Computers and Structures 157 (2015) 142–152

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Computers and Structures

journal homepage: www.elsevier .com/locate/compstruc

Topology optimization of continuum structures under bucklingconstraints

http://dx.doi.org/10.1016/j.compstruc.2015.05.0200045-7949/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: gaoxj85@126.com (X. Gao), maht@scut.edu.cn (H. Ma).

Xingjun Gao, Haitao Ma ⇑State Key Laboratory of Subtropical Building Science, Department of Civil Engineering, South China University of Technology, Guangzhou 510640, China

a r t i c l e i n f o

Article history:Received 15 January 2015Accepted 19 May 2015Available online 8 June 2015

Keywords:Topology optimizationBuckling constraintsPseudo buckling modesPseudo mode identificationTwo-phase optimization algorithms

a b s t r a c t

This paper presents a study on topology optimization of continuum structures under bucklingconstraints. New algorithms are developed for minimization of structural compliance consideringconstraints on volume and buckling load factors. The SIMP (Solid Isotropic Material with Penalization)material model is employed and nodal relative densities are used as topology design variables. A newapproach based on the eigenvalue shift and pseudo mode identification is proposed for eliminating theeffect of pseudo buckling modes. Two-phase optimization algorithms are also proposed for achieving bet-ter optimized designs. Numerical examples are presented to illustrate the effectiveness of the newmethods.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Structural strength, stiffness, and stability are three of theimportant factors considered for assessing the design of a struc-ture. Naturally, it is important to consider structural stability inthe optimization process. Recently, buckling optimization hasdrawn more research attention.

For trusses, frames, and other built-up structures consisting ofbars and beams, much work has been done to consider the stabilityrequirements in structural optimization, such as size optimizationof trusses and frames [1], shape optimization of columns, truss orbuilt-up structures [2–4] and topology optimization of truss struc-tures [5–7].

Neves et al. [8] and Min and Kikuchi [9] have considered struc-tural stability in topology optimization of continuum structures.They investigated the reinforcement of a structure to increase itsoverall stability. Neves et al.[10] extended their earlier work tothe buckling optimization of periodic material micro-structures.Geometrically nonlinear models have also been introduced intothe optimization of continuum structures against buckling[11–15]. In addition, optimization of composite structures wasconsidered by Lindgaard and Lund [16,17].

A common problem in topology optimization using the SIMPmaterial model is the appearance of pseudo buckling modes inlow-density regions. Neves et al. [8] suggested ignoring the

geometrical stiffness matrices of elements with densities and prin-cipal stresses smaller than predefined threshold values.Meanwhile, they indicated that the predefined values might havea significant influence on optimization results. Bendsøe andSigmund [18] pointed out that doing this might cause solutionoscillations due to abrupt changes of objective functions and sen-sitivities. In order to avoid the discontinuity caused by such acut-off method, they suggested the use of different penalizationschemes for element stiffness matrix and geometric stiffnessmatrix. Currently this method appears to be a standard solutionfor this problem and has been used by many researchers, e.g.Lindgaard and Lund [19]. However, Zhou [20] showed that it mightbe difficult to select an appropriate parameter value for the expres-sion of penalization in calculating accurate buckling load factors.

Pseudo eigenmodes may also appear in the optimization ofeigenfrequencies in vibration problems [21]. To eliminate thesepseudo modes, some methods of modifying element stiffnessmatrix and/or mass matrix in low-density regions have been pro-posed and details of these methods can be found in the research lit-erature, e.g. [21–23]. A topology optimization problem consideringbuckling differs from the one considering vibration modes and ismore complex as element geometrical stiffness matrices aredependent on element stresses, which depend both on the struc-ture itself and on the loading condition. In contrast, element massmatrices used in frequency analysis are dependent on material dis-tribution only.

In this paper, the pseudo buckling mode problem is investigatedand a new method combining eigenvalue shift and pseudo modeidentification is proposed. An optimization formulation for

X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152 143

minimizing the structural compliance under material volume andbuckling load factor constraints is used in the study.

This paper is organized as follows: In Section 2 the optimizationformulation and material model used are presented. In Section 3,the finite element model employed for the structural analysis isintroduced. In Section 4, expressions for the sensitivity of con-straint functions and objective function are derived. In Section 5,some of the existing methods for dealing with pseudo bucklingmodes are briefly discussed and a new approach is proposed. InSection 6, new optimization algorithms are developed. InSection 7, two numerical examples are presented to demonstrateeffectiveness of the proposed methods. Finally, concluding remarksare made.

2. Problem formulation and material interpolation scheme

2.1. Optimization problem formulation

The topology optimization of continuum structure may oftengenerate designs with slender components when the allowedmaterial volume fraction is small. If compressive stresses occurin these structural components, structural buckling may presentserious safety concerns. Therefore, structural stability require-ments should be considered in the optimization. The mathematicalformulation of the compliance minimization problem of contin-uum structures with constraints on the material volume and buck-ling load factors can be stated as

find q ¼ q1;q2; � � � ;qNf gmin C ¼ FTU ¼ UTKUs:t: KU ¼ F

minj2J

kj

�� ��P k > 0

V qð Þ 6 V0

0 < q 6 qi 6 1 i ¼ 1;2; . . . ;N

ð1Þ

where qi ði ¼ 1;2; . . . ;NÞ are design variables of relative materialdensity; N is the number of design variables; C is the structuralcompliance; U and F are the global displacement and force vectors;kj is the jth buckling load factor corresponding to the given loadcases; J is a set of indices of the buckling mode considered in theoptimization; k denotes the lower bound of buckling load factors;V qð Þ is the total material volume of the structure; V0 is the upperbound of material volume; and q is the lower bound of design vari-ables, e.g. q ¼ 0:001.

Through the introduction of an explicit constraint condition onbuckling load factors, designs that fail to satisfy stability require-ments will be excluded from the feasible solution set.Theoretically, different levels of safety margins can be achievedby using different lower bound values. For example, if k ¼ 1, theoptimized structure will be at a critical state under normal serviceconditions; if k > 1, the structure will be stable under normal ser-vice conditions with a bigger safety margin for a bigger k; if0 < k < 1, the structure may buckle under normal service condi-tions, but cannot be a mechanism.

The buckling mode index set J is introduced for two reasons.Firstly, when an applied load always points in the same direction,negative loading factors are meaningless and in this case, set Jshould contain only the modes with positive load factors.Secondly, when pseudo modes are among the calculated bucklingmodes, the corresponding mode indices must be excluded fromset J as these modes are not real and should be ignored.

2.2. Material interpolation

It is possible to obtain continuous material distributions byusing nodal relative densities as topology design variables [24]. Itis noted that Kang and Wang [25] have presented a more generaldensity interpolation strategy for topology optimization usingnodal design variables and Shepard interpolation. In this study, amore conventional interpolation scheme based on element nodalvalues and shape functions is used. Within the eth element, the rel-ative density distribution is expressed as

qeðx; yÞ ¼XNN

k¼1

Nk x; yð Þqek ð2Þ

where qek denotes nodal density value at the kth node of the ele-

ment, NN is the number of nodes in the element, and Nkðx; yÞ isthe element shape function for the kth node.

Using the SIMP material model, the elasticity matrix at pointðx; yÞ is expressed in terms of material relative density qeðx; yÞE x; yð Þ ¼ qe x; yð Þ½ �pE0 ð3Þ

where E0 is the elasticity matrix of the isotropic solid elastic mate-rial, and p P 1 is a penalization exponent number.

3. Finite element analysis methods

In this section, the finite element model for structural analysesand the computation of buckling load factors using hybrid stresselement is briefly introduced.

3.1. Finite element model

When the nodal design variable is employed, the checkerboardpatterns can be avoided naturally. However, a ‘‘layering’’ or ‘‘is-landing’’ phenomenon of black and white regions in the designdomain may appear [24]. Deng et al. [26] showed that this problemcould be effectively avoided by replacing the conventionalfour-node displacement-based quadrilateral element with a hybridstress element. The same approach is taken in this study, and inthis section, the basic theory and formulation of the hybrid stresselement to be used will be summarized.

Pian and Sumihara [27] developed a four-node hybrid stressfinite element for homogeneous plane problems. Independent ele-ment stress and displacement fields are defined and can beexpressed as

r ¼ rx;ry; sxy� �T ¼ Ub ð4Þ

u ¼ ux;uy� �T ¼ Nd ð5Þ

where rx;ry and sxy are stress components, ux and uy are displace-ment components, U and N are interpolation matrices for elementstress and displacement fields, respectively, b is an element stressparameter vector, and d is the nodal displacement vector.

Based on the Hellinger–Reissner variational principle, the fol-lowing expressions for element stiffness matrix Ke and the stressparameter vector b can be derived

Ke ¼ GTe H�1

e Ge ð6Þ

b ¼ H�1e Ged ð7Þ

where matrices Ge and He are defined as

Ge ¼Z 1

�1

Z 1

�1UTB Jj jdndg ð8Þ

He ¼Z 1

�1

Z 1

�1UTS0U

Jj jqeðn;gÞ½ �pt0

dndg ð9Þ

144 X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152

where J is Jacobian matrix, Jj j is the determinant of J;B is the strain–displacement matrix, S0 is the inverse of elasticity matrix E0 , i.e.S0 ¼ E�1

0 , and t0 is the thickness of the structure.

3.2. Element geometric stiffness matrix

For a plane continuum structure, the element geometric matrixcan be expressed as [28]

KGe ¼Z 1

�1

Z 1

�1gT S 0

0 S

" #g Jj jdndg ð10Þ

where submatrix S and matrix g are defined as

S ¼rx sxy

sxy ry

� �ð11Þ

g ¼ C M1 M2 M3 M4½ � ð12Þ

in which C and Mi ði ¼ 1;2;3;4Þ are matrices defined below

C ¼ J�1 00 J�1

" #ð13Þ

Mi ¼@Ni@n

@Ni@g 0 0

0 0 @Ni@n

@Ni@g

24

35

T

ð14Þ

From Eqs. (4) and (7), the stress vector can be calculated as follows

re ¼ UH�1e Ged ð15Þ

Now, with a mapping function H : R3# R4�4 defined as

H

rx

ry

sxy

264

375

0B@

1CA :¼

rx sxy 0 0sxy ry 0 00 0 rx sxy

0 0 sxy ry

26664

37775 ð16Þ

the element geometric matrix in Eq. (10) can be rewritten as

KGe ¼Z 1

�1

Z 1

�1gTH reð Þg Jj jdndg ð17Þ

3.3. Determination of buckling load factors

The linear buckling load factors can be calculated from the fol-lowing equation

Kþ kjKG� �

wj ¼ 0 ð18Þ

where K is the global stiffness matrix of the structure, KG is the glo-bal geometric stiffness matrix, and kj and wj are the jth bucklingload factor and the corresponding buckling mode vector.

Normally the buckling modes are ordered according to the mag-nitudes of buckling load factors and k1 will be the smallest.

4. Sensitivity analysis

In this section, the sensitivities of compliance, buckling loadfactors and volume with respect to nodal design variables arederived.

4.1. Sensitivity of compliance

When the applied loads are independent of design variables, thesensitivity of structural compliance can be expressed as

@C@qi¼ �UT @K

@qiU ¼ �

XNe

e¼1

ueð ÞT@Ke

@qiue ð19Þ

where Ke and ue are the stiffness matrix and the displacement vec-tor of element e, respectively, Ne is the number of elements used todiscrete the design domain.

From Eq. (6), we get

@Ke

@qi¼ Ge

@H�1e

@qiGe ð20Þ

Then differentiating the identical equation HeH�1e ¼ I with respect

to qi , we can get

@H�1e

@qi¼ �H�1

e@He

@qiH�1

e ð21Þ

From Eq. (9), we can get the following expression

@He

@qi¼ �

Z 1

�1

Z 1

�1UTS0U

p � Jj jqeðn;gÞ½ �pþ1t0

@qeðn;gÞ@qi

dndg ð22Þ

where

@qeðn;gÞ@qi

¼0 if i is not adjacent to element e

Nkðn;gÞ if i is the kth node of element e

ð23Þ

Then using Eqs. (19)–(22), we can obtain the sensitivity of the com-pliance with respect to the nodal design variables.

4.2. Sensitivity of buckling load factors

Introducing an auxiliary variable jj ¼ �1=kj [29], Eq. (18) can berewritten as

KG q;Uð Þ � jjK qð Þ� �

wj ¼ 0 ð24Þ

Note that for the current optimization problem, both of the two glo-bal stiffness matrices are functions of the current design q and theglobal geometric stiffness matrix depends also on U, the nodal dis-placement vector for the given loading case.

If the eigenvalue is unimodal, the sensitivity of auxiliary vari-able jj with respect to variable qi can be expressed as [29]

@jj

@qi¼ ~wT

j@KG

@qi� 1

kj

@K@qi

�~wj � vT

j@K@qi

U ð25Þ

where ~wj is the eigenvector, and vj is the adjoint displacement vec-tor. Adjoint displacement vectors are determined by solving the fol-lowing equation

Kvj ¼ Pj ð26Þ

where

Pj ¼ ~wTj@KG

@U~wj ¼

~wTj@KG@u1

~wj

..

.

~wTj@KG@ud

~wj

8>>><>>>:

9>>>=>>>;

ð27Þ

in which d is the number of degrees of freedom of the structure.Note that the eigenvectors must satisfy the orthonormalization con-dition, i.e. ~wT

j K~wk ¼ djk; djk is Kronecker’s delta.By differentiating Eq. (17), we get

@KGe

@qi¼Z 1

�1

Z 1

�1gT @H reð Þ

@qig Jj jdndg ð28Þ

Since mapping H is linear, the derivatives of H reð Þ with respect toqi can be expressed as

W

H

F

Fig. 1. Portal frame.

X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152 145

@H reð Þ@qi

¼@H UH�1

e Geue

� @qi

¼ H U@H�1

e

@qiGeue

!ð29Þ

The sensitivity of the global geometrical stiffness matrix can then begiven as

@KG

@qi¼XNe

e¼1

@KGe

@qi¼XNe

e¼1

Z 1

�1

Z 1

�1gT @H reð Þ

@qig Jj jdndg

¼XNe

e¼1

Z 1

�1

Z 1

�1gTH U

@H�1e

@qiGeue

!g Jj jdndg ð30Þ

Using Eqs. (21) and (30), we can calculate @KG=@qi.Following the same procedure, the sensitivity of the elemental

geometrical stiffness matrix with respect to displacement compo-nent ujðj ¼ 1;2; . . . ; dÞ can be calculated

@KGe

@uj¼Z 1

�1

Z 1

�1gT @H reð Þ

@ujg Jj jdndg

¼Z 1

�1

Z 1

�1gTH UH�1

e Ge@ue

@uj

�g Jj jdndg

¼Z 1

�1

Z 1

�1gTH UH�1

e GeI�

g Jj jdndg ð31Þ

where I is an 8� 1 vector. If uj is not a nodal displacement of the ethelement, then I is a zero vector. Otherwise I is a unit vector and thekth component is one if the uj is the kth nodal displacement compo-nent of the eth element.

Once the sensitivities of the introduced auxiliary variablesjj ¼ �1=kj are obtained, we can use the chain rule to calculatethe sensitivities of eigenvalues kj as follows

@kj

@qi¼ @kj

@jj

@jj

@qi¼ 1

j2j

@jj

@qi¼ k2

j@jj

@qið32Þ

If the eigenvalue is multimodal with multiplicity m greater thanone, individual eigenvalues may no longer be differentiable func-tions of the design variables. In such situations, the method pro-posed by Gravesen et al. [30] can be used. If j1 ¼ j2 , we cancalculate sensitivities of functions j1 þ j2 and j1j2 using the fol-lowing expressions

@ j1 þ j2ð Þ@qi

¼ ~wT1@KG

@qi� 1

k1

@K@qi

�~w1 � vT

1@K@qi

U

þ ~wT2@KG

@qi� 1

k2

@K@qi

�~w2 � vT

2@K@qi

U ð33Þ

@ j1j2ð Þ@qi

¼ j2~wT

1@KG

@qi� 1

k1

@K@qi

�~w1 � vT

1@K@qi

U �

þ j1~wT

2@KG

@qi� 1

k2

@K@qi

�~w2 � vT

2@K@qi

U �

ð34Þ

Other methods for dealing with multiple eigenvalues can befound in the research literature [8,10,23,31].

4.3. Sensitivity of total material volume

The total volume of material used in the structure can be calcu-lated as

V ¼XNe

e¼1

Ve ¼ t0

XNe

e¼1

Z 1

�1

Z 1

�1qe n;gð Þ Jj jdndg ð35Þ

Its sensitivity can then be expressed as

@V@qi¼XNe

e¼1

@Ve

@qi¼ t0

XNe

e¼1

Z 1

�1

Z 1

�1

@qeðn;gÞ@qi

Jj jdndg ð36Þ

where @qeðn;gÞ=@qi is given in Eq. (23).

5. Methods for dealing with pseudo buckling modes

Pseudo buckling modes usually appear in low-density regionsduring the optimization process. In this section, some existingmethods for dealing with this problem are first investigatedthrough a simple example, and then a new approach is proposed.

5.1. A discussion on existing methods for dealing with pseudo bucklingmodes

For an investigation of existing methods for dealing with localpseudo buckling modes, the 2D portal frame structure shown inFig. 1 is considered. Its first buckling load factor is to be calculated.The frame with a thickness of 10 mm is clamped at the bottom, andis modeled with a regular 40 � 20 mesh of four-node plane stresselements. A downward concentrated force F is applied at the centeron the top edge. The outer frame with a thickness of 5 mm has fullydense material q ¼ 1:0 and the inner region has a densityq ¼ 0:001.

For a real structural design, the low-density region should beignored and only the buckling modes occurring in the solid partsor the regions with a high relative density value should be consid-ered. In order to distinguish these modes from the pseudo modes,we call them ‘‘real’’ buckling modes, and their correspondingeigenvalues ‘‘real’’ buckling load factors. Thus, the first real buck-ling load factor is calculated by using a model with elements ofthe low density excluded, and a buckling load factor of 2.41 isobtained. This is taken as a reference solution.

For most of the existing methods for suppressing pseudomodes, different schemes are used for choosing modulus for stiff-ness matrix ðEK

e Þ and geometric stiffness matrix ðEKGe Þ. In this study,

the three schemes in Table 1 are considered.The results are summarized in Table 2. Quite different results

have been obtained. Even with the same method, we may stillobtain very different results when different parameter values areused.

Method (a) uses a material model without penalization onintermediate density. For this problem, it give a result 156.3%higher than the reference solution. The significantly higher buck-ling load factor is due to the contribution of the inner region,which may act as a thin membrane. As the material modulusfor the low density region is not scaled down with a small factor,say using penalization exponent number p > 1, pseudo bucklingmodes can be avoided. However, this simple treatment may leadto optimized designs with many gray regions because no penal-ization is applied. The existence of these gray regions may leadto the over-estimation of buckling load factors, as shown in thisexample (see Zhou [20]). Therefore, much care should be takenwhen using this method.

Table 1Three methods used for calculating the first buckling load factor of the portal frame.

Method Scheme for modulus calculation Reference

(a) EKe ¼ EKG

e ¼ qeE0 Zhou [20]

(b)EK

e ¼ qpe E0; EKG

e ¼0 if qe < qlqp

e E0 if qe P ql

Neves et al. [8]

(c) EKe ¼ ql þ 1� qlð Þqp

e� �

E0; EKGe ¼ qp

e E0 Bendsøe and Sigmund [18]

Note: ql is a predefined parameter.

146 X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152

Although method (b) produces a similar result as the referencesolution, it may cause solution oscillations in an optimization pro-cess [18]. Besides parameter ql , this method requires an additionalparameter of critical normalized stress value rc. Geometrical stiff-ness matrices of low-density elements with normal stress less thanrc will be ignored in the optimization process [8,10]. Generallyspeaking, the critical normalized stress rc is difficult to predefine,and an improper value may result in different topology results [8].

Method (c) can avoid the discontinuity problem of method (b).However, as the introduced parameter ql plays a critical role inavoiding the appearance of pseudo modes, using an improper valuemay lead to erroneous results [19,20]. A large value of ql mayresult in unrealistically high stiffness of void elements as shownin this example (e.g. ql P 10�5). On the other hand, a small value

of ql , such as ql 6 10�9 for this example, is insufficient for avoidingthe pseudo buckling mode. Hence, the difficulty of this method isto choose a proper value ql to calculate accurate real buckling loadfactors. Although the lower bound of relative density (say, 0.001) isoften used for ql, such a value may still be too large and as a result,the structural stability can be overestimated.

On the other hand, if buckling load factors of the portal frameare calculated by using the standard SIMP model with p ¼ 3 andincluding all the elements in the low-density region, the first 148buckling modes obtained are pseudo modes with very small eigen-values and the 149th mode with a load factor of 2.41 is the first realmode. Therefore, it is important to find reliable methods for deal-ing with these pseudo modes.

As similar situations may occur for other topology optimizationproblems, the development of effective methods for dealing withpseudo modes is an important research topic. A new approach isproposed in the following sections.

5.2. Methods for eliminating the effects of pseudo modes

An investigation into the occurrence and characteristics ofpseudo buckling modes reveals that pseudo modes have the fol-lowing two important features:

Table 2Results for the first buckling load factor (p ¼ 3).

Method ql k1 Error (%)

(a) – 6.19 156.3

(b) P 10�3 2.41 0.0

10�4 0.99 �58.9

(c) 10�3 34.37 1323.8

10�4 6.56 171.9

10�5 2.73 13.3

10�6 2.45 1.3

10�7 2.42 0.1

10�8 2.41 0.0

10�9 1.98 �17.8

10�10 1.09 �54.8

(1) Some pseudo buckling modes have eigenvalues close to zero,or more accurately, much smaller than those of the realmodes. As there may be many such pseudo modes, a largenumber of eigenvalues have to be determined to includesome of the real modes.

(2) The deformation mainly occurs in low-density regions, andas a result, the modal strain energy in low-density regionsmakes a major contribution to the total modal strain energy.

Based on these two observations, a new strategy is proposed.

5.2.1. Calculation of candidate buckling modesIn linear buckling analysis, it is a common practice to calculate

only the first several eigenmodes to reduce the computational cost.If a model has many pseudo buckling modes, it will be necessary tocalculate a large number of eigenvalues. Otherwise, it is possiblethat all those calculated are pseudo modes, which are useless.Thus, the existence of pseudo modes may cause a dramatic costincrease in eigenvalue calculation.

However, if we know the range of real buckling load factors, thisissue could be easily resolved by applying the eigenvalue-shifttechnique [32]. By applying an appropriate shift value, reallow-order buckling modes can be calculated at a reasonable costas pseudo models with very small eigenvalues are excluded fromthe calculation.

When a uniform material distribution is used as the initialdesign, no pseudo buckling modes will appear at the early stageof the optimization, because there are no low-density regions. Inthis case, the range of real buckling load factors is easily available.When a non-uniform initial design is used, the range of real eigen-values can still be determined, for example, by using the methodproposed by Neves et al. [8]. One can also calculate a relativelylarge number of buckling load factors in the first iteration and findreal low-order buckling modes using the identification techniquepresented in Section 5.2.2. Therefore, it is always possible to deter-mine the real low-order buckling modes and calculate the corre-sponding load factors for the initial design.

Let kðk�1Þ be the first real buckling load factor for the last design.Assuming that the design will not change dramatically, we can usekðk�1Þ as the shift to calculate M buckling modes of a new, modifieddesign, and then pick the smallest eigenvalue of the real modes.Obviously, it is more likely that we can find the real first bucklingmode if a larger M is used. However, considering that it requiresmore computer resources and time to compute more eigenvaluesthan necessary, a smaller number is preferred. Based on theassumption that optimized designs change only moderatelybetween iterations, it is possible to choose a reasonably smallnumber for M. In this study, M ¼ 50 is used for all examples, whichmeans that only 50 eigenmodes are determined in the eigenvalueextraction.

Therefore, by applying an eigenvalue shift, possible low-orderpseudo modes can be excluded from the eigenvalue extractioncomputation.

5.2.2. Pseudo buckling modes identificationAside from low-order pseudo models with very small eigenval-

ues, other pseudo modes among the M eigenmodes may still exist.These high-order pseudo modes must be identified to avoid a situ-ation where a wrong constraint is introduced. We can make use ofthe second feature of pseudo buckling modes to determinewhether a mode is real or not. As low-density elements (regions)make the major contribution to the total modal strain energy ofa pseudo mode, we can divide the modal strain energy of a modeinto two parts, one from the low-density regions and the other

1. Define optimization model and choose optimization parameters:

2. Determine by FE analysis

3. Calculate the sensitivities of constraint functions and objective function

4. Update design variables using optimization solver (MMA)

Converged?

Start

End

No

Yes

0 , , ,MW , , , ,ll CV p Mρλ ρ ε ε

5. Determine by FE analysis, , , ), (j j C Vλ ψ U ρ�

, , , ), (j j C Vλ ψ U ρ�

Fig. 2. Flow chart of the iterative solution procedure.

Fig. 3. 2-D continuum structure under distributed load (a) problem description and(b) topology from compliance minimization subjected to volume constraints.

X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152 147

from the rest of the structure. We can then identify this modebased on the energy contributions of these two parts.

First, a threshold parameter ql is introduced, and all of thenodes can be divided into two sets based on nodal relative densityvalues, i.e. nodes with low densities Nl ¼ i qi 6 ql; 1 6 i 6 Njf g andnodes with high densities Nh ¼ i qi > ql; 1 6 i 6 Njf g.

Then, the degrees of freedom can be divided into two groupsbased on the grouping of nodes [33], and a buckling mode vectorcan be decomposed into two vectors:

Wj ¼ Wlj þWhj ð37Þ

where Wlj is composed of displacement components for degrees offreedom of nodes in set Nl and zero elements, and Whj is composedof the displacement components for degrees of freedom of nodes inset Nh and zero elements.

Hence, the modal strain energy ratio of low-density regions rlj,

defined as the ratio of contribution from the nodes in Nl to the totalmodal strain energy, is given by

rlj ¼

WTljKWlj þWT

ljKWhj

WTj KWj

¼WT

ljKWj

WTj KWj

ð38Þ

The pseudo buckling mode identification criterion can be stated as

rlj P MWl ð39Þ

where MWl is a predefined parameter with a value between 0 and1. If the modal strain energy ratio of low-density regions is biggerthan the value, the corresponding mode is regarded as pseudomode; otherwise, the mode is treated as real.

A region with a relative density less than ql is usually regardedbeing of low-density, and all numerical tests conducted in thisstudy have shown that the threshold value ql ¼ 0:1 is appropriate.For a clear black-white design in which the design variables areequal to either one or the lower bound q, the modal strain energyratio for a pseudo mode can be bigger than 0.98, as demonstratedby the example in Section 5.1. During the optimization iteration, itis very likely that a topology design contains gray regions. For apseudo mode, deformation in these regions may be small but isnot zero, causing a reduction in the modal strain energy ratio thatmay have an effect on a pseudo mode check. In order to enhancethe reliability of the mode identification results, it is necessary todecrease the value of MWl . In this study, the selected parametervalues ql ¼ 0:1 and MWl 2 ½0:6;0:7� are used and prove to beappropriate for all of the conducted numerical tests.

The proposed identification method is used on the 150 calcu-lated buckling modes of the example structure in Section 5.1,and all the modes are correctly identified. The pseudo mode iden-tification is an important component of the optimization algorithmproposed in the next section, and numerical experiments showthat the identification method is very effective and reliable.

6. Optimization algorithm

Based on the finite element analysis and sensitivity analysis,topology optimization problem (see Eq. (1)) can now be solvedby using a gradient-based optimization algorithm. Optimizationalgorithms based on the well-known MMA optimization solver[34] are developed. Different optimization strategies are also pro-posed for achieving high-quality local solutions.

6.1. Computational procedure

Compared with the conventional problem of compliance mini-mization under material volume constraint, the optimizationmodel in Eq. (1) has an additional buckling constraint on the

smallest buckling load factor. It is both simple and natural to justadd this new constraint equation and then use an existing opti-mization algorithm to solve the new problem. Based on this idea,the iterative solution procedure in Fig. 2 is proposed for the solu-tion of the topology optimization problem in Eq. (1). To start with,the design domains are discretized and the optimization parame-ters are defined. In the FE analysis of the initial design in Step 2and the current design in Step 5, linear static responses, the ‘real’buckling load factors kj and corresponding mode vectors ~wj, thestructural compliance C and the volume of material used in thedesign V are all determined. In Step 3, the sensitivities of the object

148 X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152

and constraint functions with respect to the nodal design variablesare computed. In addition, we will check the multiplicity of thefirst eigenvalue. If the multiplicity m is bigger than one, we willuse multimodality formulation to calculate its sensitivities. InStep 4, the design is modified by using optimization solver MMA.To check the solution convergence, the following criteria are used:

(a) the maximum change in the design variables between twoconsecutive iterations is smaller than a predefined toleranceeq , i.e. qðkþ1Þ � qðkÞ

�� ��1 6 eq;

(b) the change in the objective function between two consecu-tive iterations is smaller than a predefined tolerance eC , i.e.

Table 3Optimized topologies for different buckling constraints.

Algorithm k ¼ 0:6 k ¼ 0:8 k ¼ 1:0

A

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 150

5

10

15

20

25

30

35

40

B

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 150

5

10

15

20

25

30

35

40

C

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 150

5

10

15

20

25

30

35

40

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.66.8

7.0

7.2

7.4

7.6

7.8

8.0

8.28.05

8.00

7.08

7.70

7.50

7.46

7.17

7.21

7.387.32

7.28

7.25

Com

plia

nce

Lower bound of buckling load factor

Algorithm AAlgorithm BAlgorithm C

x10-2

7.02

Fig. 4. Compliance vs the lower bound of buckling load factor.

Cðkþ1Þ � CðkÞ��� ���=CðkÞ 6 eC;

(c) all of the constraint conditions are satisfied.

6.2. Improved optimization strategies

While the algorithm presented earlier is applicable to the buck-ling optimization problem, as will be shown through numericalexamples in the next section, it does not always work well. Asthere is usually a confliction between the requirements for struc-tural stiffness and stability, the buckling constraint should not betreated just like the one in the material volume, implying thatrefined algorithms are required to achieve more optimized designs.It is proposed that the optimization process is separated into twophases and at each phase, different optimization models and/ormaterial models are employed. In the present paper, two improvedalgorithms are given and compared with the simple one phasealgorithm shown in Fig. 2. The initial value for nodal design vari-ables in all of the three algorithms is uniform with the volume frac-tion. The three algorithms investigated are as follows:

(a) Algorithm A: solve optimization problem (Eq. (1)) by follow-ing the procedure shown in Fig. 2 with no changes.

(b) Algorithm B: separate the optimization process into twophases and use different optimization models. In the firstphase, solve the conventional problem of compliance mini-mization under the volume constraint by ignoring the buck-ling constraints. In the second phase, solve the originalproblem (Eq. (1)) with the solution from the first phase asthe initial design.

k ¼ 1:2 k ¼ 1:4 k ¼ 1:6

20 0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

20 0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

20 0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

0 5 10 15 200

5

10

15

20

25

30

35

40

X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152 149

(c) Algorithm C: separate the optimization process into twophases and use different material penalization for calculat-ing buckling load factors. In the first phase, use normalmaterial penalization for calculating compliance and its sen-sitivity, but do not penalize the material modulus for calcu-lating the buckling load factor (i.e. use p ¼ 1). In the secondphase, use the normal material penalization to solve theoriginal problem (Eq. (1)) with the solution from the firstphase as the initial design.

It is noteworthy that, for both algorithms B and C, a ‘pre-solve’phase is added and the second phase is the same as the solution ofthe original problem by using algorithm A, but with a different ini-tial design from the first phase. In the first phase, the buckling con-straints are either ignored as in algorithm B or considered but in amodified form as in algorithm C. In contrast, the initial design foralgorithm A is a uniform distribution in the design domain.

The numerical examples in the next section will show that byseparating the optimization procedure into two phases, the solu-tion may converge in fewer iterations and better performance ofthe optimized structures can be achieved.

7. Numerical examples

Two examples are considered in this section and the designsobtained by using the three algorithms are compared.

a

b

c

Fig. 5. Iteration history of compliance and first buckling load factor with k ¼ 1:0 forthree algorithms. (a) Algorithm A; (b) Algorithm B; and (c) Algorithm C.

7.1. Example 1

The first example is the optimization of a 2D continuumcolumn-like structure under distributed loads. The load and sup-port conditions are shown in Fig. 3(a). The design domain is a rect-angular area of unit thickness with height H = 40 and width W = 20.A distributed load q = 0.05 is applied at the top edge with a widthd = 2/3. The design domain is discretized into 60 � 120equally-sized square four-node elements. The material constantsused are Young’s modulus E ¼ 1:0 and Poisson’s ratio m ¼ 0:3.The prescribed material volume fraction number is set to 0.35.

The initial value of all nodal design variables are set to the vol-ume fraction. The optimized design of a conventional minimumcompliance problem is shown in Fig. 3(b) and its correspondingfirst buckling load factor is 0.45. When considering buckling con-straints, the parameters for convergence criterion are ec ¼ 0:01and eq ¼ 0:001 and the move limit on design variables ismq ¼ 0:003.

This optimization problem has been solved using the threealgorithms for different lower bounds of buckling load factors.The curves of compliance versus the lower bound of buckling loadfactor are presented in Fig. 4, while the obtained topologies areshown in Table 3. It can be seen that the compliance increases asthe lower bound of buckling load factor increases. This means thatwith a fixed amount of material, the improvement of structuralstability can be achieved only by a reduction in structural stiffness.A comparison of the designs obtained with buckling constraintsand those obtained with the volume constraint only reveals thatthe requirement on the structural stability tends to distribute theavailable material over a larger area. In contrast, the maximizationof structural stiffness causes the material to distribute along theload transfer path.

From Fig. 4, it is found that the compliance values for the threesolutions are very close for all of the considered lower bound val-ues. However, the topologies obtained have some minor differ-ences. This indicates that very likely, the solutions are just localoptima, and for this particular problem, different local solutionshave similar topologies and compliance values.

The iteration history curves in Fig. 5 show that algorithm Arequires considerably more iterations than algorithms B and Cfor convergence. From the topological changes shown in Fig. 6, itcan be found that with algorithm A, modulus penalization makesthe buckling constraint harder to be satisfied and causes the mate-rial to distribute over a much larger region than required at theearly stage of the optimization. At the later stage, the optimizationalgorithm will guide the design to change and produce betterdesigns. However, the intermediate designs could be so differentfrom the final design that the solution requires a large number ofiterations to converge. In Fig. 5(b), we can see that in iterations167–480, the objective function increases first and then decreasesuntil the solution terminates. This is because different constraintsare considered in the two phases of algorithm B. In the first phase,the buckling constraint is not considered at all, thus the optimizedtopology structure has high stiffness but the buckling constraintsmay not be satisfied. In the second phase, the optimization algo-rithm will steer the design to improve structural stability andmay cause a reduction in stiffness. However, once the bucklingconstraints are satisfied, compliance will begin to decrease as opti-mization proceeds. Fig. 5(c) shows a drop in first buckling load fac-tor at iteration 300. This sudden change is due to the switch of thesolution phase. From this iteration onwards, the normal materialpenalization will be employed for buckling analysis. It can be seenfrom Fig. 6 that for algorithm B and C, at the first stage, the compli-ance of topological designs is relatively small and material is

Iter=200 2

1 5.8 018,00.1 Cλ −= = ×Iter=400

21 0.8 017,00.1 Cλ −= = ×

Iter=600 2

1 8.7 016,00.1 Cλ −= = ×

Iter=800 2

1 5.7 017,00.1 Cλ −= = ×Iter=1000

21 3.7 018,00.1 Cλ −= = ×

Iter=1200 2

1 3.7 012,00.1 Cλ −= = ×

(a) Topological changes during optimization for Algorithm A

Iter=250 2

1 4.7 019,00.1 Cλ −= = ×Iter=350

21 3.7 013,00.1 Cλ −= = ×

Iter=450 2

1 3.7 012,00.1 Cλ −= = ×(b) Topological changes during optimization for Algorithm B

Iter=300 2

1 4.7 017,16.0 Cλ −= = ×Iter=500

21 1.8 017,58.0 Cλ −= = ×

Iter=7002

1 5.7 015,00.1 Cλ −= = ×

(c) Topological changes during optimization for Algorithm C

Fig. 6. Topological changes during optimization with a low bound of buckling loadfactor of k ¼ 1:0.

Table 4The first three buckling modes of optimized designs at k ¼ 1:0.

Algorithm A B C

1st mode

2nd mode

3rd mode

150 X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152

mainly distributed along the load transfer path with a relativelylow stability. At the later stage, some material is moved away fromthe load transfer path to improve structural stability.

The first three buckling modes of optimized structures fork ¼ 1:0 are shown in Table 4. The first real buckling load factorsfor the three algorithms are all bimodal. This validates that themultimode sensitivity analysis methods presented at the end ofSection 4.2 is effective.

7.2. Example 2

The short cantilever beam shown in Fig. 7(a) is considered. Thedesign domain is a rectangular area of unit thickness with heightH = 2 and width W = 1. A concentrated load F ¼ 0:005 is appliedto the center of the right edge. The design domain is discretizedinto 40 � 80 equally-sized square four-node elements. The mate-rial constants used are Young’s modulus E ¼ 1:0 and Poisson’s ratiom ¼ 0:3. The prescribed material volume fraction is set to 0.15.

The initial value of all nodal design variables are set to the vol-ume fraction. The optimized design of a conventional minimumcompliance problem is shown in Fig. 7(b) and the correspondingfirst buckling load factor is equal to 0.22.

The optimized designs obtained by employing three algorithmsfor different lower bounds of buckling load factors are shown inTable 5. It can be clearly seen that as the lower bound of bucklingload factor increases, the structural member in compressionbecomes shorter and wider, resulting in stability improvement.At the same time, some gray regions appear. The appearance ofgray region means that it is impossible to obtain a clearblack-white design to satisfy the constraint.

Table 5Optimized topology configuration subjected to different constraints using threealgorithms.

Algorithm k ¼ 0:4 k ¼ 0:6 k ¼ 0:8 k ¼ 1:0

A

B

C

Lower bound of buckling load factor0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4.55.05.5

6.06.57.07.5

8.08.59.0

6.246.01

5.585.45

5.84

5.42

5.64

5.27

5.48

5.25

5.31

5.02

5.074.974.84

4.80

8.53

7.047.026.786.76

7.95

5.96

Com

plia

nce

Algorithm AAlgorithm BAlgorithm C

x10-4

5.16

Fig. 8. Compliance vs the lower bound of buckling load using three algorithms.W

H

F

(a) (b) Fig. 7. Short cantilever beam under concentrated load (a) problem description and(b) topology of compliance minimization subjected to volume constraint.

X. Gao, H. Ma / Computers and Structures 157 (2015) 142–152 151

Compliance values for different optimized designs are shown inFig. 8. From these results, one can see that the stiffness of the opti-mized structure reduces in order to meet a higher stability require-ment. This is the same as the observations from the last example.On the other hand, the results obtained by using different methodsare quite different in the example. It is easily seen that, for all thelower bounds of buckling load factors considered, the two-phaseoptimization algorithms B and C produce designs with much smal-ler compliance values. This is a significant improvement due to theintroduction of a first phase in the optimization algorithm. It canalso be seen that algorithms B and C produce similar designs onlywhen k is less than 0.6. When k is bigger than 0.6, algorithms C pro-duce better designs. These observations clearly show that the threealgorithms may produce significantly different solutions. For thisproblem, the proposed two-phase algorithms are better than the

one-phase algorithm and algorithm C is the best of the three. Inaddition, this example again validates the effectiveness of thenew approach combining pseudo buckling mode identificationand eigenvalue shift in dealing with pseudo buckling modes.

8. Conclusions

The compliance minimization problem under volume and sta-bility constraints is considered. A new approach has been proposedfor dealing with the well-known pseudo buckling mode problem.In addition, in consideration of the non-linear and non-convex nat-ure of the problem, two-phase algorithms are suggested for achiev-ing better local optimization solutions. Numerical examples arepresented to show the effectiveness of the new algorithms.

It should be pointed out that the proposed approach for dealingwith the pseudo buckling mode problem can be applied to vibra-tion optimization problems [35]. Further, the two-phase optimiza-tion strategies are effective for improving the performance ofoptimized designs. The same idea could be useful for other engi-neering optimization problems, which may also be highlynon-linear and non-convex.

Acknowledgement

The authors would like to acknowledge the support from StateKey Laboratory of Structural Analysis for Industrial Equipment,Dalian University of Technology, Dalian 116024, P.R. China(Project No. GZ1305).

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