Topology optimization of periodic microstructures with a penalization of highly localized buckling...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 54:809–834 (DOI: 10.1002/nme.449)

Topology optimization of periodic microstructureswith a penalization of highly localized buckling modes

Miguel M. Neves1, Ole Sigmund2 and Martin P. BendsHe3;∗;†

1Instituto de Engenharia Mecanica; Instituto Superior T�ecnico; Av. Rovisco Pais; 1; 1049-001 Lisboa; Portugal2Department of Mechanical Engineering; Section of Solid Mechanics; Technical University of Denmark;

Niels Koppels All�e 404; DK-2800 Lyngby; Denmark3Department of Mathematics; Technical University of Denmark; Matematiktorvet;

B. 303; DK-2800 Lyngby; Denmark

SUMMARY

The problem of determining highly localized buckling modes in perfectly periodic cellular microstruc-tures of in�nite extent is addressed. A double scale asymptotic technique is applied to the linearizedstability problem for a periodic structure built from linearly elastic microstructures. The obtained stabil-ity condition for the microscale level is then used to establish a comparative analysis between di�erentmaterial distributions in the base cell subjected to the same strain �eld at the macroscale level. Theidea is illustrated by some two-dimensional �nite element examples and used to design materials withoptimal elastic properties that are less prone to localized instability in the form of local buckling modesat the scale of the microstructure. Copyright ? 2002 John Wiley & Sons, Ltd.

KEY WORDS: topology optimization; periodic microstructures; linearized elastic buckling; homo-genization

1. INTRODUCTION

Spatially periodic microstructures can be obtained by a periodic repetition of a base cell (alsotermed unit cell). The respective ‘averaged’, homogenized or e�ective elastic properties can befound by the mathematical theory of homogenization (see, e.g. Reference [1]). These elastic

∗Correspondence to: Martin P. BendsHe; Department of Mathematics; Technical University of Denmark;Matematiktorvet; B. 303; DK-2800 Lyngby; Denmark

†E-mail: [email protected]

Contract=grant sponsor: Danish Research AcademyContract=grant sponsor: European Research Training Network (HMS2000); contract=grant number: RTM1-1999-00040Contract=grant sponsor: IDMEC=ISTContract=grant sponsor: FCT (Portugal); contract=grant number: PBIC=C=TPR=2404=95Contract=grant sponsor: STVF (Denmark)Contract=grant sponsor: SNF (Denmark)

Received 14 February 2001Copyright ? 2002 John Wiley & Sons, Ltd. Revised 13 August 2001

810 M. M. NEVES, O. SIGMUND AND M. P. BENDSIE

properties can be optimized by varying the size, the shape or the topology of the base cellusing the techniques of structural topology optimization. Such an ‘inverse homogenization’can be found for example in the work by Sigmund and Silva and co-workers [2–4]. Moreover,Sigmund and co-workers [5–9] have employed this computational method for the identi�cationof periodic materials attaining the theoretical bounds on elastic properties.The use of ‘inverse homogenization’ has so far been based solely on a linearized elasticity

analysis with only geometrical restrictions on the slenderness of the cell members. A typicalfeature of a broad range of the optimal periodic microstructures obtained, as exempli�ed bythe new class of extremal composites proposed in Reference [7], is that they are extremein the sense of linear elasticity but will fail by local buckling at a microscale level. It isthe appearance of slender elements in the base cell that makes these solids vulnerable tohighly localized buckling which implies that the optimal elastic properties are only validfor fairly small loads. However, it is characteristic for the ‘inverse homogenization’ problemthat it is possible to �nd several distinct periodic materials that represent equally optimalelastic properties. Thus, the material design process should allow for an improvement of thebuckling performance of the base cell while maintaining the optimal elastic properties of thehomogenized material, and it is this problem we address in this work.The importance of studying the buckling criteria e�ect on the optimality of microstructures

has been pointed out in a discussion on the optimality of bone structure [9] where it is ques-tioned why bone usually consists of open-walled cells, whereas sti�ness optimized cells areclose-walled cells, and this issue partially motivated this work. However, further investigationsin three dimensions are necessary for a better understanding of the buckling criteria e�ect onthe optimality of bone microstructures.The elastic buckling phenomena at the microscale level of a periodic material is similar

to that observed for structures with slender members, and for the macroscale level, topologyoptimization of continuum structures with a global buckling criterion has been consideredin Reference [10]. Moreover, the inclusion of geometrically non-linear response in topologyoptimization of structures and mechanisms can be found in References [11–13]. The lattertwo papers only consider buckling indirectly, while Reference [11] includes the non-linearstability load directly in the optimization formulation. For truss design, standard practice isto examine the local buckling of individual elements [14, 15, and references therein], butrecently global buckling for truss structures has been considered in References [16; 17].For structures built from materials with periodic microstructure it is necessary to consider

also instability at the microscale level. The wavelengths of the buckling modes can haveseveral di�erent length scales depending on the geometry, dimensions and loading of themedium as illustrated in References [18–20]. The models developed by these authors arebased on Bloch wave techniques applied to a cell geometry that is prede�ned and relativelysimple. The implementation of the Bloch waves method in topology design is computationallyexpensive but is subject of current studies [21]. As an alternative, one can consider the use ofa model for a periodic solid of �nite extent with a large number of cells that can be treatednumerically (for example using multiresolution schemes [22, and references therein]).In the present paper, we simplify the analysis problem by using a model for an in�nite

periodic medium restricted to periodic deformations at the scale of the base cell, thus coveringonly the case of highly localized modes. In this model, three main assumptions regarding thebuckling at microscale are enforced. First, the linearized elastic buckling model is based on anEuler (eigenvalue) type of elastic buckling where the displacements prior to the �rst critical

Copyright ? 2002 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 54:809–834

TOPOLOGY OPTIMIZATION OF PERIODIC MICROSTRUCTURES 811

load at both macro- and microscale are assumed to be small and in the linear elastic range.This simple model captures the essence of the local instability phenomenon and results ina tractable topology optimization problem. This model can then be used as a basis for thedevelopment of more elaborated models that should include the general case of non-linearelastic buckling. Second, the asymptotic model is based on the limit of in�nitely small scale,i.e. the cell characteristic size is assumed to be much smaller than the characteristic size ofthe structure. Third, in the derivation of microscale buckling equation it is assumed that themodes are Y-periodic, i.e. with the same periodicity as that of the base cell. This is related toour main concern of selecting an elastic buckling measure that characterizes if a cell is moreor less prone to highly localized buckling modes. This buckling measure is subsequently usedto introduce a local buckling load control as a constraint in the topology optimization of thebase cell of periodic materials. A generalization of this technique to include other instabilitymodes would be a natural extension but is beyond the scope of this paper (one possibleapproach is to use a Bloch wave technique [21]).This text is organized as follows. Section 2 describes the linearized buckling model, while

in Section 3 the �nite element model used to solve the problem computationally is presented.In Section 4, the optimal design problem is stated, and the respective design sensitivity gra-dients are presented in Section 5. Generalized gradients are used to deal with the non-smoothcharacter of the optimization problem for the case where repeated eigenvalues occur. Finally,in Section 6, details of the computational model are described, and in Section 7, numericalexamples are presented to illustrate the application of the local stability model in materialdesign. These examples illustrate the behaviour of the chosen measure for cell buckling, andhow highly localized buckling modes can be penalized in the topology optimization of periodicmaterials. Finally, the use of a mesh independent �ltering technique resulted in optimized cellswhich have elastic properties close to the theoretical bounds and which are simultaneouslyless prone to highly localized instabilities.

2. LINEARIZED STABILITY PROBLEM

In general terms, homogenization theory applied to linear problems establishes macroscopicproperties without any quanti�able size scale parameter. For non-linear cases such as elasticstability problems, the situation is more complex and it is expected that a size scale parametershould be present.In this section, we derive a stability condition that is a local condition for an in�nite

medium. Thus it does not involve a quanti�able size scale parameter, as would be the casefor a �nite cellular medium. In the literature on the stability of periodic microstructures ofin�nite extent [19; 20; 23], a stability parameter (related to the coercivity constants) is givenwhich measures coercivity with respect to long-wavelength deformations or coercivity withrespect to Y-periodic and possibly highly localized deformations (periodic buckling modesexisting at a length scale comparable to the periodicity of the cell). Indeed, to quantify thebuckling performance of a �nite solid with a periodic microstructure, a general non-lineartheory should be applied. Here, for a periodic solid of in�nite extent, we apply a double scaleasymptotic technique to the linearized stability problem for a structure built of linear elasticmicrostructures.



Figure 1. Schematic representation for the macrostructure, perfectly periodicmicrostructure and respective base cell.

For the case of linearized stability, the only non-linear e�ect considered is the stress sti�-ening. The structure is �xed on the boundary �u and is subject to a proportional load traction(quasi-static) on the boundary �t that is a function of a load factor parameter P; i.e. �t=Pt (seeFigure 1). As the load factor P is progressively increased from zero, the displacement �eldremains unique up to a critical load factor, Pcr, above which alternate equilibrium displace-ment solutions are possible. For P¡Pcr we assume a small displacement u0�. The superscript� identi�es the dependence on the microstructure.At P=Pcr the alternative equilibrium positions are given by u�= u0� + �u1� where u0� is

the initial small displacement of the elastostatic analysis, � is an in�nitesimal real parameterand u1� identi�es the shift displacements when the material points shifts from the initialto the new equilibrium position. Using an asymptotic expansion as is standard in homo-genization,

u0�(x; y) = u00(x; y) + �u01(x; y) + �2u02(x; y) + · · ·u1�(x; y) = u10(x; y) + �u11(x; y) + �2u12(x; y) + · · ·

(1)

we can identify three linearized problems of interest (see Appendix A for details). One is thehomogenized problem for computing the macroscopic displacement u00(x) prior to instability,which has the standard form∫

�EHijkm(x)

@u00k@xm

@vi@xj

d�=∫�ttivi d�; ∀v∈V� (2)

where the kinematically admissible global displacements at macroscale level are denoted byV� and t is the reference load. As is well known, the homogenized properties EHijkm are givenby information on the base cell Y :

EHijkm=1|Y |

∫Y

(Eijkm(y)− Eijpq(y)

@�kmp@yq

)dY (3)



Here the periodic �elds �km are the ‘k; m’ solutions of the following local elastostatic problemsat cell level: ∫

YEijpq(y)

@�kmp@yq

@vi@yjdY =

∫YEijkm(y)

@vi@yjdY; ∀v∈VY (4)

where the Y-periodic admissible displacements at the microscale level are denoted by VY .The second problem describes local buckling at the microscale, i.e. the problem of �nding

the �rst local and Y-periodic eigenmode at the length scale of the base cell (highly localizedbuckling mode):

∫YEijkm(y)

@u10i@yj

@wk@ym

dY + PY∫Y

(Eijkm(y)− Eijpq(y)

@�kmp@yq

)@u00i@xj

@u10c@yk

@wc@ym

dY =0 ∀w∈VY

(5)

where PY is a local scalar critical load factor (stability parameter), u10(y) is the correspondentlocal Y-periodic eigenmode, and u00(x) and �km are the solutions of the homogenizationproblems (2) and (4), respectively, which together describe the level and distribution ofstress sti�ening in the cell. Notice that no homogenization is involved in the derivation ofexpression (5) except in the de�nition of macroscopic strain expressed via the homogenizedequations for u00(x). Although the local stability condition for Y-periodic modes is extendedby periodicity, this local criterion ignores all other non-local e�ects.Finally, the third problem de�nes a global (macroscopic) mode for the structure with

homogenized material properties:

∫�EHijkm(x)

@u10i@xj

@vk@xm

d� + P�∫�EHijkm(x)

@u00i@xj

@u10c@xk

@vc@xm

dY=0; ∀v∈V� (6)

where P� is the critical load of the structure for macroscopic instability modes u10(x).The model admits a useful representation of local buckling for the material design procedure

described below. The model certainly presents signi�cant restrictions, due to the simpli�ca-tions introduced to obtain the macroscale level and microscale level behaviour. These are, asindicated in Appendix A, an assumption of linearized buckling corresponding to the standardrestriction of the strain �elds; use of a limit �→ 0 where � is the ratio of base cell charac-teristic size ‘d’ to the structure characteristic size ‘D’ (as in homogenization theory); and ahypothesis that the displacements at microscale level are Y-periodic. The last one is the mostcritical for the stability because it restricts the mode shapes.

3. FINITE ELEMENT MODELS

Using the �nite element approach for the elastostatic equation (2) gives the following linearproblem:

Ku00=f (7)



which requires that the homogenized properties are �rst evaluated by solving computationallythe ‘k; m’ local problems (4) at the microscale level, i.e.

KY �km=f kmY (8)

In the previous approximation, K, f and KY ; f kmY are, respectively, the sti�ness matrix andload vector for the homogenized structure and the base cell problem.The local critical load at the cell level and the respective Y-periodic eigenmodes (PY and

�r) of problem (5) are obtained by solving the following eigenvalue problem:

(KY − PYr KGY )�r=0 (9)

where KGY is the geometric sti�ness matrix for the Y-periodic base cell.The homogenized stability problem at the macroscale level is expressed by the following

eigenvalue problem:

(K − P�r KG)�r=0 (10)

where KG is the homogenized geometric sti�ness matrix for the structure. However, forthe material design problem we consider, the equation of interest is the cell eigenvalueproblem (9).

4. TOPOLOGY DESIGN OF MATERIALS WITH A LOCALSTABILITY CONSTRAINT

The topology design problem is stated as the search for an optimal distribution of a limitedamount of material in the base cell domain, which maximizes a given linear combination ofthe homogenized elastic properties. To assure a reasonable local buckling performance, weintroduce a lower bound on the local critical load value and assume that all buckling loadfactors are positive.Thus, the design problem is stated as

min�

−�ijkmEHijkm + �∫Y�(y)(1− �(y)) dY (11a)

s:t: PY¿Pmin (11b)∫Y�(y) dY=V0 (11c)

0¡�min6�6�max=1 (11d)

where the constant tensor � de�nes the weighting of the material properties to be optimizedand Pmin is a lower bound on the local critical load value given by Equation (9). The designvariables are the local material densities represented by the vector � in the base cell and thetotal amount of material is V0. Since |Y | is the geometric volume of the cell, the relativedensity of the material is �=V0=|Y |. The term �

∫Y �(1 − �) dY represents a penalization of

intermediate densities imposed to obtain material distributions that are nearly black and whitedesigns, i.e. designs with no intermediate density at the microlevel.



For the calculation of EHijkm as well as PY one uses expressions (3) and (9), with the local

material properties Eijkm(�; y) in the base cell expressed, e.g. via the solid isotropic materialwith penalization (SIMP) model, in terms of the local density � [24]; thus Eijkm(�; y)= [�(y)]p

E0ijkm for given material parameters E0ijkm and with an exponent p on the density (with p¿1).

5. SENSITIVITIES

In this work, we solve problem (11) by a derivative-based mathematical programming method.Thus, sensitivities for the topology optimization are presented next.The sensitivities of the objective function (11a) are evaluated as

@@�e

[−�ijkmEHijkm + �

∫Y�(1− �) dY

]

=−�ijkm|Y |∫Ye

[@Epqrs@�e

(�pk�qm − @�kmp

@yq

)(�ri�sj − @�ijr

@ys

)]dY + �

∫Ye(1− 2�e) dY (12)

while the material volume constraint (11c) sensitivity is obtained by

@@�e

∫Y� dY=|Ye| (13)

where the local material density is constant in each �nite element ‘e’ and given by �e, |Ye|is the geometric volume of the �nite element and �ij are the solutions of problem (4).The local buckling constraint is a non-smooth function if the critical load factor is a multiple

eigenvalue. To overcome this problem, the concept of the generalized gradient is applied [25].We use the inverse of the critical load factor as described in References [10; 25] and assumethat all PY are positive

1PY= max

�

�tKGY��tKY�

(14)

For each speci�c instability mode � a unimodal gradient is de�ned by (see Appendix Bfor details)

@(1=PY )@�

= �t@KGY@�

�− 1PY

(�t@KY@��)+ vkm

t(@KY@�

�km − @f kmY@�

)(15)

where the eigenmodes are normalized with respect to KY ; i.e. �tKY �=1 (this equation holdsfor the case where the material volume and design variable bound constraints are both strictlysatis�ed). In expression (15), vkm represents the ‘k; m’ adjoint variables associated with thelocal cell problem (8).For a critical load with multiplicity greater than one, a design change to decrease the con-

straint function 1=PYcr in the neighbourhood of the current design, if there is one, should bebased on information on the generalized gradient which is given as the following



convex hull [26]:

@�e

(1PY

)= co

{z : ze= �p�q

(�tp@KGY@�e

�q − 1PY

(�tp@KY@�e

�q

)+ vkm

t

pq

(@KY@�e

�km − @f kmY@�e

))}

(16)

where e=1; : : : ;Nel; with ‘Nel’ as the number of �nite elements, ‘co’ is the convex hull, �are components of a unitary vector, p; q=1; : : : ; m are the indexes of the multiple eigenvectorsassociated with PY and vpq are obtained from the following adjoint equation:

KY vkmpq =−�tp@KGY@�km

�q (17)

6. COMPUTATIONAL MODEL

The computational results presented in the following section are based on a �nite elementform of problem (11), using the expression given in Section 3, together with the sensitivityinformation derived above. For the optimization we use the sequentially convex approximationmethod of moving asymptotes (MMA) [27]. This has proven itself as an extremely e�cientand reliable mathematical programming method for topology optimization in general; see forexample References [7; 12; 28] for applications in structural, mechanism and material design.The di�culties due to non-smoothness of the local critical load function are handled by

introducing an auxiliary routine. This is based on the subgradient concept of non-smoothoptimization and is used to localize a direction of descent for 1=PY or a possible stationarypoint. For a general iteration ‘k’ with design �k ; the displacements, the local critical loadfactor and the eigenvectors �p=1; : : : ; m are �rst determined. The question is then to de�ne adesign change decreasing the function 1=PY at least in a neighbourhood of the current design,i.e. to choose a direction of descent of 1=PY . To characterize this direction, let �¿0 be asmall number de�ned by the user (for e.g. 0:01) and let m�; which we call the �-multiplicityof PY; be equal to the number of eigenvalues satisfying the inequality (PYi − PY )6�PY . Letalso dp; p=1; 2; : : : ; m� be the vectors with components

dpe = �tp@KGY@�e

�p −1PY

(�tp@KY@�e

�p

)+ vkm

t

pp

(@KY@�e

�km − @f kmY@�e

)(18)

Using these m� vectors, let D be the set obtained by convex combinations of the di�erentdp; i.e.

D=co

{m�∑p=1!pdp: !p¿0 and

m�∑p=1!p=1

}(19)

A direction of descent d∗ can then be de�ned as the negative of the vector in D withminimum norm, i.e. d∗ that solves the following minimization problem [29; 30]:

‖d∗‖2 = mind∈D

‖d‖2 (20)



Note that if 0∈D one has d∗= 0 and there is no descent direction in D since it is a stationarypoint. If d∗ �= 0 it does not imply that we are not at a stationary point since D is an approx-imation of the generalized gradient set and, in general, strictly contained in it. In the case ofmultiplicity m�=1; the set D has only one element, and d∗ is the vector that corresponds tothe negative of that gradient.For the buckling analysis of the base cell, the appearance of low-density regions may

result in non-physical localized modes in the low-density regions, which are an artefact of theinclusion of these low-density regions that represent void in the analysis (see also example7.1 below). In order to deal with this computational problem, a stress �ltering is requiredto identify the physically relevant modes and their respective unimodal derivatives [10]. Oneway to accomplish this is to modify the stress in the �nite elements of the base cell. Thus, forelements in which the local volume fraction is smaller than a prede�ned value (e.g. �e¡0:1),the stress is reduced to an insigni�cant stress value, for example 10−15. This modi�cationis applied when assembling KGY and when evaluating unimodal derivatives. Other recentdevelopments on techniques of avoiding localized modes in low-density regions for topologyoptimization problems involving eigenfrequencies=eigenmodes can be found in Reference [31].Finally, in order to control the geometric complexity in the optimized base cell, a mesh

independent �lter (cf. References [2; 32]) has been applied for some examples. It allows acontrol of the length scale of the details in the base cell by a modi�cation of the local deriva-tive of each element for a weighted sum of derivatives of neighbouring elements. Moreover,this mesh-independent �lter typically improves the optimization history by apparently avoidingmany local optimum points.

7. EXAMPLES

7.1. Example 1

This �rst example illustrates the necessity of stress �ltering in low-density regions in order toremove localized modes. In the �gures for this and all subsequent examples, a black regionsigni�es a region with maximum local density while a white region means void.Figure 2 shows the local buckling mode of a base cell with a solid outer rim and low

density in the interior. We see from Figure 2 that a �ltering of the contribution of the stressto the geometric sti�ening components of KGY is absolutely necessary in order to obtain therelevant instability mode, which is shown in Figure 2(b). If no stress �ltering is used we obtainlocalized modes in the low-density regions (Figure 2(a)), that are consequences of representingempty regions (holes) by low-density regions, i.e. the ‘�ctitious domain method’. The use oflocal low-density materials for representing void regions in the design domain is traditional intopology optimization and is used to allow for the reintroduction of material in such regionsas well as to avoid changing the �nite element model during the optimization cycles.

7.2. Example 2

The next example illustrates the ability of the analysis model to compare the local bucklingperformance of di�erent distributions of the same amount of material in the same base cell size,i.e. a comparison at the microscale level of di�erent designs subjected to the same normalizedmacroscopic strain �eld �0. In this particular example it is given by �0 = {−1 0 0}t .



Figure 2. First instability modes for a macroscopic strain �eld given by �0 = {�011 �022 �012}t = {−1 0 0}t :(a) without stress �ltering and (b) after stress �ltering.

Consider as the initial case the square base cell with a centred square hole, modelledcomputationally by a �nite element mesh with 10× 10 four node isoparametric �nite elementsof plane elasticity and with Y-periodic conditions imposed at the boundary nodes. The initialcase has a relative material density � of 0.36, and the hole is modeled as a material ofnegligible Young modulus by a low local density (�=10−4).To de�ne what we call the ‘uniform’ case, we add material uniformly in the hole of

the initial cell, so that the relative material density increases to �=0:52. Here, grey regionscharacterize local intermediate densities. The same amount of material is then added to the holebut with di�erent distributions as seen for cases 1–4 in Table I. For the given macroscopicstrain �eld, the best distribution (case 1) corresponds to a concentration of the additionalmaterial at the members aligned with the direction of the non-zero strain component. Asexpected, the material distribution in case 2 represents a local buckling performance that islower than that of case 1 and also of the initial case. The material distributions in cases 3and 4 can be considered as distributions including ‘chains of one-point connections’. Thesechains appear frequently in the optimal topologies of periodic materials when coarse meshes orfour node �nite elements are used without regularization techniques. As presented in Table I,the local buckling performance for these distributions is zero. This comparison of the localbuckling performance of di�erent cells is the basis for the optimization examples presentedin the following.

7.3. Example 3

This example considers the maximization of a given linear combination of components of thehomogenized elastic properties tensor, i.e. �ijkmEHijkm with �ijkm= �

0ij�0km, for the speci�c macro-

scopic strain �eld represented by �0 = {−1 −1 0}t . This problem corresponds to a maximiza-tion of the homogenized in-plane bulk modulus kH of a composite material.



Table I. Comparison of di�erent cells (Y-periodic mode, E is theYoung modulus of the base material) for a macroscopic strain �eld

given by �0 = {−1 0 0}t .

The initial design is the cell shown in Figure 3(a) and corresponds to a uniform distributionof the available material in the design region (in grey) and an outside frame (in black) thatis the �xed domain. The base cell is modeled computationally using a �nite element meshwith 40× 40 four-node isoparametric �nite elements for plane elasticity, with the Y-periodicdisplacement conditions imposed at boundary nodes.Figure 3(b) presents the optimized solution obtained without a local buckling constraint,

and Figures 3(c), 3(d) and 3(e) present the optimized solutions obtained when consideringthis constraint for Pmin =0:10; 0:15 and 0:20; respectively. The obtained numerical values arepresented in Table II. Notice that no mesh-independent �lter was used for these results as it



Figure 3. (a) Initial design; (b) solution obtained without local buckling constraint, and solutionsobtained with buckling constraint of: (c) Pmin = 0:10; (d) Pmin = 0:15; and (e) Pmin = 0:20.

Table II. Obtained values for the designs from Figure 3.

Density EH1111 EH1122 E

H1212

Figure and Pmin (square symmetry) PY=E kH

3(a) �=0:4540× 40 FE — 0:1028 0:0038 0:0244 0.063 0.053

3(b) �=0:4540× 40 FE — 0:2471 0:0509 0:0898 0.04 0.149

3(c) �=0:4540× 40 FE Pmin = 0:1 0:2287 0:0575 0:0410 0.128 0.143

3(d) �=0:4540× 40 FE Pmin = 0:15 0:2130 0:0445 0:0316 0.149 0.128

3(e) �=0:4540× 40 FE Pmin = 0:20 0:2212 0:0512 0:0353 0.199 0.136

allows to clearly illustrate how the constraint on the minimum buckling performance resultsin a penalization of ‘chains of one point connections’ as well as checkerboard patterns. Thisshould be expected on the basis of the �ndings of Example 2 and is clearly seen from acomparison between Figures 3(b) and 3(c) (for a further discussion and references on thecheckerboard issue in topology design see Reference [33]). Although the checkerboards andchains are removed by the buckling constraint the designs may still being mesh-dependent.We note that here, and in the results below, the buckling constraint is not necessarily active

at the computed (locally) optimal design. This is the case for small Pmin values, while highervalues of Pmin does imply that the buckling constraint is active. Even when not active at theoptimum, the constraint does in�uence the result; it is active at the initial steps of the iterativeoptimization procedure and thus ‘steers’ the computational procedure to a local optimum withbuckling performance better than speci�ed. This involved behaviour should be attributed tothe fact that there exist many microstructures with optimal bulk properties.The geometric complexity or the mesh-dependency of the resulting designs can be controlled

through the use of a mesh-independent algorithm (�lter), as mentioned in Section 6. Usingthis algorithm, small scale variations in the cell can be removed and Figure 4(b)–4(e) showexamples of base cells obtained with this strategy for di�erent values of Pmin. The homogenizedin-plane bulk modulus obtained (see Table III) are near the analytical Hashin–Shtrikman limit[34], k=0:159 for the relative density of 0.45, and this ensures that the results are close to the



Figure 4. Solutions obtained with both mesh independent algorithm and buckling constraint for:(a) Pmin = 0:00; (b) Pmin = 0:10; (c) Pmin = 0:15; (d) Pmin = 0:20; and (e) Pmin = 0:225.

Table III. Obtained values for the designs from Figure 4.

Density EH1111 EH1122 E

H1212

Figure and Pmin (square symmetry) PY =E kH

4(a) �=0:4540× 40 FE Pmin = 0:00 0.2707 0.0335 0.0164 0.200 0.152

4(b) �=0:4540× 40 FE Pmin = 0:10 0.2688 0.0358 0.0206 0.176 0.152

4(c) �=0:4540× 40 FE Pmin = 0:15 0.2742 0.0365 0.0217 0.192 0.155

4(d) �=0:4540× 40 FE Pmin = 0:20 0.2702 0.0397 0.0219 0.203 0.155

4(e) �=0:4540× 40 FE Pmin = 0:225 0.2671 0.0396 0.0202 0.223 0.153

global optimum in this case. It is seen that the solutions obtained with the mesh independentalgorithm, Figures 4(a)–4(e) are better than the correspondent ones obtained without this�lter, Figures 3(c)–3(e), underlining that these latter are local optima.

8. CONCLUDING REMARKS

In this work we have used an eigenvalue buckling criteria at the microscale level to charac-terize if a certain base cell of a periodic medium is more or less prone to highly localized andY-periodic buckling modes. This criterion is a condition obtained from the double scale asymp-totic expansion technique applied to the linearized elastic stability problem of structures builtwith periodic microstructures. The examples presented illustrate how the model evaluates thelocal buckling performance of di�erent distributions with the same amount of material and voidat the microscale, i.e. di�erent designs, when subjected to the same macroscopic strain �eld.This characterization of the local buckling instabilities at the base cell level is then used

for the topology optimization of periodic microstructures. When applied as a minimal localbuckling performance requirement, it improves the original model for optimizing linear elas-tic material properties because the stress sti�ening e�ect penalizes the presence of slendermembers, checkerboard pattern regions and ‘chains of one-point connections’.



Like other optimization methods, there is no guarantee of convergence to a global minimum,but by using the mesh independent algorithm the theoretical bound on the elastic propertiescan be achieved with simple topologies that have high local elastic stability properties. Theresults obtained when designing for optimal bulk modulus point to the fact that the so-calledVigdergauz type of microstructures (cf. Figure 4(a)) [35] also yield a comparatively highlocal buckling performance. On the opposite end of the spectrum of high bulk modulusmicrostructures one �nds base cells of layered type for which the layered regions are moreprone to localized buckling.As it is known, there are many di�erent possible cell designs with the same maximum

e�ective elastic properties, and some of them are more prone to highly localized bucklingthan others. The e�ect of the local buckling constraint is to restrict the space of admissiblesolutions to designs with the required minimum local buckling performance. This restrictionis more evident in the �rst steps of the topology optimization when the optimized materialdistribution starts to develop, although for small values of Pmin it is possible that the constraintis not active at the converged solution. From an optimization point of view it means that thebuckling constraint introduces a method of avoiding premature convergence to cells with abuckling performance below a speci�ed value. The value of the critical load Pmin was chosento penalize the development of undesirable highly localized modes characterized by very lowcritical load factors (stability parameters). Thus, a buckling performance measure can eliminatemany sti�ness ‘optimal’ but undesirable topologies.The computational e�ort involved and the complexity of the model are a disadvantage but

the results obtained underline the importance of the buckling e�ects for the optimized com-posites. An easier way to obtain these microstructures that are less prone to highly localizedbuckling is by using a mesh-independent algorithm that is a indirect way of controlling highlylocalized buckling.Important issues remain to be analysed. For future work, one can consider other macroscopic

loading cases and a di�erent parameterization of design in terms of structural elements, e.g.frame element models. Moreover, one should investigate the e�ect of evaluating the localcritical load PY using more cells of the periodic medium. Here, the e�ort was concentratedon characterizing instabilities inside the base cell.The use of Bloch waves techniques together with topology optimization of periodic materials

is subject of current studies [21]. Also of interest is the study of periodic solids of �nite extentwhere one cell is retained as the design domain in order to maintain the material periodicity.For solids of �nite extent, a general non-linear theory as presented in Reference [23] is a possi-bility. The limitation of the present model in terms of its removal of �nite scale e�ects does notexist in these higher order gradient models. However, they do require a much more complexmodel than the one presented here, and its use in an optimization context will be a challenge.

APPENDIX A: ASYMPTOTIC MODEL FOR LINEARIZED BUCKLINGOF STRUCTURES BUILT OF PERIODIC MATERIALS

Consider an elastic structure built of a porous material obtained from the introduction ofvery small voids of any shape into an isotropic base material. The structure is �xed on theboundary �u and subjected to a conservative proportional surface loading on the boundary �t(see Figure 1).



No body forces and no internal pressure in the voids are considered. By proportional loadingit is meant that the applied load is a function of one parameter, i.e. Pt, where P is the loadfactor parameter and t is the reference load. When this load factor is lower than a criticalload factor, i.e. P¡Pcr ; we assume that the structural response is linear and elastic witha unique and small displacement u0�. In the current text, the superscript �; where �=d=Dis the microstructure size parameter, identi�es the displacement dependence on the materialmicrostructure.For P=Pcr ; there are two in�nitely close equilibrium positions: the initial with displacement

u0� and a secondary position with displacements u�. For the second position, let us denote itsdisplacement by

u�= u0� + �u1� (A1)

where � is an in�nitesimal real parameter that vanish for P¡Pcr and u1� identi�es the displace-ment �eld in the structure when it shifts from the initial to the second equilibrium position[36].These equilibrium positions are characterized by the stationarity conditions of the total

potential energy functional

�(u�)=A(u�)− R(u�) (A2)

where A(u�) is the elastic strain energy

A(u�)=12

∫��ij(u�)eij(u�) d� (A3)

and R(u�) is the following force potential:

R(u�)=P∫�ttiu�i d� (A4)

Assuming in Equation (A3) a linearly elastic material that follows the Hooke’s law, we obtain

A(u�)=12

∫��Eijkmekm(u�)eij(u�) d� (A5)

where

eij(u�)=12

(@u�i@xj

+@u�j@xi

)+12

(@u�k@xj

@u�k@xi

)(A6)

By substituting (A1) into (A6) and considering the linearized elastic stability assumptions—i.e. that the deformations before instability are in�nitesimal and linear elastic such that the12 (@u

0�k =@xi @u

0�k =@xj) term is small relative to the 1

2 (@u0�k =@xi + @u

0�k =@xj) term, and that due to

the characteristic sudden con�guration change of linearized elastic instabilities, the non-linearterms of �{ 12 (@u0�k =@xj @u1�k =@xi)+ 1

2(@u1�k =@xj @u

0�k =@xi)} are assumed to be zero—we obtain the

following strain tensor for the problem:

eij(u�)≈ 12

(@u0�i@xj

+@u0�j@xi

)+�2

(@u1�i@xj

+@u1�j@xi

)+�2

2

(@u1�k@xj

@u1�k@xi

)(A7)



Since the displacements depend on the microscale level, analysing directly the structurewith a periodic and highly heterogeneous geometry is impossible due the huge amount ofinformation and geometric complexity involved. A way to simplify the problem is to use atwo-scale homogenization method for a periodic medium. In this case, the displacements arerepresented as an asymptotic expansions in terms of the cell size parameter �:

u0�(x; y)= u00(x; y) + �u01(x; y) + �2u02(x; y) + · · · ; y=x�

(A8a)

and

u1�(x; y)= u10(x; y) + �u11(x; y) + �2u12(x; y) + · · · ; y=x�

(A8b)

In the previous expressions ((A8a) and (A8b)), the di�erent expansion terms uab aresu�ciently regular functions of the macroscopic variable x and periodic functions of themicroscopic variable y [37].Using (A8a) and (A8b) in (A7) together with the di�erentiation rule

dF(x;x=�)dxj

=@F(x; y)@xj

+1�@F(x; y)@yj

the strain �eld (A7) can be expressed in terms of � powers as

eij(u�)= e0ij(uab) + �eIij(u

ab) + �2eIIij(uab) (A9)

where a=0; 1; and b=0; 1; 2; : : : ; and e0, eI and eII are, respectively,

e0ij(uab) =

1�

{12

(@u00i@yj

+@u00j@yi

)}+12

(@u00i@xj

+@u00j@xi

)+12

(@u01i@yj

+@u01j@yi

)

+�

{12

(@u01i@xj

+@u01j@xi

)+12

(@u02i@yj

+@u02j@yi

)}+ �2

{12

(@u02i@xj

+@u02j@xi

)+ · · ·

}+ · · ·

(A10a)

eIij(uab) =

1�

{12

(@u10i@yj

+@u10j@yi

)}+12

(@u10i@xj

+@u10j@xi

)+12

(@u11i@yj

+@u11j@yi

)

+ �

{12

(@u11i@xj

+@u11j@xi

)+12

(@u12i@yj

+@u12j@yi

)}+ �2

{12

(@u12i@xj

+@u12j@xi

)+ · · ·

}+ · · ·

(A10b)

eIIij(uab) =

1�2

{12

(@u10k@yi

@u10k@yj

)}+1�

{(@u10k@xi

@u10k@yj

)+12

(@u10k@yi

@u11k@yj

)+12

(@u10k@yj

@u11k@yi

)}

+{12

(@u10k@xi

@u10k@xj

)+12

(@u11k@yi

@u10k@xj

)+12

(@u11k@yj

@u10k@xi

)+12

(@u11k@xj

@u10k@yi

)



+12

(@u10k@yj

@u11k@xi

)+12

(@u11k@yi

@u11k@yj

)}

+ �{12

(@u11k@yi

@u11k@xj

)+12

(@u11k@xi

@u11k@yj

)+12

(@u11k@xi

@u10k@yj

)+12

(@u10k@xi

@u11k@xj

)

+12

(@u10k@xi

@u11k@yj

)+12

(@u10k@xi

@u12k@yj

)+12

(@u10k@yi

@u12k@xj

)+12

(@u12k@xi

@u10k@yj

)

+12

(@u12k@yi

@u10k@xj

)+12

(@u11k@yi

@u12k@yj

)+12

(@u12k@yi

@u11k@yj

)}

+ �2{12

(@u11k@xi

@u11k@xj

)+12

(@u10k@xi

@u12k@xj

)+12

(@u12k@xi

@u10k@xj

)+12

(@u11k@yi

@u12k@xj

)

+12

(@u12k@yi

@u11k@xj

)+12

(@u11k@xi

@u12k@yj

)+12

(@u12k@xi

@u11k@yj

)+12

(@u12k@yi

@u12k@yj

)}+ · · ·(A10c)

In Reference [10] only non-linear contributions from u10 were considered for eij and u10 isa function of the macroscopic variable x only. Here, u10 is allowed to be a function of themacroscopic variable x and of the microscopic variable y (but Y-periodic in y), and thereforeother non-linear contributions are present.

A.1. First variation of the total potential energy

The displacement �elds u0� and u1� of (A1) are characterized by the stationarity of the totalpotential energy with respect to a displacement perturbation around u0�, i.e.

��(u�)= �A(u�)− �R(u�)=0 (A11)

For a known u0�, the mentioned variation is de�ned around u0� by �u�= �[v10(x;x=�) +�v11(x;x=�) + �2v12(x;x=�) + · · ·] with

vab∈V�×Y = {v(x; y): v|�u = 0; v su�ciently regular and Y -periodic in y}With this variation we obtain from condition (A11), equating to zero each coe�cient of

power of �; two conditions (with no index summation for ‘ab’):

�∫��Eijkm[e0ij(u

ab)eIkm(vab) + e0ij(v

ab)eIkm(uab)] d�− �P

∫�ttivi d�=0; ∀vab∈V�×Y

(A12a)

and

�2∫��Eijkm{e0ij(uab)eIIkm(vab) + e0ij(vab)eIIkm(uab) + eIij(uab)eIkm(vab) + eIij(vab)eIkm(uab)} d�=0;

∀vab∈V�×Y (A12b)



A.2. Linearized elasticity for structures built of materials with periodic microstructure

Now let us assume that the microstructure size parameter � is in�nitesimal, i.e. the materialheterogeneities have a characteristic dimension ‘d’ much smaller than the global dimensionof the structure ‘D’ (see Figure 1). The equilibrium conditions are obtained by taking therespective limit as �→ 0 for each power in � of (A12a). From this we obtain exactly the sameequilibrium equations for the linearized elasticity problem that are known from homogenizationtheory for periodic materials [37; 38]. The main equations are indicated in the following fornotational proposes.The characteristic Y-periodic displacements �km of a base cell Y are the solution of the

following local problem at microscale level:

∫YEijpq

@�kmp@yq

@vi@yj

dY =∫YEijkm

@vi@yj

dY; ∀v∈VY (A13)

with VY = {v: v is su�ciently regular and Y-periodic in y}.Then, denoting the homogenized elasticity tensor EHijkm by

EHijkm=1|Y |

∫Y

(Eijkm − Eijpq

@�kmp@yq

)dY (A14)

the following elastostatic equation for the homogenized medium at macroscale level isobtained: ∫

�EHijkm

@u00k@xm

@v10i@xj

d�=∫�ttiv10i d�; ∀v10∈V� (A15)

where V� = {vab(x): vab|�u = 0; vab regular}.

A.3. Linearized elastic stability for structures built of materials with periodic microstructure

Equating to zero each coe�cient of power of � in the second stationarity condition (A12b)gives us the homogenized equilibrium equations for the linearized elastic stability problem.It is important to note that in order to apply the asymptotic method both displacements

�elds, u0� and u1�, have to be periodic in y and the parameter ‘�’ has to be small. Forthe displacements u0� the periodicity is a good approximation of the physical behaviour,while for u1� the periodicity is a restriction that we enforce over many possible instabilitymodes.Equating to zero the coe�cient of �−3 in (A12b) we obtain one equation which is satis�ed

for u00(x; y)= u00(x).For the term of �−2 in (A12b), taking the limit �→ 0 we obtain

∫�

1|Y |

∫YEijkm

@u10i@yj

@v10k@ym

dY d� +∫�

1|Y |

∫Y

(Eijkm − Eijpq

@�kmp@xq

)@u00i@xj

@u10c@yk

@v10c@ym

dY d�=0;

∀v10∈V�×Y (A16)



Note that no homogenization is involved for this equation, although homogenization isinvolved at the macroscale level to obtain u00 (see expression (A15)). Also, note that thestress 0 in each point of the cell is only the �rst approximation of the stress since �ij=0ij+�1ij + �

2(: : :) [37].Equation (A16) can also be expressed in the following way:

∫YEijkm

@u10i@yj

@v10k@ym

dY +∫Y0km

@u10c@yk

@v10c@ym

dY =0; ∀v10∈VY (A17)

which is an eigenvalue and eigenvector problem like in structural eigenvalue buckling but withperiodic boundary displacements. If u10 is a global mode and not a function of y, then thesolution to this equation is trivial. Otherwise, it is a local stability problem for the Y-periodiceigenmode.For �−1 in (A12b) and choosing v10 = 0; we obtain Equation (A17) expressed in terms

of v11.We notice that the local stability condition (A17), together with the equations presented in

Appendix A.2 are su�cient for the numerical purposes of the paper. However, as far as weknow, a model derivation of the linearized elastic buckling of periodic materials has not beenpreviously addressed in the literature, and therefore is presented in the following.Let us continue choosing alternatively v11= 0 for the term with �−1 in (A12b) and we

obtain

1�

∫��Eijkm

(@u11i@yj

@v10k@ym

)d� +

1�

∫��Eijkm

(@u00i@xj

+@u01i@yj

)(@u11c@yk

@v10c@ym

)d�

+1�

∫��Eijkm

(@u10i@yj

@v10k@xm

+@u10i@xj

@v10k@ym

)d� +

1�

∫��Eijkm

@u00i@xj

(@u10c@xk

@v10c@ym

+@u10c@ym

@v10c@xk

)d�

+1�

∫��Eijkm

@u01i@xj

(@u10c@yk

@v10c@ym

)d� +

1�

∫��Eijkm

@u01i@yj

(@u10c@yk

@v10c@xm

+@u10c@xk

@v10c@ym

)d�=0;

∀v10∈V�×Y (A18)

From all v10 let us choose v10 = v10(x); and so we obtain the following equation:

∫�

(1|Y |

∫YEijkm

@u10i@yj

dY)@v10k@xm

d� +∫�

[1|Y |

∫Y

(Eijkm − Eijpq

@�kmp@yq

)@u00i@xj

@u10c@yk

dY

]

×@v10c

@xmd�=0; ∀v10∈V� (A19)

Equation (A19) relates simultaneous instabilities at microscale and at macroscale level.



By only choosing v10= v10(y) from all v10 we obtain the following equation :

1�

∫��Eijkm

(@u11i@yj

+@u10i@xj

)@v10k@ym

d� +1�

∫��Eijkm

(@u00i@xj

+@u01i@yj

)(@u11c@yk

+@u10c@xk

)@v10c@ym

d�

+1�

∫��Eijkm

@u01i@xj

(@u10c@yk

@v10c@ym

)d�=0; ∀v10∈V�×Y (A20)

Now, assuming that u10 is an instability only at the macroscale level and neglecting thenon-linear terms of (A22), we obtain a relation between u11 and u10 given by∫

�

1|Y |

∫YEijkm

(@u11i@yj

+@u10i@xj

)@v10k@ym

dY d�=0; ∀v10∈V�×Y (A21a)

With the mentioned assumptions, the following relation is obtained from (A21a):

u11i (x; y)=−�kmi (y)@u10k (x)@xm

+ Ctei (x) (A21b)

For �0 in (A12b) and choosing v10= 0 we obtain:

�0∫��Eijkm

(@u10i@xj

@v11k@ym

+@u11i@yj

@v11k@ym

+@u10i@yj

@v11k@xm

)d�

+ �0∫��Eijkm

@u00i@xj

(@u10c@xk

@v11c@ym

+@u11c@yk

@v11c@ym

+@u10c@yk

@v11c@xm

)d�


@u01i@xj

(@u10c@yk

@v11c@ym

)d�


@u01i@yj

(@u10c@xk

@v11c@ym

+@u10c@yk

@v11c@xm

+@u11c@yk

@v11c@ym

)d�


(@u10i@yj

@v12k@ym

)d� + �0

∫��Eijkm

(@u00i@xj

+@u01i@yj

)(@u10c@yk

@v12c@ym

)d�=0

∀v11; v12∈V�×Y (A22)

Assuming that only v12 is non-zero gives (A17). All others terms in (A22) can be rearrangedin order to the respective derivatives of v11 with respect to x and y.

�0∫��Eijkm

(@u10i@xj

+@u11i@yj

)@v11k@ym

d� + �0∫��Eijkm

(@u00i@xj

+@u01i@yj

)(@u10c@xk

+@u11c@yk

)@v11c@ym

d�




(@u10i@yj

@v11k@xm

)d� + �0

∫��Eijkm

(@u00i@xj

+@u01i@xj

)(@u10c@yk

@v11c@xm

)d�


@u01i@yj

(@u11c@yk

@v11c@xm

)d�=0; ∀v11∈V�×Y (A23)

By choosing v11= v11(y) and neglecting the non-linear derivative terms, we evaluate a problemequivalent to (A21a).By selecting v11= 0 for the term �0 in (A12b), we obtain

�0∫��Eijkm

{(@u10i@xj

+@u11i@yj

)@v10k@xm

+@u11i@xj

@v10k@ym

}d�


@u00i@xj

(@u10c@xk

@v10c@xm

+@u11c@yk

@v10c@xm

+@u11c@xk

@v10c@ym

)d�


@u01i@xj

((@u10c@xk

+@u11c@yk

)@v10c@ym

+@u10c@yk

@v10c@xm

)d�


@u01i@yj

((@u10c@xk

+@u11c@yk

)@v10c@xm

+@u11c@xk

@v10c@ym

)d�


(@u10i@yj

@v12k@ym

+@u12i@yj

@v10k@ym

)d�


(@u01i@yj

+@u00i@xj

)(@u10c@yk

@v12c@ym

+@u12c@yk

@v10c@ym

)d�=0; ∀v10∈V�×Y

(A24)

For the particular case where all instability modes are at the macroscale level, i.e. u10 =u10(x) and v10 = v10(x); this equation gives that

∫��Eijkm

(@u10i@xj

+@u11i@yj

)@v10k@xm

d� +∫��Eijkm

(@u00i@xj

+@u01i@yj

)(@u10c@xk

@v10c@xm

+@u11c@yk

@v10c@xm

)d�=0

∀v10∈V�×Y (A25)

Neglecting the non-linear terms involving derivatives of u11 and introducing the result(A21b), we obtain the homogenized bifurcation model given by

∫�

1|Y |

∫Y

(Eijkm − Eijpq

@�kmp@yq

)dY@u10i@xj

@v10k@xm

d�



+∫�

1|Y |

∫Y

(Eijkm − Eijpq

@�kmp@yq

)dY@u00i@xj

@u10c@xk

@v10c@xm

d�=0; ∀v10∈V� (A25a)

or

∫�EHijkm

@u10i@xj

@v10k@xm

d� +∫�

(EHijkm

@u00i@xj

)@u10c@xk

@v10c@xm

d�=0; ∀v10∈V� (A25b)

This is the homogenized eigenvalue buckling problem for the structure, at the macro level.

APPENDIX B: SENSITIVITY ANALYSIS FOR THE LOCALCRITICAL LOAD FUNCTION

This appendix presents a derivation of the sensitivities for the local critical load function withrespect to the design variables. Expressions (15)–(17) presented in Section 5 are given in acompact and non-standard notation and it requires a formal derivation here.The unimodal derivative of the function of expression (14), the inverse of the Raleygh–

Ritz quotient (inverse of the critical load), is here derived on the basis of the continuumformulation for which we use index notation. The inverse of the Rayleygh quotient presentedin expression (14) is now given by

1PY= max

�

−��s∫Y

(Eijkm − Eijpq

@�kmp@yq

)�0km@N�

@yi@N�

@yjdY��s

��i

∫YEijkm

@N�

@yi@N�

@ymdY��k

(B1)

Writing a unimodal critical load as PY =PY (�; �;), where k =N��k , the derivativeswith respect to the design variables (here the local density at each �nite element �e; e=1;: : : ;Nel) are given as

dd�e

(1

PY (�)

)=

[−��s

∫Y

(@Eijkm@�e

− @Eijpq@�e

@�kmp@yq

)�0km@N�

@yi@N�

@yjdY��s A

+��s

∫Y

(Eijkm − Eijpq

@�kmp@yq

)�0km@N�

@yi@N�

@yjdY

×��s(��i

∫Y

@Eijkm@�e

@N�

@yj@N�

@ymdY��k

)]/A2

−[��s

∫Y

(Eijkm − Eijpq @N

@yq

)�0km

@N�

@yi@N�

@yjdY��s

@�km

p

@�e

]/A



+

[−2@�

�s

@�e

∫Y

(Eijkm − Eijpq

@�kmp@yq

)�0km

@N�

@yi@N�

@yjdY��s A

+��s

(∫Y

(Eijkm − Eijpq

@�kmp@yq

)�0km

@N�

@yi@N�

@yjdY

)

×��s(2@��i@�e

∫YEijkm

@N�

@yj@N�

@ymdY��k

)]/A (B2)

where A=(��i∫Y Eijkm(@N

�=@yj)(@N�=@ym) dY��k ) for a given �.

The characteristic local problems at the microscale level (8) give us the following result:

(@KY@�

�km +KY@�km

@�

)=@f kmY@�

⇒ @�km

@�=K−1

Y

(@f kmY@�

− @KY@��km)

Then introducing this result in the term of (B2) involving @�kmp =@�e, we get for this speci�cterm

[−��s

∫Y

(Eijkm − Eijpq @N

@yq

)�0km

@N�

@yi@N�

@yjdY��s

]@�km

p

@�e

/A

=[−��s

∫Y

(Eijkm − Eijpq @N

@yq

)�0km@N�

@yi@N�

@yjdY��s

]/A

×(∫

Y

@Eijkm@�

@N

@yjdY −

∫YEijpq

@N

@yj

@�kmp@yq

dY

)/(∫YEijpq

@N�

@yj@N�

@yqdY)

(B3)

One can avoid the inversion of the matrix KY by using an adjoint variable method. Thisintroduces an adjoint variable vkm;

vkm

i =−��s∫Y

(Eijkm − Eijpq @N

@yq

)�0km

(@N�

@yi@N�

@yj

)dY��s

/(A∫YEijpq

@N

@yj@N�

@yqdY)

(B4)

where no index summation is done in the ‘k’ and ‘m’ indexes. Written in compact form vkm

is obtained from the following problem, where the eigenmodes are normalized with respectto the sti�ness matrix, i.e. �tKY �=1

KY vkm=−�t @KGY@�km

�



The right-hand side of the last expression represents the adjoint force vector given by:

��kmtKY vkm =−�t @KGY

@�km��km�=−

∫ @0ij@�km

@s@yi

@s@yj��km dY = ��kmp�F

adjkm

p�

Fadjkm

p� =∫ (

Eijpq@N�

@yq

)�0km@s@yi

@s@yj

dY

Notice that it is not necessary to evaluate @�=@� since this term in (B2) multiplies by zero,as expressed by the product (@�=@�)(KGY − 1

PY KY )�.Thus the derivative of a unimodal critical load in index notation is given by

dd�e

(1

PY (�)

)=

[−��s

∫Y

(@Eijkm@�e

− @Eijpq@�e

@�kmp@yq

)�0km

@N�

@yi@N�

@yjdY ��s

− 1PY��i

∫Y

@Eijkm@�e

@N�

@yj@N�

@ymdY ��k

]

+ vkm�

i

(∫YEijpq

@N�

@yj

@�kmp@yq

dY −∫Y

@Eijkm@�e

@N�

@yjdY

); e=1; : : : ;Nel

(B5)

ACKNOWLEDGEMENTS

The authors are grateful to Prof. Krister Svanberg from the Royal Institute of Technology, Stock-holm, for having supplied his MMA code, and to Alejandro D�az from Michigan State University,Michigan, and the topology optimization groups at the Technical University of Denmark (DTU) andat Instituto Superior Tecnico (IST), Lisbon, for fruitful discussions. Very useful comments from theanonymous referees have also been incorporated in the manuscript. Part of this work was carried outwhile the �rst author was a visiting researcher at the ‘DCAMM International Graduate Research Schoolin Applied Mechanics’, at the DTU (sponsored by the Danish Research Academy), and, later, as a post-doc at DTU, sponsored by the European Research Training Network, ‘Homogenization and MultipleScales’ (HMS2000), Project RTMI-1999-00040. Support from IDMEC=IST (MMN); FCT (Portugal)Project PBIC=C=TPR=2404=95; STVF (Denmark) THOR=Talent project ‘Design of MicroElectroMe-chanical Systems (MEMS) (OS); SNF (Denmark) (MPB) is also gratefully acknowledged.

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Topology optimization of periodic microstructures with a penalization of highly localized buckling...

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