A MAC based excitation frequency increasing method forstructural topology optimization under harmonic excitations

Tao Liu1, Ji-Hong Zhu1,*, Fei He1, Hua Zhao1, Qinlin Liu2, and Chong Yang2

1 Laboratory of Engineering Simulation & Aerospace Computing-ESAC, Northwestern Polytechnical University, 710072 Xian,Shaanxi, China

2 China Academy of Engineering Physics, 621900 Mianyang, Sichuan, China

Received 2 September 2016 / Accepted 15 November 2016

Abstract – This work is focused on the topology optimization of structures that are subjected to harmonic forceexcitation with prescribed frequency and amplitude. As an important objective of such a design problem, the naturalresonance frequency of the structure is driven far away from the prescribed excitation frequency for the purpose ofavoiding resonance and reducing the vibration level. Therefore when the excitation frequency is higher than thenatural resonance frequency of the structure, the natural resonance frequency will decrease, then the optimumtopology configuration will be distorted with large amount of gray elements. A MAC (Modal Assurance Criteria)based excitation frequency increasing method is proposed to obtain a desired configuration. MAC is adopted hereto track the natural resonance frequency which can provide the baseline reference for the current excitation frequencyduring the optimum iterative procedure. Then the excitation frequency increases progressively up to its originallyprescribed value. By means of numerical examples, the proposed formulation can generate effective topology config-urations which can avoid resonance.

Key words: Topology optimization, Harmonic response, Resonant mode shape, Distorted configuration, MAC.

1 Introduction

Topology optimization has been recognized as an effectiveapproach to figure out the structure layout during theconceptual design phase since the original idea of homogeniza-tion-based design method was proposed [1]. Up to now,topology optimization has received remarkable success in boththeoretical studies and practical applications [2–4], wheretopology optimization of continuum structures under dynamicloading is one challenge. Correlative researches in the fieldwere mainly focused on two types of problems: one is topologyoptimization related to dynamic characteristics, the other isrelated to dynamic responses.

Natural frequencies and mode shapes are two majoroptimization objects in topology optimization related todynamic characteristics. It follows some general objectives:The first objective is maximization of the specified structuraleigenvalues such as fundamental or high-order natural frequen-cies [5]; secondly the gaps between the specified eigenvaluesand frequencies must be maximized [6]. The final objective

is to optimize a structure to obtain prescribed desired naturalfrequencies and mode shapes [7].

Another type of problems is to minimize the dynamicresponses such as displacement, velocity and accelerationamplitude or dynamic compliance in topology optimizationrelated to dynamic responses. This paper mainly deals withthe second case where the displacement amplitude underharmonic loads is involved. Physically, minimization ofdynamic responses under harmonic loads is of great impor-tance since harmonic vibration may be easily caused byrotating or reciprocating components. Besides, complicatedperiodic excitations can be transformed into a set of harmonicexcitations by use of the Fourier series.

Relevant researches have already be carried out abouttopology optimization related to dynamic responses.Ma et al. [8] minimized structure dynamic compliance underharmonic loads using the homogenization method. Shu et al.[9] studied the topology optimization for minimizingfrequency response using level set method. Yoon [10]conducted a comprehensive investigation for three modelreduction schemes namely mode displacement method(MDM), Ritz vector method and quasi-static Ritz vectormethod in topology optimization under harmonic loads.*e-mail: Jh.zhu@nwpu.edu.cn

Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4� T. Liu et al., Published by EDP Sciences, 2017DOI: 10.1051/smdo/2016012

Available online at:www.ijsmdo.org

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

OPEN ACCESSRESEARCH ARTICLE

Xiang et al. [11] introduced a differentiable q-norm form ofresponse function’s peak value in dynamics response in trusssizing and shape optimization under bandwidth frequencyexcitation. Yang and Li [12] adopted mode tracking techniqueto minimize structural dynamic compliance at resonancefrequencies in thermal environments. Liu et al. [13] made acomparative study about the accuracy and efficiency ofharmonic responses calculating using mode displacementmethod, mode acceleration method and full method.

Although many considerable efforts have been made aboutthe structure topology optimization under harmonic loads, theproblem of distorted configuration with high-frequencyexcitation has not been well solved yet, as well as the boundaryof low-frequency and high-frequency is not clear. Jog [14]presented that the topology optimization for minimizingdisplacement amplitude at a user-defined point in the structuremay result in distorted configurations with large amount ofintermediate density elements. Olhoff and Du [15] minimizedthe dynamic compliance of structure subject to harmonicloads, optimization results show that when the natural reso-nance frequency of the initial structure is less than the givenexcitation frequency, the natural resonance frequency willdecrease and the static compliance of the structure willincrease very quickly. Then he started out the design with asmall value of excitation frequency and sequentially increasedthe value up to its prescribed value. It works well when thenatural resonance frequency of the structure is always the sameas the fundamental eigenfrequency. But unfortunately, thenatural resonance frequency of most structures are not thesame, and mode switching may occur during the optimizationprocedure which makes it more difficult.

This work focuses on the topology optimization with theexcitation frequency higher than the natural resonancefrequency. In order to obtain a clear anti-resonance configura-tion, a MAC based excitation frequency increasing method isproposed. In this way, the limit value of the natural resonancefrequency can be estimated to distinguish low-frequency andhigh-frequency.

2 Optimization problem

2.1 Harmonic response analysis

As is known, the dynamic equilibrium equation of adiscretized n-DOF (degree of freedom) structure underharmonic loads can be written as:

M€uðtÞ þ C _uðtÞ þKuðtÞ ¼ PðtÞ ð1Þwhere M, C, K are the mass matrix, damping matrix andstiffness matrix respectively. u(t) denotes the displacementvector, P(t) represents the harmonic external loading vectorwhich can be expressed as PðtÞ ¼ Pejxptðj2 ¼ �1Þ, P andxp denote the magnitude vector of harmonic force and thegiven excitation frequency.

To implement the mode superposition method, the eigen-frequencies and eigenvectors are firstly obtained by solvingthe corresponding equation of free vibration as follows:

M€uðtÞ þKuðtÞ ¼ 0 ð2Þ

The ith circular eigenfrequency xi and eigenvector Ui canbe obtained by solving the free vibration system characteristicequation:

K� x2i M

� �Ui ¼ 0 ð3Þ

The mode shape matrix U = [U1, U2,. . . Un] is normal-ized by mass matrix, so the following relations hold:

UT MU ¼ I

UT KU ¼ diag x2i

� �

UT CU ¼ diag 2nixið Þ

8><

>:ð4Þ

where ni denotes the ith damping ratio, Rayleigh dampingleads to the following relation between damping ratio andfrequency:

ni ¼aþ bx2

i

2xið5Þ

a and b are two Rayleigh damping factors.By coordinating transformation,

uðtÞ ¼ UyðtÞ ð6Þwhere y(t) is the vector of generalized coordinates. By substi-tuting equation (6) into equation (7) and premultiplying UT,we obtain a set of n uncoupled equations of motion.

x2i yiðtÞ þ 2nixi _yiðtÞ þ €yiðtÞ ¼ UT

i PðtÞ ð7Þ

The solution of the above equation can be expressed as:

yiðtÞ ¼ ðx2i þ 2jnixixp � x2

p�1UT

i PðtÞ ð8Þ

Thus, the displacement response can be obtained bysubstituting equation (8) into equation (6):

u tð Þ ¼Xn

i¼1

Uiðx2i þ 2jnixixp � x2

p�1UT

i PðtÞ ð9Þ

As for the above displacement response expression, if all nmodes are taken into account, the result would the exactsolution. However, considering the computing efficiency, onlylower l modes are used to calculate the displacement responseapproximately, which is called mode displacement method(MDM).

u tð Þ ¼Xl

i¼1

Uiyi tð Þ l << n ð10Þ

When it comes to large-scale problems, a small l willgenerate big truncation errors, which may lead to an undesir-able result. While, the mode acceleration method (MAM) isable to reduce errors by including the effects of the truncatedmodes using a static analysis [16].

The solution of the uncoupled motion equations inequation (7) can be rewritten as

yi tð Þ ¼ UTi P tð Þxi

2� 2ni _yi tð Þ

xi� €yi tð Þ

xi2

ð11Þ

2 T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4

The use of equation (11) in equation (10) results in

u tð Þ ¼Xl

i¼1

UiUTi P tð Þ

xi2� 2Uini _yi tð Þ

xi�Ui€yi tð Þ

xi2

� �ð12Þ

It is noted that, the inverse of the stiffness matrix can berepresented by using all eigenmodes [17]:

K�1 ¼Xn

i¼1

UiUTi

x2i

ð13Þ

Notice that in MDM, just lower l modes are used for all thethree terms in equation (12). But in MAM, we can obtain thefirst part exactly by solving a corresponding static problem andusing equation (13) to include all n modes. As for the secondand third parts in equation (12), it can be expressed as equation(14) according to equation (11):

Xl

i¼1

� 2Uini _yi tð Þxi

�Ui€yi tð Þxi

2

� �

¼Xl

i¼1

Uiyi tð Þ �UiUTi P tð Þ

xi2

� �ð14Þ

Hence, the displacement response approximated by MAMcan be obtained through substituting equation (13) intoequation (12) and combining with equation (14):

u tð Þ ¼ K�1P tð Þ þXl

i¼1

Uiyi tð Þ �UiUTi P tð Þ

xi2

� �ð15Þ

An equivalent form of equation (15) can be obtained byusing equation (13) in equation (15):

u tð Þ ¼Xl

i¼1

Uiyi tð Þ þXn

i¼lþ1

UiUTi P tð Þ

xi2

� �ð16Þ

Comparing between equation (10) and equation (16), thesecond term of equation (16) related to MAM takes the effectsof higher order modes into account to some degree, so that thecomputing accuracy is improved apparently. Besselink et al.[17] made comparisons between MDM and MAM, he foundthat MAM outperformed the MDM in all cases. Hence, weadopt the MAM to calculate the displacement response.

2.2 Mode tracking

The MAC (Modal Assurance Criteria) has been usedintensively in experimental modal analysis to measure therelevance between tested modes and calculated modes. It isadopted here to track the resonance mode successfully eventhough Mode switching occurs during the optimizationprocedure. The definition of MAC is (see [18]):

MAC ¼ ðUTref �Ucur;iÞ2

ðUTref �Uref ÞðUT

cur;i �Ucur;iÞð17Þ

In equation (17), Uref denotes the reference eigenvectorwhich is the resonance mode shape of the initial structure;Ucur,i refers to the ith eigenvector of the structure incurrent iteration step. The value of MAC varies from 0 to 1,when the two eigenvectors are orthogonal to each other, itequals 0; when it equals 1, the two vectors represent exactlythe same mode shape. Generally the two modes cannot bethe same, and the mode with the highest value of MAC isidentified as the resonance mode shape, and the correspondingfrequency is the natural resonance frequency in currentiteration step.

3 MAC based structural topology optimizationunder harmonic loads

3.1 Topology optimization formulation

The topology optimization problem for minimizingthe displacement amplitude of concerned DOFs can bestated as:

Find : g ¼ ghjh ¼ 1; 2; � � � ;N df gmin : jjusðtÞjj

s:t: :

M€u tð Þ þ C _u tð Þ þKu tð Þ ¼ P tð ÞV � V U

g0 � gi < 1

8><

>:

ð18Þ

where gh is pseudo-density variable of element h, Nd

denotes the number of the design elements; g0 is thelower bound of the pseudo-density variables, which is usedto prevent the mass, stiffness and damping matrices frombecoming singular, here g0 = 0.001; ||us(t)|| denotes thedisplacement amplitude of concerned DOFs. V and VU

are the solid volume fraction and its upper bound,respectively.

It is known that in topological optimization of contin-uum structures subjected to dynamic or inertial loads,the interpolation scheme of Solid Isotropic Microstructurewith Penalization (SIMP) would lead to localized modesbecause of the mismatch between element stiffness andmass when pseudo-density variable gi takes a small value.In fact, a variety of interpolation schemes have beenproposed to eliminate the localized modes in some degree,such as Rational Approximation of Material Properties(RAMP) (Stolpe and Svanberg [19]), and PolynomialInterpolation Scheme (PIS) (Zhu et al. [20]). PIS isadopted here.

mh ¼ ghmh0

kh ¼15g5

hþgh

16 kh0

(

ð19Þ

where, mh and kh denote the mass matrix and stiffness matrixof element h respectively. mh0 and kh0 are the mass andstiffness matrix of a corresponding element h with fully solidmaterial.

T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4 3

3.2 Sensitivity analysis

Substitution of equation (8) into equation (15) gives

u tð Þ ¼ K�1P tð Þ

þXl

i¼1

Ui ðx2i � x2

p þ 2jnixixpÞ � x�2h i

UTi P tð Þ

ð20Þ

Suppose

Zi ¼ x2i � x2

p þ 2jnixixp

� ��1

� x�2i ð21Þ

Thus, the displacements can be express as

u tð Þ ¼ K�1P tð Þ þXl

i¼1

UiZiUTi P tð Þ ð22Þ

The sensitivity of displacement response can be written as

ou tð Þogh

¼(

o K�1P� �

oghþXl

i¼1

oUi

oghZiU

Ti

þUioZi

oghUT

i þUiZioUT

i

ogh

!

P

)

ejxpt ð23Þ

where, P represents the design independent externalharmonic loads, K�1P is the static displacement of thestructure, which can be obtain by implementing an additionalstatic analysis, its derivatives can obtained by means ofadjoint method [21]. As for the second term, it requires thesensitivities of eigenfrequencies and eigenvectors.

Differentiating the free vibration system characteristicequation in equation (3) with respect to the design variablegh and premultiplying UT

i yields

UTi

oKogh�M

ox2i

ogh� x2

i

oMogh

� �Ui

þ UTi K� x2

i M� �oUi

ogh¼ 0 ð24Þ

Since the mass matrix and stiffness matrix are symmetric,the second part equals zero according to equation (3). Thus, wecan obtain the sensitivity of eigenfrequency:

oxi

ogh¼

UTi

oKogh� x2

ioMogh

� �Ui

2xið25Þ

The derivatives of the eigenvectors can be written asfollows [22]

oUi

ogh¼Xl

r¼1

birUr ð26Þ

where bir is calculated as follows:

bir ¼UT

roKogh�x2

ioMogh

� �Ui

x2i �x2

rr 6¼ i

� 12 UT

ioMogh

Ui r ¼ i

8><

>:ð27Þ

The displacement amplitude of concerned DOFs instructure with damping yields

jjus tð Þjj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRe ðX sÞ2 þ Im ðX sÞ2

qð28Þ

where Xs denotes the complex displacement. Then, we canderive the sensitivity of displacement amplitude throughthe chain rule:

ojjus tð Þjjogh

¼

Re X sð Þ � ReoX S

ogh

� �

þ Im X Sð Þ � Im oX S

ogh

� �!

jjuS tð Þjj�1 ð29Þ

3.3 Distorted configuration

A 3D cantilever beam is designed to illustrate thephenomenon of distorted configuration under high excitationfrequency. The structure has a size of 0.8 m · 0.4 m ·0.04 mm which is clamped at the left side, as shown inFigure 1. The design domain is meshed with 80 · 40 · 4 solidelements. A harmonic force is applied at the middle node ofthe edge in the lower right corner with the amplitude of5000 N. The parameters of material properties are as follows:E0 = 2.1 · 1011 Pa, l = 0.3, q0 = 7800 Kg/m3, Rayleighdamping factors a = 10�2 b = 10�5. The optimizationobjective is to minimize the vertical displacement amplitudeat the loading position and the volume fraction is constrainedto be less than 50%.

At first, we set the initial value of all variables as 0.5, thenmodal analysis is implemented to obtain the resonantfrequency and reference eigenvector. The first four modeshapes and its national frequencies are shown in Figure 2,where fni is the ith national frequency. As for the first three

Figure 1. 3D cantilever beam subjected to a harmonic force.

4 T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4

mode shapes, they vibrate out of XY plane so that those naturalmode will not be a resonance one. On the contrary, the fourthmode shape swings along the vertical direction, which makesmost contributions to the target vertical displacementamplitude. As a conclusion, the fourth mode shape is the firstresonance mode, corresponding national frequency fn4 is theinitial resonance frequency fr0, which can be also identifythrough frequency response analysis (Figure 3).

Two harmonic forces with different level of givenexcitation frequency namely fp = 150 Hz < fr0 and fp =160 Hz > fr0 are applied on the structure separately. The twooptimized structures and their iteration history are presented

in Figure 4. As we can see in Figure 4c, when the excitationfrequency is less than the natural resonance frequency,minimization of the displacement amplitude drives the naturalfrequency away from the resonance point by increasing lownational frequencies especially the natural resonancefrequency, so that its static displacement ||us||(fP = 0) decreases,which means that the structure becomes stiffer. On thecontrary, high excitation frequency results in decreasing oflow national frequencies especially the natural resonancefrequency to drive the natural resonance frequency away fromthe resonance point (Figure 4d). As a consequence, the opti-mization objective decreases a lot, but the static displacementincreases, which means a weaker structure is obtained. It isknown that a typical stiffness design is to find the optimallay-out of a structure with the purpose of maximizing theperformance of limit material. So that all materials distributeamong the loading path, finally a clear and strong structureconfiguration is obtained such as Figure 4a, since its stiffnessincreases also. In contrast, because of the decrease of stiffnessunder high excitation frequency, optimization cannot motivatethe material to distribute among the loading path, thereforeit failed to generate a clear configuration (Figure 4b). Similardistorted configurations can be found in references [14, 15].

3.4 MAC based excitation frequency increasingmethod

As is mentioned above, in order to obtain a clear configu-ration, it is critical to avoid the decrease of stiffness duringoptimization procedure. Because of the relation between themovement of natural resonance frequency and excitation

Figure 3. Displacement amplitude of initial structure.

(a) (b)

(c) (d)

Figure 2. The first four mode shapes and natural frequencies of initial structure. (a) First mode shape, fn1 = 18.6 Hz. (b) Second modeshape, fn2 = 77.3 Hz. (c) Third mode shape, fn3 = 115.0 Hz. (d) Fourth mode shape, fn4 = 156.4 Hz.

T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4 5

frequency, it is important to guarantee the value of excitationfrequency below the natural resonance frequency.

The implementation of MAC based excitation frequencyincreasing method can be stated as follows:

Step 1, harmonic response analysis and modal analysis areimplemented to determine the resonance frequency and

extract the resonance mode as reference eigenvector withinitial structure.Step 2, MAC of each mode shape is evaluated to identifythe resonance mode with the highest value of MAC, andthe corresponding frequency is the natural resonancefrequency fri in ith iteration step.Step 3, the excitation frequency fei in ith iteration step canbe determined according to the value of fri: if the naturalresonance frequency is equal or lesser than the prescribedfrequency plus a little increment namely fri � fp + Df (f

0ri

in Figure 5), then fei = fri � Df; otherwise fri > fp + Df(f 00ri in Figure 5), then fei = fp.Finally, fei increases along with the raising of the naturalresonance frequency fri to the prescribed value. The com-pleted optimization strategy is represented in Figure 6.

Since the excitation frequency is always lower than the nat-ural resonance frequency during the optimization procedure,the final optimized configuration is fine without distortion.

4. Numerical examples

The same 3D cantilever beam (Figure 1) under excitationfrequency fp = 480 Hz > fr0 is designed to illustrate thevalidity of the proposed design strategy in this section.As shown in Figure 7, a clear optimized configuration isobtained after 128 iterations. The optimization procedure can

Figure 5. Excitation frequency depends on resonance frequency.

0 20 40 60 80 100 120 1400

100

200

300

400

500

fn1

fn2

fn3

fn4

||us||(f

P=0) ||u

s||

Iteration number

Freq

uenc

y/H

z

0

1

2

3

4

5

Dis

epla

cem

ent a

mpl

itude

/mm

0 20 40 60 80 100 120 1400

20

40

60

80

100

120

140

160 fn1

fn1

fn1

fn1

||us||(f

P=0) ||u

s||

Iteration number

Freq

uenc

y/H

z

0

1

2

3

4

5

6

7

Dis

epla

cem

ent a

mpl

itude

/mm

(a) (b)

(d)(c)

Figure 4. The optimized structures and iteration history under different excitation frequencies. (a) Optimized configuration (fp = 150 Hz).(b) Optimized configuration (fp = 160 Hz). (c) Iteration history (fp = 150 Hz). (d) Iteration history (fp = 160 Hz).

6 T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4

be divided into two stages as shown in Figure 8: in the firststage, the excitation frequency increases rapidly along withthe natural resonance frequency to the prescribed value, thedisplacement amplitude moves irregularly due to the changeof excitation frequency. A small frequency increment Df accel-erate the increase of natural resonance frequency, which meansthat raising the value of natural resonance frequency to the pre-scribed one needs fewer iteration steps. Moreover it will notaffect the final optimization results. Here Df is set at the valueof 15 Hz. In the second one, the excitation frequency remainthe prescribed value unchanged, after a small fluctuation, thenatural resonance frequency increases smoothly, the displace-ment amplitude decrease smoothly also.

Figure 8. Iteration history of the displacement amplitude theexcitation frequency along with the first resonance frequency.

Figure 9. Iteration history of the resonance mode order and themax value of MAC.

Start

Initialize design variablesdecide on excitation frequency fp

Modal analysis Harmonic response analysisModal analysis

PIS model

extraction of the referenceeigenmode Φref

Mode track

Decide on the current excitationfrequency fei

Evaluation of objective (||us(t)||)

Sensitivity analysis

Update of design variables

ConvergenceNo Yes

Stop

MAM

Computational derivatives

GCMMA algorithm

Results output

Figure 6. Flow chart of the optimization strategy.

Figure 7. Optimized configuration with given excitation frequency(fp = 480 Hz).

T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4 7

Figure 9 shows the iteration history of the resonance modeorder and the max value of MAC. As we can see, the resonancemode order changes during the optimization procedure, but itis tracked successfully by identifying the max value ofMAC. The max value of MAC holds a value above 0.9, whichindicates that large deformation will not occur on theresonance mode shape.

For the purpose of comparison, the same structure isdesigned with different optimization objectives. The first oneis the integral of displacement amplitude in a frequencyinterval (fA, fB) written as equation (30). A clear optimizedconfiguration (Figure 10a) is obtained by means of the sameintegral calculation method in reference [13]. The second is

the static compliance under same displacement amplitudeforce, Figure 10b shows the results. Compared to the initialstructure, the static displacements of three optimized structuresall decrease dramatically (Table 1), which demonstrate thesignificance of structural stiffness in attaining a clearconfiguration.

Min :

Z fB

fA

jjusjjdf ð30Þ

Figure 11 shows the displacement response curves of threedifferent configuration. As we can see, the optimizedconfiguration in static condition is failed to avoid resonancethat a major peak value will appear in the frequency interval

(a) (b)

Figure 10. Optimized configurations with different optimization objectives. (a) Optimized configuration in a frequency interval.(b) Optimized configuration in static condition.

Table 1. The displacement values of different structures.

Structures ||us(t)||(fP = 0) ||us(t)||(fP = 480)R 480

0 jjusjj dfInitial structure 0.5562 mm 0.1885 mm 2.321 m (rad/s)Static compliance 0.0506 mm 0.0753 mm 0.355 m (rad/s)A frequency interval 0.0599 mm 0.0718 mm 0.211 m (rad/s)MAC based method 0.0619 mm 0.1191 mm 0.224 m (rad/s)

Figure 11. Comparison of displacement amplitudes of threeoptimization structures.

Figure 12. Iteration history of the resonance frequency.

8 T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4

(0, 480 Hz). As for the optimized configuration in a frequencyinterval, the value of the optimization objectives ||us(t)|| andR 480

0 jjusjjdf are both the minimum one in all the threeoptimized configurations (Table 1). It is mainly because thatthe peak values of displacement decrease and troughs appearin the frequency interval. Even though, resonance willoccur when external excitation reach the exact value. Luckilythe optimized configurations using the proposed method canavoid resonance successfully by increasing the naturalresonance frequency larger than the prescribed value ofexcitation frequency.

It should be noted that the natural resonance frequency hasa limit value, when the prescribed excitation frequency exceedsthis limit, the proposed method is inapplicable. To detect thislimit value, we keep the excitation frequency always lower thanthe natural resonance frequency in the optimization procedure.Figure 12 shows the iteration history of the natural resonancefrequency, it is observed that when the excitation frequencyis closed to the limit value, optimization objective ofdisplacement amplitude can decrease only by reducing theresonance frequency, since it is the only one path to drivethe excitation frequency away from the natural resonancefrequency. Then the natural resonance frequency will increaseagain along with the increment of the excitation frequency.Finally, the resonance frequency fluctuates around the limitvalue. Therefore we can obtain this limit value approximately.This significant value can be used as the boundary of low-frequency and high-frequency excitation problems. In thisway, the range of low-frequency extends a lot compared withthe traditional criterion which just simply sets the naturalresonance frequency of initial structure as the boundary line.It means that we can obtain clear optimum anti-resonancestructures in a greater scope by employing the proposedmethod. As for the high-frequency excitation problems, its opti-mization objective may need a change or additional constraintsshould be taken into account to obtain desired results.

5 Conclusions

Topology optimization of structures with damping that aresubjected to harmonic force excitation with prescribedfrequency higher than its natural resonance frequency iscarried out in this paper. A MAC based excitation frequencyincreasing method is proposed to avoid distorted configuration.Compared with other two configurations which use the integralof displacement amplitude in a frequency interval or the samedisplacement amplitude under static condition as optimizationobjective, this configuration has a significant advantage ofavoiding resonance since its natural frequency increases abovethe given excitation frequency. In addition, the limit value ofthe natural resonance frequency can be calculated approxi-mately to distinguish low-frequency and high-frequencyexcitation problems so that the range of low-frequency extendsa lot. As a result, we can obtain clear optimum anti-resonantstructures in a greater scope by employing the proposedmethod.

Acknowledgements. This work is supported by National NaturalScience Foundation of China (11432011, 51521061, NSAF

U1330124), the 111 Project (B07050), the Fundamental ResearchFunds for the Central Universities (3102014JC02020505).

References

1. Bendsøe MP, Kikuchi N. 1988. Generating optimal topologiesin structural design using homogenization. Computer Methodsin Applied Mechanics and Engineering, 71, 197–224.

2. Maute K, Allen M. 2004. Conceptual design of aeroelasticstructures by topology optimization. Structural and Multidisci-plinary Optimization, 27, 27–42.

3. Guo X, Cheng GD. 2010. Recent development in structuraldesign and optimization. Acta Mechanica Sinica, 26(6),807–823.

4. Zhu JH, Zhang WH, Xia L. 2016. Topology optimization inaircraft and aerospace structures design. Archives of Compu-tational Methods in Engineering, 23, 595–622.

5. Pedersen NL. 2000. Maximization of eigenvalues usingtopology optimization. Structural and MultidisciplinaryOptimization, 20, 2–11.

6. Du JB, Olhoff N. 2007. Topological design of freely vibratingcontinuum structures for maximum values of simple andmultiple eigenfrequencies and frequency gaps. Structural andMultidisciplinary Optimization, 34(2), 91–110.

7. Tsai TD, Cheng CC. 2013. Structural design for desiredeigenfrequencies and mode shapes using topology optimiza-tion. Structural and Multidisciplinary Optimization, 47(5),673–686.

8. Ma ZD, Kikuchi N, Cheng HC. 1995. Topological design forvibrating structures. Computer Methods in Applied Mechanicsand Engineering, 121(s1–4), 259–280.

9. Shu L, Wang MY, Fang Z, et al. 2011. Level set based structuraltopology optimization for minimizing frequency response.Journal of Sound and Vibration, 330(24), 5820–5834.

10. Yoon GH. 2010. Structural topology optimization for frequencyresponse problem using model reduction schemes. ComputerMethods in Applied Mechanics and Engineering, 199(25–28),1744–1763.

11. Xiang LI, Wang H, Chen LF. 2012. Structural frequencyresponse optimization under bandwidth excitation. Journal ofVibration and Shock, 31(9), 113–117.

12. Yang X, Li Y. 2014. Structural topology optimization ondynamic compliance at resonance frequency in thermalenvironments. Structural and Multidisciplinary Optimization,49(1), 81–91.

13. Liu H, Zhang WH, Gao T. 2015. A comparative study ofdynamic analysis methods for structural topology optimiza-tion under harmonic force excitations. Structural andMultidisciplinary Optimization, 51(6), 1321–1333.

14. Jog CS. 2002. Topology design of structures subjected toperiodic loading. Journal of Sound and Vibration, 253(3),687–709.

15. Olhoff N, Du JB. 2005. Topological design of continuumstructures subjected to forced vibration, in Proc. of 6th WorldCongresses of Structural & Multidisciplinary Optimization(WCSMO-6), 30 May–03 June 2005, Rio de Janeiro, Brazil.

16. Cornwell RE, Craig RR, Johnson CP. 1983. On the applicationof the mode-acceleration method to structural engineeringproblems. Earthquake Engineering & Structural Dynamics,11(5), 679–688.

T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4 9

17. Besselink B, Tabak U, Lutowska A, et al. 2013. A comparisonof model reduction techniques from structural dynamics,numerical mathematics and systems and control. Journal ofSound and Vibration, 332(19), 4403–4422.

18. Ewins DJ. 1984. Modal testing: theory and practice. ResearchStudies Press, Wiley: New York.

19. Stolpe M, Svanberg K. 2001. An alternative interpolationscheme for minimum compliance topology optimization.Structural and Multidisciplinary Optimization, 22(2), 116–124.

20. Zhu JH, Beckers P, Zhang WH. 2010. On the multi-componentlayout design with inertial force. Journal of Computational andApplied Mathematics, 234(7), 2222–2230.

21. Akguacute MA, Haftka RT, Wu KC, et al. 2001. Efficientstructural optimization for multiple load cases using adjointsensitivities. AIAA Journal, 39(3), 511–516.

22. Adhikari S. 2012. Rates of change of eigenvalues andeigenvectors in damped dynamic system. AIAA Journal,37(37), 1452–1458.

Cite this article as: Liu T, Zhu J-H, He F, Zhao H, Liu Q & Yang C: A MAC based excitation frequency increasing method for structuraltopology optimization under harmonic excitations. Int. J. Simul. Multisci. Des. Optim., 2017, 8, A4.

10 T. Liu et al.: Int. J. Simul. Multisci. Des. Optim. 2017, 8, A4