 24 2233 6124.

E-mail address: [email protected] (S.M. Giusti). 0022-5096/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2008.11.008

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constitutive theory for elasticity problems (Germain et al., 1983; Michel et al., 1999) where the macroscopic strain and stress tensors at each point of the macroscopic continuum are defined as the volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. The proposed sensitivity is a symmetric fourth order tensor field over the RVE that measures how the macroscopic elasticity constants estimated within the multi-scale framework changes when a small circular void is introduced at the microscale. Its analytical formula is derived by making use of the concepts of topological asymptotic expansion and topological derivative (Ce´a et al., 2000; Sokolowski and Zochowski, 1999) within a variational formulation of the adopted multi-scale theory. The (relatively new) mathematical notions of topological asymptotic expansion and topological derivative allow the closed form exact calculation of the sensitivity of a given shape functional with respect to infinitesimal domain perturbations such as the insertion of voids, inclusions or source terms. Their use in the context of solid mechanics, topological optimisation of load bearing structures and inverse problems is reported in a number of recent publications (Bonnet, 2006; Feijo´o et al., 2003; Garreau et al., 2001; Guzina and Chikichev, 2007; Lewinski and Sokolowski, 2003; Novotny et al., 2005). In the present context, the variational setting for the multi-scale modelling methodology as described in de Souza Neto and Feijo´o (2006) is found to be particularly well suited for the application of the topological derivative formalism. The final format of the proposed analytical formula is strikingly simple and can be potentially used in applications such as the synthesis and optimal design of microstructures to meet a specified macroscopic behaviour. The paper is organised as follows. Section 2 briefly describes the multi-scale constitutive framework adopted in the estimation of the macroscopic elasticity tensor. The modelling approach is cast within the variational setting described in de Souza Neto and Feijo´o (2006). The main contribution of the paper—the closed formula for the sensitivity of the macroscopic elasticity tensor to topological microstructural perturbations—is presented in Section 3. Here, an overview of the topological derivative concept is given, followed by its application to the problem in question. This leads to the identification of the required sensitivity tensor field. Numerical confirmation of the derived analytical formula is provided by means a simple finite element-based example. Finally, some concluding remarks are made in Section 4. 2. Multi-scale modelling This section describes a homogenisation-based variational multi-scale framework for classical elasticity problems which allows the macroscopic elasticity tensor to be estimated from the given geometrical and elastic properties of a local RVE of material. This constitutive modelling approach has been proposed by Germain et al. (1983) and has been exploited in the computational context, among others, by Michel et al. (1999) and Miehe et al. (1999). Its variational structure is described in detail in de Souza Neto and Feijo´o (2006). For related multi-scale approaches based on asymptotic analysis, the reader may refer to Ammari and Kang (2007) and Sanchez-Palencia (1980). The starting point of the multi-scale constitutive theory adopted to estimate the macroscopic elastic properties of the continuum is the assumption that any material point x of the macroscopic continuum (refer to Fig. 1) is associated to a local RVE whose domain Om has characteristic length lm , much smaller than the characteristic length l of the macrocontinuum domain, O. For the purposes of the analysis conducted in this paper, we shall consider the RVE to consist of a matrix, i denoted by domain Om m , containing inclusions of different materials occupying a domain Om , see Fig. 1. Crucial to the present approach is the assumption that the macroscopic strain tensor E at a point x of the macroscopic continuum is the volume average of its microscopic counterpart Em over the domain of the RVE: Z 1 Em dOm , (1) E V m Om

macro-scale

micro-scale

x y

l << l

l Fig. 1. Macroscopic continuum with a locally attached microstructure.

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i m i where V m is a total volume of the RVE, i.e. V m ¼ V m m þ V m , with V m and V m denoting the matrix and inclusion volume of the RVE and

Em  rs um ðyÞ,

(2)

with um denoting the microscopic displacement field of the RVE. The use of Green’s Theorem in definition (1) and (2) gives the following equivalent expression for E: Z 1 E¼ um s n dqOm , (3) V m qOm where n is the outward unit normal to the boundary of the RVE and s denotes the usual symmetric tensor product of vectors. Without loss of generality, it is possible to split um into a sum ˜ m ðyÞ um ðyÞ ¼ uðxÞ þ Ey þ u

(4)

of a constant (rigid) RVE displacement coinciding with the macrodisplacement uðxÞ, a homogeneous strain displacement ˜ m ðyÞ. field, linear in y, and a displacement fluctuation field u Following (4) the microscopic strain field (2) can be expressed as a sum s ˜m Em ¼ E þ r u

(5) s

˜ m corresponding to of a homogeneous strain (uniform over the RVE) coinciding with the macroscopic strain, and a field r u a fluctuation of the microscopic strain about the homogenised (average) value. 2.1. Admissible and virtual microscopic displacement fields Assumption (1) and (2) places a constraint on the admissible displacement fields of the RVE. That is, only fields um that satisfy (1) and (2) can be said to be kinematically admissible. This condition can be formally expressed by requiring the (as yet not defined) set Km of kinematically admissible displacements of the RVE to satisfy ( ) Z Z

Km  Km 

v 2 ½H1 ðOm Þ2 :

v dOm ¼ V m u;

Om

qOm

vs n dqOm ¼ V m E; 1vU ¼ 0 on qOim ,

(6)

where Km is named the minimally constrained set of kinematically admissible RVE displacement fields—the most general set of microscopic displacement fields compatible with the strain averaging assumption—and 1vU denotes the jump of function v across the matrix/inclusion interface qOim : 1ðÞU  ðÞjm  ðÞji ,

(7)

with subscripts m and i associated, respectively, with quantity values on the matrix and inclusion sides of the interface. In view of (4) and assuming that the origin of the coordinate system coincides with the centroid of the RVE, constraint ˜ m of admissible displacement (6) can, without loss of generality, be made equivalent to requiring that the space K ˜ : fluctuations of the RVE be a subspace of the minimally constrained space of displacement fluctuations, K m ( ) Z Z ˜   v 2 ½H1 ðOm Þ2 : ˜ mK K v dOm ¼ 0; vs n dqOm ¼ 0; 1vU ¼ 0 on qOi . (8) m

Om

qOm

m

Trivially, we have that the space of virtual displacement of the RVE, defined as

Vm ¼ Vm ðOm Þ  fg 2 ½H1 ðOm Þ2 : g ¼ v1  v2 ; 8v1 ; v2 2 Km g,

(9)

˜ m. coincides with the space of microscopic displacement fluctuations, i.e. Vm ¼ K 2.2. Macroscopic stress and the Hill–Mandel principle Similarly to the macroscopic strain tensor (1), the macroscopic stress tensor, T, is defined as the volume average of the microscopic stress field Tm over the RVE: Z 1 Tm dOm . (10) T V m Om Let us consider a generic RVE with body force field bm ¼ bm ðyÞ in Om and an external traction field qm ¼ qm ðyÞ on qOm . The Hill–Mandel principle of macrohomogeneity (Hill, 1965; Mandel, 1971) establishes that the power of the macroscopic stress tensor at an arbitrary point of the macrocontinuum must be equal to the volume average of the power of the microscopic stress over the RVE associated with that point for any kinematically admissible motion of the RVE. As shown in de Souza Neto and Feijo´o (2006), as a consequence of the Hill–Mandel principle, bm and qm must satisfy the variational

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equations Z

bm  g dOm ¼ 0

and

Z

Om

qOm

qm  g dqOm ¼ 0 8g 2 Vm .

(11)

That is, the body force and external traction fields of the RVE belong to the functional space orthogonal to the chosen Vm —they are reactions to the constraints imposed upon the possible displacement fields of the RVE, embedded in the choice of Vm (or Km ). Hence, once the space Vm of microscopic displacement fluctuations is chosen, the space of admissible body force and external traction fields is automatically defined so that these fields cannot be independently prescribed (de Souza Neto and Feijo´o, 2006). 2.3. The RVE equilibrium problem In the present analysis, we shall assume the materials of the RVE matrix and inclusions to satisfy the classical linear elastic constitutive law: Tm ¼ Cm Em ,

(12)

where Cm is the fourth order elasticity tensor defined as 8 < Cm if y 2 Om m; Cm ¼ Cm ðyÞ ¼ : Ci if y 2 Oim ;

(13)

with Cm and Ci denoting, respectively, the elasticity constitutive tensors of the matrix and inclusion parts of the RVE:

Cm ¼

Em ½ð1  nm ÞII þ nm ðI  IÞ; 1  n2m

Ci ¼

Ei ½ð1  ni ÞII þ ni ðI  IÞ. 1  n2i

(14)

In the above, we have assumed the matrix and inclusion materials to be isotropic. The parameters Em and Ei are Young’s moduli of the matrix and inclusions, nm and ni the corresponding Poisson’s ratios and I and II are the second and fourth order identity tensors, respectively. If the RVE has more than one inclusion, the parameters Ei and ni are constant within each inclusion, but may differ between inclusions. The linearity of (12) together with the additive decomposition (5), allows the microscopic stress field to be split as Tm ¼ T¯ m þ T˜ m , (15) s ˜ ˜ ˜ m ; and T¯ m is the microscopic stress field ˜ m ðyÞ, i.e. Tm ¼ Cm r u where Tm is the stress fluctuation field associated with u induced by the uniform strain E, i.e. T¯ m ¼ Cm E. Taking into account (15) and in view of (11), this leads to the definition of the RVE equilibrium problem which consists of ˜ m 2 Vm , such that finding, for a given macroscopic strain E, an admissible microscopic displacement fluctuation field u Z Z ˜ m Þ  rs g dOm ¼  T˜ m ðu Cm E  rs g dOm 8g 2 Vm . (16) Om

Om

By means of standard arguments, it can be shown that the above variational form leads to the following Euler–Lagrange equations: 8 ˜ m Þ ¼ 0 in Om ; > div T˜ m ðu > > > > s > ˜ ˜ m in Om ; ˜ > Tm ðum Þ ¼ Cm r u > > R > > ˜ > < Om um dOm ¼ 0; R (17) ˜ m s n dqOm ¼ 0; u > > > qOm > > > ˜ m U ¼ 0 on qOim ; > > 1u > > > > ˜ m ÞUn ¼ 1Cm EUn on qOim : : 1T˜ m ðu ˜ m according to (8). Also note that, trivially, the Remark 1. Condition (17)4 is naturally satisfied due to the choice of space K mean value of the right-hand side of (17)6 vanishes. Then from (17)3 and the Lax–Milgram Lemma we have that there exists an unique solution for (16). 2.4. Classes of multi-scale constitutive models ˜  of variations of To completely define a constitutive model of the present type, the choice of a space Vm  K m admissible displacement must be made. We list below four classical possible choices: (a) Taylor model or rule of mixtures (homogeneous strain over the RVE). This class of models is obtained by simply defining

Vm ¼ VT m  f0g.

(18)

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In this case, the strain is homogeneous over the RVE, i.e. Em ¼ E in Om . The reactive RVE body force and external traction ? fields, ðqm ; bm Þ 2 ðVT m Þ , may be arbitrary functions. (b) Linear boundary displacement model. For this class of models the choice is 

˜ :u ˜m 2 K Vm ¼ VL m  fu m ˜ m ðyÞ ¼ 0 8y 2 qOm g.

(19) L

The only possible reactive body force over Om orthogonal to Vm is bm ¼ 0. That is, only a zero microscopic body force is ? compatible with this class of models. On qOm , the resulting reactive external traction, qm 2 ðVL m Þ , may be any function. (c) Periodic boundary fluctuations model. This class of constitutive models is appropriated to represent the behaviour of materials with periodic microstructure. Typical examples of periodic RVEs in two dimensions are square and hexagonal cells (see Fig. 2). In this case the RVE boundary is composed of N pairs of equally sized sets of sides

qOm ¼

N [

 ðGþ j ; Gj Þ,

(20)

j¼1  þ   such that, each point yþ 2 Gþ j has a corresponding point y 2 Gj , and that the normal vectors to the sides ðGj ; Gj Þ at the points ðyþ ; y Þ satisfy  nþ j ¼ nj .

(21)

The space of displacement fluctuations is defined as 

þ ˜ :u ˜m 2 K ˜ m ðy Þ 8pair ðyþ ; y Þ 2 qOm g. Vm ¼ VP m  fu m ˜ m ðy Þ ¼ u

(22)

Again, only the zero body force field is orthogonal to the chosen space of fluctuations. The external traction fields—in this case orthogonal to VP m —satisfy qm ðyþ Þ ¼ qm ðy Þ

8pair ðyþ ; y Þ 2 qOm .

(23)

That is, the external traction is anti-periodic. (d) Minimally constrained or uniform RVE boundary traction model. In this case, we chose 

˜ Vm ¼ VU m  Km .

(24)

Again only the zero body force field is orthogonal to the chosen space. The boundary traction orthogonal to the space of fluctuations in this case can be shown (refer to de Souza Neto and Feijo´o, 2006, for instance) to satisfy the uniform boundary traction condition: qm ðyÞ ¼ TnðyÞ

8y 2 qOm ,

(25)

where T is the macroscopic stress tensor (10) at point x of the macrocontinuum.

Note that the spaces of displacement fluctuations (and virtual displacement) listed above satisfy L P U VT m  Vm  Vm  Vm .

(26)

The variational framework adopted in the estimation of the macroscopic response allows different predictions (including an upper and a lower bound) of macroscopic behaviour to be obtained according to the constraints imposed upon the chosen functional space displacement fluctuations of the RVE.

Fig. 2. RVE geometries for periodic media. Square and hexagonal cells.

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2.5. The homogenised elasticity tensor Crucial to the developments presented in Section 3, which form the main contribution of the present paper, is the derivation of formulae for the macroscopic elasticity tensors obtained by means of the multi-scale modelling procedure discussed in the above. This is addressed in the following. With the notation introduced in (12), the variational problem defined by (16) can be equivalently written as Z Z Cm rs u˜ m  rs g dOm ¼  Cm E  rs g dOm 8g 2 Vm . (27) Om

Om

To derive a compact expression of the macroscopic elasticity tensor it is convenient to re-write (27) as a superposition of linear problems associated with the individual Cartesian components of the macroscopic strain tensor as suggested by Michel et al. (1999). We start by writing the macroscopic strain in Cartesian component form E ¼ ðEÞij ei  ej ,

(28)

where fei g is an orthonormal basis of the two-dimensional Euclidean space and the scalars ðEÞij are the corresponding ˜ m 2 Vm can be Cartesian components of the macroscopic strain, i.e. ðEÞij ¼ E  ðei  ej Þ. Since (27) is linear, its solution u constructed as a linear combination of the components of the macroscopic strain, ðEÞij , as ˜ m ¼ ðEÞij u ˜ mij , u

(29)

˜ mij 2 Vm are the solutions to the linear variational equations where the vector fields u Z Z Cm rs u˜ mij  rs g dOm ¼  Cm ðei  ej Þ  rs g dOm 8g 2 Vm , Om

(30)

Om

for i; j ¼ 1; 2 (in the two-dimensional case). The above equation is obtained by combining (27)–(29). By combining (5), (28) and (29) we can write the microscopic strain as a linear combination of the Cartesian components of the macroscopic strain ˜ mij Þ ¼ ðEÞij Emij . Em ¼ ðEÞij ðei  ej þ rs u

(31)

Then we can define the canonical stress and strain tensors as Tmij ¼ Cm Emij

s ˜ mij . with Emij ¼ ei  ej þ r u

(32)

With the introduction of the additive decomposition (15) and the constitutive law (12) into (10) and by making use of (29), the macroscopic stress tensor T can be written as ! Z 1 T ¼ CT E þ Cm rs u˜ mij dOm ðEÞij , (33) V m Om where CT is the homogenised (volume average) macroscopic elasticity tensor associated with the Taylor model, given by Z 1 CT ¼ Cm dOm . (34) V m Om s ˜ mij can be expressed as Now, note that the second order tensor Cm r u

Cm rs u˜ mij ¼ ðCm Þklpq ðrs u˜ mij Þpq ðek  el Þ.

(35)

Also, let us assume that, at the macroscopic level, the constitutive law for linear elasticity reads T ¼ CH E, H

where C

(36)

can be recognised as the homogenised elasticity tensor. In fact, by replacing the above into (33), we obtain

˜ E, CH E ¼ CT E þ C ˜ is given by where C " Z # ˜  1 ˜ mkl Þpq dOm ðei  ej  ek  el Þ. C ðCm Þijpq ðrs u V m Om

(37)

(38)

Finally, with the above at hand, we arrive at the following compact canonical mathematical expression for the homogenised elasticity tensor: ˜. CH ¼ CT þ C

(39)

˜ depends on the choice of space Vm (the solutions u ˜ mij of (30) taking part in (38) Note that only the contribution of C ˜ ¼ 0 and CH ¼ CT . ˜ mij ¼ 0 for all i; j. In this case, C depend of this choice). Obviously, under Taylor assumption u

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3. The topological sensitivity of the homogenised elasticity tensor This section presents the main result of this paper. Here, we derive a closed formula for the sensitivity of the homogenised elasticity tensor (39) to the introduction of a circular hole centred at an arbitrary point of the RVE domain. The result is valid for two-dimensional problems. The sensitivity in the present case is the fourth order tensor field over Om (the original domain of the RVE, without the hole) given by

SðyÞ ¼ 

1 ðHTmij  Tmkl Þei  ej  ek  el EðyÞ

8y 2 Om ,

(40)

where Tmij is defined in (32)1 and H is the fourth order tensor

H ¼ 4II  I  I.

(41) H

The tensor SðyÞ is the (topological) sensitivity of the macroscopic elasticity tensor C change of the RVE produced by the introduction of a hole centred at point y 2 Om . To gain some insight into the meaning of S, let dCH  denote the difference H H dCH  ¼ C  C

with respect to the topological

(42)

between the homogenised elasticity tensor CH  of the RVE topologically perturbed by the introduction of a hole of radius  and the elasticity tensor CH of the unperturbed (original) RVE. The approximation to dCH  linear in the volume fraction p2 =V m of perturbation is given by

dCH  ¼

p2 Vm

S þ oð2 Þ.

(43)

The sensitivity tensor (40) provides a first order sensitivity of how the macroscopic elasticity tensor varies when a topological perturbation is added to the RVE. Each Cartesian component Sijkl ðyÞ represents the derivative of the component ijkl of the macroscopic elasticity tensor with respect to the volume fraction p2 =V m of a circular hole of radius  inserted at an arbitrary point y of the RVE. ˜ mij Remark 2. The remarkable simplicity of the closed form sensitivity given by (40) is to be noted. Once the vector fields u have been obtained as solutions of (30) for the original RVE domain, the sensitivity tensor can be trivially assembled. The information provided by (40) can be potentially used in a number of practical applications such as, for example, the design of microstructures to match a specified macroscopic constitutive response. The derivation of (40)–(43) is based on the mathematical concept of topological derivative (Ce´a et al., 2000; Garreau et al., 2001; Nazarov and Sokolowski, 2003; Sokolowski and Zochowski, 1999) and is presented in detail in Sections 3.3–3.5. Before proceeding to the derivation, however, we find convenient to present below some background material on this relatively new topic which should be particularly helpful to those unfamiliar with the topological derivative concept. 3.1. Topological derivative. Preliminaries Let c be a functional that depends on a given domain and let it have sufficient regularity so that the following expansion is possible:

cðÞ ¼ cð0Þ þ f ðÞDT c þ oðf ðÞÞ,

(44)

where cð0Þ is the functional evaluated for the given original domain and cðÞ denotes the functional evaluated for a domain obtained by introducing a topological perturbation in the original domain. The parameter  is a small positive scalar defining the size of the topological perturbation, so that the original domain is retrieved when  ¼ 0. In addition, f ðÞ is a regularising function defined such that lim f ðÞ ¼ 0,

!0þ

(45)

and oðf ðÞÞ contains all terms of higher order in f ðÞ. Expression (44) is named the topological asymptotic expansion of c. The term DT c is defined as the topological derivative of c at the unperturbed (original) RVE domain. The term f ðÞDT c is a correction of first order in f ðÞ to the functional cð0Þ, evaluated for the original domain, to obtain cðÞ—the functional value for the perturbed domain. Analogously to (42) let dc denote the difference

dc ¼ cðÞ  cð0Þ,

(46)

then, similar to (43), we have the linear approximation

dc ¼ f ðÞDT c þ oðf ðÞÞ.

(47)

The concept of topological derivative is an extension of the classical notion of derivative. It has been rigorously introduced in 1999 by Sokolowski and Zochowski (1999) in the context of shape optimisation for two-dimensional heat conduction

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and elasticity problems (for an introduction to the shape optimisation concept see Murat and Simon, 1976; Sokolowski and Zole´sio, 1992). In their pioneering paper, these authors have considered domains topologically perturbed by the introduction of a hole subjected to homogeneous Neumann boundary condition. Since then, the notion of topological derivative has proved extremely useful in the treatment of a wide range of problems in mechanics, optimisation, inverse analysis and image processing and has become a subject of intensive research (Ce´a et al., 2000; Garreau et al., 2001; Guillaume and Sid Idris, 2002, 2004; Novotny et al., 2003; Feijo´o et al., 2003; Nazarov and Sokolowski, 2003, 2004; Lewinski and Sokolowski, 2003; Samet et al., 2003; Sokolowski and Zochowski, 2003, 2005; Burger et al., 2004; Feijo´o, 2004; Amstutz, 2005; Amstutz et al., 2005; Hintermu¨ller, 2005; Masmoudi et al., 2005; Amstutz and Andra¨, 2006; Bonnet, 2006; Auroux et al., 2007). More recent developments include the use of the topological derivative in two- and three-dimensional optimisation of elastic structures (Giusti et al., 2008; Novotny et al., 2007), image processing (Belaid et al., 2008; Larrabide et al., 2008) with application to breast cancer diagnosis (Guzina and Chikichev, 2007) and the extension of the original concept to the definition of a second order topological derivative (Rocha de Faria et al., 2007). 3.2. Application to the multi-scale elasticity model Our purpose here is to derive the closed formula for the topological sensitivity of the macroscopic elasticity tensor. To this end, it is appropriate to define the following functional:

cðÞ  V m T  E ) cð0Þ ¼ V m T  E,

(48)



where T denotes the macroscopic stress tensor resulting from a macroscopic strain E at a point of the macrocontinuum associated with a RVE topologically perturbed by a small hole defined by H and T denotes the macroscopic stress tensor associated to the original (unperturbed) domain Om . More precisely, the perturbed RVE domain Om is defined as follows i (refer to Fig. 3). The hole H of radius  is introduced at an arbitrary point y^ 2 Om of the original RVE domain Om ¼ Om m [ Om of Section 2. The topologically perturbed domain is then defined as

Om ¼ Om nH .

(49)

The asymptotic topological expansion of the functional (48) reads T  E ¼ T  E þ

1 f ðÞDT c þ oðf ðÞÞ, Vm

(50)

H or, equivalently, by making use of the macroscopic constitutive law used in Section 2.5 ðT ¼ CH  E; T ¼ C EÞ and definition (42),

dCH  EE¼

1 f ðÞDT c þ oðf ðÞÞ. Vm

(51)

The sensitivity tensor will be determined as follows. Once the asymptotic expansion of c leading to an explicit closed form for (51) has been carried out, the sensitivity tensor will be identified by comparing the resulting expression with (43). 3.3. Topological derivative calculation In order to obtain a closed form expression of the asymptotic expansion (50), we start here by deriving a closed formula for the associated topological derivative DT c. To this end, we define the functional Z s cðÞ  JOm ðum Þ ¼ Tm ðum Þ  r um dOm , (52) 

Om 

where ˜ m um ¼ u þ Ey þ u

(53)

x

RVE

perturbed RVE

y^

Fig. 3. Microstructure perturbed with a hole H .

hole

ε

macro-continumm

y^

n

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˜ m is the corresponding is the microscopic displacement field that solves the equilibrium problem for the perturbed RVE, u displacement fluctuation and Tm —also a functional of um —is the microscopic stress field, that is with Em ¼ rs um .

Tm ¼ Cm Em

(54)

˜ m 2 Vm is the solution of the following variational equation: In particular, u Z Z ˜ m Þ  rs g dOm ¼  T˜ m ðu Cm E  rs g dOm 8g 2 Vm , Om 

(55)

Om

where Vm ¼ Vm ðOm Þ is the chosen space of kinematically admissible displacement fluctuations of the perturbed RVE. The Euler–Lagrange equation associated with the variational form (55) is given by the following boundary value problem: 8 ˜ m Þ ¼ 0 in Om nH ; div T˜ m ðu > > > > s > ˜ m ðu ˜ m in Om nH ; ˜ > T Þ m ¼ Cm r u >  > R > > ˜ > u d O ¼ 0; m m > Om > > > ˜ > > 1um U ¼ 0 on qOim ; > > > > > > ˜ m ÞUn ¼ 1Cm EUn on qOim ; 1T˜ m ðu > > > > : T˜ ðu ˜ Þn ¼ ðC EÞn on qH : m

m



m

Remark 3. Once again (refer to Remark 1), condition (56)4 is naturally satisfied by the choice of the space Vm , compatible with the choice for Vm. In addition, since the mean value of the right-hand sides of (56)6,7 vanish, from (56)3 and the Lax–Milgram Lemma we have that there exists an unique solution for (55). Finally, from the coercivity of the left-hand side of (16) and (55) we have that the following estimate holds for the two-dimensional case under analysis (Ammari and Kang, 2004; Amstutz, 2006; Sokolowski and Zochowski, 2001): ˜ m  u ˜ m kVm ¼ OðÞ. ku 

(57)

It should be noted that the functional (52) depends explicitly and implicitly on the domain Om . Its implicit dependence ˜ m is the solution of the RVE equilibrium problem (55), for the stems from the fact that the displacement fluctuation field u perturbed RVE domain. Among the methods for calculation of the topological derivative currently available in the literature, here we shall adopt the methodology proposed by Sokolowski and Zochowski (2001) and further developed by Novotny et al. (2003), whereby the topological derivative is obtained as the limit DT c ¼ lim

1 0

d

!0 f ðÞ d

(58)

JOm ðum Þ.

The derivative of the functional JOm ðum Þ with respect to the perturbation parameter  can be seen as the sensitivity of  JOm , in the classical sense (Murat and Simon, 1976), to the change in shape produced by a uniform expansion of the hole. Accordingly, we define a sufficiently regular shape change velocity field, v, over Om , such that ( v¼0 on qOm m; (59) v ¼ n on qH :

3.3.1. Rule of mixtures Let us start by dealing with the simplest class of multi-scale models described in Section 2.4—the rule of mixtures (or Taylor) model. In this case, we combine (52) and the result presented in the Case (a) of Section 2.4 for the perturbed RVE to derive Z d d T¯ m  Em dOm JOm ðum Þ ¼ d d  Om  Z d ¼ T¯ m  E dOm d Om nH Z Z ¼ T¯ m  E ðv  nÞ dqOm þ T¯ m  E ðv  nÞ dqH qOm

qH

¼  2pT¯ m  E.

(60)

By substituting (60) into definition (58) of the topological derivative and identifying function f ðÞ as f ðÞ ¼ p2 ,

(61)

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we find that, for the rule of mixtures model, ¯ DT T c ¼ Tm  E.

(62)

Note that f ðÞ represents the hole area. 3.3.2. Other classes of multi-scale models The derivation presented in the above is relatively simple due to the trivial definition (18) of the space of variations of admissible displacement fields for the rule of mixtures model. For the other models of Section 2.4, the derivation is ˜ m is the solution of the variational considerably more elaborated as due account needs to be taken of the fact that u equilibrium problem (55) for the perturbed RVE domain. In order to proceed, it is convenient to introduce an analogy to classical continuum mechanics (Gurtin, 1981) whereby the RVE shape change velocity field (59) is identified with the classical velocity field of a deforming continuum and  is identified as a time parameter (refer to Haug et al., 1986; Sokolowski and Zole´sio, 1992 for analogies of this type in the context of shape sensitivity analysis). Proposition 4. Let JOm ðum Þ be the functional defined by (52). Then, the derivative of the functional JOm ðum Þ with respect to   the small parameter  is given by Z d JOm ðum Þ ¼ Rm  rv dOm , (63) d Om where v is the RVE shape change velocity field defined in Om and Rm is a generalisation of the classical Eshelby momentumenergy tensor (Eshelby, 1975; Gurtin, 2000) of the RVE, given by

Rm ¼ ðTm  Em ÞI  2ðru˜ m ÞT Tm .

(64)

Proof. By making use of Reynolds’ Transport Theorem (Gurtin, 1981; Sokolowski and Zole´sio, 1992), we obtain the identity Z d d JOm ðum Þ ¼ ðTm  Em Þ þ Tm  Em div v dOm . (65) d d Om  Next, by using the concept of material derivative of a spatial field (Gurtin, 1981; Sokolowski and Zole´sio, 1992), we find that the first term of the above right-hand side integral can be written as d ðTm  Em Þ ¼ 2Tm  E_ m , d

(66)

where the superimposed dot denotes the (total) material derivative with respect to . Further, note that the relation ˜ m Em ¼ E þ rs u

(67)

s ˜ m Þ  , E_ m ¼ ðr u

(68)

gives

which, after some manipulations exploring the relations between the material derivatives of spatial quantities and their gradients, results in s_ ˜ m  ðru ˜ m rvÞs . E_ m ¼ r u

(69)

Then, by introducing the above expression into (66) we obtain d s_ ˜ m  2Tm  ðru ˜ m rvÞs , ðTm  Em Þ ¼ 2Tm  r u d which, substituted in (65), gives Z d s_ ˜ m  2Tm  ðru ˜ m rvÞs þ ðTm  Em ÞI  rv dOm , JOm ðum Þ ¼ 2Tm  r u d Om

(70)

(71)

where we have made use of the identity, div v ¼ I  rv. Now, note that by definition of the spaces of displacement _˜ m 2 Vm . This, together with the equilibrium equation (55), implies that the first term of (71) variations, we have u   vanishes. & Proposition 5. Let JOm ðum Þ be the functional defined by (52). Then, the derivative of the functional JOm ðum Þ with respect to   the small parameter  can be written as Z d JOm ðum Þ ¼ Rm n  v dqOm , (72) d qOm where v is the RVE shape change velocity field and Rm is given by (64).

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565

Proof. Let us compute the shape derivative of the functional JOm using the following version for the Reynolds’ Transport  Theorem (Gurtin, 1981; Sokolowski and Zole´sio, 1992): Z Z d q JOm ðum Þ ¼ ðTm  Em Þ dOm þ ðTm  Em Þv  n dqOm . (73) d Om  q  qOm Next, by using the concept of spatial derivative (Gurtin, 1981; Sokolowski and Zole´sio, 1992) and (12), we find that the first term of the above right-hand side integral can be written as

q ðT  E Þ ¼ 2Tm  E0m , q m m

(74)

where the prime denotes the (partial) spatial derivative with respect to . Further, note that relation (67) gives _˜ m  ru ˜ 0 ¼ rs ½u ˜ m v. E0 ¼ rs u m

m





(75)

Then, by introducing the above expression into (74) we obtain

q s_ ˜ m  2Tm  rs ðru ˜ m vÞ. ðT  E Þ ¼ 2Tm  r u q m m With the above result, the sensitivity of the functional JOm reads  Z Z Z d s_ s ˜ m dOm  ˜ m vÞ dOm þ JOm ðum Þ ¼ 2 Tm  r u 2Tm  r ðru ðTm  Em Þv  n dqOm . d Om Om  qOm

(76)

(77)

_˜ m 2 Vm . This, together with the Now, note that by definition of the spaces of displacement variations, we have u   equilibrium equation (55), implies that the first term of (77) vanishes. Then, we obtain Z Z d ˜ m vÞ dOm þ JOm ðum Þ ¼  2Tm  rs ðru ðTm  Em Þv  n dqOm . (78) d Om qOm Using the tensor relation ˜ m ÞvÞ ¼ T  rs ½ðru ˜ m Þv þ divðTm Þ  ðru ˜ m Þv, divðTTm ½ðru and the divergence theorem, expression (78) can be written as Z Z Z d ˜ m Þv dOm  2 ˜ m ÞT Tm n  v dqOm þ JOm ðum Þ ¼ 2 div Tm  ðru ðru ðTm  Em Þv  n dqOm . d Om qOm qOm

(79)

(80)

Since the stress field Tm is in equilibrium, from Euler–Lagrange equation (56)1, we have that div Tm ¼ 0 in Om . Therefore, a straightforward rearrangement of the above yields (72). & Corollary 6. By applying the divergence theorem to the right-hand side of (63), we obtain Z Z d JOm ðum Þ ¼ Rm n  v dqOm  divðRm Þ  v dOm . d qOm Om Since (72) and (81) remain valid for all velocity fields v of Om , we have that Z div ðRm Þ  v dOm ¼ 0 8v 2 Om ) divðRm Þ ¼ 0 in Om ,

(81)

(82)

Om 

i.e. Rm is a divergence-free field. Then, from result (72) and taking definition (59) into account, together with the definition of qOm , we finally arrive at the following expression for the sensitivity of JOm exclusively in terms of integrals over the boundary qH of the hole  Z d J ðum Þ ¼  Rm n  n dqH . (83) d Om qH We now proceed to derive an explicit expression for the integrand on the right-hand side of (83). Then, consider a curvilinear coordinate system n2t along qH , characterised by the orthonormal vectors n and t. For convenience we decompose the stress tensor Tm ðum Þ and the strain tensor Em ðum Þ on the boundary qH as follows: nt tn tt Tm jqH ¼ Tnn m ðn  nÞ þ Tm ðn  tÞ þ Tm ðt  nÞ þ Tm ðt  tÞ, nt tn tt Em jqH ¼ Enn m ðn  nÞ þ Em ðn  tÞ þ Em ðt  nÞ þ Em ðt  tÞ.

(84)

Now note that the Neumann condition along qH gives T˜ m njqH ¼ T¯ m n ) Tm njqH ¼ 0,

(85)

so that decomposition (84)1 reads tn nn tn Tm njqH ¼ Tnn m n þ Tm t ¼ 0 ) Tm ¼ Tm ¼ 0 on qH .

(86)

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With decomposition (84) and boundary condition (86), the normal flux of the Eshelby tensor (64) through qH can be written as

Rm n  n ¼ ðTm  Em Þ  2Tm n  ðru˜ m Þn nn nt nt tt tt ¼ Tnn m Em þ 2Tm Em þ Tm Em tt ¼ Ttt m Em ,

(87)

or, equivalently (using the inverse constitutive relation),

Rm n  n ¼

1 tt tt 1 ˜ tt 2 ¯ tt T ðT  nðyÞTnn m Þ ¼ EðyÞ ðT m þ Tm Þ , EðyÞ m m

(88)

tt

tt

where T¯ m and T˜ m are the constant and fluctuation part, respectively, of component Ttt m of the stress tensor Tm jqH , and the parameters EðyÞ and nðyÞ are defined as 8 8 < Em if y 2 Om < nm if y 2 Om m; m; nðyÞ ¼ EðyÞ ¼ (89) i : Ei : ni if y 2 Om ; if y 2 Oim : In order to obtain an analytical formula for the boundary integral of (83) we make use of a polar coordinate system ðr; yÞ centred at point y^ and the classical asymptotic analysis described in Appendix A, see for instance Ammari and Kang (2004) and Kozlov et al. (1999). The substitution of (A.4) into (88) and then into (83) allows the integral of the normal flux of the Eshelby tensor across qH to be analytically solved, resulting in d 2p J ðum Þ ¼  ½4Tm  Tm  ðtr Tm Þ2  þ oðÞ, ^ d Om EðyÞ

(90)

where trðÞ denotes the trace of ðÞ. The substitution of the above expression for the derivative of JOm into (58) allows the function f ðÞ to be promptly  identified in the same way as (61). Finally, by taking the limit of the resulting formula for  ! 0, we obtain the explicit closed form expression for the topological derivative of c: DT c ¼ 

1 HTm  Tm , ^ EðyÞ

(91)

where H is given by (41). Remark 7. The topological derivative of c given by (91) is a scalar field over Om that depends only on the material properties of matrix (or inclusions) and on the solution um for the original unperturbed domain Om . The striking simplicity of the exact formula (91) is to be noted. 3.4. Numerical verification In direct analogy with classical finite difference-based methods for the numerical approximation of the derivative of a generic function, a first order topological finite difference formula based on (47) to approximate numerically the value of DT c at the unperturbed RVE configuration can be defined as dT c 

cðÞ  cð0Þ f ðÞ

,

(92)

with finite . The above satisfies lim dT c ¼ DT c. !0

(93)

If for a given RVE we calculate cð0Þ and its perturbed counterpart cðÞ for a sequence of decreasing (sufficiently small) inclusion radii , the use of formula (92) will provide an asymptotic approximation to the analytical value of DT c given by (91). Here such a procedure is used to provide a numerical validation of (91). The required values of the function c are computed numerically by means of a finite element procedure specially suited to handle the kinematical constraints of the present multi-scale theory (refer, for instance, to Giusti et al., 2009). The unperturbed RVE considered (refer to Fig. 4) consists of a unit square containing a circular inclusion of radius 0.1 centred at the point with coordinates (0.35,0.75)—with the origin of the Cartesian coordinate system located at the bottom left hand corner of the RVE. For the computation of the values of cðÞ, a sequence of finite element analyses are carried out for perturbed RVEs obtained by introducing in the original RVE a circular hole of radii

 2 f0:160; 0:080; 0:040; 0:020; 0:010; 0:005g,

(94)

centred at y^ ¼ ð0:5; 0:5Þ. Finite element meshes of six-noded isoparametric triangles are used to discretise the perturbed domains. All meshes are built such that the hole boundary of radius  has 80 elements. For example, the mesh with  ¼ 0:16 (shown in Fig. 4(b)) contains 1864 elements with a total number of 5593 nodes. The macroscopic strain tensor (which can

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567

y^

Fig. 4. Numerical verification. (a) RVE geometry and (b) finite element mesh for  ¼ 0:16.

be chosen arbitrarily) considered to compute c in the analyses is   1:00 0:05 E¼ . 0:05 2:00

(95)

The study is conducted for two different sets of material properties for the matrix and inclusion:

Case A: Em ¼ 50:0, nm ¼ 13, Ei ¼ 5:0 and ni ¼ 13. Case B: Em ¼ 50:0, nm ¼ 15, Ei ¼ 5:0 and ni ¼ 15. In each case, the numerical verification is carried out under the assumptions of: (a) linear boundary displacement; (b) periodic boundary displacement fluctuations; and (c) uniform boundary traction. The rule of mixtures (or Taylor) model is not considered. For this model, the solution is trivial and does not require a finite element analysis. If one insisted in undertaking the present verification for the rule of mixtures model, the only difference between the exact topological derivative (62), which does not depend on , and its numerical counterpart would be the result of the geometrical approximation of the inclusion domain by the relevant assembly of finite element subdomains. The results of the analyses are plotted in Fig. 5. They show the analytical topological derivative and the corresponding numerical approximation for each value of  for all classes of multi-scale model considered. The convergence of the numerical topological derivatives to their corresponding analytical values with decreasing  is obvious in all cases and confirms the estimate (57) and also the correctness of formula (91). 3.5. The sensitivity of the macroscopic elasticity tensor From (91) and (50) we have the explicit expression for the topological expansion of c: T  E ¼ T  E 

vðÞ HTm  Tm þ oð2 Þ, ^ EðyÞ

(96)

where vðÞ ¼ p2 =V m

(97)

is the RVE volume fraction occupied by the perturbation. Introducing (31) into (12), the microscopic stress tensor Tm can be written as Tm ¼ ðEÞij Cm Emij ¼ ðTmij  ei  ej ÞE,

(98)

where Tmij is defined in (32)1 . With the above expressions at hand, we see after straightforward manipulations that the topological derivative of c given by (91) can be represented as DT c ¼ DT m E  E,

(99)

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890.0 885.0 880.0 875.0 870.0 865.0 860.0 855.0 850.0 0

25

50

75

100

125

150

175

200

0

25

50

75

100

125

150

175

200

712.0 707.0 702.0 697.0 692.0 687.0 682.0

Fig. 5. Numerical verification. Convergence of numerical topological derivative to analytical value. (a) Case A and (b) Case B.

where DT m is the fourth order symmetric tensor field over Om defined by

DT m ¼

1 ðHTmij  Tmkl Þei  ej  ek  el , ^ EðyÞ

(100)

with i; j; k; l ¼ 1; 2. Then, by replacing (99), (100) and (61) into (51) we obtain the explicit form

dCH  EE¼

p2 Vm

DT m E  E þ oð2 Þ.

(101)

Finally, the sensitivity tensor (40),

S ¼ DT m ,

(102)

can be promptly identified by comparing (101) with the linear approximation (43) that defines the sensitivity. 4. Conclusion An analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological microstructural changes of the underlying material has been proposed in this paper. The derivation of the proposed fundamental formula relied on the concept of topological derivative, applied within a variational multi-scale constitutive framework for linear elasticity problems where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a RVE of material associated with that point. The derived sensitivity—a symmetric fourth order tensor field over the RVE domain—measures how the estimated macroscopic elasticity tensor changes when a small circular hole is introduced at the microscale. This crucial information can be potentially used in a number of applications of practical interest such as, for instance, the design and optimisation of microstructures to achieve a specified macroscopic behaviour. The successful application of analogous ideas in the context of design and optimisation of load bearing (macroscopic) structures is reported in Novotny et al. (2005, 2007), for instance.

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Finally, we remark that the derivation of analogous formulae for the three-dimensional case as well as for the sensitivity to the introduction of inclusions in the RVE is currently under way and should be the subject of a future publication.

Acknowledgements This research was partly supported by CNPq (Brazilian Research Council) and FAPERJ (Research Foundation of the State of Rio de Janeiro) under Grants 472182/2007-2 and E-26/171.099/2006. S.M. Giusti was supported by CAPES (Brazilian Higher Education Staff Training Agency). This support is gratefully acknowledged. Appendix A. Asymptotic analysis This appendix presents the derivation of the asymptotic formula used in the topological sensitivity analysis developed in Section 3.3. We start by considering the following expansion of the stress fluctuation field associated with the solution ˜ m to problem (56), see Sokolowski and Zochowski (1999): u 1 ˜ m Þ ¼ T˜ m ðu ˜ m Þ þ oðÞ, T˜ m ðu  1

(A.1) 2

1

where T˜ m denotes the solution of the elasticity system (56) in the infinite domain R nH , such that the stresses T˜ m tend to a constant value when kyk ! 1. Then, the exterior problem can be written as 8 1 ˜ m Þ ¼ 0 in R2 nH ; > div T˜ m ðu > > < 1  T˜ m ! T˜ m at 1; (A.2) > > > ˜1 : ¯ Tm n ¼ Tm n on qH ; where n denotes the outward unit normal to the boundary qH, T˜ m is the solution of the unperturbed problem (17) and T¯ m is that defined in (15). In a polar coordinate system ðr; yÞ having its origin at the centre of the hole H and with the angle y measured with respect to one of the principal directions of T˜ m , the components of the solution of the partial differential equation (A.2), see Obert and Duvall (1967) and Little (1973), are given by      4  1 2 2 ˜ 2 4  2 ¯ 3 4 1  4 2 þ 3 4 cos 2y þ D (A.3) cos 2ðy þ jÞ, ðT˜ m Þrr ¼ S˜ 1  2  S¯ 2 þ D 4 2 r r r r r r     4 2 2 4 1 ˜ 1 þ 3  cos 2y  3D ¯ cos 2ðy þ jÞ, (A.4) ðT˜ m Þyy ¼ S¯ 2 þ S˜ 1 þ 2  D 4 r r r r4     2 4 4 2 1 ˜ 1 þ 2   3  sin 2y þ D ¯ 3 2 ðT˜ m Þry ¼ D (A.5) sin 2ðy þ jÞ, r2 r4 r4 r2 where j denotes the angle between principal stress directions associated to the stress fields T¯ m and T˜ m . In addition, we denote

s¯ m ðuÞ s¯ ðuÞ ¯ þ s¯ m2 ðuÞ ¯ ¯  s¯ m2 ðuÞ ¯ ¯ ¼ m1 ; D , S¯ ¼ 1 2 2 s˜ m ðu˜ m Þ þ s˜ m2 ðu˜ m Þ s˜ ðu˜ Þ  s˜ m2 ðu˜ m Þ ˜ ¼ m1 m ; D , S˜ ¼ 1 2 2

(A.6) (A.7)

˜ m Þ are the principal stresses associated with the displacement fields u:¼Ey ˜ m of the original ˜ m1;2 ðu where s¯ m1;2 ðuÞ and u ¯ and s ¯ (unperturbed) domain Om . References Almgreen, R.F., 1985. An isotropic three-dimensional structure with Poisson ratio—1. Journal of Elasticity 15, 427–430. Ammari, H., Kang, H., 2004. Reconstruction of Small Inhomogeneities from Boundary Measurements. In: Lecture Notes in Mathematics, vol. 1846. Springer, Berlin. Ammari, H., Kang, H., 2007. Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory. In: Applied Mathematical Sciences, vol. 162. Springer, New York. Amstutz, S., 2005. The topological asymptotic for the Navier–Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations 11 (3), 401–425. Amstutz, S., 2006. Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Analysis 49 (1–2), 87–108. Amstutz, S., Andra¨, H., 2006. A new algorithm for topology optimization using a level-set method. Journal of Computational Physics 216 (2), 573–588. Amstutz, S., Horchani, I., Masmoudi, M., 2005. Crack detection by the topological gradient method. Control and Cybernetics 34 (1), 81–101. Auroux, D., Masmoudi, M., Belaid, L., 2007. Image restoration and classification by topological asymptotic expansion. In: Variational Formulations in Mechanics: Theory and Applications. CIMNE, Barcelona, Spain. Belaid, L.J., Jaoua, M., Masmoudi, M., Siala, L., 2008. Application of the topological gradient to image restoration and edge detection. Engineering Analysis with Boundary Elements 32 (11), 891–899. Bendsoe, M.P., Kikuchi, N., 1988. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering 71 (2), 197–224.

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