Evolutionary Piezoelectric Actuators Design Optimisation

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Evolutionary piezoelectric actuators design optimisationfor static shape control of smart plates

Quan Nguyen a, Liyong Tong a,*, Yuanxian Gu b

a School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australiab Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China

Received 25 February 2005; received in revised form 6 November 2006; accepted 5 July 2007Available online 31 August 2007

Abstract

This paper presents a new evolutionary algorithm to solve various structural shape control problems of smart composite plate struc-tures with active piezoelectric actuators. The linear least square (LLS) method and the features of evolutionary strategies are employed tofind the applied voltages and shapes for the active piezoelectric actuators, respectively, in order to achieve the desired structural shapes bygradually removing the active piezoelectric material part of the element based on the error function sensitivity number. In the finite ele-ment (FE) analysis, an error function sensitivity number, including electro-mechanical effect, is one derived to compute the change inerror functions that are defined in terms of least square difference between calculated and desired structural shapes. The evolutionarypiezoelectric actuator design optimisation (EPADO) is proposed here to optimise the active piezoelectric actuator shape at a givenapplied voltage. Finally, several numerical examples are presented to verify that the proposed algorithm improves structural shape con-trol by reducing the error function. 2007 Elsevier B.V. All rights reserved.

Keywords: Shape control; Piezoelectric actuators/sensors; Composite plates; Finite element analysis; Structural optimisation; Sensitivity analysis

1. Introduction

Shape control of a structure can be achieved throughoptimally selecting the loci, shapes and sizes of the piezo-electric actuators attached to the structure and choosingthe electric fields applied to the actuators. Shape controlcan be categorised as either static or dynamic shape con-trol. Whether it is transient or gradual change, static or

dynamic shape control, both aim to determine the configu-ration of piezoelectric actuators, their applied electricalfields and loci such that the desired shape is achievedeffectively.

For quasi-static shape control, the configuration of pie-zoelectric actuators and applied voltage are importantdesign variables for achieving the desired structural shape.However, most of the research on static shape control

focuses on finding optimum values of the electric fieldsapplied to rectangular-shaped piezoelectric actuators forachieving the desired structural shape [1]. In this case, theelectric fields in the actuators are the design variables andtheir optimum values are determined by minimising the dif-ference between the actuated and desired structural shapes.Koconis et al. [2] developed analytical methods for deter-mining the optimum values of the applied electrical fields

to fixed rectangular-shaped actuators for achieving thespecified shapes for sandwich plates and shells. Cheeet al. [3,4] employed the third-order plate theory [5] tomodel mechanical deformation and the layer-wise theory[6] for modelling the electric field in the finite element for-mulation for shape control of smart composite plate struc-tures with rectangular actuators. They developed theperturbation build up voltage distribution (BVD) algo-rithm to find the optimal voltage distribution in actuators.

An iterative approach for shape control of a structureimproves the smoothness of the controlled surfaces by

0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.07.018

* Corresponding author. Fax: +61 2 93514841.E-mail address: [email protected] (L. Tong).

www.elsevier.com/locate/cma

Available online at www.sciencedirect.com

Comput. Methods Appl. Mech. Engrg. 197 (2007) 4760
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employing different criteria. Chee et al. [79] used slope andcurvature criteria in addition to the commonly used dis-placement based objective function to find voltage distribu-tion. Chen et al. [10] employed displacementstress dualcriteria. This approach is based on normal displacementcontrol where stress modification is considered in the entire

optimisation process to minimise stress in the local domain.In the majority of research available on static structuralshape control, the shape and location of an actuator areseldom treated as design variables, which can lead tohigh-energy consumption, since high voltages may berequired. Bruch et al. [11] minimising the maximum deflec-tion of a beam by selecting the locations and lengths of theactuator patch and voltage distribution in the beam struc-tures under general loading condition. Donthireddy andChandrashekhara [12]; and Eisenberger and Abramovich[13] presented studies obtained by changing the actuatorsize, location and actuator voltages in composite beams.Lim [14] and Hac and Lui [15] found the optimal location

for sensors and actuators in flexible structures. Onoda andHanawa [16] used genetic algorithms (GA) and simulatedannealing algorithms for the optimal location of actuatorsin the shape control of space trusses. Gaudenzi et al. [17]employed GA to find the actuator distribution for shapecontrol of beam structures.

To date, there exists limited work that optimises theactuator shape by removing excessive piezoelectric materi-als based on selected criterion. Merkhujee and Joshi [18]presented a gradientless iterative technique to find the actu-ators shape by gradually removing the piezoelectric mate-rial based on the residual voltage of elements for shape

control. The residual voltage is calculated by subtractionof normalised voltages of initial and current designs. Theelements, which have negative residual voltage, are thenmoved. But in engineering applications of structural designoptimisation, there are many optimisation methods thathave been developed based on material removal. Evolu-tionary structural optimisation (ESO) is one of the gradi-entless algorithms proposed by Xie and Steven [19],which is widely used to find the optimal structural configu-ration, subjected to various criterion conditions, by remov-ing structural elements based on a selected sensitivitynumber. Chu et al. [20] developed an ESO based on FEanalysis to minimise the structural weight with stiffnessconstraint. Li et al. [21,22] presented an ESO algorithmfor problems of variable thickness whilst minimising themaximum stress in a structure. The features of ESO havenot been extensively used in structural shape control prob-lems to optimise piezoelectric actuators.

In this paper, a new algorithm for evolutionary piezo-electric actuator design optimisation, for structural shapecontrol problem, is developed by employing the featuresof ESO. On the basis of FE analysis, where the mechanicaldeformation is modelled using a third-order plate theory,and a layer-wise theory is used to model the electric field,the error function sensitivity number including electro-

mechanical effect is derived to compute the change in the

error function. The error function is defined in terms ofthe least square difference between the calculated anddesired structural shapes, due to the removal of active pie-zoelectric materials. An algorithm for evolutionary piezo-electric actuator design optimisation (EPADO) isproposed. Numerical results are presented to demonstrate

that the algorithm shows improvement in terms of reducingthe error function and enhancing the efficiency of piezoelec-tric material usage for shape control problems.

2. Problem formulation

The shape control problem considered in this paperfocuses on evolutionary piezoelectric actuator design opti-misation (EPADO) in terms of finding optimum values forapplied voltages and actuator geometries. When the geo-metrical parameter vector of piezoelectric actuators S andthe applied voltage vector / are considered as design vari-

ables, the quasi-static shape control problem can bedefined, in the context of optimisation formulation [25],as follows:

Find V= [S,/]T to

Minimise fV LnmS;/

PNi1

wdi

wci

S;/ 2

wdmax2

4 N1

Subject to gV 0 ) KuuSugS;/ Ku/S/g;

2

where in the context of general optimisation formulation,V is the design variable vector; f(V) is the objective func-

tion; and g(V) is the performance constraint vector. Con-sidering the EPADO problem, the design variable vectorVhas two components namely, the geometric variable vec-tor S and the applied voltage variable vector /. In this pa-per, the geometric variables in vector Sare chosen to be thedistribution of selected active piezoelectric actuator mate-rial, whereas the voltage variables in vector / are the elec-trical potentials applied across the thickness direction ofeach actuator. The objective function f(V) in Eq. (1) is de-fined as the error estimate function Lnm(S,/), which is aweighted sum of all square differences in displacements atselected nodes between the desired and calculated struc-tural shapes. wd

i

and wci

S;/ are the desired and actuated(or calculated) displacements of the ith node, respectively,and wdmax is the maximum displacement in the desired struc-tural shape and N is the total number of nodes considered.In the case of plates considered in this paper, wdi andwci S;/ are the transverse displacements which is the[(i 1) * 11 + 3]th displacement component in the globaldisplacement vector ug(S,/). The constraint equations inEq. (2) are the equilibrium equations that govern the struc-tural behavior when the structure is only subjected to elec-trical loading and are formulated using the finite elementmethod. [/g] is global vector of nodal voltages. [Kuu(S)]and [Ku/(S)] are the global stiffness matrix and the global

piezoelectric linearly coupling matrix, respectively [23]. It

48 Q. Nguyen et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 4760
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is worth noting that structural shape control is an inverseproblem. It is possible that the relative error Lnm betweenthe desired and actuated shapes can be small in some re-gions and relatively large in others. However, the fitnessof the actuated structural shape is controlled by the squareerror function defined in Eq. (1) [7].

3. Linear least squares (LLS) method

When actuator configurations are given, the LLSmethod [4] has been used successfully to determine theoptimal voltage distribution required to achieve the desiredstructural shape. Eqs. (1) and (2) become

Find / to

Minimise Lnm/

PNi1

wdi

wci

/ 2

wdmax2

4 N3

Subject to ug/ Kuu1Ku//

g

: 4

By minimising the least square error function between theactuated and the desired shapes, defined by the transversenodal displacements for the case of plate structures. Eq.(3) can then be simplified as follows:

Lnm/ wd wc/2 0: 5

Eq. (4) indicates that there exists a linear relationshipbetween the voltage and displacement variables. The linearrelationship between the displacements and the voltagesallows the former to be expressed as a linear combinationof the latter. Eq. (4) can be used to calculate the influence

coefficient matrix [Cw

]. The kth column [Cw

]k of [Cw

] can bedetermined as the displacement vector wci ] obtained byapplying a unit voltage to the kth patch, with / = 1.0 Vand 0.0 V on all other patches. The entire influence coeffi-cient matrix [Cw] can be constructed by applying a unitvoltage to one patch at a time namely

wc/ XNi1

wci / XNi1

Cwi1/1 Cwi2/2 C

wiNp/Np

XNi1

XNpk1

Cwik/k or

wc/ XNp

k1Cwk/k Cw/: 6

By substituting Eq. (6) into Eq. (5), the solution of opti-mum voltage can be expressed in terms of the influencecoefficient matrix and the desired displacement vector[wd] as follows:

/ CwTCw1CwTwd: 7

4. Sensitivity analysis

In piezoelectric actuator design, shape of a piezoelectric

actuator is an important aspect in improving a desired

structural shape. The optimal shape of piezoelectric actua-tors in terms of piezoelectric material distribution can befound by removing or adding piezoelectric actuator ele-ments that lead to reduction of the error function. The sen-sitivity analysis approach is employed to evaluate thecontribution of a change in volume of piezoelectric actua-

tors to the variation of the error function. This section pre-sents the derivation of the sensitivity equations for theobjective function of single and multiple displacement com-ponents (error function sensitivity number) including theelectro-mechanical effect for a given applied voltage distri-bution. The applied voltage could be given a specific valueor obtained via the LLS solution using Eq. (7). For sensi-tivity analysis, Eqs. (1) and (2) can be rewritten as

Find S to

Minimise LnmS

PNi1

wdi

wci

S2

wdmax2

4 N8

Subject to KuuSugS Ku/S/g: 9

4.1. Objective function of single displacement component

For the shape control problem, the jth element consistsof a host material and adaptive (piezoelectric) materials.The adaptive materials can be set as active or inactive.When the active piezoelectric material part of the jth ele-ment, which has volume ofDv, is removed, the change inthe global mechanical and electro-mechanical stiffnessmatrices are defined as

DbKjuu bKjuuv bKjuuv Dv andDbKju/ bKju/v bKju/v Dv;where bKjuuv; bKju/v and bKjuuv Dv; bKju/v Dvare the set of global mechanical and electro-mechanicalstiffness matrices before and after removal of the active pie-zoelectric material part of the jth element, respectively. Thechange in the global displacement vector [Dug] can be deter-mined by considering the equilibrium condition before andafter removal of the element as given in Eqs. (9) and (10),respectively. The reason for a positive [Dug] in Eq. (10) isthat the structural stiffness is reduced when active piezo-electric material part is removed:

KuuS DbKjuuugS Dug Ku/S DbKju//g: 10

By expanding Eq. (10), then subtracting by Eq. (9) andignoring the higher order terms. The change in the globaldisplacement vector Dug can be written as

Dug KuuS1DbKjuuug DbKju//g: 11

To find the change in the ith nodal transverse displace-ment for the case of the plate element [23], which is the[(i 1) * dof + 3]th displacement component of ug, a vir-

tual load vector fi is considered where its [(i 1) * dof +

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3]th component is set equal to 1 and zero for all others. Thedof term represents the mechanical nodal degree of free-dom, which is 11 in FE formulation [23]. Multiplying Eq.(11) by the transpose offi, the change Duig of the ith nodaltransverse displacement due to the piezoelectric materialvolume change in jth element is rewritten as

Dwci Duig f

iTKuuS1DbKjuuug DbKju//g orDuig u

iTDbKjuuug DbKju//g; 12where [ui]T is the displacement vector solution of the virtualsystem [Kuu(S)][u

i] = [fi].Similarly to Xie and Steven [19], Eq. (12) can be con-

ducted at the elemental level as

Dwci ujiTDKjuuu

jg DK

j

u//jg; 13

where [uji] and ujg are the entries from global solution vec-tors [ug(S)] and u

i, respectively. DKjuu and DKju/ are the

change in the mechanical and electro-mechanical element

stiffness matrices, respectively, due to the removal of activepiezoelectric material volume related to the jth element.

In the shape control problem, the term of [ Ku/(S)][/g] isrepresentative of an electrical force. Therefore, it is obviousthat the removal of active piezoelectric material in the ele-ment changes the equivalent applied nodal voltage inducedforce to the nodes of this element. Eq. (13) indicates thechange of in nodal transverse displacement due to theremoval of the active piezoelectric material part in the jthelement. It is also defined as the displacement sensitivitynumber (aSC) of the smart structure. It can be positive ornegative, which implies that the transverse displacement

may increase or decrease, respectively.Finally, the displacement sensitivity number for objec-tive function of a single displacement component includingelectro-mechanical effect can be written as

aiSC uji

TDKjuuu

jg DK

j

u//jg: 14

4.2. Error function sensitivity number

The above displacement sensitivity number can be useddirectly for problems with an objective function with a sin-gle displacement. But in the shape control problem, theobjective function (error function Lnm) is defined in termsof a sum of least square error of all nodal transverse dis-placements between the calculated and desired shapes.Therefore, to deal with this problem, the effectiveness ofthe active piezoelectric material volume change of an ele-ment needs to be evaluated using this form of error func-tion. The general form for an error function includingmultiple displacement components is given by

LnmS hwc1;wc2; . . . ;w

ci ; . . . ;w

cN: 15

The change in error function due to the removal of an ac-tive piezoelectric part of an element is defined in terms ofthe summation of the productions of the change in dis-

placements and differentiation of the function h with re-

spect to all these displacement components, as given inEq. (16). This is known as the error function sensitivity(DLnm):

DLnmj XNi1

oh

owciDwci

: 16

Substituting Eq. (13) into Eq. (16), it becomes

DLnmj rh aSC; 17

where rh ohowc

1S

;oh

owc2

S; . . . ;

ohowc

iS

; . . . ;oh

owcN

S

aSC Dw

c1;Dw

c2; . . . ;Dw

ci ; . . . ;Dw

cN

a1SC; a2SC; . . . ; a

iSC; . . . ; a

NSC

T:

Then refer to Eq. (8) and Appendix A, Eq. (18) gives theerror function differentiation with respect to ith displace-ment components

oh

owci

wci S w

di

2 N wdmax : 18Finally, the displacement sensitivity number for error

function including multiple displacement components canbe written as

DLnmj XNi1

oh

owci Suji

TDKjuuu

jg DK

j

u//jg

or

DLnmj XNi1

oh

owci SujiT

DKjuuu

jg DK

j

u//j

g:

19

In addition, an alternative form for finding the change inerror function including multiple displacement componentsis employed by Haftka and Gurdal [24] and Li et al. [22].This takes advantage of one virtual system instead of mul-tiple virtual systems, which correspond to the number ofdisplacement components. This significantly reduces thecomputational cost. Hence the error function sensitivitynumber due to removal active piezoelectric material ofthe jth element can be rewritten as (refer to Appendix B).

DLnmj ujT

DKjuuujg DK

j

u//jg

; 20

where uj are the entries from solution vectors u KuuS

1 f and f PN

i1

ohow

ci S

fi.

5. Optimisation procedures

This section presents a new optimisation scheme thatemploys evolutionary structural optimisation strategies[19] in the application of shape control of smart compositeplate structures. This is referred to as the evolutionary pie-zoelectric actuator design optimisation (EPADO), i.e. tooptimise the active piezoelectric actuator distribution atgiven applied voltage. The method aims to minimise theerror function in achieving a desired structural shape.The method is based on the proposed evolutionary crite-

rion of error function sensitivity number. At the end of

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each iteration, a selected number of active piezoelectricmaterial parts of the elements are gradually removed fromthe initial design for the active actuator patch. To achievethe goal of reducing the error function, removal of anactive piezoelectric material element must have the lowestvalue of DLnm such that Lnm1 > Lnm2 > > Lnmiter

(iter is the number of iterations). The selected number isreferred to as removal material rate (RMR), which isdefined in terms of the ratio es/er, the number of removalpiezoelectric material pieces (es) and the number of activepiezoelectric material pieces in the current design (er).RMR should be small to ensure a smooth change (see[19]). This is because the error function sensitivity is formu-lated by neglecting the higher order term, which is onlyvalid when the changes in both stiffness and displacementvectors are small. Furthermore, RMR cannot be too smallsince the computational cost will increase. Setting RMR tozero (RMR = 0) is equivalent to an active piezoelectricpart of a single element being selected for removal during

the iteration. If there is more than one element that hasthe same smallest error function sensitivity number thenthey will be removed simultaneously. The evolutionary iter-ation procedure of EPADO is given below:

Step 1: Obtain input parameter RMR and applied voltage/.

Step 2: Define the initial active piezoelectric actuatorshape using a dense finite element (FE) mesh.

Step 3: Perform FEA to calculate the nodal transverse dis-placement of the structure and find out gradientvector $h of the objective displacement function.

Step 4: Calculate the error function Lnmb and introduce avirtual system.

Step 5: Perform FEA for the virtual system to obtain vir-tual nodal transverse displacement.

Step 6: Compute the error function sensitivity numberDLnm for all elements.

Step 7: Remove active piezoelectric material part for ele-ment that have lowest value ofDLnm and calculatethe error function Lnma.

Step 8: Perform error functions check, which are calcu-lated before (Lnmb) and after (Lnma) removingthe piezoelectric material part of an element. If(Lnma < Lnmb) then update active piezoelectricactuator configuration and error function by mak-ing Lnmb = Lnma and return to step 3 to continueoptimisation process. Otherwise proceed to step 9.

Step 9: Terminate the optimisation process. The results oflast iteration are considered as an optimal solution.

And the flowchart for EPADO process is shown inFig. 1.

6. Numerical examples

The examples are presented to demonstrate the featuresof the EPADO method based on the proposed evolution-ary criterion for error function sensitivity number includingelectro-mechanical effect for shape control problem ofsmart plates. The results are then compared with resultsobtained from the LLS method to show the improvement

Start Program

Structural, EPADO Input Parameters and Specify Applied Voltages

Compute Nodal Displacement of The Structure by FEA and Error Function Lnmb

Specify The Initial Design for Active Piezoelectric Patch as an Design Domain

Compute Error Function Sensitivity Lnm

Yes

No

End Program

Update Design of Active Piezoelectric Actuators and Lnmb = Lnma

Perform Error Function Checked (Lnma < Lnmb)

Remove Active Piezoelectric Part of an Element Which Has Smallest ofLnm

Compute Gradient Vectorh of The Error Function and Set-up Virtual Force Vector

Compute Virtual Nodal Displacement via Virtual System

Compute Nodal Displacement of The Structure by FEA and Error Function of Current Design Lnma

Start Program

Structural, EPADO Input Parameters and Specify Applied Voltages

Compute Nodal Displacement of The Structure by FEA and Error Function Lnmb

Specify The Initial Design for Active Piezoelectric Patch as an Design Domain

Compute Error Function Sensitivity Lnm

Yes

No

End Program

Update Design of Active Piezoelectric Actuators and Lnmb = Lnma

Perform Error Function Checked (Lnma < Lnmb)

Remove Active Piezoelectric Part of an Element Which Has Smallest ofLnm

Compute Gradient Vectorh of The Error Function and Set-up Virtual Force Vector

Compute Virtual Nodal Displacement via Virtual System

Compute Nodal Displacement of The Structure by FEA and Error Function of Current Design Lnma

Fig. 1. Flowchart for EPADO algorithm.



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in achieving the desired structural shape in terms of thereduction in the value of the error function.

6.1. Model description

Consider a cantilever plate, which is clamped at its left

edge and subject to non-applied mechanical load. The platehas a length ofL meters; width ofWmeters and consists oftwo layers of 0.01 m thickness each. The piezoelectric actu-ators are attached on at the top and bottom surfaces of theplate, which have a thickness of 0.001 m each. Only the topactuator patches are activated, which is chosen as a designdomain where the piezoelectric material can be removed asshown in Fig. 2. The plate material has the followingstiffness constants: c11 = c22 = c33 = 92.9 GPa, c12 = c13 =c23 = 39.8 GPa, c44 = c55 = c66 = 26.5 GPa. The patchproperties are: c11 = c22 = c33 = 113.0 GPa, c12 = c13 =c23 = 55.5 GPa, c44 = c55 = c66 = 28.6 GPa; the v11 =

v22 = 15.3

10

9

F/M, v33 = 15.0

10

9

F/M and d31 =254 pm/V, d32 = 254 pm/V, d33 = 374 pm/V, d24 =584 pm/V, d15 = 584 pm/V. The above properties wereused in all calculations.

6.2. Example 1

This section will examine the performance of shape con-trol in terms of the error function using EPADO algorithm,where the active piezoelectric actuator shape is optimised ata given applied voltage. A plate has dimensions L = 0.15 mand W= 0.06 m. Initially, the active piezoelectric actuator

was assumed to be distributed over the entire plate asshown in Fig. 2. In this section, only the bending structuralshape will be tested for all test cases. The bending desiredstructural shape is defined as wd(x,y) = 105cos(10x 1).

The first test case investigates the improvement on theresulting of error function and piezoelectric material usageof the EPADO algorithm compared to the LLS methodusing with several FE mesh sizes 10 10, 10 20 and20 40 finite elements. It is worth noting that a half-platemodel could be used to reduce computational cost. Consid-ering RMR = 0 which is equivalent to an active piezoelec-

tric part of single element being selected to be removed at atime during the EPADO iteration. It should be noted thatthere could be more than one of these elements havingidentical smallest value of error function sensitivity num-ber. Furthermore, the voltage distribution (/ = /LLS)obtained from the LLS method is used as a given applied

voltage for the EPADO procedure. The applied voltagefor the 10 10, 10 20 and 20 40 FE mesh are / =/LLS = 253.43 V, 255.24 V and 255.96 V, respectively.The convergence histories and final piezoelectric actuatordesigns are shown in Figs. 3 and 4, respectively. Fig. 4depicts similar locations of unused active piezoelectricmaterial. The numerical results obtained from LLS andEPADO, which respectively optimise the voltage distribu-tion and the number of active piezoelectric actuator ele-ments, are shown in Table 1. The third row of Table 1shows the number of active piezoelectric elements. Thereare reductions in both the error function and the volumeof active piezoelectric material. It is noted that all error

functions have reduced by from 12% to 19% and the reduc-tion in volume of active piezoelectric material has reducedby from 2.5% to 4%. The effect of the result due to the dif-ferent FE mesh sizes cannot be clearly seen here. This isbecause the voltages distribution obtained from LLSmethod are different. The results show that there is notmuch change in reduction of error function and volumeof active piezoelectric material when /LLS was consideredas an applied voltage for EPADO. This test case is furtherexamined with different applied voltages, which are lessthan the voltage distribution obtained by the LLS method(/ < /LLS). The results show that there are no active piezo-

electric actuator elements removed when (/ < /LLS). Fromall the results in this test case, it can be understood that theoptimal voltage distribution obtained by the LLS method(/LLS) could be used as the desired voltage to find the opti-mal shape of piezoelectric actuators using error functionminimisation via EPADO. In other words, the appliedvoltage must be higher than the voltage distributionobtained by the LLS method (/P /LLS) to yield improve-ment in error function and reduction in piezoelectric mate-rial usage.

X

Y

X

Z

W

L

Fig. 2. Cantilever plate in XY- and XZ-plane.

-4.06

-4.04

-4.02

-4.00

-3.98

-3.96

-3.94

-3.92

-3.90

-3.88

0 10

Iteration number

Log10

(Lnm)

10x10 FE mesh

10x20 FE mesh

20x40 FE mesh

8642

Fig. 3. Convergence histories for different FE mesh variations at / =

/LLS.



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The second test case is designed to demonstrate theinfluence of FEA mesh sizes on the error function andthe evolutionary piezoelectric actuator design by runningthe first test case again with an applied voltage of / =300 V and RMR = 0. The numerical results are given inTable 2. It can be observed that there is a remarkablereduction in both error function and volume of piezoelec-

tric material due to an increase in the applied voltage by

about 25% of /LLS. The 10 10 FE mesh model yieldsmore than a 66% reduction in the error function and a20% reduction in the active piezoelectric material volume.The 10 20 and 20 40 FE mesh models have similarresults, namely, with a 90% reduction in the error functionand 27% reduction in the active piezoelectric material vol-ume. Fig. 5 shows the effects of FE mesh size on the piezo-

electric actuator design. In Fig. 5b and c, it can be observed

Fig. 4. Effect of FE mesh variations on the piezoelectric actuator design at / = /LLS.

Table 1Voltage and error function when using LLS and EPADO at / = /LLS

FE mesh 10 10 10 20 20 40

LLS EPADO Error (%) LLS EPADO Error (%) LLS EPADO Error (%)

Voltages 253.43 255.24 255.96 Lnm 1.29 104 1.14 104 11.58 1.14 104 9.32 105 18.50 1.09 104 9.15 105 16.16Number of elements 100 96 4 200 194 3 800 780 2.5Number of iterations 2 3 10



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that the final shapes of piezoelectric actuator patches aresimilar. The curve in piezoelectric actuator design can beseen in Fig. 5c. The convergence histories of the error func-tion for different FE meshes are shown in Fig. 6. The testresults show that the optimised shapes of piezoelectricactuators have a smooth boundary and an optimal errorfunction when the finest mesh is considered. In addition,

an investigation into the evolutionary histories of active

piezoelectric actuator design for the 20 40 FE meshmodel, as shown in Fig. 7, shows that there were two piecesof piezoelectric material that were removed at each itera-tion. This is because the transverse displacements (z-dis-placement) are symmetrical about the centre horizontalline (x-direction) and due to setting of RMR = 0. The opti-mal solution is reached after 106 iterations, and 212 ele-

ments of active piezoelectric material were removed.

Table 2Voltage and error function when using LLS and EPADO at / = 300 V

FE mesh 10 10 10 20 20 40

LLS EPADO Error (%) LLS EPADO Error (%) LLS EPADO Error (%)

Voltages 253.43 300 18.38 255.24 300 17.54 255.96 300 17.21Lnm 1.29 104 4.35 105 66.35 1.14 104 1.28 105 88.83 1.09 104 1.04 105 90.46

Number of elements 100 80 20 200 146 27 800 588 26.5Number of iterations 10 27 106

Fig. 5. Effect of FE mesh variations on the piezoelectric actuator design at / = 300 V.



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Third test case investigates the effect of varying RMR onthe improvement of error function. As given in Table 3, the10 20 FE mesh model with an applied voltage of / =

300 V was considered for five cases of RMR (0.0,0.02499, 0.03499, 0.04499 and 0.05499). The numbers ofelement of piezoelectric materials removed are respectively2, 4, 6, 8 and 10 at each iteration. The third row of Table 3shows the relative errors of final error functions withrespect to their initial values for each case of RMR. Itcan be seen that they all yield more than 96% the improve-ment in the error function. The fourth row of Table 3shows the relative errors of the final error functions withrespect to the final error functions for the case ofRMR = 0. Amongst all four non-zero RMR cases, onlythe case of RMR = 0.03499 yields a 29% improvement in

error function, 33% reduction in active piezoelectric mate-rial usage and about 50% reduction in computational cost.Fig. 8 shows the convergence histories for the cases withdifferent RMR variations. The effect of RMR variationson the piezoelectric actuator design is also shown inFig. 9. The results show that the performance and effective-

ness of the optimisation procedure is affected by RMR var-iation. Therefore, it is important to choose reasonable

value of RMR to expedite the optimisation process to con-verge to the optimal solution that best fits the desired struc-tural shape.

The purpose of the forth test case is to observe the effectof applied voltage on the resulting piezoelectric materialdistribution and error function through using the 10 20

-5.00

-4.50

-4.00

-3.50

-3.00

-2.50

0 20 40 60 80 100

Iteration number

Lo

g10(Lnm)

10x10 FE mesh

10x20 FE mesh

20x40 FE mesh

Fig. 6. Convergence histories for different FE mesh variations at / =300 V.

Fig. 7. Evolution histories of piezoelectric actuator design for 20 40 FE mesh model at / = 300 V.

Table 3Performance with different RMR variations for 10 20 FE mesh model at/ = 300 V

RMR 0 0.02499 0.03499 0.04499 0.05499

Initial 1.91 103 1.91 103 1.91 103 1.91 103 1.91 103

Final 1.28 105 3.17 105 9.08 106 2.72 105 5.96 105

Error (%) 99.33 98.33 99.52 98.57 96.87

Error (%) 148.37 28.86 113.03 366.59Number of

iteration27 13 14 9 4

-5.50

-5.00

-4.50

-4.00

-3.50

-3.00

-2.50

-2.00

0 5 10 15 20 25 30

Iteration number

Log10(Lnm)

RMR = 0

RMR = 0.02499

RMR = 0.03499

RMR = 0.04499

RMR = 0.05499

Fig. 8. Evolution histories for 10 20 FE mesh model with different RMRat / = 300 V.



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FE mesh model. Twelve different applied voltages, rangingfrom 255.24 V to 650 V, are investigated. Fig. 10 shows the

relationship between the error function and applied voltageusing the EPADO. As the evolution progresses, it can beseen that the error functions for different applied voltagesreduces by approximately 2090%. It is shown that theredoes not exist a linear relationship between the error func-tion and the applied voltage. Fig. 10 shows that the lowesterror function happens at an applied voltage of 300 V.Fig. 11 shows the final piezoelectric actuator design forthe four different applied voltage variations, and alsodepicts the effect of electric field on the results of the piezo-electric actuator design. As the electric field increases, lesspiezoelectric material is used. The results show that the

lowest error function could be obtained when the correctapplied voltage is chosen. Results in this example demon-strate that the EPADO algorithm is capable of optimisingthe piezoelectric actuator distribution for best achieving thedesired structural shape with the smallest error function ata given applied voltage.

Fig. 9. Effect of RMR variations on the piezoelectric actuators design for 10 20 FE mesh model at / = 300 V.

-5.10

-4.90

-4.70

-4.50

-4.30

-4.10

-3.90

-3.70

200 300 400 500 600 700

Applied voltages (Volts)

Log10(Lnm)

Fig. 10. Effect of applied voltage variations on the final error function for10 20 FE mesh model.

Fig. 11. Effect of applied voltage variations on the piezoelectric actuators design for 10 20 FE mesh model.



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It is believed that the piezoelectric materials distributednear the free sides of the cantilever plate considered in thisexample are to offset its free edge effect because they areassumed to transversely isotropic in-plane.

6.3. Example 2

This section will examine the performance of shape con-trol in terms of error function using the EPADO algorithmfor several desired structural shapes such as bending andtwisting desired shapes. The bending and twisting desiredshapes are defined as wd(x,y) = 105cos(10x 1) andwd(x,y) = 105(cosh(x) 1) * sin(y), respectively, where0 6 x 6 0.15 and 0.075 6 y 6 0.075. A square plate isassumed to have dimensions L = 0.15 m and W= 0.15 m.Initially, the active piezoelectric actuators are assumed tobe distributed over the entire plate. Both top and bottomactuators are subdivided into two separate actuators as

shown in Fig. 12. Only top actuator patches are activated.The lower and upper patches are specified as patch number1 and 2, respectively. In the EPADO calculation, theparameter RMR is taken as 0.021.

For the bending case, the desired structural shape isshown in Fig. 13a. The calculated structural shape usingLLS voltage optimisation only, is shown in Fig. 13b.The applied voltage distribution and error function are/1 = /2 = 243.24 V and Lnm = 5.15 10

4, respectively.When using EPADO with applied voltages of /1 = /2 = 243.24 V, the optimal solution is obtained after11 iterations and the minimum value of the error function(Lnm) is 2.09 104. It is shown that the EPADO algo-

rithm yields a remarkable 59% reduction in the minimumvalue of the error function and a 19% decrease in the vol-ume of active piezoelectric materials compared to the LLSsolutions. In addition, for further tests with applied volt-

ages of /1 = /2 = 300 V, Fig. 13c shows the final calcu-lated structural shapes. With initial piezoelectric actuatordesign, the error function is 3.66 103. The final piezo-electric material distribution, which is the shaded area inFig. 13c, was achieved after 25 iterations with a minimalerror function of 5.18 105. By comparing this with the

LLS results, the error function has been improved byapproximately 90% in conjunction with an increase in theelectric field in each actuator by about 20%. It should also

X

Y

Z

X

1

2

Y

Z

L

W

1

2

Fig. 12. Square cantilever plate in XY-, YZ- and XZ-plane.

Fig. 13. Bending case: (a) desired structural shape; (b) calculatedstructural shape using LLS method voltage optimisation only; (c)

calculated structural shape with final actuators design using EPADO.



12/14

be noted that the active piezoelectric material usage is 37%less than the LLS method without performing the piezo-electric actuator optimisation.

For the twisting test case, the desired structural shape isshown in Fig. 14a. The calculated structural shape usingLLS voltage optimisation only, is shown in Fig. 14b.

The voltage distribution and error function are /1 = 0.92,/2 = 0.92 V and Lnm = 5.71 10

3, respectively. Whenthe EPADO algorithm is used with the considered appliedvoltages of/1 = 0.92, /2 = 0.92, the optimal solution isreached after 28 iterations and the minimum value of theerror function (Lnm) is 6.39 104. The error function pre-

dicted by EPADO decreases by about 89% in addition to areduction in active piezoelectric material usage of about40%. Fig. 14c depicts the final calculated structural shapesfor applied voltages of /1 = 1.10 V and /2 = 1.10 V,where the applied voltage in both patches are increasedby about 20% compared to /LLS, using the EPADO algo-rithm. The error function (Lnm) of initial piezoelectricactuator design is 6.18 103. The optimal solution wasreached after 32 iterations. The shaded region in Fig. 14cshows the optimal piezoelectric actuator design. The finalerror function is 5.06 104, which represents an improve-ment of about 91% in the error function and a 44% reduc-tion in the active piezoelectric material usage compared to

the LLS solution. In particular, Fig. 14c shows that the pie-zoelectric material distribution populates around the edgesof the host plate except along the tip loci. In other words, alarge number of bumps will occur in the middle of the hostplate when such a distribution of unused piezoelectricmaterial occurs as shown in Fig. 14b.

Finally, the results demonstrated that the EPADO algo-rithm is capable of finding optimal piezoelectric actuatorshapes to achieve the desired structural shape, whether itbe simple or complex. Bruch et al. [11] mentioned thatmore actuators are required if the desired structural shapeis more complex. In this section, the twisting case example

shows that EPADO can reduce the number of actuators fora complex shape. As Bruch et al. [11] noticed actuatorsmust be subdivided into several pieces (more than two) toreduce the waves in the resultant structural shape usingthe LLS method for the twisting case above. In engineeringdesign applications, it can be difficult to apply a voltagefrom multiple power sources due to non-uniform voltagedistributions when there are many actuators.

7. Conclusion

For the shape control problem of minimising the errorfunction between the actuated and desired structuralshapes of smart structures, an evolutionary piezoelectricactuator design optimisation (EPADO) algorithm at agiven applied voltage has been developed. Based on a sen-sitivity analysis with respect to discrete piezoelectric mate-rial layer, an error function sensitivity number is presentedto estimate the effect of the piezoelectric material layer dis-tribution on the error function. The approach of this algo-rithm is to gradually remove the active piezoelectricmaterial layer in elements to optimise the piezoelectricactuator shape by minimising of the error function. Illus-trative examples are presented and discussed. It is clear thatthe error functions obtained by the proposed algorithm

have been improved compared to the results obtained by

Fig. 14. Twisting case: (a) desired structural shape; (b) calculatedstructural shape using LLS method voltage optimisation only; (c)

calculated structural shape with final actuators design using EPADO.



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the LLS (voltage optimisation only). In other words, thelowest error function can be achieved in combination withreduced piezoelectric material usage. The removal materialrate (RMR) and finite element mesh is found to be effectivein the influencing the error function and computationalcost. The shape of the piezoelectric actuator can be refined

when finer finite element meshes are considered. Finally,EPADO is capable of solving for a variety of desired struc-tural shapes from simple (bending) cases to complex (twist-ing) cases with less number of actuator patches than theLLS methods.

Acknowledgements

The authors are grateful to the support of AustralianResearch Council (Grant DP0210716 and LX0348548)and National Natural Science Foundation of China tothe ARC Linkage International Award.

Appendix A

Appendix B

The solution for ith virtual system

KuuSuji fi:

Multiply the gradient component ohowc

iS

to equation aboveyields

KuuSoh

owci Suji

oh

owci Sfi:

Adding all such N virtual equilibrium equations as givenabove. Hence

KuuSXNi1

oh

owci Suji

XNi1

oh

owci Sfi;

KuuSu

f;

where

u XNi1

oh

owci Suji and f

XNi1

oh

owci Sfi;

) u KuuS1f KuuS

1XNi1

oh

owci Sfi:

Hence the error function sensitivity number due toremoving active piezoelectric part of the jth element is

DLnmj ujT

DKjuuujg DK

j

u//jg:

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LnmS

PNi1

wdi

wci

S2

wdmax2

4 N;

LnmS wd1 w

c1S

2 wd2 wc2S

2 wdi wci S

2 wdN wcNS

2

4 N wdmax2

;

wci S wdi

2 wci S

2 2wci Sw

di w

di

2;

o

wc

i S wd

i

2

owci S 2w

c

i S 2wd

i 4 N wdmax

2 w

c

i S wd

i 2 N wdmax

2;

)oh

owci S

wci S wdi

2 N wdmax2

:



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Evolutionary Piezoelectric Actuators Design Optimisation