Struct Multidisc Optim (2014) 49:595–608DOI 10.1007/s00158-013-0994-6

RESEARCH PAPER

Topology optimization of micro-structure for compositesapplying a decoupling multi-scale analysis

Junji Kato · Daishun Yachi · Kenjiro Terada ·Takashi Kyoya

Received: 29 December 2012 / Revised: 30 July 2013 / Accepted: 25 August 2013 / Published online: 13 September 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract The present study proposes topology optimiza-tion of a micro-structure for composites considering the ma-cro-scopic structural response, applying a decoupling multi-scale analysis based on a homogenization approach. In thisstudy, it is assumed that topology of macro-structure isunchanged and that topology of micro-structure is uniqueover the macro-structure. The stiffness of the macro-structure is maximized with a prescribed material volumeof constituents under linear elastic regime. A gradient-basedoptimization strategy is applied and an analytical sensitivityapproach based on numerical material tests is introduced.It was verified from a series of numerical examples that theproposed method has great potential for advanced materialdesign.

Keywords Topology optimization · Decouplingmulti-scale analysis · Micro-structures · Homogenization

1 Introduction

It is well known that the mechanical behavior of a compos-ite material mainly depends on the geometric properties ofthe micro-structure, such as material distribution, shape orsize of the material, and that the dependency will be remark-ably increased in the non-linear regime in which material

J. Kato (�) · K. TeradaInternational Research Institute of Disaster Science,Tohoku University 6-6, Aoba, Aramaki, Aoba-ku,SENDAI 980-8579, Japane-mail: jkato@civil.tohoku.ac.jp

D. Yachi T. KyoyaMechanics of Materials Laboratory, Tohoku University 6-6,Aoba, Aramaki, Aoba-ku, SENDAI 980-8579, Japan

reaches fracture. It is becoming known that non-linear mate-rial behavior always closely relates to its micro-structure,for instance, in order to improve strength or toughness of ametallic material, crystal micro-structure enabling improve-ment of strength or toughness will be surveyed, and in orderto improve energy-absorption capacity or wear resistanceof synthetic rubber, micro-composition is investigated. It issaid that what these composites have in common is that itwill be possible to maximize the mechanical performanceof a macro-structure, if ‘types or kinds of materials to bemixed or the combinations thereof’ are optimized and ‘thegeometric properties of micro-structure’ are optimized.

As production technologies enabling control of materialproperties of micro-structure will be realized in the nearfuture, this study proposes a method to maximize structuralperformance of macro-structure by optimizing the materialdistribution, topology in this study, in the micro-structure.

So far, study on topology optimization most often hasbeen mainly developed for topology of macro-structure.With regard to the preceding studies on topology opti-mization of the micro-structure, for example, Sigmund(1994) proposes a method, named ‘inverse-homogenizationmethod’, to determine topology of micro-structure thatenables exhibiting stiffness equivalent to the prescribedmaterial stiffness CH. Sigmund and Torquato (1997) intro-duce a method, as its application, to determine topology ofmicro-structure that enables exhibiting a thermal expansioncoefficient equivalent to the prescribed thermal expansioncoefficient, and Larsen et al. (1997) also introduce topol-ogy of micro-structure able to exhibit negative Poisson’sratio. Amstutz et al. (2010) also propose an algorithm foroptimization of micro-structures based on an exact formulafor the topological derivative of the macroscopic elasticitytensor and a level set domain representation, where maxi-mization of shear modulus and maximization/minimization

596 Kato et al.

of Poisson’s ratio are described. However, these prominantstudies focus on the micro-structure only, that is, governingequations consisting of a micro-scale boundary value prob-lem (BVP) only, and do not basically consider behavior ofmacro-structure.

For the optimization problem dealing with both micro-and macro-structures, several methodologies have beendeveloped. For example, Rodrigues et al. (2002) propose ahierarchical approach to optimize topologies of the micro-and macro-structure simultaneously, by considering behav-iors of both structures. Niu et al. (2009) optimize topologiesof both micro- and macro-structures for maximizing fun-damental frequency based on the general homogenizationtheory, where topology of microstructure is assumed to beunique over the macro-structure. Su and Liu (2010) alsopropose an optimal micro-structure design considering theinfluence of macro-structural behavior based on the couple-stress theory. However, these applications are limited tolinearly elastic structural problems although they are easy toimplement.

In the meanwhile it is necessary to introduce a multi-scale analysis based on the homogenization method in orderto solve the above-mentioned micro-macro coupling BVP.With regard to multi-scale analysis based on the homog-enization method, many studies have been reported, andvarious analytical methods considering materially and/orgeometrically nonlinear mechanical behavior have beenproposed by, for example, Feyel and Chaboche (2000),Smit et al. (1998), Terada and Kikuchi (2001), Wieckowski(2000), and Zheng et al. (2000). As those methods solveall micro- and macro-scale BVPs simultaneously by recip-rocal exchange in order to achieve higher precision on amicro-macro two-scale BVP, they are considered theoret-ically established and reliable. However, those analyticalapproaches are rarely applied to actual designing becausethey are theoretically difficult to understand and the compu-tational costs are enormous.

The reason for it is briefly described as follows. Firstly,the macroscopic constitutive equation is an implicit functionof the solutions of the micro-scale BVP and, thus, the micro-scale BVP indirectly represents the macroscopic materialresponse. That is, it is not until the micro-scale equilibratedstress is determined that the macroscopic stress can be cal-culated. Therefore, if the two-scale coupling analysis is per-formed by the finite element method, the micro-scale BVPmust be associated with an integration point located in amacro-scale finite element model and solved for the micro-scale equilibrated stress to evaluate the macro-scale stressby the averaging relation (1) introduced later, which mustsatisfy the macro-scale BVP at the same time. In particu-lar, when an implicit and incremental solution method witha Newton–Raphson type iterative procedure is employedto solve the two- scale BVP, the micro-scale BVP is to

be solved in every iteration to attain the macro-scale equi-librium state at every loading step. This type of solutionscheme, i.e. the micro–macro coupling scheme, requires asignificant amount of computational cost.

Taking this problem into account, (Terada et al. 2008,2013; Watanabe and Terada 2010) propose a new methodcalled decoupling multi-scale analysis to solve the micro-macro two-scale BVP by decoupling. This method isintended to reduce the computational costs by introducingan approximate approach called numerical material tests(NMTs), for the problems including numerically expen-sive calculations such as micro-macro two-scale BVPs withmaterial and/or geometrical nonlinearity. Furthermore, asthis approach is intended to solve a micro-scale BVP and amacro–scale BVP independently, this is a theoretcally-clear-approximate approach, and is superior in general-purposeuse, because the method is applicable to various materialsusing the same framework. The details including the neces-sity and numerical accuracy of the decoupling multi-scaleanalysis compared to the micro-macro coupling multi-scaleanalysis are exclusively discussed in Watanabe and Terada(2010). Taking these characteristics into account, the decou-pling of micro- and macro-scale BVPs is “indispensable”for applying the two-scale approach based on homogeniza-tion to various nonlinear structural problems encountered inpractice.

As our final goal is to develop an optimization schemefor material designs considering materially and/or geomet-rically nonlinear structural behavior, this study applies thedecoupling multi-scale analysis for optimization problems,specifically topology optimization of micro-structure con-sidering the structural response of macro-scale structuralanalysis. However, as the present study stands only at thebasic stage of introduction of the decoupling multi-scaleanalysis to optimization problems, this paper handles onlytopology optimization of micro-structure to maximize stiff-ness (to minimize compliance) of the macro-structure onthe assumption of simply applying a two-dimensional planestrain problem in a linear elastic regime. Extension ofour method to more realistic and complex nonlinear struc-tural problems will be challenged in our next step. In thatsense, the present paper is placed at the introductory mate-rial for the decoupling multiscale topology optimizationconsidering nonlinear material models. Of course, for themicro–macro two–scale BVPs under an assumption of lin-ear elasticity, no distinct advantage and difference existsbetween the coupling and decoupling schemes in computa-tional costs.

In this study, we try to determine an optimal uniquetopology of a micro-structure to maximize stiffness of amacro-structure with keeping the initial topology of themacro-structure as a range of feasibility for actual pro-duction. The reason for not changing the topology of the

Topology optimization of micro-structure 597

macro-structure is to consider actual design circumstancesin which the topology of a macro-structure is to be fixed toalmost one by various conditions, such as in designing tiresof automobiles using a synthetic rubber.

In the following, outline of the decoupling multi-scaleanalysis is described first, and then material models usedin this study and the process of formulating the optimiza-tion problem are described. With regard to the algorithm foroptimization, we use the optimality criteria method (Patnaiket al. 1995), hereinafter shown as OC method, on the basisof the gradient-based method that is effective in numericalanalysis. In this paper, an analytical method on the basis ofthe adjoint method is proposed as the derivation of sensitiv-ity, and the process of formulation is explained. Finally, theoptimization method proposed in this paper is verified by aseries of numerical examples.

2 Decoupling multi-scale analysis basedon homogenization approach

2.1 Outline

Decoupling multi-scale analysis (Terada et al. 2013;Watanabe and Terada 2010) differs from a general approachto solve both parts of a micro- and macro-two-scale BVPsimultaneously by maintaining coupling, but to solve thesame problem as two independent boundary value prob-lems by decoupling the boundary value problems. First,with regard to the micro-scale BVP, we extract a periodicmicro-structure called ‘unit cell’, identified based on thehomogeneous method, and then simulate the material testson the micro-structure, regarding it as a numerical experi-ment. Then, by converting the results of micro-analysis tothe macro-material parameters, we regard the conversionas measuring the material response of the macro-structure.A series of processes to identify macro-material behaviorthrough numerical analysis on a unit cell like this is calleda ‘numerical material test’. In this study, as we assumelinearly elastic material, we calculate a macro-stress �

from the micro-stress σ obtained from micro-analysis, andthen convert it into macro-material stiffness C

H. Then, themacro-scale BVP is solved by directly using the macro-material stiffness.

In the meantime, conventional linear multi-scale analysiscalculates macro-material stiffness CH by introducing char-acteristic displacement (generally expressed as χ ) on theassumption that the prescribed macro–strain E given to theunit cell has a linear relation with fluctuation displacement.However, the decoupling multi-scale analysis neither needsexistence of characteristic displacement nor the assumptionbetween the prescribed strain and the fluctuation displace-ment. As the method identifies the macro-stiffness from

results of the numerical material test on the unit cell,it differs theoretically from the conventional micro-macrocoupling multi-scale analysis. In the following, decouplingmulti-scale analysis for the linear elastic model is outlined.Refer to the literatures (Terada et al. 2013; Watanabe andTerada 2010) for details of the decoupling scheme appliedfor nonlinear structural problems.

2.2 Micro-scale BVPs

Macro-structure is defined as a homogeneous body withforce equivalent to the non-homogeneous elastic body hav-ing periodic micro-structure in the state of force equilib-rium. Here, force equilibrium means that macro-stress �

at an arbitrary point x in a macro-structure depends ona periodic micro-structure (unit cell) characterizing non-homogeneity, and is obtained by average volume of micro-stress σ distributed in the unit cell as shown in the followingequation,

� = 1

|Y |∫

Y

σdy = 〈σ 〉 . (1)

Here, Y means the periodic micro-structure field, and y,called a micro-scale variable, shows a position vector ofan arbitrary point in a micro-structure. Similarly, the rela-tion between macro-strain E and micro-strain ε is shown asfollows:

E = 1

|Y |∫

Y

εdy = 〈ε〉 . (2)

Micro-strain ε in (2) is defined by micro-displacementfield w(x, y) in a unit cell as follows:

ε = ∇symy w, (3)

where w is assumed to be divided into two segments asshown in (4): (i) E · y, a term distributed linearly in propor-tionate to macro-strain, and (ii) u∗, fluctuation displacementwhich shows a gap from the linear distribution caused bythe linear displacement field and heterogeneity,

w = Ey + u∗. (4)

Here, as usual in the computational homogenization, thefluctuation displacement u∗ is restricted to be periodic onthe boundary ∂Y of the unit cell, so that

u∗|∂Y [k] = u∗|∂Y [−k] , for k = 1, 2 on ∂Y. (5)

As shown in Fig. 1, if the unit cell is a rectangularparallelepiped and its boundary is placed parallel to the

598 Kato et al.

Fig. 1 Traction force vector t of a unit cell and macro-stress vector t

coordinate axis, ∂Y [k] is a boundary which is parallel withY [k] line, i.e., the boundary on which the orthonormal basisvector ek , is to be the normal vector. From the periodicityof the fluctuation displacement, the following restriction isobtained for the actual displacement:

w[k] − w[−k] = EL[k]. (6)

In other words, this is a formula for constraint with respectto the relative displacement vector between boundary sidesof periodicity to be coupled. For simplicity, we set asw[k] := w|∂Y [k] . Also, L[k] is called a side vector of a unitcell that couples corresponding material points on a pair ofboundaries in ek-axial direction of a rectangular unit cell,

L[k] := Y |∂Y [k] − Y |∂Y [k−1] . (7)

The other periodic boundary condition of a unit cell isthat micro-traction force vector t [n] = σ · n on a bound-ary having a unit vector n is to be imposed anti-periodiccondition on the boundary of the unit cell to be coupled,

t [k] + t [−k] = 0, (8)

where we set t [±k] := t [±ek ] for simplicity. By integrat-ing and averaging the micro-traction force vector t on theperiodic boundary at the boundary of the unit cell, themacro-stress vector t can be obtained as the followingequation (see Figs. 1 and 2),

t[k] = � · ek

= 1

|∂Y [k]|∫

∂Y [k]

σ · ekdy = 1

|∂Y [k]|∫

∂Y [k]

t [k]dy. (9)

By an equation adding the micro-scale equilibrium equa-tion along with the constitutive law for the linearly elasticmaterials with elastic moduli C to the abovementioned

external material points

q k1

q k2 R k

2

R k1

(a)

(b)

Fig. 2 a Concept of external material points having degrees offreedom for relative displacement vector and macro-stress vector, bdegrees of freedom of relative displacement and the correspondingreaction force at an external material point in 2D

equation, a micro-scale BVP is defined. These equations arere-written after rearrangement as follows:

∇y · σ = 0

σ = C : εε = ∇sym

y w

� = 〈σ 〉

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

in Y, (10)

t[k] = 1

|∂Y [k]|∫

∂Y [k]

t [k]dy

w[k] −w[−k] = EL[k]

⎫⎪⎬⎪⎭ on ∂Y [k]. (11)

2.3 Extended system of a micro-scale BVP considering theexternal material points

Here, according to the literatures (Terada et al. 2013;Watanabe and Terada 2010), a formula employing a conceptof external material point is introduced in contrast to bound-ary conditions about an aforementioned micro-scale BVP.First, a constraint condition to relative displacement aroundperiodic boundary of a unit cell is written as follows:

w[k] − w[−k] = q [k], (12)

with

q [k] = EL[k], (13)

where q[k] means relative displacement vector betweenboundary sides of periodicity to be coupled. Also, as this

Topology optimization of micro-structure 599

is a two-dimensional problem, L[k] can be written asfollows:

L[1] ={l[1] 0}T

and L[2] ={

0 l[2]}T

, (14)

where l[1], l[2] show length of boundary sides of a rectangu-lar parallel to e1, e2 axes respectively, namely, l[1] = |∂Y [2]|and l[2] = |∂Y [1]|.

Terada et al. (2013) and Watanabe and Terada (2010),as shown in Fig. 2, placed arbitrary material points foreach periodic boundary side ∂Y [k] (k = 1, 2) to the direc-tion of boundary normal lines respectively, and defined itas an ‘external material point’, and provided two degreesof freedom (DOFs) on each material point in parallel withe1, e2 axes, and allocated components of relative displace-ment vector q [k] to the DOFs of the external material points.Namely, (12) is a formula to control amount of relative dis-placement, which is calculated by the actual displacementvector of two points on the periodic boundaries to be cou-pled. In other words, the external material point is simplyintroduced as a control point for relative displacement andhas no physical meanings. As even the coordinates of theexternal material points are arbitrary, the points are sim-ply located near the corresponding boundaries as shown inFig. 2.

Accordingly and consequently, as shown in (13), in orderto provide arbitrary components of macro-strain E to theunit cell in the numerical material test, it is enough to controlthe components of relative displacement q [k]j at the externalmaterial point (Fig. 3). Here, in order to make the mean-ing of (12) and (13) clear, the deformation of body andthe corresponding macro-strin E, together with relative dis-placements q are depicted in Fig. 4, where the parentheseson E indicates the tensile direction performed in the numer-ical material tests and l[1] and l[2] are assumed to be unitlength for simplicity.

Now, if the relative displacement q[k]j in (12) is given

as a known vector, it just means that relative displacementw[k]j − w

[−k]j is given. The surface traction force vector t [k]j

on the boundary ∂Y [k] will turn out to be unknown. Con-sequently, the macro-stress vector t

[k]j , the average of the

surface traction force vector t[k]j over the boundary ∂Y [k]

will also turn out to be unknown.However, if the reaction force of an external material

point corresponding to known components of relative dis-placement vector q [k]j is expressed as R[k]

j , it is nothing but

the definite integral of traction force vector t [k]j linearly overthe corresponding periodic boundary. Namely,

R[k] =∫

∂Y [k]

t [k]dy. (15)

(wa, w

b)

Y[1]

Y[-1]

q[1]= wc-w

a

wd-w

b

(wp, w

q) (w

r, w

s)

= wr-w

p

ws-w

q

= .........

...

(wc, w

d)

pair of two constrained points

Fig. 3 Pair of constrained points and relative displacement vector

Consequently, in relation to (9), it is understood that thevalue of reaction of an external material point, (15), dividedby a length of boundary line of the unit cell |∂Y [k]| comesout to be an unknown macro-stress component �jk . That is,

�jk = t[k]j = R

[k]j

|∂Y [k]| . (16)

At this stage, when tensile numerical material tests are per-formed to three directions (11), (22), (12) independently,the components of the macro-material stiffness CH can bedetermined as follows:

CHpqrs = �(rs)

pq (17)

For implementation, (16) and (17) can be specificallyexpressed using the discrete formulation as follows:

� =⎧⎨⎩�11

�22

�12

⎫⎬⎭ =

⎧⎪⎨⎪⎩

t[1]1t[2]2

t[1]2 = t

[2]1

⎫⎪⎬⎪⎭ , (18)

q[1]=1

0q[2]=

0

0

q[1]=0

0q[2]=

0

1

q[1]=0

1q[2]=

1

0

E = 10

00

(11)

(22)

(12)

E = 00

01

E = 0 0.500.5

Fig. 4 Original and deformed homogenized bodies of unit cells (left)and its relation between macro-strain E and relative displacementsq[k](k = 1, 2) for 2D case, where l[1] and l[2] are assumed to be unitlength for simplicity

600 Kato et al.

and

CH =⎡⎣C

H11 C

H12 C

H13

CH12 C

H22 C

H23

CH13 C

H23 C

H33

⎤⎦ =⎡⎢⎣�

(11)11 �

(22)11 �

(12)11

�(11)22 �

(22)22 �

(12)22

�(11)12 �

(22)12 �

(12)12

⎤⎥⎦ . (19)

As a results, by using the macro-material stiffness CH

given by the above numerical material test, it becomespossible to solve the macro-scale BVP.

This means resolving a micro-scale BVP including DOFsof newly introduced artificial external material points byproviding relative displacement at the periodic boundaryside to be coupled gives us the reaction force R

[k]j corre-

sponding to the DOFs. This determines the macro-stressvector t [k]j by dividing by the length of the correspondingboundary. The macro-material physical properties necessaryfor macro structural analysis are to be obtained by directlyusing the macro-stress vector t [k]j .

From this regard, if a micro-scale BVP is once resolved,it is possible to induce physical properties of macro-materialfrom the response obtained, irrespective of material modelsto be used. That is to say, macro-material physical prop-erties even having nonlinear behavior can be estimated oridentified using a similar framework.

For the details of the finite element analysis of a unit cellwith external material points, it is referred to Appendix.

3 Definition of design variable and material modelfor micro-structure

3.1 Definition of design variable

This section defines design variables for optimization anddescribes a micro-material model for composites. The mate-rial to be used in this study is a linearly elastic materialexcluding voids made of two-phase composite material con-sisting of two solid materials in micro-scale field, as shownin Fig. 5 (left, above). In this study, we use the finite elementmethod to solve the micro-scale BVP, and define the design

microstructure

i

i

Fig. 5 Concept of two-phase material optimization

variable as volume fraction of constituent materials of eachfinite element in a unit cell,

si = ri

r0. (20)

In the above formula, si means design variable, and isdefined as a continuously varying function between 0 ≤si ≤ 1 similar to in the cases of the general topology opti-mization. Subscript i (= 1, .., nele) means the i-th finiteelement and the subscript nele is the number of elements in aunit cell. r0 and ri show height of an arbitrary finite elementand phase-2 material in a unit cell respectively, as shown inFig. 5 (left, below).

Accordingly, in case si = 0, phase-1 occupies the ele-ment, and in contrast, in case si = 1, phase-2 occupies it. Inthe case of 0 < si < 1, it is considered the mixture of bothphases.

3.2 Material model for micro-structure

In this study as a material model for composite micro-structure, multiphase material model in Kato et al. (2009)assumed to be isotropic, linearly elastic material is used.The multiphase material model is what the concept ofSIMP method (Zhou and Rozvany 1991) (Solid IsotropicMicro-structure with Penalization of intermediate densi-ties), widely used to single porous material, is enlarged tocomposite and is the same as the multiple material model(Bendsøe and Sigmund 1999) in linear elastic case. Theeffective material stiffness is defined as follows:

C = (1 − sηi

)C1 + s

ηi C2. (21)

Here, C is a material stiffness in the linear elastic regime,and is identical to that in (10). As clearly seen from theequation, it is understood that the material stiffness coeffi-cient C explicitly depends on design variable si . C1 and C2

are known specific material stiffness of phase-1 and phase-2respectively, and remain unchanged during the optimiza-tion. η is the power-law factor that does not guarantee thephysical meaning.

4 Setting of optimization problem

Optimization problems are generally defined by objec-tive function f (s), equality constraint condition h(s), andinequality constraint condition g (s). The small bold italicletter s represents an array containing design variable siarranged in a row, i.e., it means design variable vector.

Hereafter, we present the process of formulation of anoptimization problem handled in this study. As the objec-tive is to maximize the stiffness of macro-structure, thefollowing equations are formulated on the assumption that

Topology optimization of micro-structure 601

maximizing stiffness of the macro-structure is equivalent tominimizing the compliance. With regard to the constraintcondition, we provide an equality constraint condition forthe volume of phase-2 in the unit cell not to vary duringoptimization. In this optimization problem, as we set theonly two kinds of material to exist, it means that the vol-ume of phase-1 will not vary simultaneously as a whole unitcell. Furthermore, as the whole structure co-occupies oneunit cell, it is self-evident that the volume of each individ-ual material will not vary as a whole macro-structure. Then,the optimization problems expressed by the matrix form areshown as follows:

min f (s) = F Td, (22)

h (s) =∫

Y

si dY − V = 0, (23)

sL ≤ si ≤ sU i = 1, ..., ns. (24)

Here, F and d are external force vector and nodal dis-placement vector of the whole system of macro-structure,respectively. sL and sU are the lower and the upperbounds of design variables, ns shows the number of design

yes

: design variabless Start

Initialize FEM

Numerical material tests (NMTs)

FE analysis of macro-structure

Sensitivity analysis

Optimization

End

converged?

Fig. 6 Flowchart of optimization procedure for the proposed method

variables, which corresponds to the number of finite ele-ments in the unit cell nele here. V is the prescribed totalvolume of phase-2 material in the unit cell.

As the optimization algorithm by the gradient-basedmethod is used in this study, sensitivities of the objectivefunction and constraint with respect to the design variablesi , ∂f/∂si and ∂h/∂si respectively, should be obtained afterresolving the two-scale BVP. By incorporating the thus-obtained sensitivity into the optimality criteria method, theoptimized solution at that time is obtained. Then, the solu-tion is subjected to repeated calculations until the solutionis converged (Fig. 6). In the following section, derivationof sensitivity of objective function is explained. As the sen-sitivity of constraint with respect to the design variable isderived as the same approach as that of the general topol-ogy optimization, the explanation for it is eliminated in thispaper.

5 Sensitivity analysis

5.1 Sensitivity of objective function

In this section, the process to obtain sensitivity of objectivefunction f with respect to design variable si is explained.Specifically, the sensitivity is derived by the adjoint method,more accurately, the analytical adjoint method. First, theobjective function f is converted to equivalent objectivefunction f with a constraint condition of discretized for-mulation of equilibrium equation Kd = F , K as globalstiffness matrix of macro-structure,

f (s) = dTKd − dT(Kd − F ) , (25)

where d is an adjoint vector. Next, the above (25)is differentiated by design variable si and arranged asfollows:

∂f

∂si=⎛⎝dTK︸︷︷︸

FT

−dTK

⎞⎠ ∂d

∂si− d

T ∂K

∂sid. (26)

Then, it should be noticed that the adjoint vector d

does not depend on design variable si , because the adjointvector d is arbitrary. In this equation, the term unableto be obtained explicitly is a differential term regardingdisplacement ∂d/∂si . Then, when the adjoint vector d isplaced as d = d to make the amount in the parenthe-sis of the first term in the right side zero, the implicitdifferential term disappears. By rearranging (26) again,the equation is transformed to the form that is able to

602 Kato et al.

obtain a solution explicitely and easily, as shown in thefirst line of the following (27). Furthermore, by revers-ing the equation to the notation according to the finiteelement level, as shown in the second line, the sensitiv-ity of the objective function can be easily obtained, if thederivative of macro-material stiffness matrix ∂CH/∂si iscalculated as,

∂f

∂si

(= ∂f

∂si

)= −dT ∂K

∂sid

= −∫

�

ET ∂CH

∂siEd�, (27)

where E denotes the macro strain obtained by macro-scale BVP, and not the given macro-strain used in numer-ical material tests. Thus, the macro-scale sensitivity canbe separately obtained if the remaining term, ∂CH/∂si ,is calculated. In the following section, we propose thederivation of the sensitivity, ∂CH/∂si , in an analyticalformulation.

5.2 Derivation of sensitivity of macro-materialstiffness tensor

Recalling the tensor formulation of (17), we transform itscomponent into the following expression in terms of (1),

CHpqrs = �(rs)

pq

= 1

|Y |∫

Y

σpq

(ε(rs))

dy

= 1

|Y |∫

Y

σ(ε(rs)): E(pq) dy

= 1

|Y |∫

Y

C : ε(rs) :(ε(pq) − ε∗(pq)

)dy

= 1

|Y |∫

Y

C : ε(pq) : ε(rs) dy, (28)

where we rewrite the strain field of fluctuation as ε∗ =∇symy u∗. In the fourth equilibrium, the relation ε(pq) =

E(pq) + ε∗(pq) is employed and in the fifth equilibrium, theterm with fluctuation strain ε∗ is eliminated in terms of (33),where the fluctuation strain ε∗ is used instead of the virtualfluctuation strain field δε∗ for its arbitrariness.

Differentiating (28) with respect to design variable siyields

∂CHpqrs

∂si= 1

|Y |∫

Y

∂C

∂si: ε(pq) : ε(rs) dy

+ 1

|Y |∫

Y

C : ∂ε∗(pq)

∂si:(E(rs) + ε∗(rs)

)dy

+ 1

|Y |∫

Y

C :(E(pq) + ε∗(pq)

): ∂ε∗(rs)

∂sidy. (29)

Here, we replace the virtual strain field δε∗ in (33) by∂ε∗/∂si considering its arbitrariness and periodicity, andinsert the modified equation (33) into (29). This procedureeliminates the second and third summations of (29), thus(29) is simply rewritten as follows:

∂CHpqrs

∂si= 1

|Y |∫

Y

∂C

∂si: ε(pq) : ε(rs) dy. (30)

Finally, for implementation, the discretized formulationof (30) is written in the following expression:

∂CHαβ

∂si= 1

|Y |∫

Y

weα

TBT ∂C

∂siBwe

β dy

with∂C

∂si= ηs

η−1i (C2 − C1) , (31)

where we denotes the nodal micro-displacement vector ofan element in a unit cell, and its discretization procedure isreferred to Appendix. Although the symbols for the direc-tion of the numerical material tests, i.e. (pq) and (rs),have been abbreviated in (31) for simplicity , they are stilleffective. As described in Appendix, the nodal displacementvector we is obtained as the result of the numerical materialtests.

In the solution procedure, the global nodal displacementvector for each loading direction, i.e. w(11), w(22) and w(12),or the condensed version of the displacement, i.e. w(11),w(22) and w(12) introduced in (40), should be stored in orderto calculate each component of ∂CH

αβ/∂si , which is obtainedfrom combination of the above three kinds of displacementvectors. For reference, the entire procedure of the proposedmethod is described in Fig. 3.

6 Verification of the proposed method by numericalexamples

6.1 Common problems

In this section, the proposed method, topology optimiza-tion of micro-structure by decoupling multi-scale analysis,

Topology optimization of micro-structure 603

is verified using a series of numerical examples. The point tobe cleared in this verification is whether or not this approachcan obtain topology of micro-structure which characterizingthe emechanical behavior of macro-structuref properly. Forthis purpose, it is designed to easily assess the examples ofverification in the former half by providing basic uniformdeformation to a simple macro-structure model constructedby a 4-node rectangular element in order to avoid difficultyin assessing the results of optimization. In the numericalexamples in the latter half, comparative verification was per-formed among the macro-structures with increased numberof finite elements and with conflicting geometric properties.In this verification, it should be noticed that any exampleof optimization is ‘non-uniqueness’ that means any solutionfor optimization is not unified. This means that mathe-matically plural solutions for optimization may exist forthe same objective function. Therefore, it is possible forthe same topology optimization problem to lead to a dif-ferent solution due to different computers used, and bothresults are the optimum solutions. Obtaining a differenttopology in the same unit cell might be caused by minorerrors in the numerical analysis, e.g., intrinsic faint errorsin a computer or in calculation program; the phenomenonitself is numerically correct as introduced in Sigmund andPettersson (1998).

This situation for the optimization solution not to be uni-fied is caused by a ‘uniform deformation’ which occurs ina unit cell. Such deformation occurs when boundary con-ditions of a unit cell have periodicity and all the finiteelements in a unit cell are provided with the same physicalproperties. A situation like this is often seen in the initialstructure before starting the optimization calculation. At thistime, when the sensitivity of the objective function ∂f /∂si(vector) is calculated, all the elements show the same valueof sensitivity, and the contribution of each element to theobjective function is assessed as equal. Accordingly, fromthe theoretical viewpoint, further optimization will not becalculated.

But, in this case, despite the existence of inevitable minorerrors as mentioned above, we are able to continue theoptimization calculation by tracing the minimal differenceamong elements of ∂f /∂si . In such cases, the authors haveconfirmed through our experience that three scenarios willdevelop as follows: (a) topology remains unchanged, almostkeeping the same initial stage, (b) topology disturbed by theabovementioned difference becomes disturbed and mean-ingless due to dependency on the initial value of designvariable, characteristic of the gradient-based method, or(c) topology stagnates showing checkerboard pattern intro-duced in Diaz and Sigmund (1995) and Jog and Haber(1996).

In this study, as one of the countermeasures to avoid thisproblem, we employed the mesh-independent filter method

proposed by Bendsøe and Sigmund (2003), Sigmund andPettersson (1998), evaluated as the most effective, as fol-lows:

∂f

∂si= 1

si

N∑j=1

Hj

N∑j=1

Hj sj∂f

∂sj

with Hj = rmin − dist(i, j). (32)

In the above equation, dist(i, j) shows the center-to-center distance of ith and jth finite elements, rmin, calleda filter radius, determines a range of elements to befiltered.

In the numerical examples here, we tried to obtain thefinal topology with two phases distinguished as clearly aspossible by using a filter with rather large radius rmin at thebeginning of optimization to avoid locally stagnant topol-ogy and tuning the radius gradually smaller. In this study,the minimum value of rmin is set to be a slightly largervalue than the center-to-center distance between adjacentelements to avoid numerical instability.

6.2 Case of a simple structure discretized with a 4-noderectangular element

This numerical example is to verify whether or not a reason-able topology is obtained when a macro-structure, modeledwith a simple structure discretized with a 4-node rectangu-lar element, is provided with a uniform deformation. Themacro-structure of a square with a side of 100 mm is mod-eled using a plane strain element. The material used insidethe micro-structure is assumed to be of two kinds (two-phase composite material consisting of two solid materialsin micro-field) excluding voids. Here, the material stiffnessof phase-2 (black) is set larger than that of phase-1 (white).All the material used is assumed to be a linearly elasticmodel. The material properties used are shown in Table 1.With regard to the power-law factor η, shown in (21), weused as η = 5 in every case.

The shape of the unit cell is set to as a square, and thelength of a side is normalized and set to the unit length.Finite elements used were 8-node rectangular elements, andthe number of elements was 400 (20 x 20). At the initialstage before optimization, we assumed that every element

Table 1 Material data

Young’s modulus (MPa) Poisson’s ratio

phase–1 10 0.3

phase–2 10000 0.3

604 Kato et al.

Fig. 7 a Conceptual diagramsof uniform tension/compressiondeformation of macro-structure,(b), (c) the optimized topologyof micro-structure (a unit celland its patch); b Case, the radiusof a filter rmin is 0.051(theminimum value) until thenumber of optimization stepsreaches 100 and thereafter thefilter was eliminated, c rmin isreduced from 0.251 to 0.051(the minimum value) by degrees

(a) (c) (b)

was included in each phase-1 and phase-2 by 50 % respec-tively. Thus, the initial value of the design variable was setto si = 0.5 in every element. The total volume of materialincluded in the structure was set to remain unchanged dur-ing the optimization. In this connection, we used the sameassumption for all the unit cells used in this paper.

Hereafter, results of optimization obtained in the numer-ical examples are explained. First, Fig. 7a shows a con-ceptual diagram of deformation provided to the macro-structure. We provided 10 % strain as uniaxial tensile orcompression deformation. Figure 7b shows the result ofoptimization in a case that the radius of the filter rmin wasset to be a slightly larger value (0.051) than the center-to-center distance (0.05) between adjacent elements until thenumber of optimization steps reached 100, and thereafterthe filter was eliminated. On the other hand, Fig. 7c showsthe results of optimization with a thicker layer of phase-2than in the previous cases, because we reduced the radius ofthe filter gradually according to the increase of the numberof optimization steps for longer distance initially set among5 elements (initial rmin = 0.05 × 5 + small value 0.01 =0.251). This is due to the problem of non-uniqueness, whichconfirmed that different topologies were obtained accord-ing to the setting of the filtering radius. As far as thesetwo results are concerned, either exhibits the most suitablestructure to increase stiffness for either direction tensile orcompression.

Next, Fig. 8 shows the result of optimization when themacro-structure is subjected to a simple shear deforma-tion. The topology obtained shows the material distributionwith the inclination of 45 degrees from the horizontal line,which is the result we expected. In some cases, topol-ogy turned around 180 degrees with respect to e2-axis, i.e.topology with the inclination of -45 degrees, was obtained.This result also introduces the non-uniqueness problem andthe obtained both macro-material stiffness matrices CH areidentical.

From the above examples, it is confirmed that the pro-posed method can determine a mechanically reasonabletopology for behavior of a simple macro-structure.

6.3 Cases for various macro-structures

Considering the verification results in the previous section,the proposed method for several representative behaviorsof macro-structure, including bending deformation, is to beverified in this section. In the following examples, we useda macro-structure with 8-node rectangular elements, andassumed plane strain condition.

Figure 9 shows the result of optimization when a uni-form load of 1.0kN/mm was applied to a beam 200 mmlong and 100 mm high. Figure 9 top shows the distribu-tion of the horizontal stress σxx (MPa) with deformationand the middle shows the distribution of shear stress σxy

(MPa). From Fig. 9 it is understood that the horizontal

Fig. 8 Conceptual diagrams of uniform shear deformation of macro-structure and the optimized topology of micro-structure (unit cell andits patch)

Topology optimization of micro-structure 605

Fig. 9 Result of topology optimization of micro-structure in case of amacro-cantilever beam: deformation and horizontal stress σxx diagram(top), deformation and shear stress σxy diagram (middle), optimizedmicro-structure (bottom)

stresses dominate at both the top and bottom ends on theleft boundary of the beam, and shear stresses are shown toalmost the whole area of the macro-structure, although theshear stresses are rather small compared to the maximumhorizontal stress. The optimized micro-structure shows thattopology with a thick layer of the phase-2 arranged horizon-tally as a reinforcement for horizontal stresses and diagonalreinforcement against shear stresses is obtained. From theabove results, it is considered that the optimization structureis reasonable as concerns structural mechanics.

Figure 10 shows the result of optimization of a macro-structure that is modeled from a right half-section of asimple slender beam. In this example, a uniformly dis-tributed load of 6.0kN/mm was applied on the upper surfaceof the beam. The upper diagram in Fig. 10 shows itsdeformation and horizontal stress distribution. As clearlynoticed from the diagram, the macro-structure is a struc-ture in which the horizontal stress dominates and large shear

Fig. 10 Results of optimization of micro-structure in case of a macro-slender beam: deformation and horizontal stress σxx diagram (top) andoptimized micro-structure (bottom)

stress does not occur because of low beam height (Dia-gram of shear stress is omitted). As the result, the optimizedmicro-structure shows that most of the phase-2 material isarranged to the direction of the horizontal axis, and thetopology shows slightly diagonal to reinforce against theshear stresses. Here, looking at deeper inside of Fig. 10,one may notice that the slightly diagonal thin layer in theobtained topology is one-node hinged layout. Although theobtained topology is structural reasonable as mentionedabove, this kind of layout may be cured by using moreadvanced filter method such as Morpho–logy–based blackand white filter developed by Sigmund (2007).

Figure 11 shows the results of optimization using amacro-structure of a deep beam under the same structuralcondition. As the stress distribution of the macro-structureconsiderably differs because of difference in height of abeam, this is to verify whether or not optimization ofmicro-structure properly recognizing the difference of stressdistribution is obtained. Figure 11 left shows distribution ofshear stress σxy (MPa), and Fig. 11 center shows distribu-tion of vertical stress σyy (MPa). As the absolute value ofhorizontal stress is so small, a diagram of horizontal stressσxx is omitted. From the diagram it is understood that thedistribution of shear stress dominates on the line connect-ing the loading points to the supporting point. With regardto the distribution of vertical stress, large compression stressoccurs adjacent to a loading point and vertical support-ing point, and especially at the vertical supporting point,large stress concentration exceeding -250 (MPa) is seen.As the result, topology reinforced by diagonal direction for

606 Kato et al.

Fig. 11 Results of topologyoptimization of micro-structurein case of a macro-deep beam:shear stress σxy (left) andoptimized micro-structure(right), vertical stress σyy(middle) and optimizedmicro-structure (right)

shear stress and upright direction for vertical stress appearedon the optimized micro-structure. Consequently, it is con-sidered that the topology of micro-structure consideringcharacteristics of macro-structure is obtained.

From all these optimization calculations, it is verified thatthe approach proposed in this study can optimize topologyof a micro-structure by reflecting mechanical behavior ofthe macro-structure in the linear elastic regime.

6.4 Conclusion

The purpose of this study is to propose an approach to max-imize the performance of macro-structure by optimizing thetopology of micro-structure of composite material. In thisstudy, an optimization method that is intended to maximizethe stiffness of a macro-structure under the new frameworkof introducing decoupling multi-scale analysis into topol-ogy optimization is proposed, and the proposed method wasverified through various numerical calculations.

Main results of this study are as follows:

• It was verified that the proposed method reflectsthe mechanical behavior of macro-structure with highfidelity and that the method optimizes topology ofmicro-structure by all the optimization examples.

• An analytical sensitivity approach was formulatedconsidering both micro- and macroscopic structuralresponse. In this methodology, no extra assumption isnecessary which is concerned with the linear relationbetween micro- and macro–scale deformation definedby the characteristic displacement.

• In this paper, we implemented topology optimizationof microstructure for the micro–macro two–scale BVPsunder an assumption of linear elasticity. As mentionedin the Introduction, no distinct advantage and differenceexists between the coupling and decoupling schemes in

computational costs under the linear elastic regime. Itis expected that this approach will be extended to non-linear structural problems in considering the essentialeffect of decoupling multi-scale analysis.

Acknowledgments This work was supported by MEXT KAKENHIGrant Numbers 23560561, 23656285. These supports are gratefullyacknowledged.

Appendix: Finite element analysis for a unit cellwith external material points

For preparation of the finite element analysis (FEA) forthe micro-scale BVP, the spatial domain of the unit cell isdiscretized to generate its FE mesh. The principle of vir-tual work for micro-structure is formulated considering thefirst equation in (10) and anti-periodicity condition (8) withsome mathematical rearrangement as follows:

∫

Y

δε∗ : σ dy =∫

Y

∇symy δu∗ : σ dy = 0, (33)

where δu∗ and δε∗ denote the virtual fluctuation displace-ment field and its strain field satisfying the periodic condi-tion, respectively. The virtual work expression (33) is dis-cretized in the finite element sense assuming the followingapproximation:

w =nnode∑α=1

Nαweα or w = Nwe

, (34)

δw =nnode∑α=1

Nαδweα or δw = Nδwe

, (35)

Topology optimization of micro-structure 607

ε =nnode∑α=1

Bαweα or ε = Bwe

, (36)

δε =nnode∑α=1

Bαδweα or δε = Bwe

, (37)

where N is the general shape function and B the B-operator,respectively. w

e indicates the nodal micro-displacementvector of an element in a unit cell. Furthermore, we sim-ilarly discretize δu∗ and δε∗ as δu∗ = N

(δd

∗e)

and

δε∗ = B(δd

∗e)

. The discretized formulation of (33) can

be written by inserting these equations as follows:nele∑e=1

{(δd

∗e)T ∫

Ye

BTCBdy(we)} = 0. (38)

As the virtual fluctuation displacement δd∗e is arbitrary, the

discretized formulation of (33) is expressed by assembling(38) over the unit cell as:

Kmw = 0 with Km =nele∑e=1

∫

Ye

BTCBdy, (39)

where Km is the global stiffness matrix of a unit cell and w

is the global nodal micro-displacement vector.At this stage the boundary conditions (12) and (13) have

not been included in (39) yet. In order to establish theextended micro-scale BVP aforementioned, the relative dis-placement q is embedded to (39) as constraints by replacingpairs of DOFs in w. For this purpose, each external mate-rial point is also ‘discretized’ to an element with a singlenode which has two DOFs and no mass. Since the externalmaterial points enable us to control the components of themacro-scale stress and deformation, as explained above, thenode corresponding to an external material point is referredto as a control node in this study. Thus, we obtain anextended system of FE-discretized equations involving fouradditional DOFs of two control nodes. In the following, weintroduce some specific usages of the two control nodes tosolve the extended system.

First, the macro-scale strain is assumed to be known; thatis, all the components of the macro-strain E are given asdata. Using (13), we obtain all the components of the nodalrelative displacement vector q [k] at the two control nodes.Then, given all the components q [k]i , we solve the extendedsystem of FE equations with the appropriate number of‘two-point’ constraints realized by (12).

This procedure starts from applying a transformationmatrix � such as,

w = �w, (40)

where � is an operator which transforms w to w and w isthe nodal displacement vector in which some components

are replaced by the components of q [k]. For example, con-centrating only on q[1], we impose the corresponding twopoints on the boundaries ∂Y [−1] and ∂Y [1] whose DOFs aredefined as (wa, wb) and (wc, wd), respectively, to be peri-odically constrained. Then, we can establish w from w asshown in Fig. 3, namely

w = {w1 w2, ..., wa wb, ..., wc wd, ..., wN

}(41)

and

w ={w1 w2, ..., wa wb, ..., q

[1]1 q

[1]2 , ..., wN

}(42)

with wc − wa = q[1]1 and wd −wb = q

[1]2 . (43)

where N is the total number of DOFs in the unit cell with-out the number of DOFs of the external material points.Of course, as the above example is simply for one set ofthe constrained points on the ∂Y [−1] and ∂Y [1], all otherconstrained points have to be considered in w at the sametime. Note that one set of q [k](k = 1 or 2) can control allconstrained points staying on ∂Y [−k] and ∂Y [k], see againFig. 3.

Then, inserting (40) into (39) and pre-multiplying theoperator � by both sides of (39) yields,

Kmw = 0 with K

m = �Km�T. (44)

This linear equations can be condensed depending on thedistribution of components of q [k] in w. After solving thislinear equation, unknown components in w are determined.The results of the FEA contain not only the micro-scale dis-placement w, strain ε and stress σ , but also the reactionforce R[k] as aforementioned. This means that R[k] can beobtained by (15) since the traction force vector t [k] on ∂Y [k]is obtained by the Cauchy law t [k] = σ ek . Therefore, themacro-stress �jk can be computed from (16), without per-forming a numerical integration on (1). Finally, as shownin (19), the macro-material stiffness CH can be obtainedby computing the macro-stress �jk separately three timesaccording to the corresponding relative displacements.

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