An Inhomogeneous Cell-Based Smoothed Finite Element Method … · 2019. 7. 30. · An Inhomogeneous Cell-Based Smoothed Finite Element Method for Free Vibration Calculation of Functionally

Research ArticleAn Inhomogeneous Cell-Based Smoothed Finite ElementMethod for Free Vibration Calculation of Functionally GradedMagnetoelectroelastic Structures

Yan Cai , Guangwei Meng , and Liming Zhou

School of Mechanical Science and Engineering, Jilin University, Changchun 130025, China

Correspondence should be addressed to Liming Zhou; [email protected]

Received 27 October 2017; Revised 23 December 2017; Accepted 15 January 2018; Published 7 March 2018

Academic Editor: Marco Alfano

Copyright © 2018 Yan Cai et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

To overcome the overstiffness and imprecise magnetoelectroelastic coupling effects of finite element method (FEM), we present aninhomogeneous cell-based smoothed FEM (ICS-FEM) of functionally gradedmagnetoelectroelastic (FGMEE) structures.Then theICS-FEM formulations for free vibration calculation of FGMEE structures were deduced. In FGMEE structures, the true parametersat the Gaussian integration point were adopted directly to replace the homogenization in an element. The ICS-FEM providesa continuous system with a close-to-exact stiffness, which could be automatically and more easily generated for complicateddomains, thus significantly decreasing the numerical error. To verify the accuracy and trustworthiness of ICS-FEM, we investigatedseveral numerical examples and found that ICS-FEM simulated more accurately than the standard FEM. Also the effects of variousequivalent stiffness matrices and the gradient function on the inherent frequency of FGMEE beams were studied.

1. Introduction

Functionally graded magnetoelectroelastic (FGMEE) mate-rials are generally multiphase composites with continuouslyvarying mechanical properties. FGMEE materials can con-vert magnetic, electric, andmechanical energy from one typeinto another and have received wide attention recently dueto their electroelastic, magnetoelastic, and electromagneticcoupling effects [1, 2].Therefore, FGMEEmaterials have beenadopted in various smart structures, such as magnetic fieldprobes, smart vibration sensors, optoelectronic devices, andmedical ultrasonic transducers [3, 4]. The smart FGMEEstructures are commonly fabricated in the beam pattern.However, to better apply FGMEE beams, researchers mustanalyse the statics and free vibration, and to predict thecoupled response of FGMEEbeams for practical applications,they should accurately calculate the properties of free vibra-tions.

Several computational techniques were proposed toinvestigate the electroelastic, magnetoelastic, and electro-magnetic coupling effects of smart structures, such as finiteelement method (FEM), mesh-free method, and scaled

boundary FEM [5–10]. Bhangale and Ganesan analyzed thestatic behaviors of linear anisotropic FGMEE plates usingsemianalytical FEM and investigated the free vibration ofFGMEE plates and cylindrical shells [11, 12]. A layerwisepartial mixed FEM was proposed to model MEE plates [13].Phoenix et al. analysed the static and dynamic behaviorsof coupled MEE plates using FEM with the Reissner mixedvariational theorem [14]. Buchanan used FEM to study thefree vibrations of infinite magnetoelectroelastic cylinders[15]. However, these FEMs overestimated the stiffness ofsolid structures and were limited by low accuracy. Therefore,Sladek et al. proposed amesh-freemethod tomore accuratelystudy the static behavior of a circular FGMEE plate [16],but this method reduced the computational effectiveness.Recently, Liu et al. solved the deformations of a nonuniformMEEplate using scaled boundary FEM [17]. By incorporatingthe nonlocal theory into scaled boundary FEM, Ke andWangmore accurately and effectively studied the free vibrationsof MEE beams [18]. However, the effectiveness of scaledboundary FEM is still low and should be improved.

FEM, as a powerful computational tool to investigateMEE coupling behaviors, yet overestimates the stiffness of

HindawiShock and VibrationVolume 2018, Article ID 5141060, 17 pageshttps://doi.org/10.1155/2018/5141060

http://orcid.org/0000-0002-9997-108Xhttp://orcid.org/0000-0001-7880-4648http://orcid.org/0000-0002-5238-3803https://doi.org/10.1155/2018/5141060

2 Shock and Vibration

z

zn...

...

zk+1zk

z1

sL

ℎ

x,

Figure 1: FGMEE beams: Cartesian coordinate system and geometric parameters.

FGMEE structures, which may result in locking behav-ior and inaccurate eigenvalue solutions [19]. To overcomethese limitations, a series of cell-based smoothed FEM(CS-FEM) [20–26] and node-based, edge-based, or face-based smoothed FEMs [27–32] were proposed. In recentyears, many CS-FEM-based formulations were proposed.Moreover, CS-FEM does not require the shape functionderivatives or high generosity of program and is insensitiveto mesh distortion because of the absence of isoparametricmapping.

CS-FEM has been successfully extended into dynamicalcontrol of piezoelectric sensors and actuators, topologicaloptimization of linear piezoelectric micromotor, and analysisof static behaviors, frequency, and defects of piezoelectricstructures [33–43]. Due to its versatility, CS-FEM becomes asimple and effective numerical tool to solve numerous electricand mechanical physical problems. However, the applicationof CS-FEM to investigate MEE properties is still a challenge.

In this work, free vibrations of FGMEE structures werestudied. Inhomogeneous CS-FEM (ICS-FEM) for FGMEEmaterials was formulated by incorporating gradient smooth-ing into the standard FEM for the multi-physics field ofFGMEE. Then the equations of free vibration computationwere deduced under the multi-physics coupling field forFGMEE materials. Finally, FGMEE beams in the func-tional gradient exponential form or power law form werecalculated under different boundary conditions. ICS-FEMoutperformed FEM when compared with the referencesolution.

This paper is organized as follows: Section 2 introducesthe basic formulations for FGMEEmaterials. Section 3 brieflydescribes the properties of the FGMEE materials. Section 4contains the detailed formulation of ICS-FEM. In Section 5,two numerical examples and the model of typical MEMS-based FGMEE energy harvester are investigated in detail.Final conclusions from the numerical results are drawn inSection 6.

2. Basic Formulations

The material properties of a functionally graded material(FGM) plate vary continuously, which is an advantage overthe discontinuity across adjoining layers in a laminatedplate. The wide range of engineering applications of FGMhas attracted many scientists to investigate the behaviors ofFGM.

Considering the transverse isotropy of the FGMEEmedium [9, 44] and for the plane stress problem, we set

stress components 𝜎𝑦 = 𝜎𝑦𝑧 = 𝜏𝑥𝑦 = 0, electric displacementcomponent 𝐷𝑦 = 0, and magnetic induction component 𝐵𝑦= 0. The geometric parameters and the chosen Cartesiancoordinate system (𝑥, 𝑦, 𝑧) are illustrated in Figure 1.

The basic formulations for MEE materials include equi-librium equations, geometric equations, and constitutiveequations. The equilibrium equations are

𝜕𝜎𝑥𝜕𝑥 + 𝜕𝜏𝑥𝑧𝜕𝑧 = 0,𝜕𝜏𝑥𝑧𝜕𝑥 + 𝜕𝜎𝑧𝜕𝑧 = 0,𝜕𝐷𝑥𝜕𝑥 + 𝜕𝐷𝑧𝜕𝑧 = 0,𝜕𝐵𝑥𝜕𝑥 + 𝜕𝐵𝑧𝜕𝑧 = 0,(1)

where 𝜎𝑥, 𝜎𝑧, and 𝜏𝑥𝑧 denote stress components; 𝐷𝑥 and𝐷𝑧 are electric displacement components; 𝐵𝑥 and 𝐵𝑧 aremagnetic induction components.

The geometric equations are

𝑆𝑥 = 𝜕𝑢𝜕𝑥 ,𝑆𝑧 = 𝜕𝑤𝜕𝑧 ,𝑆𝑥𝑧 = 𝜕𝑤𝜕𝑥 + 𝜕𝑢𝜕𝑧 ,𝐸𝑥 = −𝜕Φ𝜕𝑥 ,𝐸𝑧 = −𝜕Φ𝜕𝑧 ,𝐻𝑥 = −𝜕Ψ𝜕𝑥 ,𝐻𝑧 = −𝜕Ψ𝜕𝑧 ,

(2)

where 𝑆𝑥, 𝑆𝑧, and 𝑆𝑥𝑧 denote strain components; 𝑢 and𝑤 are displacement components; 𝐸𝑥 and 𝐸𝑧 are electricfield components; Φ is electrical potential; 𝐻𝑥 and 𝐻𝑧 aremagnetic field components; Ψ is magnetic potential.

Shock and Vibration 3

The constitutive equations are

{{{{{𝜎𝑥𝜎𝑧𝜏𝑥𝑧

}}}}} = [[[𝐶11 𝐶13 0𝐶13 𝐶33 00 0 𝐶44

]]][[[𝑆𝑥𝑆𝑧𝑆𝑥𝑧

]]] + [[[0 𝑒310 𝑒33𝑒15 0

]]]{𝐸𝑥𝐸𝑧}

+ [[[0 𝑞310 𝑞33𝑞15 0

]]]{𝐻𝑥𝐻𝑧} ,

{𝐷𝑥𝐷𝑧} = [ 0 0 𝑒15𝑒31 𝑒33 0 ][[[𝑆𝑥𝑆𝑧𝑆𝑥𝑧

]]] + [𝜀11 00 𝜀33]{𝐸𝑥𝐸𝑧}

+ [𝑚11 00 𝑚33]{𝐻𝑥𝐻𝑧} ,{𝐵𝑥𝐵𝑧} = [ 0 0 𝑞15𝑞31 𝑞33 0 ][[[

𝑆𝑥𝑆𝑧𝑆𝑥𝑧]]] + [

𝑚11 00 𝑚33]{𝐸𝑥𝐸𝑧}+ [𝜇11 00 𝜇33]{𝐻𝑥𝐻𝑧} ,

(3)

where 𝐶𝑖𝑗, 𝜀𝑖𝑗, and 𝜇𝑖𝑗 are the elastic, dielectric, and magneticpermeability coefficients, respectively; 𝑒𝑖𝑗, 𝑞𝑖𝑗, and 𝑚𝑖𝑗 arepiezo-electric, piezomagnetic, and magnetoelectric coeffi-cients, respectively. For FGMEE materials, we have𝐶𝑖𝑗 = 𝐶0𝑖𝑗𝑓 (𝑧) ,𝜀𝑖𝑗 = 𝜀0𝑖𝑗𝑓 (𝑧) ,𝜇𝑖𝑗 = 𝜇0𝑖𝑗𝑓 (𝑧) ,𝑒𝑖𝑗 = 𝑒0𝑖𝑗𝑓 (𝑧) ,𝑞𝑖𝑗 = 𝑞0𝑖𝑗𝑓 (𝑧) ,𝑚𝑖𝑗 = 𝑚0𝑖𝑗𝑓 (𝑧) ,

(4)

where𝑓(𝑧) is an arbitrary function;𝐶0𝑖𝑗 and 𝜀0𝑖𝑗, 𝜇0𝑖𝑗, 𝑒0𝑖𝑗, 𝑞0𝑖𝑗, and𝑚0𝑖𝑗 are the values on the plane 𝑧 = 0.3. FGMEE Materials

An FGMEE structure is characterized by the high hetero-geneity of material properties with a distribution prescribingthe volume fractions of constituent phases. For particu-lar analysis, it is functional to idealize them as continuawith smooth gradual variation of material properties inthe spatial coordinates. Hence, the proper micromechanicalmodel should be able to characterize the material propertydistribution of a system in accurate sense.

Previous literatures focus on two types of gradationmethods widely applied to solve many problems. Among

various methods for composites, some are also used forFGMEE materials, including the exponential and Voigt ruleof mixture scheme.

For FGMEE materials with exponential variation in thethickness direction (𝑧-direction), (4) can be rewritten as𝐶𝑖𝑗 = 𝐶0𝑖𝑗𝑒𝜂𝑧/ℎ,𝜀𝑖𝑗 = 𝜀0𝑖𝑗𝑒𝜂𝑧/ℎ,𝜇𝑖𝑗 = 𝜇0𝑖𝑗𝑒𝜂𝑧/ℎ,𝑒𝑖𝑗 = 𝑒0𝑖𝑗𝑒𝜂𝑧/ℎ,𝑞𝑖𝑗 = 𝑞0𝑖𝑗𝑒𝜂𝑧/ℎ,𝑚𝑖𝑗 = 𝑚0𝑖𝑗𝑒𝜂𝑧/ℎ,

(5)

where 𝜂 is the exponential factor governing the degreeof 𝑧-direction gradient, ℎ is the thickness, the superscript0 indicates the 𝑧-independent coefficients, and 𝜂 = 0 inhomogeneous MEE materials.

The volume fraction of an FGMEE structure across thethickness direction is assumed as a simple power law type asfollows:

𝑉𝐵 = (2𝑧 + ℎ2ℎ )𝑛 , (6)where −ℎ/2 ≤ 𝑧 ≤ ℎ/2 and 𝑛 is the power law index. Thebottom surface of the material (𝑧 = −ℎ/2) is 𝑉𝐶 whereasthe top surface (𝑧 = ℎ/2) is 𝑉𝐵. The total volume of theconstituents should be 𝑉𝐶 + 𝑉𝐵 = 1. (7)

Based on (6) and (7), the effective material property isdefined as follows:(MC)eff = (MC)top 𝑉𝐵 + (MC)bottom 𝑉𝐶, (8)where “MC” is general notation for material property. With(3), the effective coefficients can be written as𝐶eff = (𝐶𝐵 − 𝐶𝐶)𝑉𝐵 + 𝐶𝐶,𝜀eff = (𝜀𝐵 − 𝜀𝐶)𝑉𝐵 + 𝜀𝐶,𝜇eff = (𝜇𝐵 − 𝜇𝐶)𝑉𝐵 + 𝜇𝐶,𝑒eff = (𝑒𝐵 − 𝑒𝐶)𝑉𝐵 + 𝑒𝐶,𝑞eff = (𝑞𝐵 − 𝑞𝐶)𝑉𝐵 + 𝑞𝐶,𝑚eff = (𝑚𝐵 − 𝑚𝐶)𝑉𝐵 + 𝑚𝐶,

(9)

where “eff” stands for effective properties correspondingto a specific value of 𝑛. Material coefficients of piezoelec-tric BaTiO3, magnetostrictive CoFe2O4, and MEE BatiO3-CoFe2O4 are given in Table 1. Figure 2 depicts the through-the-thickness distribution of the volume fraction changingwith different values of 𝑛. For 𝑛 = 1.0, the variation of effectivematerial property is linear.


Table 1: Magnetoelectroelastic coefficients of material properties[6].

Material constants CoFe2O4 BatiO3-CoFe2O4 BatiO3𝐶11 109N/m2 286 200 166𝐶12 109N/m2 173 110 77𝐶13 109N/m2 170 110 78𝐶33 109N/m2 269.5 190 162𝐶44 109N/m2 45.3 45 43𝑒31 C/m2 0 −3.5 −4.4𝑒33 C/m2 0 11 18.6𝑒15 C/m2 0 0 11.6𝜀11 10−9 C/Vm 0.08 0.9 11.2𝜀33 10−9 C/Vm 0.093 7.5 12.6𝜇11 10−4Ns2/C2 −5.9 −1.5 0.05𝜇33 10−4Ns2/C2 1.57 0.75 0.1𝑞31 N/Am 580 200 0𝑞33 N/Am 700 260 0𝑞15 N/Am 560 180 0𝑚11 10−12Ns/VC 0 6.0 0𝑚33 10−12Ns/VC 0 2500 0𝜌 kgm−3 5730 5730 5730

0.00.10.20.30.40.50.60.70.80.91.01.1

n = 15.0n = 5.0

n = 2.0

n = 1.0

n = 0.5

n = 0.2

Vc

n = 0.0

0.30.20.10.0 0.4 0.5−0.2−0.3−0.4 −0.1−0.5z/h

Figure 2: Variation of the volume fraction function versus the non-dimensional thickness 𝑧/ℎ with varying 𝑛.4. ICS-FEM

The solution domain Ω is discretized into 𝑛𝑝 elementscontaining𝑁𝑛 nodes, the approximation displacement u, theapproximation electrical potentialΦ, and the approximationmagnetic potentialΨ. For FGMEE materials, we have

u = 𝑛𝑝∑𝑖=1

𝑁𝑢𝑖 𝑢𝑖 = N𝑢u,Φ = 𝑛𝑝∑𝑖=1

𝑁Φ𝑖 Φ𝑖 = NΦΦ,

Ψ = 𝑛𝑝∑𝑖=1

𝑁Ψ𝑖 Ψ𝑖 = NΨΨ,(10)

where u, Φ, and Ψ are the vectors of node displacement,node electrical potential, and node magnetic potential,respectively;N𝑢,NΦ, andNΨ are displacement shape, electri-cal potential shape, and magnetic potential shape functionsof ICS-FEM, respectively. N𝑢, NΦ, and NΨ were expressed insimilar shape functions. Four-node element divided into foursmoothing subdomains [27], field nodes, edge smoothingnodes, center smoothing nodes, edge Gaussian point, outernormal vector distribution, and shape function values areshown in Figure 3.

At any point x𝑘 in the smoothing subdomain Ω𝑘𝑖 , thesmoothed strain 𝜀(x𝑘), smoothed electric field E(x𝑘), andsmoothed magnetic fieldH(x𝑘) are

S (x𝑘) = ∫Ω𝑘𝑖

S (x) 𝜅 (x − x𝑘) 𝑑Ω,E (x𝑘) = ∫

Ω𝑘𝑖

E (x) 𝜅 (x − x𝑘) 𝑑Ω,H (x𝑘) = ∫

Ω𝑘𝑖

H (x) 𝜅 (x − x𝑘) 𝑑Ω,(11)

where S(x), E(x), and H(x) are the strain, electric field, andmagnetic field in FEM, respectively; 𝜅(x − x𝑘) is the constantfunction:

𝜅 (x − x𝑘) = {{{1𝐴𝑘𝑖 x ∈ Ω𝑘𝑖0 x ∉ Ω𝑘𝑖 , (12)

where

𝐴𝑘𝑖 = ∫Ω𝑘𝑖

𝑑Ω. (13)Substituting (12) into (11), we get

S (x𝑘) = 1𝐴𝑘𝑖 ∫Γ𝑘𝑖 n𝑘𝑢u 𝑑Γ,E (x𝑘) = 1𝐴𝑘𝑖 ∫Γ𝑘𝑖 n𝑘ΦΦ 𝑑Γ,H (x𝑘) = 1𝐴𝑘𝑖 ∫Γ𝑘𝑖 n𝑘ΨΨ 𝑑Γ,

(14)

where Γ𝑘𝑖 is the boundary ofΩ𝑘𝑖 ; n𝑘𝑢, n𝑘Φ, and n𝑘Ψ are the outernormal vector matrices of the boundary:

n𝑘𝑢 = [[[[𝑛𝑘𝑥 00 𝑛𝑘𝑧𝑛𝑘𝑧 𝑛𝑘𝑥

]]]] ,


(1

4,1

4,1

4,1

4)

Field nodesCentroidal pointsMidside points

Gaussian pointsOutward normal vectors

Ak4Ak3

Ak2Ak1

Ωk3

Ωk2Ωk1

Ωk4

Element k

(0, 0, 0, 1) (0, 0, 1, 0)

(0, 1, 0, 0)(1, 0, 0, 0) (1/2, 1/2, 0, 0)

(0, 0, 1/2, 1/2)

(0, 1/2, 1/2, 0)(1/2, 0, 0, 1/2)

Figure 3: Smoothing subdomains and the values of shape functions.

n𝑘Φ = [𝑛𝑘𝑥𝑛𝑘𝑧] ,n𝑘Ψ = [𝑛𝑘𝑥𝑛𝑘𝑧] .

(15)

Eqs. (14) can be rewritten as

S (x𝑘) = 𝑛𝑒∑𝑖=1

B𝑖𝑢 (x𝑘) u𝑖,E (x𝑘) = − 𝑛𝑒∑

𝑖=1

B𝑖Φ (x𝑘)Φ𝑖,H (x𝑘) = − 𝑛𝑒∑

𝑖=1

B𝑖Ψ (x𝑘)Ψ𝑖,(16)

where 𝑛𝑒 is the number of smoothing elementsB𝑖𝑢 (x𝑘) = 1𝐴𝑘𝑖 ∫Γ𝑘 [[[[

𝑁𝑢𝑖 𝑛𝑘𝑥 00 𝑁𝑢𝑖 𝑛𝑘𝑧𝑁𝑢𝑖 𝑛𝑘𝑧 𝑁𝑢𝑖 𝑛𝑘𝑥]]]]𝑑Γ,

B𝑖Φ (x𝑘) = 1𝐴𝑘𝑖 ∫Γ𝑘 [𝑁Φ𝑖 𝑛𝑘𝑥𝑁Φ𝑖 𝑛𝑘𝑧]𝑑Γ,

B𝑖Ψ (x𝑘) = 1𝐴𝑘𝑖 ∫Γ𝑘 [𝑁Ψ𝑖 𝑛𝑘𝑥𝑁Ψ𝑖 𝑛𝑘𝑧]𝑑Γ.

(17)

At the Gaussian point of the smoothing boundary x𝐺𝑏 , (17)are rewritten as

B𝑖𝑢 (x𝑘) = 1𝐴𝑘𝑖𝑛𝑏∑𝑏=1

(𝑁𝑢𝑖 (x𝐺𝑏 ) 𝑛𝑘𝑥 00 𝑁𝑢𝑖 (x𝐺𝑏 ) 𝑛𝑘𝑧𝑁𝑢𝑖 (x𝐺𝑏 ) 𝑛𝑘𝑧 𝑁𝑢𝑖 (x𝐺𝑏 ) 𝑛𝑘𝑥)𝑙𝑘𝑏 ,

B𝑖Φ (x𝑘) = 1𝐴𝑘𝑖𝑛𝑏∑𝑏=1

(𝑁Φ𝑖 (x𝐺𝑏 ) 𝑛𝑘𝑥𝑁Φ𝑖 (x𝐺𝑏 ) 𝑛𝑘𝑧) 𝑙𝑘𝑏 ,B𝑖Ψ (x𝑘) = 1𝐴𝑘𝑖

𝑛𝑏∑𝑏=1

(𝑁Ψ𝑖 (x𝐺𝑏 ) 𝑛𝑘𝑥𝑁Ψ𝑖 (x𝐺𝑏 ) 𝑛𝑘𝑧) 𝑙𝑘𝑏 ,(18)

where 𝑙𝑘𝑏 is the length of the smoothing boundary; 𝑛𝑏 is thetotal number of boundaries for each smoothing subdomain.

As for the essential difference, FEM has to derive theshape function matrix of the element, but ICS-FEM avoidsthis step and simply uses the shape function at x𝐺𝑏 , whichreduces the requirement for continuity of the shape functionand improves the accuracy and convergence by the use ofgradient smoothing.

The thermodynamic potential of a 2D FGMEE problemis given as 𝐺 = 𝐺 (S,E,H) , (19)where S, E, andH are independent variables of strain, electricfield, and magnetic field, respectively.

By applying (1) into (19), we get the variational expressionof MEE plane:𝐺 = (12STCS) − (12ET𝜀E) − (12HT𝜇H) − SeE− SqH − EmH. (20)


L

H

z

x

Figure 4: Geometry of an FGMEE beam and the coordinates.

1 2 3 4 5 6 7 8 9 10 110Mode

01000020000300004000050000600007000080000

Freq

uenc

y (H

z)

ICS-FEM 60 × 4ICS-FEM 120 × 8ICS-FEM 150 × 10

ICS-FEM 30 × 2

ICS-FEM 30 × 2

= 0

= 0

= 0

= 0

= 1


ICS-FEM 60 × 4

ICS-FEM 60 × 4

= 1

= 1

= 1

= 5

= 5

ICS-FEM 120 × 8 = 5ICS-FEM 150 × 10 = 5

(a)

1 2 3 4 5 6 7 8 9 10 110Mode

01000020000300004000050000600007000080000

Freq

uenc

y (H

z)


ICS-FEM 30 × 2

ICS-FEM 30 × 2

= 0

= 0

= 0

= 0

= 1


ICS-FEM 60 × 4

ICS-FEM 60 × 4

= 1

= 1

= 1

= 5

= 5

ICS-FEM 120 × 8 = 5ICS-FEM 150 × 10 = 5

(b)

01000020000300004000050000600007000080000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


ICS-FEM 30 × 2

ICS-FEM 30 × 2

= 0

= 0

= 0

= 0

= 1


ICS-FEM 60 × 4

ICS-FEM 60 × 4

= 1

= 1

= 1

= 5

= 5

ICS-FEM 120 × 8 = 5ICS-FEM 150 × 10 = 5

(c)

1 2 3 4 5 6 7 8 9 10 110Mode

01000020000300004000050000600007000080000

Freq

uenc

y (H

z)


ICS-FEM 30 × 2

ICS-FEM 30 × 2

= 0

= 0

= 0

= 0

= 1


ICS-FEM 60 × 4

ICS-FEM 60 × 4

= 1

= 1

= 1

= 5

= 5

ICS-FEM 120 × 8 = 5ICS-FEM 150 × 10 = 5

(d)

01000020000300004000050000600007000080000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


ICS-FEM 30 × 2

ICS-FEM 30 × 2

= 0

= 0

= 0

= 0

= 1


ICS-FEM 60 × 4

ICS-FEM 60 × 4

= 1

= 1

= 1

= 5

= 5

ICS-FEM 120 × 8 = 5ICS-FEM 150 × 10 = 5

(e)

Figure 5: Structural frequencies (a) 𝑓uu; (b) 𝑓eq; (c) 𝑓eq-re; (d) 𝑓eq ΦΦ; (e) 𝑓eq ΨΨ of clamp-free BaTiO3–CoFe2O4 FGMEE beams.


Equivalentstiffnessmatrix

Mode 3Mode 2Mode 1Elements

30 × 2[KＯＯ]

60 × 4[KＯＯ]

30 × 2[K？Ｋ]

60 × 4[K？Ｋ]

30 × 2[K？Ｋ_Ｌ？＞Ｏ＝？＞]

60 × 4[K？Ｋ_Ｌ？＞Ｏ＝？＞]

30 × 2[K？Ｋ_ΦΦ]

60 × 4[K？Ｋ_ΦΦ]

30 × 2[K？Ｋ_ΨΨ]

60 × 4[K？Ｋ_ΨΨ]

Figure 6: The first- to third-order modes of clamp-free BaTiO3–CoFe2O4 FGMEE beams using different elements with exponential factor 𝜂= 1.0 by ICS-FEM.

Byminimizing (20) for nodal variables of shape functionsfor strain-displacement, electric field–electric potential, andmagnetic field–magnetic potential, we get the ICS-FEMequations for MEE plane:[[Kuu] − 𝜔2 [M]] {u} + [K𝑢Φ] {Φ} + [K𝑢Ψ] {Ψ} = 0,[K𝑢Φ]T {u} − [KΦΦ] {Φ} − [KΦΨ] {Ψ} = 0,[K𝑢Ψ]T {u} − [KΦΨ]T {Φ} − [KΨΨ] {Ψ} = 0,

(21)

where 𝜔 is the eigenvalues.Different elemental stiffness matrices used for FGMEE

beams are expressed as follows:

Kuu = 𝑛𝑐∑𝑖=1

B𝑖T𝑢 [C]B𝑖𝑢𝐴𝑘𝑖 ,K𝑢Φ = 𝑛𝑐∑

𝑖=1

B𝑖T𝑢 [e]B𝑖Φ𝐴𝑘𝑖 ,K𝑢Ψ = − 𝑛𝑐∑

𝑘=1

B𝑖T𝑢 [q]B𝑖Ψ𝐴𝑘𝑖 ,

KΦΨ = 𝑛𝑐∑𝑖=1

B𝑖TΦ [m]B𝑖Ψ𝐴𝑘𝑖 ,KΦΦ = 𝑛𝑐∑

𝑖=1

B𝑖TΦ [𝜀]B𝑖Φ𝐴𝑘𝑖 ,KΨΨ = 𝑛𝑐∑

𝑖=1

B𝑖TΨ [𝜇]B𝑖Ψ𝐴𝑘𝑖 ,M = ∑

𝑒

M𝑒,M𝑒 = diag {𝑚1, 𝑚1, 𝑚2, 𝑚2, 𝑚3, 𝑚3, 𝑚4, 𝑚4} ,

(22)

where 𝑛𝑐 = 𝑛𝑝 × 𝑛𝑒; 𝑚𝑖 = 𝜌𝑖𝑡 𝐴𝑘𝑖 (𝑖 = 1, 2, 3, 4) is the mass ofsmoothing element 𝑖; 𝑡 is the smoothing element thickness;𝜌𝑖 is density of Gaussian integration point in smoothingsubdomain 𝑖; [C], [𝜀], [𝜇], [e], [q], and [m] are thematrices ofelastic constant, dielectric coefficient, magnetic permeability,piezomagnetic coefficient, piezomagnetic coefficient, andmagnetoelectric coefficient, respectively.


Equivalentstiffnessmatrix

Mode 3Mode 2Mode 1Elements

30 × 2[KＯＯ]

60 × 4[KＯＯ]

30 × 2[K？Ｋ]

60 × 4[K？Ｋ]

30 × 2[K？Ｋ_Ｌ？＞Ｏ＝？＞]

60 × 4[K？Ｋ_Ｌ？＞Ｏ＝？＞]

30 × 2[K？Ｋ_ΦΦ]

60 × 4[K？Ｋ_ΦΦ]

30 × 2[K？Ｋ_ΨΨ]

60 × 4[K？Ｋ_ΨΨ]

Figure 7: The first- to third-order modes of clamp-free BaTiO3–CoFe2O4 FGMEE beams using different elements with exponential factor 𝜂= 5.0 by ICS-FEM.

Figure 8: Domain discretization using four-node extremely irregular elements.

The inhomogeneous smoothing element was adoptedto calculate its stiffness matrix. Because the parameters offour smoothing subdomains 𝐴𝑘𝑖 (𝑖 = 1, 2, 3, 4) differed inelement 𝑠, the actual parameters at the Gaussian integrationpoint were taken directly to simulate the changes of materialproperty in each element.

By eliminating the terms of electric and magnetic poten-tials using a condensation technique, we get the equivalentstiffness matrix [Keq]:[Keq] {u} + [M] {ü} = 0, (23)where [Keq] = [Kuu] + [K𝑢Φ] [KII]−1 [KI]+ [K𝑢Ψ] [KIV]−1 [KIII] ,

[KI] = [K𝑢Φ]T − [KΦΨ] [KΨΨ]−1 [K𝑢Ψ]T ,[KII] = [KΦΦ] − [KΦΨ] [KΨΨ]−1 [KΦΨ]T ,[KIII] = [K𝑢Ψ]T − [KΦΨ]T [KΦΦ]−1 [K𝑢Φ]T ,[KIV] = [KΨΨ] − [KΦΨ]T [KΦΦ]−1 [KΦΨ] .(24)

The eigenvectors corresponding toΦ andΨ are given as

Φ = [KII]−1 [KI] {u} ,Ψ = [KIV]−1 [KIII] {u} . (25)

To study the effect of magnetoelectric constant on systemfrequencies, we derived [Keq reduced] by neglecting the mag-netoelectric coupling effect.


1 2 3 4 5 6 7 8 9 10 110Mode

0300060009000

12000150001800021000

Freq

uenc

y (H

z)

ICS-FEM fuuFEM fuuICS-FEM feqFEM feqICS-FEM feq_reFEM feq_reICS-FEM feq_ΦΦFEM feq_ΦΦICS-FEM feq_ΨΨFEM feq_ΨΨRef. sol.

60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(a)

1 2 3 4 5 6 7 8 9 10 110Mode

0300060009000

1200015000180002100024000

Freq

uenc

y (H

z)


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(b)

0300060009000

120001500018000210002400027000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(c)

1 2 3 4 5 6 7 8 9 10 110Mode

0300060009000

120001500018000210002400027000300003300036000

Freq

uenc

y (H

z)


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(d)

0100002000030000400005000060000700008000090000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(e)

Figure 9: Structural frequencies 𝑓 of clamp-free CoFe2O4 FGMEE beams with exponential factor 𝜂 = (a) 0, (b) 0.5, (c) 1.0, (d) 2.0, and (e)5.0.

Bymaking [KΦΨ] = 0, we get the reduced cell-based finiteelement equations:

[[Kuu] − 𝜔2 [M]] {u} + [K𝑢Φ] {Φ} + [K𝑢Ψ] {Ψ} = 0,[K𝑢Φ]T {u} − [KΦΦ] {Φ} = 0,[K𝑢Ψ]T {u} − [KΨΨ] {Ψ} = 0.

(26)

The reduced stiffness matrix [Keq reduced] is[Keq reduced] = [Kuu] + [K𝑢Φ] [KΦΦ]−1 [K𝑢Φ]T+ [K𝑢Ψ] [KΨΨ]−1 [K𝑢Ψ]T . (27)

To evaluate the effect of PE phase on beam frequency, wederived the stiffness matrix [Keq ΦΦ] by setting the magneticpotential = 0:


ICS-FEMFEM

0

150

300

450

600

750

900

1050

1200

Tim

e (s)

230 460 690 920 1150 13800Elements

Figure 10: Comparison of computational efficiency.

Table 2: List of f used in the study.

Structuralfrequency Matrix used to compute the structural frequency𝑓uu [Kuu]𝑓eq [Keq]𝑓eq re [Keq reduced]𝑓eq ΦΦ [Keq ΦΦ]𝑓eq ΨΨ [Keq ΨΨ]

[Keq ΦΦ] = [Kuu] + [K𝑢Φ] [KΦΦ]−1 [K𝑢Φ]T . (28)To study the magnetic effect of PM phase on system fre-

quency, we obtained [Keq ΨΨ] by plugging electric potentialto zero in (26):

[Keq ΨΨ] = [Kuu] + [K𝑢Ψ] [KΨΨ]−1 [K𝑢Ψ]T . (29)5. Results and Discussion

5.1. C-F Beam. The free vibrations on FGMEE beams werecalculated by changing the exponential factor (Figure 4). Thematerial properties of FGMEE beams were governed by the𝑧-direction exponential variation. The following geometricalparameters were considered: length 𝐿 = 0.3m and width𝐻 = 0.02m with the assumption of plane stress. Boundaryconditions were 𝑢=𝑤=Φ=Ψ= 0 at the clamped end. Table 2gives the various structural frequencies in the study.

Firstly, the convergence of ICS-FEMwas verified by usingBaTiO3-CoFe2O4 FGMEE beams, with properties listed inTable 1. The natural frequencies of these beams were calcu-lated using ICS-FEMwith differentmeshes (30× 2, 60× 4, 120× 8, 150 × 10) (Figure 5).The simulation results with differentmeshes agree well, which prove the good convergence ofICS-FEM. The first- to third-order modes of clamp-freeBaTiO3–CoFe2O4 FGMEE beams using different elements

with exponential factor 𝜂= 1.0 and 5.0were calculated by ICS-FEM, and the results were summarized in Figures 6 and 7,respectively. It was found that the first- to third-order modesof BaTiO3–CoFe2O4 FGMEE beams in the same gradientdistribution were basically not affected by equivalent stiffnessmatrix or mesh number. the first- to third-order modes ofBaTiO3–CoFe2O4 FGMEE beams were basically not differentbetween 𝜂 = 1.0 and 5.0.

Secondly, the free vibration frequencies of CoFe2O4FGMEE beams were studied by both ICS-FEM and FEMusing extremely irregular elements, with domain discretiza-tion shown in Figure 8. The frequencies of clamp-freeCoFe2O4 FGMEE beams with different values of exponen-tial factor are shown in Figure 9; the first eleven naturalfrequencies calculated by ICS-FEM are smaller than thosecalculated by FEM. The validity of ICS-FEM is verified bythe agreements between the calculations and the referencesolutions. The shape of quadrilateral element in FEM can-not be severely distorted but was eliminated in ICS-FEM.ICS-FEM abstains from calculating the derivative of theshape functions of an element, and the area integral ofthe solution domain is converted to the boundary integral.The stiffness of FGMEE structures is improved becauseICS-FEM does not require continuity of the shape func-tion. The ICS-FEM provides a continuous system with aclose-to-exact stiffness, which could be automatically andmore easily generated for complicated domains, thus sig-nificantly decreasing the numerical error. The free vibra-tion of CoFe2O4 FGMEE beam, a pure CoFe2O4 materialwithout piezoelectric or magnetoelectric material coeffi-cients, influences structural frequency 𝑓eq because the mag-netic effect is marginally higher compared with 𝑓uu. 𝑓ΦΦcoincides with 𝑓uu since piezoelectric phase is absent inCoFe2O4. Similarly, 𝑓eq re coincides with 𝑓eq of the CoFe2O4FGMEE beam as the magnetoelectric effect is absent in pureCoFe2O4 FGMEE beam.The natural frequencies of CoFe2O4FGMEE beams increase with the rise of the exponentialfactor.


1 2 3 4 5 6 7 8 9 10 110Mode

02000400060008000

100001200014000160001800020000220002400026000

Freq

uenc

y (H

z)


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(a)

02000400060008000

100001200014000160001800020000220002400026000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(b)

1 2 3 4 5 6 7 8 9 10 110Mode

040008000

120001600020000240002800032000

Freq

uenc

y (H

z)


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(c)

040008000

12000160002000024000280003200036000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(d)

0100002000030000400005000060000700008000090000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(e)

Figure 11: Structural frequencies 𝑓 for simply supported BaTiO3-CoFe2O4 FGMEE beams with exponential factor 𝜂 = (a) 0, (b) 0.5, (c) 1.0,(d) 2.0, and (e) 5.0.

Finally, the comparison of calculation time between ICS-FEM and FEM at Intel (R) Xeon (R) CPU E3-1220 v3 @3.10GHz, 16G RAM is shown in Figure 10, with the elementnumber of 60, 240, 960, and 1500. As showed in Figure 10,the time required to solve algebraic equations by ICS-FEMis similar to that of FEM. Because the stiffness constructionof ICS-FEM is based on smoothing cells inside each element,no coupling occurs between nodal degrees-of-freedom thatare the distance of up to two elements. In other words, the

bandwidth of ICS-FEM stiffness matrix is the same as thatof FEM. Nevertheless, ICS-FEM is more effective in termsof generalized displacement (including displacement, elec-trical potential and magnetic potential) and computationalefficiency (computation time for the same accuracy).

5.2. S-S Beam. The free vibrations on BaTiO3–CoFe2O4FGMEE beams were studied by changing the gradientfunction form in Figure 4. The geometrical parameters were


02000400060008000

100001200014000160001800020000220002400026000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(a)

02000400060008000

100001200014000160001800020000220002400026000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(b)

02000400060008000

100001200014000160001800020000220002400026000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(c)

02000400060008000

100001200014000160001800020000220002400026000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(d)

02000400060008000

100001200014000160001800020000220002400026000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(e)

Figure 12: Structural frequencies 𝑓 of simply supported BaTiO3-CoFe2O4 FGMEE beams with power law index 𝑛 = (a) 0, (b) 1.0, (c) 5.0, (d)10.0, and (e) 20.0.

the same as the C-F beam. The simply supported boundaryconditions were used: 𝑢 = 𝑤 = Ψ = Φ = 0 at (𝑥 = 0, 𝑧 = ℎ/2)and 𝑤 = Ψ = Φ = 0 at (𝑥 = 𝐿, 𝑧 = ℎ/2).

Firstly, the free vibration frequencies for BaTiO3–CoFe2O4 FGMEE beams were calculated by both ICS-FEMwith 60× 4meshes and FEMwith 240× 16meshes (Figure 11).Results show the first eleven natural frequencies calculatedby ICS-FEM are closer to the reference solutions than thosecalculated by FEM, indicating ICS-FEM ismore efficient than

FEMdue to the reduced number ofmeshes.Thedifferences innatural frequencies𝑓eq,𝑓eq re, and𝑓eq ΦΦ aremarginal, so themagnetic effect only slightly impacts the natural frequenciesof FGMEE beams. The natural frequencies increase with therise of the exponential factor.

Secondly, as for BaTiO3–CoFe2O4 FGMEE beams withpower law type (𝑛 = 0, 1.0, 5.0, 10.0, 20.0), the bottom surface(𝑧 = −ℎ/2) is BaTiO3–CoFe2O4 whereas the top surface(𝑧 = ℎ/2) is CoFe2O4. The free vibration frequencies for


(a)

F

A

B

30mm

＂；４Ｃ／3

2m

m

(b)

Figure 13: Typical MEMS-based energy harvester fabricated with BaTiO3 FGMEE. (a) Model of the energy harvester; (b) simplified modelof ICS-FEM.

FGMEE beams calculated by ICS-FEM with 60 × 4 meshesand FEM with 240 × 16 meshes are shown in Figure 12.Results show the first eleven natural frequencies calculatedby ICS-FEM are closer to the reference solutions than thosecalculated by FEM. Also the differences in 𝑓eq, 𝑓eq re, and𝑓eq ΦΦ are marginal, so the magnetic effect does not largelyimpact the natural frequencies of FGMEEbeams.Meanwhile,the natural frequency of BaTiO3–CoFe2O4 FGMEE beamsis between those of BaTiO3–CoFe2O4 MEE beams andCoFe2O4 MEE beams.

5.3. Typical MEMS-Based FGMEE Energy Harvester. Themodel of FGMEE energy harvester developed by ICS-FEM isshown in Figure 13.The free vibrations on the FGMEE energyharvester were studied by changing the exponential factor.The geometrical parameters were 𝐿 = 30mm, 𝐻 = 2mm,and its structure was fabricated with BaTiO3 FGMEE.

The free vibration frequencies for the FGMEE energyharvester calculated by ICS-FEM with 60 × 4 meshes andFEM with 240 × 16 meshes are shown in Figure 14. Thefirst eleven natural frequencies calculated by ICS-FEM arecloser to the reference solutions than those calculated byFEM, indicating ICS-FEM is more efficient than FEM owingto the reduced number of meshes. The ICS-FEM does nottake the derivative of the shape functions of the element andcan be much easily generated automatically for complicateddomains, thus significantly decreasing the numerical errors.The natural frequencies 𝑓eq and 𝑓eq ΦΦ agree well with eachother since the piezomagnetic phase is absent from theBaTiO3 FGMEE energy harvester. Moreover, 𝑓eq re coincideswith 𝑓eq of the BaTiO3 FGMEE energy harvester as themagnetoelectric effect is absent in pure BaTiO3materials.The𝑓uu and 𝑓eq are very close, so the piezoelectric effect onlyslightly affects the natural frequencies of the energy harvester,which increase with the rise of the exponential factor.

TheWilson-𝜃method and the equivalent stiffness matrix[Keq] were employed to solve the dynamic response ofthe FGMEE energy harvester. The parameters were set astime step = 0.005 s, 𝜃 = 1.4; without damping; sine-wave transient load with a time period of 2 s; 4 cycles ofloading (Figure 15). The dynamic behaviors for the harvestercalculated by ICS-FEMwith 60× 4meshes and FEMwith 240× 16 meshes are shown in Figures 16 and 17, respectively. Thetemporal variations of displacement 𝑢𝑦 and electric potentialΦ calculated by ICS-FEM are closer to the reference solutionsthan those by FEM, which validate the accuracy of ICS-FEM. The temporal variations of 𝑢𝑦 and Φ of the FGMEEenergy harvester decreasewith the increase of the exponentialfactor when the material properties of the harvester are inexponential distribution.

6. Conclusions

The free vibrations on FGMEE structures were studied.Firstly, ICS-FEM for FGMEE materials was formulatedby incorporating gradient smoothing into the FEM-basedcomputation for the FGMEE multi-physics field. Then theequations of free vibration computation were deduced forthe multi-physics coupling field of FGMEEmaterials. Finally,the FGMEE beams were calculated with functional gradientexponential form or power law form under different bound-ary conditions.

(i) ICS-FEM reduced the systematic stiffness of thefinite element, which improved the computational accuracycompared with FEM under the same element number. ICS-FEM was more efficient than FEM in terms of computationtime for the same accuracy.

(ii) Due to thematerial property changes in each smooth-ing element, the true parameters at the Gaussian integration


040008000

1200016000200002400028000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(a)

1 2 3 4 5 6 7 8 9 10 110Mode

040008000

120001600020000240002800032000

Freq

uenc

y (H

z)


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(b)

1 2 3 4 5 6 7 8 9 10 110Mode

040008000

120001600020000240002800032000

Freq

uenc

y (H

z)


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(c)

040008000

120001600020000240002800032000360004000044000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(d)

0100002000030000400005000060000700008000090000

100000110000120000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110Mode


60 × 4

60 × 4240 × 1660 × 4240 × 1660 × 4240 × 16

240 × 1660 × 4240 × 16

(e)

Figure 14: Structural frequencies 𝑓 for BaTiO3 FGMEE energy harvester with exponential factor 𝜂 = (a) 0, (b) 0.5, (c) 1.0, (d) 2.0, and (e) 5.0.point were adopted directly. ICS-FEM avoided the deriva-tive of the shape functions but only transformed the areaintegral to the boundary integral in the solution domain,which omitted the requirement of continuity of the shapefunction.

(iii) The magnetic effect slightly influenced the naturalfrequencies of FGMEE beams, which increased with theincrease of the exponential factor when the material proper-ties of FGMEE beams were under exponential distribution.

Thenatural frequency of FGMEEbeams lied in between thoseof the FGMEE beams using the bottom surface of materialsand the FGMEE beams using the upper surface, when thematerial properties of FGMEE beams were under the powerlaw distribution.

(iv)The natural frequencies and general displacements ofthe FGMEE energy harvester developed by ICS-FEM weremore accurate compared with FEM, owing to the reducednumber of meshes.


0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.000.00Time (s)

−120

−90

−60

−30

0

30

60

90

120

F (N

)

F = 100 × ＭＣＨ(2/0.5 × t)

Figure 15: Sine-wave load at point 𝐴 of the BaTiO3 FGMEE energyharvester.

−2.0 × 10−7

−1.5 × 10−7

−1.0 × 10−7

−5.0 × 10−8

0.0

5.0 × 10−8

1.0 × 10−7

1.5 × 10−7

2.0 × 10−7

uy

(m)

0.5 1.0 1.5 2.0 2.50.0Time (s)

= 0

= 0

= 0.5

= 0.5

= 1.0

FEM 240 × 16ICS-FEM 60 × 4FEM 240 × 16ICS-FEM 60 × 4FEM 240 × 16ICS-FEM 60 × 4FEM 240 × 16ICS-FEM 60 × 4FEM 240 × 16ICS-FEM 60 × 4

= 1.0

= 2.0

= 2.0

= 5.0

= 5.0

Figure 16: Temporal variations of displacement 𝑢𝑦 at point 𝐵 ofBaTiO3 FGMEE energy harvester with different values of exponen-tial factor calculated by ICS-FEM and FEM.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Special thanks are due to Professor Guirong Liu for theS-FEM Source Code in http://www.ase.uc.edu/∼liugr/software.html. This work was financially supported bythe National Natural Science Foundation of China (Grant

0.5 1.0 1.5 2.0 2.50.0Time (s)

−8

−6

−4

−2

0

2

4

6

8

Φ (V

)

= 0

= 0

= 0.5

= 0.5

= 1.0

FEM 240 × 16ICS-FEM 60 × 4FEM 240 × 16ICS-FEM 60 × 4FEM 240 × 16ICS-FEM 60 × 4FEM 240 × 16ICS-FEM 60 × 4FEM 240 × 16ICS-FEM 60 × 4

= 1.0

= 2.0

= 2.0

= 5.0

= 5.0

Figure 17: Temporal variations of electric potential Φ at point𝐴 of BaTiO3 FGMEE energy harvester with different values ofexponential factor calculated by ICS-FEM and FEM.

no. 11502092), Jilin Provincial Department of Science andTechnology Fund Project (Grant nos. 20160520064JH,20170101043JC), and the Fundamental Research Funds forthe Central Universities.

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