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Homayouni-Amlashi, Abbas and Schlinquer, Thomas and Mohand-Ousaid, Abdenbi and Rakotondrabe, Micky 2D topology optimization MATLAB codes for piezoelectric actuators and energy harvesters. (2020) Structural and Multidisciplinary Optimization. ISSN 1615-147X

Official URL: https://doi.org/10.1007/s00158-020-02726-w

mailto:tech-oatao@listes-diff.inp-toulouse.frhttp://www.idref.fr/250769387http://www.idref.fr/118678698

2D topology optimization MATLAB codes for piezoelectric actuatorsand energy harvesters

Abbas Homayouni-Amlashi1,2 · Thomas Schlinquer2 · Abdenbi Mohand-Ousaid2 · Micky Rakotondrabe1

AbstractIn this paper, two separate topology optimization MATLAB codes are proposed for a piezoelectric plate in actuation and energy harvesting. The codes are written for one-layer piezoelectric plate based on 2D finite element modeling. As such, all forces and displacements are confined in the plane of the piezoelectric plate. For the material interpolation scheme, the extension of solid isotropic material with penalization approach known as PEMAP-P (piezoelectric material with penalization and polarization) which considers the density and polarization direction as optimization variables is employed. The optimality criteria and method of moving asymptotes (MMA) are utilized as optimization algorithms to update the optimization variables in each iteration. To reduce the numerical instabilities during optimization iterations, finite element equations are normalized. The efficiencies of the codes are illustrated numerically by illustrating some basic examples of actuation and energy harvesting. It is straightforward to extend the codes for various problem formulations in actuation, energy harvesting and sensing. The finite element modeling, problem formulation and MATLAB codes are explained in detail to make them appropriate for newcomers and researchers in the field of topology optimization of piezoelectric material.

Keywords Topology optimization · MATLAB code · Piezoelectric actuator · Piezoelectric energy harvester

1 Introduction

Topology optimization (TO) is a methodology to distributethe material within a design domain in an optimal waywhile there is no prior knowledge about the final layoutof the material (Bendsoe 2013). This main specificationof TO provides a great degree of freedom in termsof designing innovative structures to satisfy predefined

� Abbas Homayouni-Amlashiabbas.homayouni@femto-st.fr

1 Laboratoire Génie de Production, Nationale Schoolof Engineering in Tarbes (ENIT), Toulouse INP, Universityof Toulouse, 47, Avenue d’Azereix, Tarbes, France

2 CNRS, FEMTO-ST Institute, Université BourgogneFranche-Comté, Besançon, 25000, France

engineering goals. Historically, minimization of mechanicaldeformation of a structure under application of differentloading conditions was a classical engineering goal (Schmit1960). Aiming for this goal, the work of Bendsoe andKikuchi (1988) paves the way for a methodology knowntoday as topology optimization. The general idea of thismethodology is the combination of finite element methodand optimization to maximize or minimize an objectivefunction. In this regard, the design domain is discretizedby a finite number of elements and design variables for theoptimization problem are the variables attributed to eachof these elements. Different approaches are introduced inthe literature to implement the TO method (Sigmund andMaute 2013). Among these approaches, homogenizationapproach was proposed to optimize the porous elements asunit cells or microstructures within the design domain toobtain the final layout (Suzuki and Kikuchi 1991; Bendsøeand Sigmund 1995). However, the number of optimizationvariables is high in homogenization method which canmake the optimization cumbersome. The other famous andpopular approach is the SIMP approach which stands forsolid isotropic material with penalization. In this approach,the elements in the design domain can have intermediatedensities (Bendsøe 1989; Bendsoe 2013). This will let the

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elements to be gray in addition to black (material) andwhite (void). However, due to practical constraints, it isdesired that the optimization finally converges to a black andwhite layout. To do so, a penalization factor is defined forintermediate densities. One of the reasons for the popularityof this approach is its simplicity of implementationin comparison with other approaches. Similar to SIMPapproach, there is evolutionary structural optimization(ESO) (Xie and Steven 1993) or the more general formbi-directional evolutionary structural optimization (BESO)(Xia et al. 2018) which is about removing or adding theelements inside the design domain during optimizationiterations. In addition to the aforementioned approaches,there are other approaches including level set method (vanDijk et al. 2013; Andreasen et al. 2020) and method ofmoving morphable components (MMC) (Guo et al. 2014;Zhang et al. 2017) which is a geometrical approach. For adetailed review and comparison between these approaches,one can refer to the following review papers: Sigmund andMaute (2013) and Deaton and Grandhi (2014).

Due to the success of TO methodology, severalimplementation codes in different software are published inthe literature. Using the SIMP approach, Sigmund (2001)published the 99 lines of MATLAB code for 2D topologyoptimization of compliance problems. Andreassen et al.(2011) published the 88 lines of MATLAB code which wasan improvement of Sigmund’s 99 lines of code while havingmuch faster speed in each iteration thanks to introducing theconnectivity matrix that facilitates the assembly procedureof elemental matrices. Liu and Tovar (2014) publishedthe TOP3D MATLAB code by extension of the 88 linesof code for topology optimization of 3D structures. Chenet al. (2019) published 213 lines of MATLAB codefor 2D topology optimization of geometrically nonlinearstructures. There are other published codes using otherTO approaches like level set method (Challis 2010; Weiet al. 2018; Yaghmaei et al. 2020), BESO (Xia et al.2018), and projection method (Smith and Norato 2020).These published codes facilitate the implementation ofTO methodology for various applications. For this reason,the application of TO can be seen in solving differentproblems including the compliance problems (Bendsoe2013), compliant mechanism problems (Zhu et al. 2020),heat conduction (Gersborg-Hansen et al. 2006), and smartmaterials in particular the piezoelectric materials (Sigmundand Torquato 1999).

Due to their electromechanical coupling effect, piezo-electric materials have applications in actuation, sensingand energy harvesting. Plenty of methods can be foundin the literature to analyze and improve the performanceof the piezoelectric structures, whether actuators, energyharvesters or sensors such as geometrical and size opti-

mization (Schlinquer et al. 2017; Bafumba Liseli andAgnus 2019; Homayouni-Amlashi et al. 2020a), shapeoptimization (Muthalif and Nordin 2015), layers numberoptimization (Rabenorosoa and et al 2015), or parame-ters sub-optimization (Rakotondrabe and Khadraoui 2013;Khadraoui et al. 2014) with interval techniques (Rakoton-drabe 2011). After development of TO methodology, it isextended to different physics (Alexandersen and Andreasen2020; Deaton and Grandhi 2014) including the piezoelec-tricity. Primarily, the homogenization approach is used(Silva et al. 1997; Sigmund et al. 1998). Afterwards, otherapproaches including SIMP (Kögl and Silva 2005), BESO(de Almeida 2019) or level set method (Chen et al. 2010) arealso explored. By defining proper objective functions, TOmethodology is applied to piezoelectric actuators (Morettiand Silva 2019; Gonċalves et al. 2018), sensors (Menuzziet al. 2018) and energy harvesters (Homayouni-Amlashiet al. 2020b; Homayouni-Amlashi 2019; Townsend et al.2019). The publications considered different types of sys-tem modeling including the static (Zheng et al. 2009),dynamic (Noh and Yoon 2012; Wein et al. 2009), modal(Wang et al. 2017) and electrical circuit coupling (Salaset al. 2018; Rupp et al. 2009). Different types of prob-lem formulation can be found as well such as optimizationwith stress constraints (Wein et al. 2013). Although theapplication of TO methodology to piezoelectric materials iswell established in the literature, no implementation code ispublished yet.

In this paper, 2D topology optimization codes areproposed for actuation and energy harvester by usingthe extension of SIMP approach known as PEMAP-P(piezoelectric material with penalization and polarization).The codes are written based on the 88 lines of MATLABcode written by Andreassen et al. (2011), except the codeis extended and modified considerably to consider theelectromechanical coupling effect of piezoelectric materialand problem formulation. In Section 2, the finite elementmodeling of one-layer piezoelectric plate is presented byusing the plane stress assumption. The finite elementequation is derived for both actuation and energy harvesting.A normalization is applied to the finite element equationwhich significantly reduces the numerical instabilities inoptimization iterations. Hence, the proposed codes inthis paper work smoothly in both actuation and energyharvesting. In Section 3, first, the material interpolationscheme for piezoelectric material is explained. Then, theoptimization problem is formulated for both actuation andenergy harvesters. The problem formulations are basic foreducational purposes. The sensitivity analysis is performed,and finally, the optimization algorithms are explained. InSection 4, the MATLAB codes are explained part by partin detail. The explanations in this part help the readers to

extend and implement the codes for their own purposes. InSection 5, different numerical examples are illustrated andthe modification to the original codes to implement thosenumerical examples are expressed as well.

2 Finite element modeling

2.1 Constitutive equation

The general linearly coupled mechanical and electrical con-stitutive equation of piezoelectric materials by neglectingthe thermal coupling can be written as (Lerch 1990)

T̄ = cES̄ − eĒD̄ = eT S̄ + εSĒ (1)

In (1), T̄ and S̄ are the vectors of mechanical stress andstrain while cE is the stiffness tensor in constant electricalfield. D̄ and Ē are the vectors of electrical displacement andelectrical field. e is the piezoelectric matrix, εS is the matrixof permittivity in constant mechanical strain and T showsthe matrix transpose.

The 4-mm tetragonal crystal class piezoelectric material(Piefort 2001) which has orthotropic anisotropy is consid-ered to derive the corresponding model. This class includesmost of the piezoelectric material in particular the well-known PZT (lead zirconate titanate) materials. By thisconsideration, the mechanical stiffness tensor, piezoelectricmatrix and permittivity for full 3D modeling are

cE =

⎡⎢⎢⎢⎢⎢⎢⎣

cE11 cE12 c

E13 0 0 0

cE12 cE11 c

E13 0 0 0

cE13 cE13 c

E33 0 0 0

0 0 0 cE44 0 00 0 0 0 cE44 00 0 0 0 0 cE66

⎤⎥⎥⎥⎥⎥⎥⎦

eT =⎡⎣

0 0 0 0 e15 00 0 0 e15 0 0

e31 e31 e33 0 0 0

⎤⎦

εS =⎡⎣

εS11 0 00 εS11 00 0 εS33

⎤⎦ (2)

Now, a piezoelectric plate sandwiched between twoelectrodes as shown in Fig. 1 is considered. Without lossof generality, several assumptions are considered for thisconfiguration.

– the thickness to length ratio of the piezoelectric plate isless than 1/10,

– the piezoelectric plate is confined to have planarmovement and it is subjected to loading only in the xyplane,

– the thickness of the electrodes is negligible in compari-son with thickness of piezoelectric plate,

– the electromechanical system is assumed to be linear,– the electrodes are perfectly conductive,– the polarization direction is perpendicular to the plate

in parallel to→z axis,

– the electrical field is uniform in the direction ofthickness aligned with the polling direction,

– the variation of the potential in the direction of thethickness is linear,

The first two assumptions let us use the plane stressassumption modeling for the piezoelectric plate (Hutton andWu 2004). In this case, any stress in the direction of z willbe zero. By considering transversely isotropic piezoelectricmaterial and considering plane stress assumption, the piezo-electric plate has in-plane isotropic behavior. Furthermore,by poling the piezoelectric material in the z direction, theonly non-zero electric field will be in the z direction. Inthis case, the reduced (2D) form of piezoelectric constitutiveequation can be derived in the following form (Junior et al.2009)

⎡⎢⎢⎣

T1T2T3D3

⎤⎥⎥⎦ =

⎡⎢⎢⎣

c∗11 c∗12 0 −e∗31c∗12 c∗11 0 −e∗310 0 c∗33 0

e∗31 e∗31 0 ε∗33

⎤⎥⎥⎦

⎡⎢⎢⎣

S1S2S3E3

⎤⎥⎥⎦ (3)

Fig. 1 Piezoelectric plate sandwiched between two electrodes. aIsometric view. b Side view

The components of the reduced constitutive equation canbe written as (Junior et al. 2009)

c∗E11 = cE11 −(cE13

)2cE33

, c∗12 = cE12 −(cE13

)2cE33

, c∗33 = c66

e∗31 = e31 − e33cE13

cE33

, ε∗33 = εS33 +e233

cE33

(4)

The obtained constitutive equation in (3) will be used inthe finite element modeling of the piezoelectric plate whichwill be discussed in the next section.

2.2 Piezoelectric finite elementmodel

In this section, to derive the finite element (FE) formulation,the piezoelectric plate is discretized by rectangular elementswhich are particular form of the more general 2D elementscalled “bilinear quadrilateral element” (Hutton and Wu2004; Kattan 2010). It should be noted that since the

thickness of electrodes is negligible in comparison with thethickness of the piezoelectric plate, its structural effects areneglected in the modeling. Hence, only the piezoelectricplate is discretized by the finite number of elements. Theschematic form of this discretization is illustrated in Fig. 2a.As can be seen in this figure, The piezoelectric plate isdiscretized as 3 by 4 elements. It is clear that a finerdiscretization will be used for the numerical optimization.It can be seen in the figure that each rectangular elementhas 4 nodes and each node has 2 in-plane mechanicaldegrees of freedom regarding the displacement in x and ydirections. The rectangular element shown in Fig. 2b canhave arbitrary length le and width we. In fact, with themethod of finite element modeling which is used to writethe optimization code, the dimensions of plate and numberof elements can be defined separately. This freedom willhave two advantages: first, for a predefined geometry ofa piezoelectric plate, higher number of elements can bedefined to have better results in terms of having small detail.

Fig. 2 Finite elementdiscretization of design domain.Panel (a) is the numberingformat inside the design domain.Panel (b) is the numberingformat inside each element.Panel (c) is the parent element inthe natural coordinates

Second, it is possible to define lower number of elementswhen reducing the computation time is necessary. As it isillustrated in Fig. 2c, the rectangular element is mapped toa parent element which is a square element with naturalcoordinates ξ and η. The displacement of every point withinthe element will be expressed by the displacement of thenodes through the interpolation functions in the followingformat (Hutton and Wu 2004).

x = x1n1 + x2n2 + x3n3 + x4n4y = y1n1 + y2n2 + y3n3 + y4n4 (5)

and the interpolation functions can be written based on thenatural coordinates

n1 = 14(1 − ξ)(1 − η), n2 = 1

4(1 + ξ)(1 − η)

n3 = 14(1 + ξ)(1 + η), n4 = 1

4(1 − ξ)(1 + η) (6)

where the matrix of interpolation function can be written as,

N =[

n1 0 n2 0 n3 0 n4 00 n1 0 n2 0 n3 0 n4

](7)

So far, for each element, just mechanical degreesof freedom are considered. However, in piezoelectricmaterial, the mechanical and electrical fields are coupled.Therefore, the electrical degree of freedom should bemodeled as well. The general approach for this case isto consider one electrical degree of freedom for eachnode in addition to mechanical degrees of freedom asexplained in Lerch (1990). However, here by assumingthat conductive electrodes are placed on top and bottom ofthe piezoelectric plates as shown in Fig. 1, the electricalpotential over each electrode is constant. This conditionis known as equipotential condition. Furthermore, byconsidering the bottom electrode as ground electrode, thewhole piezoelectric plate will have one electrical degree offreedom. On the other hand, for the purpose of elementalsensitivity analysis which will be explained in Section 3, foreach element, one electrical degree of freedom is consideredas it is shown in yellow in Fig. 2. The global equipotentialcondition will be imposed after assembling the globalmatrices.

Now, the strain and electrical field of each element can beexpressed with the help of mechanical and electrical degreesof freedom

S̄ = Buu, Ē = Bφφ (8)

In (8), u and φ are the vectors of mechanical displacementand scalar value of electric potential respectively. Bu is

the strain displacement matrix which is written as follows(Kattan 2010),

Bu = 1|J |[B1 B2 B3 B4

]

Bi =⎡⎢⎣

a∂ni∂ξ

− b ∂ni∂η

0

0 c ∂ni∂η

− d ∂ni∂ξ

c∂ni∂η

− d ∂ni∂η

a∂ni∂ξ

− b ∂ni∂η

⎤⎥⎦ (9)

and the parameters a, b, c and d are given by Kattan (2010),

a = 14[y1(ξ − 1) + y2(−1 − ξ) + y3(1 + ξ) + y4(1 − ξ)]

b = 14[y1(η − 1) + y2(1 − η) + y3(1 + η) + y4(−1 − η)]

c = 14[x1(η − 1) + x2(1 − η) + x3(1 + η) + x4(−1 − η)]

d = 14[x1(ξ − 1) + x2(−1 − ξ) + x3(1 + ξ) + x4(1 − ξ)]

(10)

where xi and yi are the coordinates of the nodes in therectangular element before mapping.

The determinant of Jacobian matrix J , which transfersthe natural coordinates to the generalized coordinates, is

|J | = 18

[x1 x2 x3 x4

]

×

⎡⎢⎢⎣

0 1 − η η − ξ ξ − 1η − 1 0 ξ + 1 −ξ − ηξ − η −ξ − 1 0 η + 11 − ξ ξ + η −η − 1 0

⎤⎥⎥⎦

⎡⎢⎢⎣

y1y2y3y4

⎤⎥⎥⎦ (11)

By considering the last two assumptions, Bφ is (Junioret al. 2009)

Bφ = 1/h (12)

where h is the thickness of the piezoelectric plate. Now byusing the Hamilton’s variational principle and neglectingthe damping effect, the linear differential equation for onesingle element can be written in the following form (Lerch1990)

[m 00 0

] [ü

φ̈

]+

[kuu kuφkφu −kφφ

] [u

φ

]=

[f

q

](13)

in which m is the mass matrix, kuu is the mechanicalstiffness matrix, kuφ is the piezoelectric coupling matrix,kφφ is the dielectric stiffness matrix, f is the external

mechanical force and q is the charge. These components oflinear differential equation are derived in the following form

kuu = h∫

A

BTu cEBu |J | dξdη

kuφ = h∫

A

BTu eT Bφ |J | dξdη

kφφ = h∫

A

BTφ εSBφ |J | dξdη

m = ρh∫

A

NT N |J | dξdη (14)

where A is the top surface area of the element and ρ is thedensity of the material. In fact, (14) illustrates the analyticalcalculations of elemental matrices. However, for numericalimplementation in MATLAB, two-point Gauss quadraturemethod (Hutton and Wu 2004) is utilized for calculation ofthe elemental matrices numerically, which gives the exactvalues. The implementation procedure is explained later inSection 4.

To have the global FEM equation for a whole piezoelec-tric plate, the elemental matrices in (13) should be assem-bled, which is a general procedure in the FEM methodol-ogy and which will also be explained in Section 4. Afterassembling the elemental matrices, the global finite elementequation for the whole design domain can be written as[

M 00 0

] [Ü

Φ̈

]+

[Kuu KuφKφu −Kφφ

] [U

Φ

]=

[F

Q

](15)

Now, for two cases of actuation and energy harvesting,the global FEM (15) can be interpreted in different ways.Here, we focus on static actuation so that the dynamics willnot be considered. Therefore, the global FEM equation forthe actuation can be written as

KuuU + KuφΦ = F (16)This equation will be used to calculate the mechanical

displacement due to applied potential.For the energy harvesting case, the external charge (Q)

is considered to be zero. In addition, the external force isconsidered to be a harmonic excitation of frequency Ω . Inthis case, by considering a linear electromechanical system,the force and the response of the system can be stated as

F = f0eiΩtU = u0eiΩt , Φ = φ0eiΩt (17)where f0, u0 and φ0 are the amplitude of harmonic force,displacement and potential. By substituting the (17) in (15),the global FEM equation for the energy harvesting case canbe written as

−Ω2[

M 00 0

] [U

Φ

]+

[Kuu KuφKφu −Kφφ

] [U

Φ

]=

[F

0

](18)

Equation (18) can also be written in following form

[Kuu − MΩ2 Kuφ

Kφu −Kφφ] [

U

Φ

]=

[F

0

](19)

To solve the FEM (19), the mechanical boundarycondition and equipotential condition should be applied.This will be explained in detail in Section 4.

2.3 Normalization

Here, the critical point is that the scale difference betweenthe piezoelectric matrices including the mechanical stiffnessmatrices (kuu) and (kuφ) and the dielectric stiffness matrix(kφφ) is huge. This huge scale difference can bringnumerical instabilities in form of singularities in solving thefinal global FEM equation during the optimization loops. Toeliminate the scale difference, a normalization is suggested(Homayouni-Amlashi 2019; Homayouni-Amlashi et al.2020b) by factorizing the highest value of each elementalmatrix which can be expressed in the following format

k̃uu = kuu/k0, k̃uφ = kuφ/α0k̃φφ = kφφ/β0, m̃ = m/m0 (20)

Starting by this normalization of elemental matrices, theactuation FEM (16) can be rewritten as

K̃uuŨ + K̃uφΦ̃ = F̃ (21)in which

F̃ = F/f0, Ũ = U/u0, Φ̃ = Φ/φ0u0 = f0/k0, φ0 = f0/α0 (22)

The same normalization can be performed on the energyharvesting FEM (19)

[K̃uu − M̃Ω̃2 K̃uφ

K̃φu −γ K̃φφ] [

Ũ

Φ̃

]=

[F̃

0

](23)

where

Ω̃2 = Ω2m0/k0, γ = k0β0/α20 (24)Here γ is a normalization factor which keeps the

solution of the system equal before and after applyingthe normalization. This normalization factor is having thescale of 101 and in this way the scale difference betweenthe piezoelectric matrices is eliminated. The proof ofnormalization is provided in the Appendix.

Now, by having the FEM equations (21) and (23), it ispossible to enter the optimization phase. In the upcomingsections, optimization of actuator and energy harvester isseparated.

3 Topology optimization

As explained before, topology optimization is aboutdistribution of the material within a design domain whilethere is no prior knowledge of the final optimized layout ofthe structures (Bendsoe 2013). There are several approachesfor topology optimization method (Maute and Sigmund2013). However, the density-based approach is chosen forthis paper since its efficiency is already established in manyresearches specially in the area of piezoelectric actuators(Kögl and Silva 2005; Ruiz et al. 2017; Moretti and Silva2019) or energy harvesters (Homayouni-Amlashi 2019;Zheng et al. 2009; Noh and Yoon 2012).

3.1 Piezoelectric material interpolation scheme

One of the famous material interpolation schemes isthe density-based approach which has been introducedto relax the optimization from the binary (void-material)problem. In this approach, for a discretized design domainby finite number of elements, the material properties ofeach element are related to element’s density through apower law interpolation function. For passive isotropicmaterial, this interpolation function is referred to as solidisotropic material with penalization (SIMP) which relatesthe element’s Young modulus of elasticity to its density. Foractive non-isotropic piezoelectric material, the interpolationfunction is the extension of SIMP scheme which can bewritten as follows (Kögl and Silva 2005)

k̃uu(x) =(Emin + xpuu(E0 − Emin)

)k̃uu

k̃uφ(x, P ) = (emin + xpuφ (e0 − emin))(2P − 1)pP k̃uφk̃φφ(x) = (εmin + xpφφ (ε0 − εmin))k̃φφ

m̃(x) = xm̃ (25)

where Emin, emin and εmin are small numbers to definethe minimum values for stiffness, coupling and dielectricmatrices while E0, e0 and ε0 are equal to 1 to define themaximum values of the respected matrices. The definitionof minimum values is provided to avoid the singularitiesduring the optimization iterations. x is the density ratio ofeach element which has a value between 0 and 1. P isthe polarization variable which also has the value between0 and 1 and determines the direction of polarization.puu, puφ , pφφ and pP are penalization coefficients forthe stiffness, coupling, dielectric matrices and polarizationvalue respectively. It is obvious that in (25), the normalizedform of piezoelectric matrices is used. However, theinterpolation function is true for non-normalized matrices aswell.

The introduced material interpolation scheme is knownas PEMAP-P (piezoelectric material with penalization and

polarization) (Kögl and Silva 2005), which is the exten-sion of the SIMP approach. Although some projectionsare defined for SIMP to have a robust topology optimiza-tion (Wang et al. 2011), there are alternative interpolationfunctions like RAMP (rational approximation of materialproperties) (Stolpe and Svanberg 2001), or newer interpola-tion function introduced in Clausen et al. (2015). This latterone is also used in topology optimization of piezoelectrictransducers (Donoso and Sigmund 2016). In fact, the studyon preference of these interpolation functions is not in thescope of this paper. But, with the proposed MATLAB code,it will be easy for implementation of other interpolationfunctions.

After establishing the material interpolation scheme, therest of this section will be divided into two parts: actuationand energy harvesting.

3.2 Actuation

3.2.1 Problem formulation

Following the classic approach for compliant mechanismsreported in Bendsoe (2013), the optimization of a pla-nar piezoelectric actuator can be defined simply as dis-placement optimization or minimization of the followingobjective function,

minimize Jact = −LT Ũ

Subject to V (x) =NE∑i=1

xivi ≤ V

0 < xi ≤ 10 ≤ Pi ≤ 1 (26)

where L is a vector with a value of 1 that corresponds tothe output displacement node and 0 otherwise. In addition,a constraint is defined on the final volume of the optimizeddesign. V is the target volume which is a fraction of theoverall volume of the design domain while vi is the volumeof each element and NE is the total number of elementswhile i is the number of each element.

3.2.2 Sensitivity analysis

For applying a gradient-based optimization, the sensitivityof objective function with respect to the optimizationvariables should be calculated. As such, the sensitivity ofobjective function with respect to the xi can be derived as

∂J

∂xi= ∂

∂xi

(−LT Ũ + ΛT

(K̃uuŨ + K̃uφΦ̃ − F̃

))

= ∂∂xi

((−LT + ΛT K̃uu

)Ũ + ΛT K̃uφΦ̃ − ΛT F̃

)

(27)

Through using the procedure known as adjoint method,Λ is introduced to avoid taking the derivative of displace-ment with respect to design variable, i.e., ∂ũi

∂x. Therefore, the

following adjoint equation should be solved

− LT + ΛT K̃uu = 0 (28)where Λ is the global adjoint vector. By solving the adjoint(28), the sensitivity values can be obtained as

∂J

∂xi= λTi

∂k̃uu

∂xiũi + λTi

∂k̃uφ

∂xiφ̃i (29)

where λi is the elemental format of global adjoint vector Λ.By using the same procedure, sensitivity analysis with

respect to polarization P is derived in the following form aswell

∂J

∂Pi= λTi

∂k̃uφ

∂Piφ̃i (30)

where λi is the same adjoint vector which is alreadycalculated in (28).

Based on (29) and (30), the derivative of piezoelectricstiffness and coupling matrices with respect to designvariables is required which can be derived with the help of(25) as

∂k̃uu

∂xi= puu(E0 − Emin)xpuu−1i k̃uu

∂k̃uφ

∂xi= puφ(e0 − emin)xpuφ−1i (2Pi − 1)pP k̃uφ (31)

∂k̃uφ

∂Pi= 2pP (e0 − emin)(2Pi − 1)pP −1xpuφi k̃uφ (32)

After performing the sensitivity analysis, and definingthe constraint, the optimization of variables should be doneby the optimization algorithms which will be discussed laterin this section.

3.3 Energy harvesting

3.3.1 Problem formulation

In the case of energy harvesting optimization, electri-cal boundary condition should be applied in addition tomechanical boundary condition. As mentioned previously,by considering perfectly conductive electrodes, the equipo-tential boundary condition can be applied by using theBoolean matrix in the following form (Cook and et al 2007)

Φ = BVp (33)in which Vp is the voltage of the top electrode while thebottom electrode considered as ground. For a general case ofmulti-layer piezoelectric plates, Boolean matrix B is havingthe dimension of Ne ×NP where Ne is the number of nodesand NP is the number of electrodes. However, in the case of

the one piezoelectric plate of this paper, B will be a vectorof ones.

It is worth to note that the equipotential condition givenin (33) will be applied only to the energy harvesting FEM(19). For the case of actuation, the equipotential conditionwill be applied automatically by defining equal appliedvoltage for all of the elements.

By applying the equipotential condition in (33), theenergy harvesting FEM (19) can be rewritten as[

Kuu Kuφ

Kφu −Kφφ] [

Ũ

Vp

]=

[F̃

0

](34)

where

Kuu =[K̃uu − M̃Ω̃2

]bc

Kuφ =[K̃uφB

]bc

Kφφ = γBT K̃φφB (35)in which ([ ]bc) shows the application of mechanicalboundary condition.

Now, the objective function for the energy harvestingapplication should be defined. Generally, the energy con-version ratio (Zheng et al. 2009; Noh and Yoon 2012) orelectromechanical coupling coefficient (de Almeida 2019)which is equivalent mathematically is chosen as the objec-tive function. However, this format of objective functionsuffers from numerical instabilities during optimization iter-ations where it is suggested to penalize the mechanicalenergy as suggested in de Almeida (2019) and Salas et al.(2018). Therefore, to avoid the numerical instabilities here,a classical format of objective function is defined. The opti-mization is defined as minimization of the weighted sum ofthe mechanical and electrical energy of the system,

minimize JEH = wjΠS − (1 − wj)ΠE

Subject to V (x) =NE∑i=1

xivi ≤ V

0 < xi ≤ 10 ≤ Pi ≤ 1 (36)

ΠE and ΠS are electrical and mechanical energiesrespectively which are defined in the following form (Nohand Yoon 2012; Zheng et al. 2009)

ΠS =(

1

2

)ŨT KuuŨ, Π

E =(

1

2

)V Tp KφφVp (37)

In optimization of (36), wj is the weighing factor whichhas the value between 0 and 1. Choosing the value of 1for wj will make the optimization problem, a minimumcompliance problem in which the goal is to minimize themechanical deflection of the system under the applied force.By decreasing the value of wj , more weight will be given tomaximize the output electrical energy. However, choosing

very small values for wj will result in mechanically unstablelayouts. Therefore, the value for wj will be found by usingtrial and error approach. The basis for choosing this valuecan be the maximum energy conversion factor of the plateunder the same force. For example, in Homayouni-Amlashiet al. (2020b), the maximum energy conversion factor foran optimized piezoelectric plate under planar excitation isfound to be 0.03 while in Noh and Yoon (2012) for a two-layer optimized piezoelectric plate under the bending forcethis ratio is 0.1. Therefore, the initial value of wj for thetrial error approach can be considered between 0.01 and0.1. The final chosen value of wj depends on the maximumstress and strain induced by the defined mechanical input tothe structure which can be revealed by the post processinganalysis. On the other hand, stress and strain constraintscan also be considered in the optimization problem as it isinvestigated by Wein et al. (2013).

In fact, the advantage of the objective function definedhere is that the optimization algorithm converges verysmoothly to the final result. However, the drawback of thisobjective function is that the obtained result can be sub-optimal depending on the chosen value for the wj . On theother hand, the problem of sub-optimal results exists inother formats of the objective function. For example, Nohand Yoon (2012) showed that by considering the energyconversion factor (ΠE/ΠS) as objective function, differentvalues of penalization factors can produce different results.

Eventually, we believe that the chosen objective functionsuits the educational purpose of this paper. Indeed, withthe help of provided MATLAB code, the readers can easilychange the code to implement other objective functions.

3.3.2 Sensitivity analysis

Similar to the actuation case, the next step after definingthe objective function is sensitivity analysis. Since theobjective function in (36) consists of mechanical andelectrical energies, the sensitivity of each energy withrespect to density ratio x can be found as (Zheng et al. 2009;Homayouni-Amlashi 2019; Homayouni-Amlashi et al.2020b)

∂ΠS

∂xi=

(1

2ũTi + λT1,i

)∂(k̃uu − m̃Ω̃2)

∂xiũi

+λT1,i∂k̃uφ

∂xiφ̃i + μT1,i

∂k̃φu

∂xiũi − μT1,i

γ ∂k̃φφ

∂xiφ̃i (38)

∂ΠE

∂xi= 1

2φ̃Ti

γ ∂k̃φφ

∂xiφ̃i − μT2,i

γ ∂k̃φφ

∂xiφ̃i

+λT2,i∂(k̃uu − m̃Ω̃2)

∂xiui + λT2,i

∂k̃uφ

∂xiφ̃i + μT2,i

∂k̃φu

∂xiũi

(39)

in which μ and λ are the elemental adjoint vectors whichare calculated by the following global coupled system[

Kuu Kuφ

Kφu −Kφφ] [

Λ1Υ1

]=

[ −KuuŨ0

]

[Kuu Kuφ

Kφu −Kφφ] [

Λ2Υ2

]=

[0

−KφφVp]

(40)

where Λ and Υ are the global adjoint vectors which need tobe disassembled to form the elemental adjoint vectors

[λ1]bc = Λ1, [λ2]bc = Λ2, [μ1] = BΥ1, [μ2] = BΥ2 (41)Now, the sensitivities with respect to polarization (P )

are calculated as well (Homayouni-Amlashi et al. 2020b;Homayouni-Amlashi 2019)

∂ΠS

∂Pi= λT1,i

∂k̃uφ

∂Piφ̃i + μT1,i

∂k̃φu

∂Piũi

∂ΠE

∂Pi= λT2,i

∂k̃uφ

∂Piφ̃i + μT2,i

∂k̃φu

∂Piũi (42)

Based on sensitivity equations in (39) and (42), thederivative of all piezoelectric matrices with respect to thedesign variables is required. The derivative of stiffnessand coupling matrices is found in (31) and (32). Here,the derivative of dielectric matrix and mass matrix is alsorequired which are

∂k̃φφ

∂xi= pφφ(ε0 − εmin)xpφφ−1i k̃φφ

∂m̃

∂xi= m̃i (43)

In addition to derivative of piezoelectric matrices withrespect to density, derivation of the piezoelectric couplingmatrix with respect to polarization variable is also required

∂k̃uφ

∂Pi= 2pP (2Pi − 1)pP −1xpuφi k̃uφ (44)

After calculation of sensitivities, the optimization vari-ables can be updated in each iteration of optimization withthe help of optimization algorithm which is the subject ofthe next section.

3.4 Optimization algorithms

For solving the optimization problem, there are severaloptimization algorithm like sequential linear programming(SLP), sequential quadratic programming (SQP), methodof moving asymptotes (MMA) or the optimality criteria(OC) method. This latter one is more historical than theother methods and its application is more simple. However,MMA is more powerful in terms of solving multi-variableand multi-constraint optimization problems. In addition, the

convergence in MMA is more assured due to considerationof two past successive iterations during optimization.

Therefore, in this paper to solve the actuation problem,the OC method is implemented so the proposed code isself-working. However, the OC method has convergenceproblem in energy harvesting code due to coupling effectof piezoelectric material. In the upcoming two sections, theOC method and MMA and their implementation codes willbe explained.

3.4.1 Optimality criteria method

Optimality criteria is a heuristic method to update thedesign variables in each element of design domain duringeach iteration of optimization. Here, there are two designvariables for each element including the density andpolarization. It is common for structural optimization withoptimality criteria that the mutual influence of the designvariables on each other and from element to element canbe ignored (Hassani and Hinton 1998). Therefore, they canbe updated separately from each other in each iteration ofoptimization. In fact, this is similar to the case of topologyoptimization by homogenization method where the length,width and rotation angle of the hole in a microstructure areoptimized by OC separately in each iteration of optimization(Suzuki and Kikuchi 1991).

Here, the densities should be chosen such a way torespect the volume constraints. In this case, the followingKarush–Kuhn–Tucker (KKT) condition for intermediatedensities (0 < xi < 1) should be satisfied to guarantee theconvergence

∂J

∂xi+ λ̄ ∂V

∂xi= 0 (45)

in which λ̄ is the Lagrange multiplier to augment the volumeconstraint in the optimization. To solve the optimizationproblem mentioned in KKT (45), the OC algorithm given byBendsøe and Sigmund (1995) and Bendsoe (2013) is usedhere which can be written as

xnewi =⎧⎨⎩

max(0, xi − move)min(1, xi + move)

xiβηi

if xiβηi ≤ max(0, xi − move)

if xiβηi ≥ min(1, xi − move)

otherwise(46)

where move parameter is the maximum amount of densitychange in each iteration of optimization, η is a numericaldamping coefficient and

βi = − ∂J∂xi

(λ̄

∂V

∂xi

)−1(47)

Similar to the classical compliant problems of passivematerial (Bendsoe 2013), the values of move and η areconsidered to be 0.2 and 0.3 respectively.

For optimization of polarization there is no volumeconstraint. In fact, since the polarization of each elementwill be optimized separately, it is not necessary to forcethe optimization algorithm to push the polarization value tozero for the elements with low density values. Therefore, theoptimization problem can be simply defined as follows

∂J

∂Pi= 0 (48)

The algorithm for this simple optimization can beobtained by modifying the OC algorithm given in (46) in thefollowing form

P newi ={

max(0, Pi − move)min(1, Pi + move)

if ∂J∂Pi

≥ 0if ∂J

∂Pi< 0

(49)

By defining the optimization algorithm for polarizationbased on (49), the polarity value (Pi) for all of the elementswill be steered to −1 or +1 even for the elements with verylow density. However, this will not affect the optimizationresults since based on (25) and with maximum value ofpolarization (P ), low density (x) will push the couplingmatrix to its minimum value.

3.4.2 The method of moving asymptotes

The method of moving asymptotes (MMA) is a structuraloptimization method proposed by Svanberg (1987). Theproblem formulation of this methodology is as follows(Svanberg 2007)

minimize f0(χ) + a0z +mc∑i=1

(ciyi

+ 12diy

2i

)

subject to fi(χ) − aiz − yi ≤ 0 i = 1, ..., mcχ ∈ X, y

i≥ 0, z ≥ 0 (50)

in which X = {χ ∈ �nvar |χmini ≤ 0 ≤ χmaxi}, while nvar is

the number of design variables, χ is the vector of all designvariables, y

iand z are the artificial optimization variables,

f0(χ) is the cost function to be minimized, mc is thenumber of constraints and a0, ai, ci , di are the coefficientswhich have to be determined to match the optimizationproblem mentioned in (50) to different types of optimizationproblems.

For the optimization problem of this paper, there is onlyone constraint defined on the maximum volume of theoptimized design. Therefore, based on the description givenin Svanberg (2007), by considering a0 = 1 and ai = 0 forall i, then z = 0 in any optimal solution, and by consideringdi = 0 and ci = “a large number,” then the variables yi = 0

for all i and the optimization problem mentioned in (50) willbe matched to the optimization problem of this paper.

Finally, the OC and MMA optimization algorithmscan update the optimization variables in each iteration ofoptimization. To do so, the implementation code for both ofOC and MMA will be given in Section 4.

3.5 Filtering

Like other problems in structural optimization, topologyoptimization of piezoelectric materials also suffers fromnumerical instabilities such as checker board problem ormesh dependency. To remedy, the solutions need to befiltered in each iteration of optimization. So far, manyfiltering methods are suggested in the literature (Sigmund2007). Among the proposed filtering methods, sensitivityfilter and density filter proved their success in overcomingthe aforementioned numerical instabilities. The sensitivityfilter is used in the 99 lines of topology optimization codewritten by Sigmund (2001) and density filter is used as analternative option in the 88 lines of topology optimizationcode written by Andreassen et al. (2011). As such, inthis paper, the density and sensitivity filters are used astwo available options. To implement these filters, the samelines of codes written in 88 lines of code by Andreassenet al. (2011) are employed. Therefore, the concepts behindthese filters will not be explained here since the relatedexplanations can be found in the mentioned references.

The important point here is that the density filter showsmore promising performance in comparison with sensitivityfilter specially in the case of energy harvesting problem withusing the MMA as the optimization algorithm.

Finally, the filters will be applied only to the densitiesand filtering of the polarization variable is not necessary.

3.6 Choosing proper penalization factors

After preparing the ingredients of topology optimizationalgorithm, the important question would be how to tunethe parameters of optimization? in addition to parameterswhich are the same for optimization of passive and activematerials like radius of filtering and volume fraction, thereare other parameters which belong to multiphysics natureof piezoelectric topology optimization like penalizationcoefficients of the piezoelectric matrices, i.e., puφ , pφφ andpP . In fact by choosing the penalization coefficients, weare pursuing two goals: (1) guaranteeing the convergenceto perfect void-material in the final obtained layout, (2)avoiding local optima. For choosing puu, it is already proventhat the value of 3 is the best choice (Bendsoe 2013) to reacha perfect void/material in the final layout.

There are several studies which focused on definingthe criteria for choosing other penalization coefficients,i.e., puφ , pφφ and pP (Kim and Shin 2013; Noh andYoon 2012; Kim et al. 2010). In particular, Noh andYoon (2012) chose the penalization factors randomly andobtained different density layouts. The final conclusionwas that the penalization factors have extreme effect onthe final topology. But, no criteria or rule is presented inthis study about the method of choosing the penalizationfactors. On the other hand, a detailed study is presentedby Kim et al. (2010) for choosing the penalization factors.Based on this study, necessary condition for choosing thepenalization factors is that the electromechanical couplingcoefficient (EMCC) should be increased when the density(x) increases and vice versa. Based on this condition, thefollowing intrinsic condition which is independent fromobjective function is proposed for choosing the penalizationfactors (Kim et al. 2010)

2puφ −(puu + pφφ

)> 0 (51)

In addition to this condition, Kim et al. (2010) proposedother objective dependent criteria for the actuation andenergy harvesting application. In particular, for the actuationobjective function in (26), the following condition isproposed (Kim et al. 2010),

puφ − puu > 0 (52)and for energy harvesting approach, when the objectivefunction is energy conversion factor then the followingconditions should be satisfied (Kim et al. 2010),

puφ − puu > 0pφφ − puu > 0 (53)It should be noted that these conditions are proposed to

guarantee the convergence of the final topology and yetthere is no study on the methods to define these penalizationfactors to avoid the local optima.

For polarization penalization, the value of 1 seems thebest choice as it is suggested by Kögl and Silva (2005).Indeed, it is not necessary to penalize the polarization.

4MATLAB implementation codes

In this section, the goal is to establish the MATLAB code forthe piezoelectric optimization methodology explained in theprevious section. Two MATLAB codes are mentioned in theAppendix of the paper. The first code is for actuation and thesecond code is for energy harvesting. Each MATLAB codeis partitioned so the readers can have a perception of the goalof each part. The important lines of codes are also labeled

to provide a connection between the line and the analyticalcalculation. Again, the rest of this section will be divided foractuation and energy harvesting codes and in each sectiondifferent parts of the codes will be explained more in detail.

4.1 Actuation

4.1.1 General definition

The first part of the code is GENERAL DEFINITION. Thispart consists of geometrical dimensions of the piezoelectricplate, the resolution of the mesh, i.e., the number of

elements in the→x and

→y axes, the penalization factors,

the type of filter, the filter radius and maximum number ofiterations and the stiffness of the attached spring are defined.It should be noted that two stopping criteria are definedby the codes. The classical criteria is the density changebetween two last successive iterations. The other criteriais the maximum number of iterations. Satisfying either ofthese will stop the optimization. The reason behind addingthe second criteria is that in contrast to pure mechanicalproblems, here we have different types of material anddifferent types of objective functions. As such, the densitychange will not stop the optimization generally or it needsvery high number of iterations while there are no significantoscillations in the objective function’s value.

4.1.2 Material properties

The second part of the code is MATERIAL PROPERTIES.In this part of the code, the properties of the chosenpiezoelectric material are given. These properties are thedensity, e∗31 coupling coefficient, �∗33 permittivity coefficientand the elements of the stiffness tensor all after applyingthe plane stress assumption as mentioned in (4). The chosenmaterial for the code is the PZT 4 which is popular in theliterature. There are PZT materials with lower and highercoupling coefficients. If one wants to investigate the resultsof other piezoelectric materials, the values of this part canbe changed.

It should be noted that choosing different PZT materialsin this section will not affect the final layout obtained byoptimization algorithm. In fact the resulted layout of theoptimization is independent from Piezoelectric coefficients.

4.1.3 Finite element model

In the section called PREPARE FINITE ELEMENTANALYSIS, the proposed code for the finite element modelof the piezoelectric plate is the extension of the MATLABcode provided by Kattan (2010) for the bilinear quadrilateralelements of the passive materials. However, instead of using

analytical calculations, elemental matrices are calculated byusing the two-point Gauss quadrature method. To do so, firstthe geometrical sizes of each element are calculated basedon the defined geometry of the design domain and desiredresolution of mesh. Then, Gauss quadrature points aredefined in matrix GP. Afterwards, the elemental matricesare found inside a loop with the help of Gauss points asfollows,

In this method of elemental matrix calculation, there is noanalytical integration, the calculation time is very fast andat the same time the code is flexible in terms of differentgeometrical dimensions and mesh resolution and aspectratios.

After calculating the elemental matrices, normalization isapplied based on (20) and normalization factors are obtainedwith the following lines of code

It should be noted that for simplicity, the non-normalizedmatrices are replaced by normalized matrices. Since thenormalization factors are saved, it is easy to find the realvalues after the calculation of the final results.

So far, the elemental matrices are calculated, normalizedand the normalization factors are derived. For assemblingthe elemental matrices, the element connectivity matrixknown as edofMat (Andreassen et al. 2011) is builtwith the help of the numbering format shown in Fig. 2.In this figure, the coarse discretization is shown withthe numbering of elements, nodes and mechanical andelectrical degrees of freedom. This numbering format issimilar to the one presented in 99 (Sigmund 2001) and88 (Andreassen et al. 2011) lines of MATLAB codes forstructural problems. The numberings of the nodes andmechanical degrees of freedom are from top to bottomand from left to right. It should be noted that insideeach element, the numbering order is different as it isshown in Fig. 2b. The numbering inside each element iscounterclockwise and it starts from bottom left and thesequence of numbers in each row of edofMat matrixis following this numbering order. Furthermore, since thesystem here is having additional degree of freedom aspotential, a vector of potential connectivity is defined asedofMatPZT. In fact, by considering one potential degreeof freedom for each node, potential connectivity would be a

matrix with rows containing the node IDs. However, due toequipotential condition, one potential degree of freedom isconsidered for each element and the potential connectivityis a vector. At last, the connectivity matrix edofMat and thepotential connectivity vector edofMatPZT is defined as

The implementation lines to create the edofMat matrixand edofMatPZT vector with the help of nodeIDs are

where nele is the number of elements, nodenrs is the matrixof node numbers and edofVec is the vector of nodeIDscontaining the first nodeID of each element. Now, withthe edofMat and edofMatPZT, it is possible to assemblethe elemental matrices and build the global matrices withfollowing lines of code

The matrices iK and jK corresponds to (i, j) entry ofstiffness matrix for each element. With the same strategy,iKup and jKup are written for the ith and jth entry of Kuφ

matrix. To avoid redundancy, there is no iKpp and jKpp. Infact, these latter are just equal to the edofMatPZT.

The material interpolation scheme mentioned in (25),is applied in sKuu and sKup matrices. xPhys(:) andpol(:) are the vectorized physical density and polarizationmatrices respectively which will be updated in each iterationof optimization.

4.1.4 Boundary condition

The mechanical boundary conditions are defined in thepart DEFINITION OF BOUNDARY CONDITION. Inthis part, fixeddofs and freedofs are defined to containthe fixed degrees of freedom and the free degrees offreedom respectively. They are defined by using the methodreported in 88 lines (Andreassen et al. 2011) and 99lines (Sigmund 2001) of code. Therefore, for applying theclamped boundary condition on the left side of the designdomain, the following lines of code can be used,

By considering that for the coarse mesh in Fig. 2, nelx= 4 and nely = 3, fixeddofs in the aforementionedimplementation line produce the numbers from 1 to 8 whichare the mechanical degrees of freedom in the left side of thedesign domain.

4.1.5 Output displacement definition

In this part of the code for actuation, the goal is todefine the particular point of the design domain where themaximization of displacement in a particular direction isdesired. To do so, the desired mechanical degree of freedomshould be defined. The variable DMDOF which is theabbreviation of desired mechanical degree of freedom isdefined for this purpose. Thereafter, the vector L is created.This vector will be used later in the objective function andsensitivity analysis for optimization.

A spring is attached at DMDOF which simulates thereaction force exerted by an imaginary object. The modeledspring will modify the piezoelectric stiffness matrix withthis line of code

where Ks is the stiffness of the modeled spring. Actually,by changing the stiffness of the modeled spring, it ispossible to determine whether more force is desired ormore displacement. Indeed, since the stiffness matrix isnormalized, the stiffness of the modeled spring can bedetermined with respect to the stiffness of the piezoelectricplate. For example, by putting the stiffness of the spring

equal to 1, the stiffness will be equal to the highestvalue of the piezoelectric stiffness matrix. In this case,the piezoelectric layout will be optimized to producemaximum possible of force. However, by defining verylow values of stiffness for the spring (i.e., 0.01), thenthe piezoelectric layout will be optimized for maximumpossible of deflection.

4.1.6 Objective function

Calculation of objective function for actuator is a routineprocedure to calculate the desired displacement in eachiteration. To do so, first the mechanical displacement due toapplied voltage should be calculated. Therefore, based on(21), the mechanical displacement vector is calculated bythe following line of code

then based on (36), the objective function can be calculatedby

4.1.7 Sensitivity analysis

For sensitivity analysis, the first step is the calculation ofadjoint vectors. It is calculated based on (28),

then the calculation of sensitivities based on (29) and (30)starts afterwards

where dc and dp are the sensitivities of the objectivefunction with respect to x and P and dv is the sensitivitiesof the volume constraint with respect to x. It is obvious from

(26) that the sensitivity of volume constraint with respect toP is zero.

As can be seen in the aforementioned lines of the code,the sensitivities containing the stiffness (kuu) and couplingmatrix (kup) are calculated separately. This will help us todefine different penalization coefficients for each of thesematrices.

4.1.8 Optimization algorithm

The implementation lines of code for the OC update ofdensities are the same as what is mentioned in 88 linesof code (Andreassen et al. 2011). In addition to densities,for updating the polarization based on algorithm (49), thereis no need for the bi-sectioning loop and just one lineof code can optimize the polarization in each iteration ofoptimization as follows

4.1.9 Plot densities and polarization

In part (PLOT DENSITIES & POLARIZATION), thedensities and polarization will be plotted in two figuresseparately. As it is common, in the density figure, white areameans no material while the black area means material.

Different color spectrum is chosen for polarizationprofile in which the red and blue color shows oppositedirection of optimization while the green color shows theneutral material with no polarization. The important pointhere is that the OC will steer the polarization value to 0 or 1even for the elements with lower density. Therefore, here toeliminate the confusion, the matrix of densities is multipliedto the polarization. In this way, for element with minimumdensity (no material), the polarization turns to green as well.

4.1.10 Filtering

The density filtering lines of code are similar to 88 linesof code (Andreassen et al. 2011). Two types of density andsensitivity filter can be chosen. In general, the density filteris more recommended. However, since the code is writtenfor both of these filters, the best choice will be up to readers.

4.2 Energy harvesting

4.2.1 General definition

In general definition part of the energy harvesting code,in addition to what is mentioned for the actuation code,

the objective function’s weighing factor (wj) and excitationfrequency (omega) are defined.

4.2.2 Material properties

This part of the energy harvesting code is similar to theactuation code. However, the results of actuation are freefrom the PZT coefficients while in the case of energyharvesting for different materials and different piezoelectriccoefficients, the results will be changed. PZT materials withhigher coupling coefficients produce more electrical energyin comparison with ones with lower coupling coefficients.At the same time, PZT materials with higher couplingcoefficients have more coupling effect which affect theoptimization during the iteration and can change the finallayout. Similar to actuation code, PZT 4 is chosen for thecode.

4.2.3 Finite element model

In comparison with FE model presented in the actuationcode, the FEM part of the energy harvesting is the same withsome additional calculations. Indeed, the elemental matricesfor mass and dielectric matrices are calculated as well in thefollowing lines of code,

For assembling the global mass matrix M , the samematrices of iK and jK as described in the actuation code canbe used. The assembling lines of mass matrix and dielectricmatrix K̃φφ can be found in energy harvesting code as

where the related material interpolation scheme is appliedin sM and sKpp.

Furthermore, the stiffness matrix is modified by thedynamic matrix -omega*M which represents (−M̃Ω̃2) in

(23). This modification is done by the following line ofcode

After building the global piezoelectric matrices, theequipotential condition is applied with the following line ofcode

where KupEqui and KppEqui are the piezoelectric cou-pling and dielectric matrices after applying the mechanicaland equipotential condition which represent Kuφ and Kφφ .

4.2.4 Objective function

To find the value of objective function, the mechanicaldisplacement and resulted potential due to applied forceshould be calculated based on (23). But, first a totalmatrix is built based on the left-hand side of (23).The symmetry is guaranteed afterward, and finally, themechanical displacements and potential are calculated

After calculation of mechanical displacement and poten-tial, for calculation of objective function, Wm and We aredefined for mechanical and electrical energies respectively.

It is important to note that in the written code,the mechanical and electrical energies are calculated bysumming the energies in each element. That is whythe mechanical displacement and electrical potentials aremultiplied to elemental stiffness matrix kuu and elementaldielectric matrix kpp. Then, the objective function for theenergy harvester is the weighted sum of the energies whileconsidering the weighting factor (wj).

4.2.5 Sensitivity analysis

Sensitivity analysis starts by calculating adjoint vectors withthe help of (40)

Here, B is the Boolean matrix as defined in (41). In fact,mu1 and mu2 are vectors of equal values for all of the elementsin the design domain due to equipotential condition.For sensitivity analysis, the adjoint vectors related tomechanical and electrical states should be separated. Forthis reason, ADJ resolved to lambda and mu.

After calculation of adjoint vectors, the sensitivityanalysis is performed by the following lines of code

The code is written in a general form to considernf number of load cases. Therefore, for each numberof load cases the mechanical displacement, potential andadjoint vectors are separated by defining the Uu i, Up i,lambda1 i, lambda2 i, mu1 i and mu2 i. In addition,mechanical displacement, potential and adjoint vectors arecalculated in global format while the sensitivity analysisshould be performed on the elemental scale. To do so,edofMat is utilized to convert the vector of mechanicaldisplacement to the matrix Uu i(edofMat) in whicheach row is related to one element while the columnsrepresent the mechanical displacement of that element.The same idea is also applied to the adjoint vectors, i.e.,lambda1 i(edofMat) and lambda2 i(edofMat). On the otherhand, due to equipotential condition, it is obvious that thepotential of each element is equal to Up, and as mentionedbefore, values of mu1 and mu2 are equal for all elements.

Similar to the actuation code, here the sensitivities con-taining each piezoelectric matrix are calculated separatelyto facilitate the definition of different penalization factors.

4.2.6 Optimization algorithm

In the energy harvesting code, the MMA optimizationalgorithm is used. However, this code cannot be executedwithout having the external MMA code. The MMAimplementation MATLAB code can be obtained bycontacting Prof. Krister Svanberg. In this way, the interestedreaders will obtain two MATLAB codes to implement theMMA optimization code which includes mmasub.m andsubsolv.m. The version 2007 of these codes is used in thispaper. Supposing that these two MATLAB codes are alreadyavailable, to implement the MMA, first initial parametersare defined in MMA Preparation part based on what ismentioned in Section 3.4.2. Then, optimization of variableswill be done in MMA OPTIMIZATION OF DESIGNVARIABLES part. The important point in this part is that,due to the scale difference between the density sensitivitiesand polarization sensitivity, it is suggested here to normalizethe polarization sensitivity with the following line of code

Then, the main lines of MMA optimization implementa-tion are

The output xmmaa is the vector of all optimizationvariables. Therefore, the upper half of this vector is theupdated density and lower half is the polarization values.

Fig. 3 Topology optimization of a piezoelectric actuator (pusher) fordifferent stiffness of the modeled spring. Panel a presents the spec-ification for a practical use and panel b is the mechanical modelfor implementation in the finite element software. Panels c, e, g, i

and d, f, h, j respectively present the density and polarization pro-file of the design for the specified output stiffness after convergence.For the polarization profile, blue, red and green represent respectivelynegative, positive and null polarization

After updating the optimization variables and plotting theresults, this iteration of optimization will be finished and theoptimization will be started from the beginning of the loopfor the next iteration.

5 Numerical examples

In this section, the goal is to investigate the performancesof the codes in different application cases of actuationand energy harvesting with different configurations. First,different examples of actuation will be investigated.Thereafter, different configurations of energy harvesting areexplored.

5.1 Actuation

5.1.1 Pusher

The first example in the actuation part is a simple pusheras can be seen in Fig. 3a. The gray area shows thedesign domain which can be optimized by the optimizationalgorithm. The actuation optimization code which ismentioned in the Appendix is written for this example. As itcan be seen from the code, the aspect ratio of the elements inthe x and y directions is following the aspect ratio betweenthe length and width of the plate which produces squareelements for discretization of the design domain. This is notmandatory, but it is known that increasing the aspect ratio of

width and length of each element increases the inaccuracyof the finite element model (Logan 2000). Therefore, itis recommended to follow the aspect ratio of the plate indefining the number of elements in x and y directions.

The chosen penalization factors are puu = 3 and puφ =4, which satisfy the conditions mentioned in (52). Thesepenalization factors are the same for all of the actuationexamples.

For having a completely symmetrical response withrespect to the horizontal dotted line in Fig. 3a and todecrease the number of elements in the design domain, thedefined design domain in the code is the upper half of thepiezoelectric plate. Therefore, in the symmetry line of thedesign domain, the roller mechanical boundary condition isapplied with the following line of code,

With this mechanical boundary condition, the nodesconnected to the symmetry line can have displacement inthe x direction but not in the y direction.

As can be seen in Fig. 3, the results of topologyoptimization for different spring stiffness are plotted. Theupper row of the figure shows the density layout while thelower row shows the polarization profile. By mirroring theobtained result with respect to the symmetry line, the resultsare illustrated for whole piezoelectric plate.

The numerical results for different spring stiffness arealso reported in Table 1. The numerical results show the

Table 1 Displacement amplification ratio of optimized actuators withrespect to full plate

Pusher Gripper

ks Amplification ks Amplification

ratio ratio

CASE (1) 1 0.76 1 3.95

CASE (2) 0.1 0.99 0.1 11.42

CASE (3) 0.01 1.54 0.01 34.04

CASE (4) 0.005 2.75 0.003 60.11

amplification ratio of the optimized design with respect tothe full plate under application of same value of voltage. Theobjective value which is reported by the code is not showingthis amplification ratio. To calculate the amplification ratio,the final value of the objective function after finishing theoptimization should be divided by the objective functionvalue of the full plate. To find the objective function of thefull plate, it is possible to define the initial values of densityequal to 1. To do so, the following line of code

put 1 instead of volfrac, then stop the code after thecalculation of objective function. In this case, the value ofobjective function for the full plate is obtained.

From the plots in Fig. 3c and d, it is clear that whenthe spring stiffness is one, the optimal layout is a verysimple lumped design with uniform polarization profile.Based on Table 1, for this case, the full plate is havingmore displacement. On the other hand, by considering verylow stiffness (Ks = 0.005) and then based on Fig. 3i andj, the density layout is more complicated and polarizationprofile is not uniform anymore. In fact, it is obvious that theblue region in the polarization profile will have extensionwhile the red part will have compression (shrinkage). Thecombination of this extension and compression will producean amplification ratio with respect to full plate equal to 2.75as reported in Table 1.

5.1.2 Gripper

The second example of piezoelectric actuation is a gripper,which is similar to the case discussed in Ruiz et al. (2017).The goal is to design a gripper to grab an object as it isshown in Fig. 4a and b. To do so, some modifications should

be done to the actuation code in the Appendix. First of all,the OUTPUT DISPLACEMENT DEFINITION part shouldbe completely changed by replacing the following lines

where L grip and W grip are the length and width of theempty box in the piezoelectric plate as shown in Fig. 4aand b.

Next, to enforce zero material in the desired box of thedesign domain, the passive elements should be defined. Thestrategy is the same as in 99 lines (Sigmund 2001) and in 88lines (Andreassen et al. 2011) of code. The following partshould be added after the part INITIALIZE ITERATIONand before the part START ITERATION

Then to apply the passive material in each iteration, thefollowing line should be added after the OC update line

Now by executing the code, the results of Fig. 4 fordifferent spring stiffness will be obtained and the numericalresults are reported in Table 1. It is interesting to note thatfor the gripper, the amplification ratios in optimized designsare much higher than the amplification ratios for optimizedpushers. Indeed, the polarization optimization plays a majorrole in designing the gripper. That is why for any chosenvalues of Ks, the polarization profile is not uniform and the

Fig. 4 Topology optimization of a piezoelectric gripper for different stiffness of the modeled spring. The panels follow the same presentation asFig.3

gripper needs the combination of expansion and retractionfor increasing the amplification ratio.

5.2 Energy harvesting

5.2.1 Lateral force

The first example of the energy harvesting code is a plateunder a lateral force excitation as it is shown in Fig. 5.The code in the Appendix is written for this case. Here,the goal is to maximize the output electrical energy whileminimizing the mechanical energy of the system. For thiscase, the problem is static and the excitation frequencyis considered to be zero. The chosen penalization factors

for the energy harvesting code in contrast to actuationpart are not the same for all cases. As such in Table 2,the penalization factors are reported for each case. But,for all cases, the chosen penalization factors are satisfyingthe conditions mentioned in (51) and (52). The reason fordifferent penalization factors for each case is that the energyharvesting optimization is more complicated than actuationdue to existence of the coupling effect. This coupling effectis highly affected by the chosen penalization factors inparticular puφ and pφφ . By choosing proper penalizationfactors, it is possible to avoid the nonsymmetric results or toimprove the convergence.

The results of optimization for different values of theweighting factor (wj ) are illustrated in Fig. 5. For the first

Fig. 5 Topology optimization of a piezoelectric energy harvester underapplication of a lateral static force for different values of wj . Panel apresents the specification for a practical use and panel b is the resultthat can be obtained through classical compliance optimization. Panels

c, e, g, and d, f, h respectively present the density and polarization pro-file of the obtained design. For the polarization profile, blue, red andgreen represent respectively negative, positive and null polarization

Table 2 Output energies of the optimized designs for different weighting factors

wj ΠS ΠE puu puφ pφφ pP Ω x{0} MMA move

Lateral force

Compliance 1 116.66 0.00 3 6 4 0 0 volfrac 1

CASE (1) 1 99.90 1.25 3 6 4 1 0 volfrac 1

CASE (2) 0.01 152.87 1.70 3 4 4 1 0 volfrac 1

CASE (3) 0.005 741.09 7.77 3 4 4 1 0 volfrac 1

CASE (4) 1 106.04 1.38 3 6 6 1 1 kHz volfrac 1

CASE (5) 0.01 123.15 1.43 3 6 6 1 1 kHz volfrac 0.1

CASE (6) 0.02 240.50 2.63 3 6 4 1 3.5 kHz volfrac 0.1

2 load case

Compliance 1 106 0.03 3 6 6 0 0 1 1

CASE (1) 0.005 168.54 1.20 3 6 6 1 0 1 1

CASE (2) 1 107.88 0.86 3 6 6 1 1 kHz 1 0.1

CASE (3) 0.005 163.43 1.04 3 6 4 1 1 kHz 1 0.1

CASE (4) 0.05 120.06 0.95 3 6 6 1 1.5 kHz 1 0.1

case, the weighting (wj ) is equal to 1. As such, the problemis now a compliance problem in which minimization ofdeflection is the target. In this case, the optimization isdone without polarization optimization. To do optimizationwithout polarization, it is possible to simply put thepenalization factor for the polarization equal to 0, i.e.,penalPol = 0 in GENERAL DEFINITIONS part of thecode. As can be seen in Fig. 5b, the obtained densitylayout for this case is similar to the results of the topologyoptimization of passive materials as reported by 99 lines(Sigmund 2001) or 88 lines (Andreassen et al. 2011) ofMATLAB code. This was expected since the PZT materialshave the plane isotropic behavior. The numerical results ofthe optimizations are given in Table 2. It is reported for theaforementioned case that the output electrical energy is zerowhich is due to the charge cancellation. In fact, lateral forceinduces tension and compression in different parts of thepiezoelectric plate which produces voltages with oppositesign on the surface of the electrode. The opposite signs ofvoltages nullify each other.

For the next case, polarization is also optimized byputting penalPol = 1. In Fig. 5c and d, it is obviousthat the density layout did not change and the polarizationprofile is not uniform anymore. By this polarizationoptimization, based on Table 2, not only the mechanicalenergy of the piezoelectric plate is reduced in comparisonwith the first case, but also the problem of chargecancellation is suppressed and we have a non-zero electricalenergy.

In the next case, the goal is to increase the electricalenergy due to the same amount of force. To do so, theweighting factor is decreased to 0.01. As can be seenin Fig. 5e and f, the density layout is changed andthe polarization profile is changed accordingly as well.By observing the obtained numerical results in Table 2,the electrical energy is increased in comparison with theprevious cases with the cost of increasing the mechanicalenergy as well. By further increasing the weighting factor,the results of Fig. 5g and h will be obtained in which ajump in both mechanical energy and electrical energy of thedesign can be seen in Table 2.

It can be noticed that in the optimized polarization profiles,there are areas with null polarity (green color) while thereare materials (non-void). This null polarity is mostly atplaces where there is a transition between the polarizationdirection. It is possible to take these null polarity areas intoconsideration in the optimization problem formulation asdiscussed in Donoso and Sigmund (2016).

In the next example, the piezoelectric plate with thesame configuration of boundary and load condition isconsidered while the force is considered to be harmonic. InFig. 6, the results of topology optimization under harmonicforce can be seen. The related numerical results are alsoreported in Table 2. In this figure, the excitation frequencyis primarily considered to be 1 kHz. At this frequency,the optimization converges to the final layout for differentvalues of wj . However, by increasing the frequency ofexcitation, convergence problems begin. The problems are

Fig. 6 Topology optimization of a piezoelectric energy harvester underapplication of a lateral harmonic force for different values of wj . Panela presents the specification for a practical use and panels b, d, f,

and c, e, g respectively present the density and polarization profile ofthe obtained design. For the polarization profile, blue, red and greenrepresent respectively negative, positive and null polarization.

parasitic effects of the material layout and the force will bedisjointed from the material. These problems are due to thefact that the excitation frequency is close to the resonanceand antiresonance frequency of the piezoelectric plate. Theresonances are the natural frequencies when the electrodesare short-circuited and antiresonances are the ones whenthe electrodes are in open-circuit condition (Lerch 1990).During the optimization, these frequencies are changing ineach iteration. Therefore, it is possible that the excitationfrequency comes close to the resonance frequency duringthe optimization which will introduce singularity in theFEM (23) and the amplitude of the displacement vectorreaches to infinity. Even by defining the damping, still thejump in the displacement vector will result in numericalinstabilities and disjoint problem in which the force willbe disconnected from material as it is reported by Nohand Yoon (2012). One solution to reduce the numericalinstabilities is to restrict the move limit of the MMAcode. Indeed, the move limit of the MMA optimizationcode is by default set to 1 which means the density ofmaterial can jump from 0 (void) to 1 (material) in a singleiteration. To modify this move limit, in the mmasub.m,one needs to change the value of move from 1 to 0.1.With this restriction on the move limit, convergence toa black and white final layout is achieved for 3.5 kHzexcitation frequencies as it is shown in Fig. 6f and g.However, by reducing the MMA move limit, the final resultcan be trapped in the local optima. This is the maximumexcitation frequency that the convergence can be achieved.After this frequency, again the force will be disjointedfrom the material and the convergence problem can beseen. To overcome the challenges of dynamic topologyoptimization of mechanical structures, several methodscan be found in the literature. For example, Olhoff andDu (2005) suggest that the excitation frequency can be

increased gradually during the optimization iterations. Liuet al. (2015) and Jensen (2007) modeled the damping in thedynamic system and optimized the structure for an intervalof frequencies including the resonance ones. For mechanicalstructures, the goal is to reduce the displacement or storedmechanical energy in the system. For piezoelectric energyharvesters in which the maximization of electrical outputregarding the mechanical input is desired, Noh and Yoon(2012) solved the disjoint problem of force and materialin dynamic topology optimization by defining a constrainton the mechanical energy of the piezoelectric structure andthey defined the objective function as maximization of theelectrical output of the system.

Reducing the weighting factor wj brings the resonancefrequency closer to the excitation frequency to someextent. However, a very low weighing factor can introduceconvergence problems as well. On the other hand, forpiezoelectric energy harvesters, matching the resonancefrequency and excitation frequency is favorable. To do so,the problem formulation should be changed. For example,Kim and Shin (2013) did eigenfrequency optimization toincrease the electromechanical coupling coefficient of thedesign and to match the resonance frequency and theexcitation frequency. Wang et al. (2017) and Nakasoneand Silva (2010) formulated the optimization problem tooptimize the eigenmodes in addition to optimization ofthe eigenfrequency. It is worth mentioning that for theseproblem formulations, alternative material interpolationfunctions should be used to avoid the artificial local modesin the low-density regions (Pedersen 2000). The alternativematerial interpolation functions can be the one introducedby Huang et al. (2010) for the stiffness matrix or the RAMPinterpolation function which is used by Nakasone and Silva(2010) in combination with the PEMAP-P interpolationfunction.

Fig. 7 Topology optimization of a piezoelectric energy harvester underapplication of 2 load cases for different values of wj and Ω . Panel apresents the specification for a practical use and panel b is the resultthat can be obtained through classical compliance optimization. Panels

c, e, g, i and d, f, h, j respectively present the density and polar-ization profile of the obtained design. For the polarization profile,blue, red and green represent respectively negative, positive and nullpolarization

5.2.2 Two loads case

In this example, the goal is to optimize piezoelectric energyharvesters for two loads case as shown in Fig. 7a. In fact,the piezoelectric plate will be optimized for in-plane forcesthat can come from different directions as it is discussed inHomayouni-Amlashi et al. (2020b) since any in-plane forcecan be decomposed to the load cases shown in Fig. 7a. Themechanical boundary condition of this figure is proposed tomake the harvested energy symmetric with respect to theforces in each direction as it is possible.

To implement the two loads case, the following changesshould be made to the energy harvesting code mentionedin the Appendix. The parameters in the GENERALDEFINITIONS part should be changed by putting Lp= 3e−2, Wp = 3e−2, nelx = 100, and nely = 100.To define the boundary condition, the DEFINITIONOF BOUNDARY CONDITION part will be changedcompletely as follows

BCTratio is the ratio of the clamped part to the totallength of the edge. To define the two load case the partFORCE DEFINITION will be changed with the followinglines:

In contrast to previous case, the initial values for thedensities (x{0}) are not equal to volume fraction (volfrac).It is observed that by changing the initial values to one,the obtained results are more symmetric. Therefore, thefollowing line of energy harvesting code is updated,

The results for the defined load and boundary conditionare illustrated in Fig. 7. Figure 7b shows the complianceresult due to static force without polarization optimization.In Fig. 7c and d the excitation frequency is zero, wjis decreased and polarization is also optimized. Othercases are optimized for different excitation frequencies andweighting factors. The numerical results are reported inTable 2. For defined geometry, load and boundary conditionof this example, the highest frequency that the convergenceto black and white is achieved is 1.5 kHz. This is due to the

fact that with the defined geometry and boundary condition,the first resonance frequency is lower than the previousexample of Fig. 6.

6 Discussion

In the actuation code, the optimization algorithm is OC.However, it is very simple to apply the MMA to theactuation code. One has just to copy the MMA Preparationpart from the energy harvesting code and paste it after theINITIALIZE ITERATION part of the actuation code. Then,one has to remove the OPTIMALITY CRITERIA UPDATEOF DESIGN VARIABLES part and substitute the MMAOPTIMIZATION OF DESIGN VARIABLES part from theenergy harvesting code. It will be observed that the obtainedresult remains the same.

It is possible to implement the OC in the energyharvesting code as well. This can be simply done bysubstituting the MMA OPTIMIZATION OF DESIGNVARIABLES part with the OPTIMALITY CRITERIAUPDATE OF DESIGN VARIABLES part of the actuationcode. However, due to coupling effect, sometimes theconvergence problem appears.

It should be noted that the results presented in thispaper are not the best results that can be obtained from thecodes. In fact by changing the penalization factors, volumefraction, filter radius and initial values, different results canbe obtained. However, finding the best results or performingthe parameters analysis on the final results is not the subjectof this educational paper.

There are similar aspects between the proposed codeshere and the 88 lines of MATLAB code (Andreassenet al. 2011), such as the use of different filtering methods,different boundary conditions, and different load cases.These aspects are not discussed here since the procedures ofimplementation are similar.

Extension of the codes for different goals such asdifferent objective functions, multi-material optimization,and 3D finite element modeling is straightforward. Forexample, the 169 lines of code for 3D topology optimization(Liu and Tovar 2014) of passive material are the extensionof the 88 lines of code. The same strategy can be used for theproposed code here to extend the code for 3D finite elementsto consider multi-layer piezoelectric plates and out of planeforces.

7 Conclusion

Two MATLAB codes are proposed for topology optimiza-tion of piezoelectric actuators and energy harvesters. The

codes are developed based on the finite element modelingof piezoelectric materials. The PEMAP-P as an extension ofSIMP approaches is used for material interpolation scheme.Optimality criteria and method of moving asymptotes areused for optimization of element’s density and polariza-tion direction. Different parts of the codes are explainedin detail to make the implementation and extension of thecodes straightforward. Some basic and general examples arechosen to show the effectiveness of the codes. The aim ofthe codes is to help the students and newcomers in the fieldof topology optimization of smart materials in particularthe piezoelectric material. While the codes have been andcan be used for energy harvesting and actuation applica-tions, perspective works include their application to opti-mization of piezoelectric sensors as well as piezoelectricsensors-actuators, also named as self-sensing (BafumbaLiseli and Agnus 2019; Rakotondrabe 2013; Aljanaidehet al. 2018).

Funding This work has been supported by the national CODE-TRACK project (ANR-17-CE05-0014-01, Control theory tools foroptimal design of piezoelectric energy harvesters devoted to birdstracking devices). This work has also been partially supported by theBourgogne Franche-Comté region project COMPACT.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict ofinterest.

Replication of results All the results presented in this work can bereproduced with the MATLAB code available within the paper and byadding reference of this latter.

Appendix: Proof of normalization

Here the goal is to prove the normalization which isproposed by authors in Homayouni-Amlashi (2019) andHomayouni-Amlashi et al. (2020b). To normalize the globalFEM equation of the piezoelectric plate

[Kuu − MΩ2 Kuφ

Kφu −Kφφ] [

U

Φ

]=

[F

0

](A.1)

the first step of normalization is to factorize the valuesdefined in (20) and (22)

Kuu = k0K̃uu, M = m0M̃, Kuφ = α0K̃uφ,Kφφ = β0K̃φφ, U = u0Ũ , Φ = φ0Φ̃, F = f0F̃

(A.2)

It should be noted that since the responses of the system,i.e., U and Φ are unknown, their factorization values will bedefined later.

With the help of factorization defined in (A.2), (A.1) canbe rewritten as follows,

[k0K̃uu − m0M̃Ω2 α0K̃uφ

α0K̃φu −β0K̃φφ] [

u0Ũ

φ0Φ̃

]=

[f0F̃

0

](A.3)

This is two linearly coupled equations. The secondequation can be written as

α0u0K̃φuŨ − β0φ0K̃φφΦ̃ = 0 (A.4)

Then, the Φ̃ will be

Φ̃ = α0u0β0φ0

K̃−1φφ K̃φuŨ (A.5)

The first linear equation from (A.3) can be written as

(k0K̃uu − m0M̃Ω2)u0Ũ + α0φ0K̃uφΦ̃ = f0F̃ (A.6)

By substituting (A.5) to (A.6) and dividing the resultedequation by k0u0, one will have

(K̃uu − m0

k0M̃Ω2

)Ũ + α0φ0

u0k0

α0u0

β0φ0K̃uφK̃

−1φφ K̃φuŨ

= f0k0u0

F̃ (A.7)

Now, it is possible to define,

Ω̃2 = m