ACouplingElectromechanicalInhomogeneousCell-Based ...downloads.hindawi.com/journals/amse/2019/2812748.pdf · in FGPMs were adopted directly to replace the homogenization parameters

Research ArticleA Coupling Electromechanical Inhomogeneous Cell-BasedSmoothed Finite Element Method for Dynamic Analysis ofFunctionally Graded Piezoelectric Beams

Bin Cai 1 and Liming Zhou 2

1School of Civil Engineering Jilin Jianzhu University Changchun Jilin China2School of Mechanical and Aerospace Engineering Jilin University Changchun Jilin China

Correspondence should be addressed to Bin Cai bincai666163com

Received 16 April 2019 Accepted 7 July 2019 Published 6 August 2019

Guest Editor Ayman E Hamada

Copyright copy 2019 Bin Cai and Liming Zhou is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

To accurately simulate the continuous property change of functionally graded piezoelectric materials (FGPMs) and overcome theoverstiffness of the finite element method (FEM) we present an electromechanical inhomogeneous cell-based smoothed FEM(ISFEM) of FGPMs Firstly ISFEM formulations were derived to calculate the transient response of FGPMs and then a modifiedWilson-θmethod was deduced to solve the integration of the FGPM systeme true parameters at the Gaussian integration pointin FGPMs were adopted directly to replace the homogenization parameters in an element ISFEM provides a close-to-exactstiffness of the continuous system which could automatically and more easily generate for complicated domains and thussignificantly decrease numerical errorse accuracy and trustworthiness of ISFEMwere verified as higher than the standard FEMby several numerical examples

1 Introduction

Because of their outstanding electromechanical propertieseasy fabricability and preparation flexibility piezoelectricmaterials are extensively applied as sensors and actuators tomonitor and modulate the response of structures [1 2]Piezoelectric actuators and sensors are innovative for mi-croscopic electromechanical systems and intelligent materialsystems particularly in aerospace and medical fields [3]Conventional piezoelectric sensors and actuators comprisemultiple layers of various piezoelectric materials [4ndash7]Moreover piezoelectric layers with uniform materialproperties are limited by large bending displacement stressconcentration creeping at high temperature and failurefrom interfacial unbounding All these phenomena are in-duced by mechanical or electric loading at layer interfaces[8]

To overcome the above limitations Zhu et al introducedand fabricated functionally graded piezoelectric material(FGPM) sensors and actuators [9 10] FGPMs change

nonstop in one or more directions without generating in-ternal stress concentration despite the production of largedisplacements Takagi et al fabricated FGPM bimorph ac-tuators by using a mixed system of lead-zirconate-titanate(PZT) and Pt [11] Nowadays FGPMs are widely used in-telligent materials for sensors and actuators in micro-structural engineering Many efforts have been made toanalyze the behaviors and staticdynamic responses ofFGPMs (eg shells beams and plates) such as the wavepropagation study of FGPM plates based on the laminatetheory [12] Moreover exact 3D analysis of FGPM rectan-gular plates was conducted by using a state-space approach[13] and to investigate the natural frequencies and modeshapes after being poled perpendicular to the middle plane[14] e above method was also applied to explore the freevibration of rectangular FGPM plates [15] e semi-analytical finite element method (FEM) was used to in-vestigate the static response of anisotropic and linearfunctionally graded magneto-electro-elastic plates [16]Lezgy-Nazargah et al [17] carried out static and dynamic

HindawiAdvances in Materials Science and EngineeringVolume 2019 Article ID 2812748 16 pageshttpsdoiorg10115520192812748

analyses on piezoelectric beams by using a refined sinusmodel e results have been found in good agreement withthe reference solutions for various electrical and mechanicalconstrained conditions Meanwhile the 3D exact state-spacesolution [18] and Peano series solution [19] were developedfor the cylindrical bending vibration of the FGPM laminatesrespectively Layerwise FEM was adopted to investigate thedisplacement and stress responses of an FGPM bimorphactuator [20] Qiu et al inhibited the vibration of a smartflexible clamped plate by using piezoelectric ceramic patchsensors and actuators [21] e Timoshenko beam theorywas used to analyze the static and dynamic responses ofFGPM actuators to thermo-electro-mechanical loading [22]e first-order shear deformation theory was used to studythe static bending free vibration and dynamic responses ofFGPM plates under electromechanical loading [23] e freeand induced vibrations of FGPM beams under thermo-electro-mechanical loading were characterized using the 3-order shear deformation beam theory [24] A high-ordertheory for FGPM shells was proposed based on the gen-eralized Hamiltonrsquos principle [8] Although the FEM (h-version) is adequate for low-frequency vibration analysis itis not well suited to the vibration analysis of medium- orhigh-frequency regimes [25] e spectral finite elementmethod (SFEM) [26 27] and the weak form quadraturemethod (QEM) [28 29] are developed for the dynamicanalysis of FGPM beams and structures

ough FEM is the most widely used and effectivenumerical approach in practical issues in research andengineering (including mechanics of vibration) it is notnecessarily fully perfect or cannot be further improved Forexample the probable overestimation of stiffness in solidstructures may lead to locking behavior and inaccuratestress-solving [30] By adding strain smoothing into FEM[31] Liu et al established a series of cell-based [32ndash36]node-based [37 38] edge-based [39ndash41] or face-based[42 43] smoothed FEMs (S-FEMs) and their combinations[44ndash47] ese S-FEMs with different properties can be usedto get desired solutions for a variety of benchmarks andpractical mechanic issues [48ndash50] e strain-smoothingoperations can reduce or alleviate the overstiffness ofstandard FEM significantly improving the accuracy of bothprimal and dual quantities [51] Moreover owing to absenceof parametric mapping the shape function derivatives andS-FEMmodels established in elasticity are not required to beinsensitive to mesh distortion [52] S-FEMs have beensuccessfully extended to analyze the dynamic control ofpiezoelectric sensors and actuators topological optimizationof linear piezoelectric micromotors statics frequency ordefects of smart materials [53ndash63] Zheng et al [64] utilizedthe cell-based smoothed finite element method with theasymptotic homogenization method to analyze the dynamicissues on micromechanics of piezoelectric composite ma-terials Zhou et al [65 66] deduced the linear and nonlinearcell-based smoothed finite element method of functionallygraded magneto-electro-elastic (MEE) structures and fur-ther examined the transient responses of MEE sensors orenergy harvest structures considering the damping factorsHowever there is little literature reported concerning the

dynamic response of FGPMs using the electromechanicalinhomogeneous cell-based smoothed finite element methodBecause of versatility S-FEMs become convenient and ef-ficient numerical approaches to address different physicalissues

Given the continuous change of the gradient of materialproperties along the thickness x3 direction and with cell-based gradient smoothing we deduced the basic formula ofISFEM and amodifiedWilson-θmethod to solve the integralsolution of the FGPM system e displacements and po-tentials of FGPM cantilever beams under sine wave loadcosine wave load step wave load and triangular wave loadwere analyzed in comparison with FEM

2 Basic Equations for Piezoelectric Materials

21 Geometry and Coordinate System Each beam has arectangular uniform cross section and is made of Ni layerseither completely or partially composed of FGPM beamse Cartesian coordinate system (x1 x2 x3) and geometricparameters are illustrated in Figure 1

22 Constitutive Equations At the k-th layer 3D linearconstitutive equations are polarized along its global co-ordinates as follows

σ11σ22σ33σ23σ13σ12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

c11 c12 c13 0 0 0

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

0 0 0 0 0 c66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k) ε11ε22ε332ε232ε132ε12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

minus

0 0 e31

0 0 e32

0 0 e33

0 e24 0

e15 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

E1

E2

E3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

(1)

D1

D2

D3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

0 0 0 0 e15 00 0 0 e24 0 0

e31 e32 e33 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

ε11ε22ε332ε232ε132ε12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

+

χ11 0 00 χ22 00 0 χ33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)E1

E2

E3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

(2)

where σij and εkl are the stress tensor and infinitesimal straintensor respectively Ei and Di are electric field and electric

2 Advances in Materials Science and Engineering

displacement vector components respectively eik ckl andXij are the piezoelectric elastic and dielectric materialconstants respectively Different from homogeneous pie-zoelectric materials the three constants are dependent oncoordinate x3 We assume that the material properties alongthe thickness direction are arbitrarily distributed as follows

cijkl x3( 1113857 c0ijklf x3( 1113857

eikl x3( 1113857 e0iklf x3( 1113857

χik x3( 1113857 χ0ikf x3( 1113857 i j k l 1 2 3

(3)

where f(x3) is an arbitrary function and c0kl e0kl and χ0kl arevalues at the plane x3 0

In an FGPM beam equations (1) and (2) are reduced to

σ Cεminus eE

D eTε + χE(4)

where

σ

σ11σ33σ13

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C

c11 c13 0

c31 c33 0

0 0 c55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

e

0 e31

0 e33

e15 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E E1

E31113890 1113891

ε

ε11ε33ε13

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D D1

D31113890 1113891

χ χ11 0

0 χ331113890 1113891

(5)

23 Weak Formulation e principle of virtual work for apiezoelectric medium of volume Ω and regular boundarysurface Γ can be written as

δΠ δUminus δW minus1113946ΩδεTσ dΩ + 1113946

ΓδuTFs dΓ

+ 1113946ΩδuTFv dΩminus1113946

ΩρδuT eurou dΩ + 1113946

ΩδETD dΩ

+ 1113946ΓQδφdΓ minus1113946

Ωqδφ dΩ 0

(6)

where Fs Fv u and φ are the vectors of surface forcemechanical body force node displacements and nodeelectrical potentials respectively q Q and ρ are theelectrical body charge surface charge and mass densityrespectively and δ is the virtual quantity

3 Electromechanical ISFEM

e solving domain Ω is discretized into np elements whichcontain Nn nodes the approximation displacement u andthe approximation electrical potential φ for the FGPMproblem can be expressed as

u 1113944

np

i1N

ui ui Nuu

φ 1113944

np

i1N

φi φi Nφφ

(7)

where Nu and Nφ are the ISFEM displacement shapefunction and electrical potential shape function respectively

Four-node element is divided into four smoothingsubdomains Field nodes edge smoothing nodes centersmoothing nodes and edge Gaussian points the outernormal vector distribution and the shape function valuesare shown in Figure 2

At any point xk in the smoothing subdomains Ωki the

smoothed strain ε(xk) and the smoothed electric field E(xk)

are

ε xk1113872 1113873 1113946

Ωki

ε(x)Φ x minus xk1113872 1113873 dΩ (8)

E xk1113872 1113873 1113946

Ωki

E(x)Φ x minus xk1113872 1113873 dΩ (9)

z1

zkzk+1

zn

helliphellip

x3

x1SV(t)

L

h

b

Figure 1 FGPM beams Cartesian coordinate system and geometric parameters length (L) width (b) and height (h)

Advances in Materials Science and Engineering 3

where ε(x) and E(x) are the strain and electric field in FEMrespectively and Φ(x minus xk) is the constant function

Φ x minus xk1113872 1113873

1Ak

i

x isin Ωki

0 x notin Ωki

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where

Aki 1113946Ωk

i

dΩ (11)

Substituting equation (10) into equations (8) and (9)then we have

ε xk1113872 1113873

1Ak

i

1113946Γk

i

nkuu dΓ (12)

E xk1113872 1113873

1Ak

i

1113946Γk

i

nkφφdΓ (13)

where Γki is the boundary of Ωki and nk

u and nkφ are the outer

normal vector matrixes of the smoothing domain boundary

nku

nkx1

0

0 nkx3

nkx3

nkx1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nkφ

nkx1

nkx3

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(14)

Equations (12) and (13) can be rewritten as

ε xk1113872 1113873 1113944

ne

i1Bi

u xk1113872 1113873ui

E xk1113872 1113873 minus1113944

ne

i1Bi

φ xk1113872 1113873φi

(15)

where ne is the number of smoothing elements

Bi

u xk1113872 1113873

1Ak

i

1113946Γk

Nui nk

x10

0 Nui nk

x3

Nui nk

x3Nu

i nkx1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dΓ (16)

Bi

φ xk1113872 1113873

1Ak

i

1113946Γk

Nφi nk

x1

Nφi nk

x3

⎡⎢⎣ ⎤⎥⎦ dΓ (17)

At the Gaussian point xGb equations (16) and (17) are

Bi

u xk1113872 1113873

1Ak

i

1113944

nb

b1

Nui xGb( 1113857nk

x10

0 Nui xGb( 1113857nk

x3

Nui xGb( 1113857nk

x3Nu

i xGb( 1113857nkx1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lkb

Bi

φ xk1113872 1113873

1Ak

i

1113944

nb

b1

Nφi xGb( 1113857nk

x1

Nφi xGb( 1113857nk

x3

⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠ l

kb

(18)

where xGb and lkb are the Gaussian point and the length of thesmoothing boundary respectively and nb is the total numberof boundaries for each smoothing subdomain As the shapefunction is linearly changed along each side of the smoothingsubdomain one Gauss point is sufficient for accurate boundaryintegration [30]

e essential distinction between ISFEM and FEM is thatFEM needs to construct the shape function matrix of theelement while ISFEM only needs to use the shape functionat the Gaussian point of the smoothing element boundaryand does not require to involve the shape function de-rivatives e above can reduce the continuity requirementof the shape function and therefore the accuracy andconvergence of the method are improved

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

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

(0 0 12 12)

(0 12 12 0)

(12 12 0 0)

(14 141414)

(12 0 0 12)

Element k

NodesCentroidal points

Midside pointsGaussian points

Node

Smoothing cells

Γc

Ak4 Ak

3

Ak2Ak

1Ωk

1

Ωk3

Ωk2

Ωk4

Figure 2 Smoothing subcells and the values of shape functions (reproduced from Zheng et al [64] under the Creative CommonsAttribution Licensepublic domain)


Table 1 Material constants

Material c11(GPa)

c12(GPa)

c13(GPa)

c33(GPa)

c44(GPa)

ρ(kgm3)

e31(Cm2)

e33(Cm2)

e15(Cm2) χ11χ0 χ33χ0 χ0 (CNmiddotm2)

PZT-4 1390 7780 7430 1150 2560 7600 minus520 1510 1270 14760 13010 8854times10minus12

PZT-5H 1260 7960 8410 1140 2330 7500 minus650 1744 2330 14689 16983 8854times10minus12

000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 05sin (2πt001)

F (N

)

t (s)

Figure 4 Sine wave load

000 001 002 003 004 005ndash15 times 10ndash7

ndash10 times 10ndash7


00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

u 3 (m

)

ISFEM 480ISFEM 800ISFEM 1200

ISFEM 1680Reference [67]

t (s)

(a)

000 001 002 003 004 005ndash3

ndash2

ndash1

0

1

2

3

φ (V

)



t (s)

(b)

Figure 5 Variations of the displacement u3 and electrical potential φ at the loading point with time when the gradient parameter was nminus5(a) Displacement (b) Electrical potential

L

h

b

F

Figure 3 Geometry of the cantilever beam






0050 times 10ndash9

10 times 10ndash8

15 times 10ndash8

20 times 10ndash8u 3

(m)



t (s)

(a)

000 001 002 003 004 005

ndash02

ndash01

00

01

02

φ (V

)



t (s)

(b)




00

40 times 10ndash9

80 times 10ndash9

12 times 10ndash8

u 3 (m

)



t (s)

(c)

000 001 002 003 004 005ndash015

ndash010

ndash005

000

005

010

015

φ (V

)



t (s)

(d)

000 001 002 003 004 005




00

20 times 10ndash9

40 times 10ndash9

60 times 10ndash9

u 3 (m

)



t (s)

(e)

000 001 002 003 004 005ndash009

ndash006

ndash003

000

003

006

009

φ (V

)



t (s)

(f )

Figure 6 Continued


e dynamic model of the FGPM electromechanicalsystem can be derived from the Hamilton principle in thefollowing form

Meuroq + Kq F (19)

where

M Muu 0

0 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

qmiddotmiddot umiddotmiddot

euroφ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

K

Kuu Kuφ

KTuφ Kφφ

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

q

u

φ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

F

F

Q

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

Muu 1113944e

Meuu

Meuu diag m1 m1 m2 m2 m3 m3 m4 m41113864 1113865

(20)

where mi piT Aki (i 1 2 3 4) is the mass of the i-th

smoothing element corresponding to node i T is thesmoothing element thickness and pi is the density ofGaussian integration point of the i-th smoothing subdomain

Kuu 1113944

nc

i1BiTu CB

i

uAki

Kuφ 1113944

nc

i1BiTu eB

i

φAki

Kφφ minus1113944

nc

i1BiTφ χB

i

φAki

F 1113946ΩNT

uFv dΩminus1113946ΓqNT

uFs dΓ

Q 1113946ΩNT

φQ dΩ + 1113946ΓqNT

φq dΓ

(21)

where nc np times ne

000 001 002 003 004 005




00

30 times 10ndash10

60 times 10ndash10


(m)



t (s)

(g)

000 001 002 003 004 005

ndash00045

ndash00030

ndash00015

00000

00015

00030

00045

φ (V

)



t (s)

(h)

Figure 6 Variations of the displacement u3 and electrical potential φ at the loading point with time when the gradient parameter wasn minus1 0 1 and 5 (a) Displacement (n minus1) (b) Electrical potential (n minus1) (c) Displacement (n 0) (d) Electrical potential (n 0)(e) Displacement (n 1) (f ) Electrical potential (n 1) (g) Displacement (n 5) (h) Electrical potential (n 5)

600 800 1000 1200 1400 1600 1800

1E ndash 3

001

Err

Number of elements

ISFEM t = 0010sISFEM t = 0002s

Figure 7 Convergence rate of ISFEM


Table 2 Displacement u3 and electrical potential φ at the loading point with the gradient parameter nminus5 minus1 0 1 and 5 of the FGPMbeambased on PZT-4

Time (s) Field variablesnminus5 nminus1 n 0 n 1 n 5

ISFEM FEM ISFEM FEM ISFEM FEM ISFEM FEM ISFEM FEM

00001 u3 times10minus10m 12852 12658 2760 2751 1707 1702 1006 1004 00756 00754φtimes 10minus2 V 11848 11912 0442 0427 0212 0207 00961 00944 000112 000110


00025 u3 times10minus8m 13345 13191 1538 1528 0917 0913 0564 0561 00882 00873φ (V) 2447 2391 0198 0191 0112 0109 00621 00605 000386 000372

0004 U3 times10minus9m 77799 77682 10154 10146 6165 6123 3727 3725 0514 0516φ (V) 1434 1406 0129 0125 00756 00740 00414 00401 000224 000221

Figure 9 Mesh of the FGPM beam

Table 3 Displacement u3 and electrical potential φ at the loading point with the gradient parameter nminus5 minus1 0 1 and 5 of the FGPMbeambased on PZT-5H





00025 u3 times10minus8m 16122 16006 2017 2003 1118 1117 0741 0736 0107 0106φ (V) 2522 2584 0219 0219 0122 0123 00691 00689 000392 000393

0004 u3 times10minus9m 76019 75052 10339 10264 6264 6261 3795 3768 0501 0496φ (V) 1402 1437 0113 0113 00649 00653 00351 00350 000179 000178

100 1000

1E ndash 3

001

01

Err

t (s)

FEMISFEM

Figure 8 Comparison of computational efficiency


e application of the inhomogeneous smoothing ele-ment is to calculate stiffness matrix of the element eparameters of four smoothing subdomains Ak

i (i 1 2 3 4)are various in the elements so the actual parameters at theGaussian integration point are taken directly in order toreflect the changes of material property in each element

4 Modified Wilson-θ Method

e modified Wilson-θ method is an important schemeand an implicit integral way to solve the dynamicsystem equations [63] If θgt 137 the solution is

unconditionally stable e detailed procedures areshowed as follows

41 Initial Calculation

(1) Formulate generalized stiffness matrix K massmatrix M and damping matrix C

(2) Calculate initial values of q _q euroq(3) Select the time step Δt and the integral constant θ

(θ14)

Table 4 Displacement u3 and electrical potential φ at the loading point with the gradient parameter n minus5 minus1 0 1 and 5 of the FGPMbeam based on PZT-4 under cosine load





00025 u3 times10minus9m minus33487 minus32845 minus12058 minus11952 minus4445 minus4345 minus4436 minus4392 minus0225 minus0221φ (V) minus0138 minus0137 minus0132 minus0128 minus00491 minus00479 minus00486 minus00470 minus000086 minus000085

0004 u3 times10minus9m minus11111 minus11501 minus0984 minus0969 minus5549 minus5616 minus0351 minus0339 minus00981 minus00913φ (V) minus1408 minus1365 minus00176 minus00174 minus00653 minus00635 000234 000229 0000596 0000579

Table 5 Displacement u3 and electrical potential φ at the loading point with the gradient parameter nminus5 minus1 0 1 and 5 of the FGPMbeambased on PZT-5H under cosine load





00025 u3 times10minus9m 117958 122357 minus4831 minus4723 minus7090 minus6942 minus1777 minus1715 0795 08136φ (V) 050863 05017 minus00407 minus00393 minus00688 minus00652 minus00151 minus00148 000338 000342


000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)

Figure 10 Cosine wave load


a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)

(4) Formulate an effective generalized stiffness matrix 1113957K1113957K K + a0M + a1C

42 For Each Time Step(1) Calculate the payload at time t+ θΔt

1113957Flowastt+θΔt Ft + θ Ft+Δt minusFt( 1113857

+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)

(2) Calculate the generalized displacement at timet+ θΔt

1113957Kqt+θΔt 1113957Flowastt+θΔt (24)

(3) Calculate the generalized acceleration generalizedspeed and generalized displacement at time t+Δt

euroqt+Δt a4 qt+θΔt minus qt( 1113857 + a5 _qt + a6 euroqt

_qt+Δt _qt + a7 euroqt+Δt + euroqt( 1113857

qt+Δt qt + Δt _qt + a8 euroqt+Δt + 2euroqt( 1113857

(25)

5 Numerical Examples

Four numerical examples were conducted under sine waveload cosine wave load step wave load and triangular waveload respectively FGPM cantilever beams of the same di-mensions (length L 40mm width h 5mm and thicknessb 1mm) were subjected to forced vibration (Figure 3) ematerial constants are shown in Table 1 And initial con-ditions were q 0 and _q 0 at t 0 moment e FGPMbeams were made of PZT-4 or PZT-5H on basis of expo-nentially graded piezoelectric materials with the followingmaterial properties

ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)

where f(x3) enx3h and n is the gradient parameter

51 SineWave Load e load F applied to the free end andthe load waveform is demonstrated in Figure 4 A con-vergence investigation with respect to meshes was firstcarried out Four smoothing subcells were used for elec-tromechanical ISFEM with ∆t 1times 10minus3 s e variations ofdisplacement u3 and electrical potential φ at the loadingpoint of the PZT-4-based FGPM beam combined with re-spect to time are shown in Figure 5e results at nminus5 withthe element number of 480 800 1200 or 1680 werecompared with the reference solution [67] e variations ofu3 and φ at the loading point combined with respect to timeat nminus1 0 1 and 5 in comparison with the referencesolution are shown in Figure 6 [67] Figure 7 illustrates thetotal energy norm Err versus the mesh density at t 0002 s

000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)

Figure 11 Step wave load


and t 001 s e simulation results are well consistentamong different numbers of meshes which demonstrate thehigh convergence of ISFEM

Figure 8 shows a comparison of calculation time betweenISFEM and FEM at Intel reg Xeon reg CPU E3-1220 v3 310GHz 16GB RAM e bandwidth of the system ma-trices for the FEM and ISFEM is identical However in theISFEM only the values of shape functions (not the de-rivatives) at the quadrature points are needed and the re-quirement of traditional coordinate transform procedure isnot necessary to perform the numerical integration

erefore the ISFEM generally needs less computationalcost than the FEM for handling the dynamic analysisproblems

e variations of u3 and φ at the loading point combinedwith t 00001 s 00004 s 00025 s and 0004 s in the PZT-4-based FGPM cantilever beam are shown in Table 2 e80times10 meshes of ISFEM at nminus5 minus1 0 1 and 5 are shownin Figure 9 and FEM is considered 160times 20 elementsClearly the results of ISFEM with 80times10 elements are thesame as the calculated results of FEM using 160times 20 ele-ments suggesting ISFEM has higher accuracy

000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)

FEM n = ndash5FEM n = 0FEM n = 5

ISFEM n = ndash5ISFEM n = 0ISFEM n = 5

(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)

Figure 12 Variations of displacement u3 and electrical potential φ at the loading point with time of the FGPM beam based on PZT-4 whenthe gradient parameter was nminus5 0 and 5 under step load (a) Displacement (b) Electrical potential

000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)

Figure 13 Variations of displacement u3 and electrical potential φ at the loading point with time of the FGPMbeam based on PZT-5Hwhenthe gradient parameter was nminus5 0 and 5 under step load (a) Displacement (b) Electrical potential


e variations of u3 and φ at the loading point combinedwith time of the PZT-5H-based FGPM cantilever beam arelisted in Table 3 Clearly when n changes from minus5 to 5 themaximum u3 and φ decrease which is consistent with thePZT-4-based FGPM cantilever beam Furthermore theresults of ISFEM with 80times10 elements are the same as thecalculated results of FEM using 160times 20 elements sug-gesting ISFEM has higher accuracy

52 CosineWave Load e cosine load F applied to the freeend and load waveform is shown in Figure 10e variationsof u3 and φ at the loading point combined with time of PZT-4- and PZT-5H-based FGPM cantilever beams are listed inTables 4 and 5 respectively e calculated results of ISFEMwith 80times10 elements are the same as those of FEM using

160times 20 elements implying that ISFEM possesses higheraccuracy

53 StepWave Load e step load F applied to the free endand load waveform is indicated in Figure 11 e variationsof u3 and φ at the loading point combined with time of PZT-4- and PZT-5H-based FGPM cantilever beams are illustratedin Figures 12 and 13 respectively It is clearly shown thatISFEM possesses higher accuracy than FEM for the calcu-lated results of ISFEM using 80times10 elements are the same asFEM using 160times 20 elements

54 TriangularWave Load e triangular load F applied tothe free end and load waveform is shown in Figure 14 e

000 001 002 003 00400

01

02

03

04

05

05ndash50(t ndash 003)003 lt t le 004

50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)

Figure 14 Triangular wave load

000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)

Figure 15 Variations of displacement u3 and electrical potential φ at the loading point with time of the FGPM beam based on PZT-4 whenthe gradient parameter was nminus5 0 and 5 under triangular load (a) Displacement (b) Electrical potential


variations of u3 and φ at the loading point combined withtime of PZT-4- and PZT-5H-based FGPM cantilever beamsare shown in Figures 15 and 16 respectively It shows thatthe solutions of CS-FEM with less elements are the same asthe solutions of FEM using more elements

6 Conclusions

An electromechanical ISFEM was proposed given thecontinuous changes of the gradient of material propertiesalong the thickness x3 direction and with cell-based gradientsmoothing e modified Wilson-θ method was deduced tosolve the integral solution of the FGPM system e dis-placements and potentials of cantilever beams combiningwith sine load cosine load step load and triangular loadwere analyzed by ISFEM in comparison with FEM

(1) ISFEM is correct and effective in solving the dynamicresponse of FGPM structures

(2) ISFEM can reduce the systematic stiffness of FEMand provides calculations closer to the true values

(3) ISFEM is more efficient than FEM and takes lesscomputation time at the same accuracy

is study indicates a possibility to select suitablegrading controlled by the power law index according to theapplication

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Authorsrsquo Contributions

Liming Zhou and Bin Cai performed the simulations BinCai contributed to the writing of the manuscript

Acknowledgments

Special thanks are due to Professor Guirong Liu for the S-FEMSource Code in httpwwwaseucedusimliugrsoftwarehtmlis work was financially supported by the National Key RampDProgram of China (grant no 2018YFF01012401-06) JilinProvincial Department of Science andTechnology FundProject(grant no 20170101043JC) Jilin Provincial Department ofEducation (grant nos JJKH20180084KJ and JJKH20170788KJ)Fundamental Research Funds for the Central UniversitiesScience and Technological Planning Project of Ministry ofHousing and UrbanndashRural Development of the Peoplersquos Re-public of China (2017-K9-047) and Graduate Innovation Fundof Jilin University (grant no 101832018C184)

References

[1] K M Liew X Q He and S Kitipornchai ldquoFinite elementmethod for the feedback control of FGM shells in the fre-quency domain via piezoelectric sensors and actuatorsrdquoComputer Methods in Applied Mechanics and Engineeringvol 193 no 3ndash5 pp 257ndash273 2004

[2] G Song V Sethi and H-N Li ldquoVibration control of civilstructures using piezoceramic smart materials a reviewrdquoEngineering Structures vol 28 no 11 pp 1513ndash1524 2006

[3] B Legrand J-P Salvetat B Walter M Faucher D eronand J-P Aime ldquoMulti-MHz micro-electro-mechanical sen-sors for atomic force microscopyrdquo Ultramicroscopy vol 175pp 46ndash57 2017

[4] F Pablo I Bruant and O Polit ldquoUse of classical plate finiteelements for the analysis of electroactive composite plates

u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)

Figure 16 Variations of the displacement u3 and electrical potential φ at the loading point with time of the FGPM beam based on PZT-5Hwhen the gradient parameter was nminus5 0 and 5 under triangular load (a) Displacement (b) Electrical potential


Numerical validationsrdquo Journal of IntelligentMaterial Systemsand Structures vol 20 no 15 pp 1861ndash1873 2009

[5] M DrsquoOttavio and O Polit ldquoSensitivity analysis of thicknessassumptions for piezoelectric plate modelsrdquo Journal of In-telligent Material Systems and Structures vol 20 no 15pp 1815ndash1834 2009

[6] P Vidal M DrsquoOttavio M Ben aıer and O Polit ldquoAnefficient finite shell element for the static response of pie-zoelectric laminatesrdquo Journal of Intelligent Material Systemsand Structures vol 22 no 7 pp 671ndash690 2011

[7] S B Beheshti-Aval M Lezgy-Nazargah P Vidal andO Polit ldquoA refined sinus finite element model for theanalysis of piezoelectric-laminated beamsrdquo Journal of In-telligent Material Systems and Structures vol 22 no 3pp 203ndash219 2011

[8] X-H Wu C Chen Y-P Shen and X-G Tian ldquoA high ordertheory for functionally graded piezoelectric shellsrdquo In-ternational Journal of Solids and Structures vol 39 no 20pp 5325ndash5344 2002

[9] X H Zhu and Z Y Meng ldquoOperational principle fabricationand displacement characteristic of a functionally gradientpiezoelectric ceramic actuatorrdquo Sensors and Actuators APhysical vol 48 no 3 pp 169ndash176 1995

[10] C C M Wu M Kahn and W Moy ldquoPiezoelectric ceramicswith functional gradients a new application in material de-signrdquo Journal of the American Ceramic Society vol 79 no 3pp 809ndash812 2005

[11] K Takagi J-F Li S Yokoyama RWatanabe A Almajid andM Taya ldquoDesign and fabrication of functionally graded PZTPt piezoelectric bimorph actuatorrdquo Science and Technology ofAdvanced Materials vol 3 no 2 pp 217ndash224 2002

[12] G R Liu and J Tani ldquoSurface waves in functionally gradientpiezoelectric platesrdquo Journal of Vibration and Acousticsvol 116 no 4 pp 440ndash448 1994

[13] Z Zhong and E T Shang ldquoree-dimensional exact analysisof a simply supported functionally gradient piezoelectricplaterdquo International Journal of Solids and Structures vol 40no 20 pp 5335ndash5352 2003

[14] C Piotr ldquoree-dimensional natural vibration analysis andenergy considerations for a piezoelectric rectangular platerdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 1093ndash1113 2005

[15] W Q Chen and H J Ding ldquoOn free vibration of afunctionally graded piezoelectric rectangular platerdquo ActaMechanica vol 153 no 3-4 pp 207ndash216 2002

[16] R K Bhangale and N Ganesan ldquoStatic analysis of simplysupported functionally graded and layered magneto-electro-elastic platesrdquo International Journal of Solids and Structuresvol 43 no 10 pp 3230ndash3253 2006

[17] M Lezgy-Nazargah P Vidal and O Polit ldquoAn efficient finiteelement model for static and dynamic analyses of functionallygraded piezoelectric beamsrdquo Composite Structures vol 104pp 71ndash84 2013

[18] M Lezgy-Nazargah ldquoA three-dimensional exact state-space solution for cylindrical bending of continuously non-homogenous piezoelectric laminated plates with arbitrarygradient compositionrdquo Archive of Mechanics vol 67pp 25ndash51 2015

[19] M Lezgy-Nazargah ldquoA three-dimensional Peano series so-lution for the vibration of functionally graded piezoelectriclaminates in cylindrical bendingrdquo Scientia Iranica vol 23no 3 pp 788ndash801 2016

[20] H-J Lee ldquoLayerwise laminate analysis of functionally gradedpiezoelectric bimorph beamsrdquo Journal of Intelligent MaterialSystems and Structures vol 16 no 4 pp 365ndash371 2005

[21] Z-C Qiu X-M Zhang H-X Wu and H-H ZhangldquoOptimal placement and active vibration control for piezo-electric smart flexible cantilever platerdquo Journal of Sound andVibration vol 301 no 3ndash5 pp 521ndash543 2007

[22] J Yang and H J Xiang ldquoermo-electro-mechanicalcharacteristics of functionally graded piezoelectric actua-torsrdquo Smart Materials and Structures vol 16 no 3pp 784ndash797 2007

[23] B Behjat M Salehi A Armin M Sadighi and M AbbasildquoStatic and dynamic analysis of functionally graded piezo-electric plates under mechanical and electrical loadingrdquoScientia Iranica vol 18 no 4 pp 986ndash994 2011

[24] A Doroushi M R Eslami and A Komeili ldquoVibrationanalysis and transient response of an FGPM beam underthermo-electro-mechanical loads using higher-order sheardeformation theoryrdquo Journal of Intelligent Material Systemsand Structures vol 22 no 3 pp 231ndash243 2011

[25] G W Wei Y B Zhao and Y Xiang ldquoA novel approach forthe analysis of high-frequency vibrationsrdquo Journal of Soundand Vibration vol 257 no 2 pp 207ndash246 2002

[26] P Kudela M Krawczuk and W Ostachowicz ldquoWavepropagation modelling in 1D structures using spectral finiteelementsrdquo Journal of Sound and Vibration vol 300 no 1-2pp 88ndash100 2007

[27] Y Kim S Ha and F-K Chang ldquoTime-domain spectral el-ement method for built-in piezoelectric-actuator-inducedlamb wave propagation analysisrdquo AIAA Journal vol 46 no 3pp 591ndash600 2008

[28] H Zhong and Z Yue ldquoAnalysis of thin plates by the weakform quadrature element methodrdquo Science China PhysicsMechanics and Astronomy vol 55 no 5 pp 861ndash871 2012

[29] X Wang Z Yuan and C Jin ldquoWeak form quadrature ele-ment method and its applications in science and engineeringa state-of-the-art reviewrdquo Applied Mechanics Reviews vol 69no 3 Article ID 030801 2017

[30] G R Liu and T Nguyen-oi Smoothed Finite ElementMethods CRC Press Taylor and Francis Group Boca RatonFL USA 2010

[31] J-S Chen C-T Wu S Yoon and Y You ldquoA stabilizedconforming nodal integration for Galerkin mesh-freemethodsrdquo International Journal for Numerical Methods inEngineering vol 50 no 2 pp 435ndash466 2001

[32] K Y Dai and G R Liu ldquoFree and forced vibration analysisusing the smoothed finite element method (SFEM)rdquo Journalof Sound and Vibration vol 301 no 3ndash5 pp 803ndash820 2007

[33] S P A Bordas and S Natarajan ldquoOn the approximation inthe smoothed finite element method (SFEM)rdquo InternationalJournal for Numerical Methods in Engineering vol 81 no 5pp 660ndash670 2010

[34] C V Le H Nguyen-Xuan H Askes S P A BordasT Rabczuk and H Nguyen-Vinh ldquoA cell-based smoothedfinite element method for kinematic limit analysisrdquo In-ternational Journal for Numerical Methods in Engineeringvol 83 no 12 pp 1651ndash1674 2010

[35] G R Liu W Zeng and H Nguyen-Xuan ldquoGeneralizedstochastic cell-based smoothed finite element method (GS_CS-FEM) for solid mechanicsrdquo Finite Elements in Analysisand Design vol 63 pp 51ndash61 2013

[36] K Nguyen-Quang H Dang-Trung V Ho-Huu H Luong-Van and T Nguyen-oi ldquoAnalysis and control of FGMplates integrated with piezoelectric sensors and actuators


using cell-based smoothed discrete shear gap method (CS-DSG3)rdquo Composite Structures vol 165 pp 115ndash129 2017

[37] Y H Bie X Y Cui and Z C Li ldquoA coupling approach ofstate-based peridynamics with node-based smoothed finiteelement methodrdquo Computer Methods in Applied Mechanicsand Engineering vol 331 pp 675ndash700 2018

[38] G Liu M Chen and M Li ldquoLower bound of vibrationmodes using the node-based smoothed finite elementmethod (NS-FEM)rdquo International Journal of Computa-tional Methods vol 14 no 4 Article ID 1750036 2016

[39] Z C He G Y Li Z H Zhong et al ldquoAn ES-FEM foraccurate analysis of 3D mid-frequency acoustics usingtetrahedron meshrdquo Computers amp Structures vol 106-107pp 125ndash134 2012

[40] X Y Cui G Wang and G Y Li ldquoA nodal integrationaxisymmetric thin shell model using linear interpolationrdquoApplied Mathematical Modelling vol 40 no 4 pp 2720ndash2742 2016

[41] X Y Cui X B Hu and Y Zeng ldquoA copula-based pertur-bation stochastic method for fiber-reinforced compositestructures with correlationsrdquo Computer Methods in AppliedMechanics and Engineering vol 322 pp 351ndash372 2017

[42] T Nguyen-oi G R Liu K Y Lam and G Y Zhang ldquoAface-based smoothed finite element method (FS-FEM) for 3Dlinear and geometrically non-linear solid mechanics problemsusing 4-node tetrahedral elementsrdquo International Journal forNumerical Methods in Engineering vol 78 no 3 pp 324ndash3532009

[43] Z C He G Y Li Z H Zhong et al ldquoAn edge-basedsmoothed tetrahedron finite element method (ES-T-FEM)for 3D static and dynamic problemsrdquo ComputationalMechanics vol 52 no 1 pp 221ndash236 2013

[44] E Li Z C He X Xu and G R Liu ldquoHybrid smoothed finiteelement method for acoustic problemsrdquo Computer Methodsin Applied Mechanics and Engineering vol 283 pp 664ndash6882015

[45] C Jiang Z-Q Zhang G R Liu X Han and W Zeng ldquoAnedge-basednode-based selective smoothed finite elementmethod using tetrahedrons for cardiovascular tissuesrdquo En-gineering Analysis with Boundary Elements vol 59 pp 62ndash772015

[46] X B Hu X Y Cui Z M Liang and G Y Li ldquoe per-formance prediction and optimization of the fiber-rein-forced composite structure with uncertain parametersrdquoComposite Structures vol 164 pp 207ndash218 2017

[47] X Cui S Li H Feng and G Li ldquoA triangular prism solid andshell interactive mapping element for electromagnetic sheetmetal forming processrdquo Journal of Computational Physicsvol 336 pp 192ndash211 2017

[48] G R Liu H Nguyen-Xuan and T Nguyen-oi ldquoA theo-retical study on the smoothed FEM (S-FEM) models prop-erties accuracy and convergence ratesrdquo International Journalfor Numerical Methods in Engineering vol 84 no 10pp 1222ndash1256 2010

[49] E Li Z C He GWang and G R Liu ldquoAn efficient algorithmto analyze wave propagation in fluidsolid and solidfluidphononic crystalsrdquo Computer Methods in Applied Mechanicsand Engineering vol 333 pp 421ndash442 2018

[50] W Zeng G R Liu D Li and X W Dong ldquoA smoothingtechnique based beta finite element method (βFEM) forcrystal plasticity modelingrdquo Computers amp Structures vol 162pp 48ndash67 2016

[51] H Nguyen-Xuan G R Liu T Nguyen-oi and C Nguyen-Tran ldquoAn edge-based smoothed finite element method for

analysis of two-dimensional piezoelectric structuresrdquo SmartMaterials and Structures vol 18 no 6 Article ID 0650152009

[52] H Nguyen-Van N Mai-Duy and T Tran-Cong ldquoAsmoothed four-node piezoelectric element for analysis of two-dimensional smart structuresrdquo Computer Modeling in Engi-neering and Sciences vol 23 no 3 pp 209ndash222 2008

[53] P Phung-Van T Nguyen-oi T Le-Dinh and H Nguyen-Xuan ldquoStatic and free vibration analyses and dynamic controlof composite plates integrated with piezoelectric sensors andactuators by the cell-based smoothed discrete shear gapmethod (CS-FEM-DSG3)rdquo Smart Materials and Structuresvol 22 no 9 Article ID 095026 2013

[54] Z C He E Li G R Liu G Y Li and A G Cheng ldquoA mass-redistributed finite element method (MR-FEM) for acousticproblems using triangular meshrdquo Journal of ComputationalPhysics vol 323 pp 149ndash170 2016

[55] W Zuo and K Saitou ldquoMulti-material topology optimizationusing ordered SIMP interpolationrdquo Structural and Multi-disciplinary Optimization vol 55 no 2 pp 477ndash491 2017

[56] E Li Z C He J Y Hu and X Y Long ldquoVolumetric lockingissue with uncertainty in the design of locally resonantacoustic metamaterialsrdquo Computer Methods in Applied Me-chanics and Engineering vol 324 pp 128ndash148 2017

[57] L Chen Y W Zhang G R Liu H Nguyen-Xuan andZ Q Zhang ldquoA stabilized finite element method for certifiedsolution with bounds in static and frequency analyses ofpiezoelectric structuresrdquo Computer Methods in Applied Me-chanics and Engineering vol 241ndash244 pp 65ndash81 2012

[58] L Zhou M Li Z Ma et al ldquoSteady-state characteristics of thecoupled magneto-electro-thermo-elastic multi-physical sys-tem based on cell-based smoothed finite element methodrdquoComposite Structures vol 219 pp 111ndash128 2019

[59] L Zhou S Ren G Meng X Li and F Cheng ldquoA multi-physics node-based smoothed radial point interpolationmethod for transient responses of magneto-electro-elasticstructuresrdquo Engineering Analysis with Boundary Elementsvol 101 pp 371ndash384 2019

[60] H Nguyen-Van N Mai-Duy and T Tran-Cong ldquoA node-based element for analysis of planar piezoelectric structuresrdquoCMES Computer Modeling in Engineering amp Sciences vol 36no 1 pp 65ndash96 2008

[61] E Li Z C He Y Jiang and B Li ldquo3D mass-redistributedfinite element method in structural-acoustic interactionproblemsrdquo Acta Mechanica vol 227 no 3 pp 857ndash8792016

[62] L Zhou M Li G Meng and H Zhao ldquoAn effective cell-basedsmoothed finite element model for the transient responses ofmagneto-electro-elastic structuresrdquo Journal of IntelligentMaterial Systems and Structures vol 29 no 14 pp 3006ndash3022 2018

[63] L Zhou M Li H Zhao and W Tian ldquoCell-basedsmoothed finite element method for the intensity factors ofpiezoelectric bimaterials with interfacial crackrdquo In-ternational Journal of Computational Methods vol 16 no7 Article ID 1850107 2019

[64] J Zheng Z Duan and L Zhou ldquoA coupling electrome-chanical cell-based smoothed finite element method basedon micromechanics for dynamic characteristics of piezo-electric composite materialsrdquo Advances in Materials Sci-ence and Engineering vol 2019 Article ID 491378416 pages 2019

[65] L Zhou S Ren C Liu and Z Ma ldquoA valid inhomogeneouscell-based smoothed finite element model for the transient


characteristics of functionally graded magneto-electro-elasticstructuresrdquo Composite Structures vol 208 pp 298ndash313 2019

[66] L Zhou M Li B Chen F Li and X Li ldquoAn in-homogeneous cell-based smoothed finite element methodfor the nonlinear transient response of functionally gradedmagneto-electro-elastic structures with damping factorsrdquoJournal of Intelligent Material Systems and Structuresvol 30 no 3 pp 416ndash437 2019

[67] R X Yao and Z F Shi ldquoSteady-state forced vibration offunctionally graded piezoelectric beamsrdquo Journal of IntelligentMaterial Systems and Structures vol 22 no 8 pp 769ndash7792011


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analyses on piezoelectric beams by using a refined sinusmodel e results have been found in good agreement withthe reference solutions for various electrical and mechanicalconstrained conditions Meanwhile the 3D exact state-spacesolution [18] and Peano series solution [19] were developedfor the cylindrical bending vibration of the FGPM laminatesrespectively Layerwise FEM was adopted to investigate thedisplacement and stress responses of an FGPM bimorphactuator [20] Qiu et al inhibited the vibration of a smartflexible clamped plate by using piezoelectric ceramic patchsensors and actuators [21] e Timoshenko beam theorywas used to analyze the static and dynamic responses ofFGPM actuators to thermo-electro-mechanical loading [22]e first-order shear deformation theory was used to studythe static bending free vibration and dynamic responses ofFGPM plates under electromechanical loading [23] e freeand induced vibrations of FGPM beams under thermo-electro-mechanical loading were characterized using the 3-order shear deformation beam theory [24] A high-ordertheory for FGPM shells was proposed based on the gen-eralized Hamiltonrsquos principle [8] Although the FEM (h-version) is adequate for low-frequency vibration analysis itis not well suited to the vibration analysis of medium- orhigh-frequency regimes [25] e spectral finite elementmethod (SFEM) [26 27] and the weak form quadraturemethod (QEM) [28 29] are developed for the dynamicanalysis of FGPM beams and structures

ough FEM is the most widely used and effectivenumerical approach in practical issues in research andengineering (including mechanics of vibration) it is notnecessarily fully perfect or cannot be further improved Forexample the probable overestimation of stiffness in solidstructures may lead to locking behavior and inaccuratestress-solving [30] By adding strain smoothing into FEM[31] Liu et al established a series of cell-based [32ndash36]node-based [37 38] edge-based [39ndash41] or face-based[42 43] smoothed FEMs (S-FEMs) and their combinations[44ndash47] ese S-FEMs with different properties can be usedto get desired solutions for a variety of benchmarks andpractical mechanic issues [48ndash50] e strain-smoothingoperations can reduce or alleviate the overstiffness ofstandard FEM significantly improving the accuracy of bothprimal and dual quantities [51] Moreover owing to absenceof parametric mapping the shape function derivatives andS-FEMmodels established in elasticity are not required to beinsensitive to mesh distortion [52] S-FEMs have beensuccessfully extended to analyze the dynamic control ofpiezoelectric sensors and actuators topological optimizationof linear piezoelectric micromotors statics frequency ordefects of smart materials [53ndash63] Zheng et al [64] utilizedthe cell-based smoothed finite element method with theasymptotic homogenization method to analyze the dynamicissues on micromechanics of piezoelectric composite ma-terials Zhou et al [65 66] deduced the linear and nonlinearcell-based smoothed finite element method of functionallygraded magneto-electro-elastic (MEE) structures and fur-ther examined the transient responses of MEE sensors orenergy harvest structures considering the damping factorsHowever there is little literature reported concerning the

dynamic response of FGPMs using the electromechanicalinhomogeneous cell-based smoothed finite element methodBecause of versatility S-FEMs become convenient and ef-ficient numerical approaches to address different physicalissues

Given the continuous change of the gradient of materialproperties along the thickness x3 direction and with cell-based gradient smoothing we deduced the basic formula ofISFEM and amodifiedWilson-θmethod to solve the integralsolution of the FGPM system e displacements and po-tentials of FGPM cantilever beams under sine wave loadcosine wave load step wave load and triangular wave loadwere analyzed in comparison with FEM

2 Basic Equations for Piezoelectric Materials

21 Geometry and Coordinate System Each beam has arectangular uniform cross section and is made of Ni layerseither completely or partially composed of FGPM beamse Cartesian coordinate system (x1 x2 x3) and geometricparameters are illustrated in Figure 1

22 Constitutive Equations At the k-th layer 3D linearconstitutive equations are polarized along its global co-ordinates as follows

σ11σ22σ33σ23σ13σ12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

c11 c12 c13 0 0 0

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

0 0 0 0 0 c66

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k) ε11ε22ε332ε232ε132ε12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

minus

0 0 e31

0 0 e32

0 0 e33

0 e24 0

e15 0 0

0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

E1

E2

E3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

(1)

D1

D2

D3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

0 0 0 0 e15 00 0 0 e24 0 0

e31 e32 e33 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

ε11ε22ε332ε232ε132ε12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

+

χ11 0 00 χ22 00 0 χ33

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)E1

E2

E3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(k)

(2)

where σij and εkl are the stress tensor and infinitesimal straintensor respectively Ei and Di are electric field and electric



cijkl x3( 1113857 c0ijklf x3( 1113857

eikl x3( 1113857 e0iklf x3( 1113857

χik x3( 1113857 χ0ikf x3( 1113857 i j k l 1 2 3

(3)



σ Cεminus eE

D eTε + χE(4)

where

σ

σ11σ33σ13

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C

c11 c13 0

c31 c33 0

0 0 c55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

e

0 e31

0 e33

e15 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E E1

E31113890 1113891

ε

ε11ε33ε13

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D D1

D31113890 1113891

χ χ11 0

0 χ331113890 1113891

(5)



ΓδuTFs dΓ



ΩδETD dΩ

+ 1113946ΓQδφdΓ minus1113946

Ωqδφ dΩ 0

(6)




u 1113944

np

i1N

ui ui Nuu

φ 1113944

np

i1N

φi φi Nφφ

(7)





are

ε xk1113872 1113873 1113946

Ωki

ε(x)Φ x minus xk1113872 1113873 dΩ (8)

E xk1113872 1113873 1113946

Ωki

E(x)Φ x minus xk1113872 1113873 dΩ (9)

z1

zkzk+1

zn

helliphellip

x3

x1SV(t)

L

h

b




Φ x minus xk1113872 1113873

1Ak

i

x isin Ωki

0 x notin Ωki

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where

Aki 1113946Ωk

i

dΩ (11)


ε xk1113872 1113873

1Ak

i

1113946Γk

i

nkuu dΓ (12)

E xk1113872 1113873

1Ak

i

1113946Γk

i

nkφφdΓ (13)




nku

nkx1

0

0 nkx3

nkx3

nkx1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nkφ

nkx1

nkx3

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(14)


ε xk1113872 1113873 1113944

ne

i1Bi

u xk1113872 1113873ui

E xk1113872 1113873 minus1113944

ne

i1Bi

φ xk1113872 1113873φi

(15)


Bi

u xk1113872 1113873

1Ak

i

1113946Γk

Nui nk

x10

0 Nui nk

x3

Nui nk

x3Nu

i nkx1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dΓ (16)

Bi

φ xk1113872 1113873

1Ak

i

1113946Γk

Nφi nk

x1

Nφi nk

x3

⎡⎢⎣ ⎤⎥⎦ dΓ (17)


Bi

u xk1113872 1113873

1Ak

i

1113944

nb

b1

Nui xGb( 1113857nk

x10


x3

Nui xGb( 1113857nk

x3Nu

i xGb( 1113857nkx1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lkb

Bi

φ xk1113872 1113873

1Ak

i

1113944

nb

b1

Nφi xGb( 1113857nk

x1

Nφi xGb( 1113857nk

x3

⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠ l

kb

(18)



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

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

(0 0 12 12)

(0 12 12 0)

(12 12 0 0)

(14 141414)

(12 0 0 12)

Element k



Node

Smoothing cells

Γc

Ak4 Ak

3

Ak2Ak

1Ωk

1

Ωk3

Ωk2

Ωk4




Material c11(GPa)

c12(GPa)

c13(GPa)

c33(GPa)

c44(GPa)

ρ(kgm3)

e31(Cm2)

e33(Cm2)




000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 05sin (2πt001)

F (N

)

t (s)





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

u 3 (m

)



t (s)

(a)

000 001 002 003 004 005ndash3

ndash2

ndash1

0

1

2

3

φ (V

)



t (s)

(b)


L

h

b

F







0050 times 10ndash9

10 times 10ndash8

15 times 10ndash8


(m)



t (s)

(a)

000 001 002 003 004 005

ndash02

ndash01

00

01

02

φ (V

)



t (s)

(b)




00

40 times 10ndash9

80 times 10ndash9

12 times 10ndash8

u 3 (m

)



t (s)

(c)

000 001 002 003 004 005ndash015

ndash010

ndash005

000

005

010

015

φ (V

)



t (s)

(d)

000 001 002 003 004 005




00

20 times 10ndash9

40 times 10ndash9

60 times 10ndash9

u 3 (m

)



t (s)

(e)

000 001 002 003 004 005ndash009

ndash006

ndash003

000

003

006

009

φ (V

)



t (s)

(f )

Figure 6 Continued



Meuroq + Kq F (19)

where

M Muu 0

0 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


euroφ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

K

Kuu Kuφ

KTuφ Kφφ

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

q

u

φ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

F

F

Q

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

Muu 1113944e

Meuu


(20)



Kuu 1113944

nc

i1BiTu CB

i

uAki

Kuφ 1113944

nc

i1BiTu eB

i

φAki

Kφφ minus1113944

nc

i1BiTφ χB

i

φAki

F 1113946ΩNT


uFs dΓ

Q 1113946ΩNT


φq dΓ

(21)


000 001 002 003 004 005




00

30 times 10ndash10

60 times 10ndash10


(m)



t (s)

(g)

000 001 002 003 004 005

ndash00045

ndash00030

ndash00015

00000

00015

00030

00045

φ (V

)



t (s)

(h)


600 800 1000 1200 1400 1600 1800

1E ndash 3

001

Err

Number of elements









00025 u3 times10minus8m 13345 13191 1538 1528 0917 0913 0564 0561 00882 00873φ (V) 2447 2391 0198 0191 0112 0109 00621 00605 000386 000372

0004 U3 times10minus9m 77799 77682 10154 10146 6165 6123 3727 3725 0514 0516φ (V) 1434 1406 0129 0125 00756 00740 00414 00401 000224 000221







00025 u3 times10minus8m 16122 16006 2017 2003 1118 1117 0741 0736 0107 0106φ (V) 2522 2584 0219 0219 0122 0123 00691 00689 000392 000393

0004 u3 times10minus9m 76019 75052 10339 10264 6264 6261 3795 3768 0501 0496φ (V) 1402 1437 0113 0113 00649 00653 00351 00350 000179 000178

100 1000

1E ndash 3

001

01

Err

t (s)

FEMISFEM











(θ14)















000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)



a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





cijkl x3( 1113857 c0ijklf x3( 1113857

eikl x3( 1113857 e0iklf x3( 1113857

χik x3( 1113857 χ0ikf x3( 1113857 i j k l 1 2 3

(3)



σ Cεminus eE

D eTε + χE(4)

where

σ

σ11σ33σ13

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C

c11 c13 0

c31 c33 0

0 0 c55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

e

0 e31

0 e33

e15 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E E1

E31113890 1113891

ε

ε11ε33ε13

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D D1

D31113890 1113891

χ χ11 0

0 χ331113890 1113891

(5)



ΓδuTFs dΓ



ΩδETD dΩ

+ 1113946ΓQδφdΓ minus1113946

Ωqδφ dΩ 0

(6)




u 1113944

np

i1N

ui ui Nuu

φ 1113944

np

i1N

φi φi Nφφ

(7)





are

ε xk1113872 1113873 1113946

Ωki

ε(x)Φ x minus xk1113872 1113873 dΩ (8)

E xk1113872 1113873 1113946

Ωki

E(x)Φ x minus xk1113872 1113873 dΩ (9)

z1

zkzk+1

zn

helliphellip

x3

x1SV(t)

L

h

b




Φ x minus xk1113872 1113873

1Ak

i

x isin Ωki

0 x notin Ωki

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where

Aki 1113946Ωk

i

dΩ (11)


ε xk1113872 1113873

1Ak

i

1113946Γk

i

nkuu dΓ (12)

E xk1113872 1113873

1Ak

i

1113946Γk

i

nkφφdΓ (13)




nku

nkx1

0

0 nkx3

nkx3

nkx1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nkφ

nkx1

nkx3

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(14)


ε xk1113872 1113873 1113944

ne

i1Bi

u xk1113872 1113873ui

E xk1113872 1113873 minus1113944

ne

i1Bi

φ xk1113872 1113873φi

(15)


Bi

u xk1113872 1113873

1Ak

i

1113946Γk

Nui nk

x10

0 Nui nk

x3

Nui nk

x3Nu

i nkx1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dΓ (16)

Bi

φ xk1113872 1113873

1Ak

i

1113946Γk

Nφi nk

x1

Nφi nk

x3

⎡⎢⎣ ⎤⎥⎦ dΓ (17)


Bi

u xk1113872 1113873

1Ak

i

1113944

nb

b1

Nui xGb( 1113857nk

x10


x3

Nui xGb( 1113857nk

x3Nu

i xGb( 1113857nkx1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lkb

Bi

φ xk1113872 1113873

1Ak

i

1113944

nb

b1

Nφi xGb( 1113857nk

x1

Nφi xGb( 1113857nk

x3

⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠ l

kb

(18)



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

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

(0 0 12 12)

(0 12 12 0)

(12 12 0 0)

(14 141414)

(12 0 0 12)

Element k



Node

Smoothing cells

Γc

Ak4 Ak

3

Ak2Ak

1Ωk

1

Ωk3

Ωk2

Ωk4




Material c11(GPa)

c12(GPa)

c13(GPa)

c33(GPa)

c44(GPa)

ρ(kgm3)

e31(Cm2)

e33(Cm2)




000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 05sin (2πt001)

F (N

)

t (s)





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

u 3 (m

)



t (s)

(a)

000 001 002 003 004 005ndash3

ndash2

ndash1

0

1

2

3

φ (V

)



t (s)

(b)


L

h

b

F







0050 times 10ndash9

10 times 10ndash8

15 times 10ndash8


(m)



t (s)

(a)

000 001 002 003 004 005

ndash02

ndash01

00

01

02

φ (V

)



t (s)

(b)




00

40 times 10ndash9

80 times 10ndash9

12 times 10ndash8

u 3 (m

)



t (s)

(c)

000 001 002 003 004 005ndash015

ndash010

ndash005

000

005

010

015

φ (V

)



t (s)

(d)

000 001 002 003 004 005




00

20 times 10ndash9

40 times 10ndash9

60 times 10ndash9

u 3 (m

)



t (s)

(e)

000 001 002 003 004 005ndash009

ndash006

ndash003

000

003

006

009

φ (V

)



t (s)

(f )

Figure 6 Continued



Meuroq + Kq F (19)

where

M Muu 0

0 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


euroφ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

K

Kuu Kuφ

KTuφ Kφφ

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

q

u

φ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

F

F

Q

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

Muu 1113944e

Meuu


(20)



Kuu 1113944

nc

i1BiTu CB

i

uAki

Kuφ 1113944

nc

i1BiTu eB

i

φAki

Kφφ minus1113944

nc

i1BiTφ χB

i

φAki

F 1113946ΩNT


uFs dΓ

Q 1113946ΩNT


φq dΓ

(21)


000 001 002 003 004 005




00

30 times 10ndash10

60 times 10ndash10


(m)



t (s)

(g)

000 001 002 003 004 005

ndash00045

ndash00030

ndash00015

00000

00015

00030

00045

φ (V

)



t (s)

(h)


600 800 1000 1200 1400 1600 1800

1E ndash 3

001

Err

Number of elements









00025 u3 times10minus8m 13345 13191 1538 1528 0917 0913 0564 0561 00882 00873φ (V) 2447 2391 0198 0191 0112 0109 00621 00605 000386 000372

0004 U3 times10minus9m 77799 77682 10154 10146 6165 6123 3727 3725 0514 0516φ (V) 1434 1406 0129 0125 00756 00740 00414 00401 000224 000221







00025 u3 times10minus8m 16122 16006 2017 2003 1118 1117 0741 0736 0107 0106φ (V) 2522 2584 0219 0219 0122 0123 00691 00689 000392 000393

0004 u3 times10minus9m 76019 75052 10339 10264 6264 6261 3795 3768 0501 0496φ (V) 1402 1437 0113 0113 00649 00653 00351 00350 000179 000178

100 1000

1E ndash 3

001

01

Err

t (s)

FEMISFEM











(θ14)















000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)



a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





Φ x minus xk1113872 1113873

1Ak

i

x isin Ωki

0 x notin Ωki

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(10)

where

Aki 1113946Ωk

i

dΩ (11)


ε xk1113872 1113873

1Ak

i

1113946Γk

i

nkuu dΓ (12)

E xk1113872 1113873

1Ak

i

1113946Γk

i

nkφφdΓ (13)




nku

nkx1

0

0 nkx3

nkx3

nkx1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nkφ

nkx1

nkx3

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(14)


ε xk1113872 1113873 1113944

ne

i1Bi

u xk1113872 1113873ui

E xk1113872 1113873 minus1113944

ne

i1Bi

φ xk1113872 1113873φi

(15)


Bi

u xk1113872 1113873

1Ak

i

1113946Γk

Nui nk

x10

0 Nui nk

x3

Nui nk

x3Nu

i nkx1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dΓ (16)

Bi

φ xk1113872 1113873

1Ak

i

1113946Γk

Nφi nk

x1

Nφi nk

x3

⎡⎢⎣ ⎤⎥⎦ dΓ (17)


Bi

u xk1113872 1113873

1Ak

i

1113944

nb

b1

Nui xGb( 1113857nk

x10


x3

Nui xGb( 1113857nk

x3Nu

i xGb( 1113857nkx1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

lkb

Bi

φ xk1113872 1113873

1Ak

i

1113944

nb

b1

Nφi xGb( 1113857nk

x1

Nφi xGb( 1113857nk

x3

⎛⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎠ l

kb

(18)



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

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

(0 0 12 12)

(0 12 12 0)

(12 12 0 0)

(14 141414)

(12 0 0 12)

Element k



Node

Smoothing cells

Γc

Ak4 Ak

3

Ak2Ak

1Ωk

1

Ωk3

Ωk2

Ωk4




Material c11(GPa)

c12(GPa)

c13(GPa)

c33(GPa)

c44(GPa)

ρ(kgm3)

e31(Cm2)

e33(Cm2)




000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 05sin (2πt001)

F (N

)

t (s)





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

u 3 (m

)



t (s)

(a)

000 001 002 003 004 005ndash3

ndash2

ndash1

0

1

2

3

φ (V

)



t (s)

(b)


L

h

b

F







0050 times 10ndash9

10 times 10ndash8

15 times 10ndash8


(m)



t (s)

(a)

000 001 002 003 004 005

ndash02

ndash01

00

01

02

φ (V

)



t (s)

(b)




00

40 times 10ndash9

80 times 10ndash9

12 times 10ndash8

u 3 (m

)



t (s)

(c)

000 001 002 003 004 005ndash015

ndash010

ndash005

000

005

010

015

φ (V

)



t (s)

(d)

000 001 002 003 004 005




00

20 times 10ndash9

40 times 10ndash9

60 times 10ndash9

u 3 (m

)



t (s)

(e)

000 001 002 003 004 005ndash009

ndash006

ndash003

000

003

006

009

φ (V

)



t (s)

(f )

Figure 6 Continued



Meuroq + Kq F (19)

where

M Muu 0

0 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


euroφ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

K

Kuu Kuφ

KTuφ Kφφ

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

q

u

φ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

F

F

Q

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

Muu 1113944e

Meuu


(20)



Kuu 1113944

nc

i1BiTu CB

i

uAki

Kuφ 1113944

nc

i1BiTu eB

i

φAki

Kφφ minus1113944

nc

i1BiTφ χB

i

φAki

F 1113946ΩNT


uFs dΓ

Q 1113946ΩNT


φq dΓ

(21)


000 001 002 003 004 005




00

30 times 10ndash10

60 times 10ndash10


(m)



t (s)

(g)

000 001 002 003 004 005

ndash00045

ndash00030

ndash00015

00000

00015

00030

00045

φ (V

)



t (s)

(h)


600 800 1000 1200 1400 1600 1800

1E ndash 3

001

Err

Number of elements









00025 u3 times10minus8m 13345 13191 1538 1528 0917 0913 0564 0561 00882 00873φ (V) 2447 2391 0198 0191 0112 0109 00621 00605 000386 000372

0004 U3 times10minus9m 77799 77682 10154 10146 6165 6123 3727 3725 0514 0516φ (V) 1434 1406 0129 0125 00756 00740 00414 00401 000224 000221







00025 u3 times10minus8m 16122 16006 2017 2003 1118 1117 0741 0736 0107 0106φ (V) 2522 2584 0219 0219 0122 0123 00691 00689 000392 000393

0004 u3 times10minus9m 76019 75052 10339 10264 6264 6261 3795 3768 0501 0496φ (V) 1402 1437 0113 0113 00649 00653 00351 00350 000179 000178

100 1000

1E ndash 3

001

01

Err

t (s)

FEMISFEM











(θ14)















000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)



a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





Material c11(GPa)

c12(GPa)

c13(GPa)

c33(GPa)

c44(GPa)

ρ(kgm3)

e31(Cm2)

e33(Cm2)




000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 05sin (2πt001)

F (N

)

t (s)





00

50 times 10ndash8

10 times 10ndash7

15 times 10ndash7

u 3 (m

)



t (s)

(a)

000 001 002 003 004 005ndash3

ndash2

ndash1

0

1

2

3

φ (V

)



t (s)

(b)


L

h

b

F







0050 times 10ndash9

10 times 10ndash8

15 times 10ndash8


(m)



t (s)

(a)

000 001 002 003 004 005

ndash02

ndash01

00

01

02

φ (V

)



t (s)

(b)




00

40 times 10ndash9

80 times 10ndash9

12 times 10ndash8

u 3 (m

)



t (s)

(c)

000 001 002 003 004 005ndash015

ndash010

ndash005

000

005

010

015

φ (V

)



t (s)

(d)

000 001 002 003 004 005




00

20 times 10ndash9

40 times 10ndash9

60 times 10ndash9

u 3 (m

)



t (s)

(e)

000 001 002 003 004 005ndash009

ndash006

ndash003

000

003

006

009

φ (V

)



t (s)

(f )

Figure 6 Continued



Meuroq + Kq F (19)

where

M Muu 0

0 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


euroφ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

K

Kuu Kuφ

KTuφ Kφφ

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

q

u

φ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

F

F

Q

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

Muu 1113944e

Meuu


(20)



Kuu 1113944

nc

i1BiTu CB

i

uAki

Kuφ 1113944

nc

i1BiTu eB

i

φAki

Kφφ minus1113944

nc

i1BiTφ χB

i

φAki

F 1113946ΩNT


uFs dΓ

Q 1113946ΩNT


φq dΓ

(21)


000 001 002 003 004 005




00

30 times 10ndash10

60 times 10ndash10


(m)



t (s)

(g)

000 001 002 003 004 005

ndash00045

ndash00030

ndash00015

00000

00015

00030

00045

φ (V

)



t (s)

(h)


600 800 1000 1200 1400 1600 1800

1E ndash 3

001

Err

Number of elements









00025 u3 times10minus8m 13345 13191 1538 1528 0917 0913 0564 0561 00882 00873φ (V) 2447 2391 0198 0191 0112 0109 00621 00605 000386 000372

0004 U3 times10minus9m 77799 77682 10154 10146 6165 6123 3727 3725 0514 0516φ (V) 1434 1406 0129 0125 00756 00740 00414 00401 000224 000221







00025 u3 times10minus8m 16122 16006 2017 2003 1118 1117 0741 0736 0107 0106φ (V) 2522 2584 0219 0219 0122 0123 00691 00689 000392 000393

0004 u3 times10minus9m 76019 75052 10339 10264 6264 6261 3795 3768 0501 0496φ (V) 1402 1437 0113 0113 00649 00653 00351 00350 000179 000178

100 1000

1E ndash 3

001

01

Err

t (s)

FEMISFEM











(θ14)















000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)



a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls








0050 times 10ndash9

10 times 10ndash8

15 times 10ndash8


(m)



t (s)

(a)

000 001 002 003 004 005

ndash02

ndash01

00

01

02

φ (V

)



t (s)

(b)




00

40 times 10ndash9

80 times 10ndash9

12 times 10ndash8

u 3 (m

)



t (s)

(c)

000 001 002 003 004 005ndash015

ndash010

ndash005

000

005

010

015

φ (V

)



t (s)

(d)

000 001 002 003 004 005




00

20 times 10ndash9

40 times 10ndash9

60 times 10ndash9

u 3 (m

)



t (s)

(e)

000 001 002 003 004 005ndash009

ndash006

ndash003

000

003

006

009

φ (V

)



t (s)

(f )

Figure 6 Continued



Meuroq + Kq F (19)

where

M Muu 0

0 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


euroφ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

K

Kuu Kuφ

KTuφ Kφφ

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

q

u

φ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

F

F

Q

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

Muu 1113944e

Meuu


(20)



Kuu 1113944

nc

i1BiTu CB

i

uAki

Kuφ 1113944

nc

i1BiTu eB

i

φAki

Kφφ minus1113944

nc

i1BiTφ χB

i

φAki

F 1113946ΩNT


uFs dΓ

Q 1113946ΩNT


φq dΓ

(21)


000 001 002 003 004 005




00

30 times 10ndash10

60 times 10ndash10


(m)



t (s)

(g)

000 001 002 003 004 005

ndash00045

ndash00030

ndash00015

00000

00015

00030

00045

φ (V

)



t (s)

(h)


600 800 1000 1200 1400 1600 1800

1E ndash 3

001

Err

Number of elements









00025 u3 times10minus8m 13345 13191 1538 1528 0917 0913 0564 0561 00882 00873φ (V) 2447 2391 0198 0191 0112 0109 00621 00605 000386 000372

0004 U3 times10minus9m 77799 77682 10154 10146 6165 6123 3727 3725 0514 0516φ (V) 1434 1406 0129 0125 00756 00740 00414 00401 000224 000221







00025 u3 times10minus8m 16122 16006 2017 2003 1118 1117 0741 0736 0107 0106φ (V) 2522 2584 0219 0219 0122 0123 00691 00689 000392 000393

0004 u3 times10minus9m 76019 75052 10339 10264 6264 6261 3795 3768 0501 0496φ (V) 1402 1437 0113 0113 00649 00653 00351 00350 000179 000178

100 1000

1E ndash 3

001

01

Err

t (s)

FEMISFEM











(θ14)















000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)



a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





Meuroq + Kq F (19)

where

M Muu 0

0 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦


euroφ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

K

Kuu Kuφ

KTuφ Kφφ

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

q

u

φ

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

F

F

Q

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

Muu 1113944e

Meuu


(20)



Kuu 1113944

nc

i1BiTu CB

i

uAki

Kuφ 1113944

nc

i1BiTu eB

i

φAki

Kφφ minus1113944

nc

i1BiTφ χB

i

φAki

F 1113946ΩNT


uFs dΓ

Q 1113946ΩNT


φq dΓ

(21)


000 001 002 003 004 005




00

30 times 10ndash10

60 times 10ndash10


(m)



t (s)

(g)

000 001 002 003 004 005

ndash00045

ndash00030

ndash00015

00000

00015

00030

00045

φ (V

)



t (s)

(h)


600 800 1000 1200 1400 1600 1800

1E ndash 3

001

Err

Number of elements









00025 u3 times10minus8m 13345 13191 1538 1528 0917 0913 0564 0561 00882 00873φ (V) 2447 2391 0198 0191 0112 0109 00621 00605 000386 000372

0004 U3 times10minus9m 77799 77682 10154 10146 6165 6123 3727 3725 0514 0516φ (V) 1434 1406 0129 0125 00756 00740 00414 00401 000224 000221







00025 u3 times10minus8m 16122 16006 2017 2003 1118 1117 0741 0736 0107 0106φ (V) 2522 2584 0219 0219 0122 0123 00691 00689 000392 000393

0004 u3 times10minus9m 76019 75052 10339 10264 6264 6261 3795 3768 0501 0496φ (V) 1402 1437 0113 0113 00649 00653 00351 00350 000179 000178

100 1000

1E ndash 3

001

01

Err

t (s)

FEMISFEM











(θ14)















000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)



a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls









00025 u3 times10minus8m 13345 13191 1538 1528 0917 0913 0564 0561 00882 00873φ (V) 2447 2391 0198 0191 0112 0109 00621 00605 000386 000372

0004 U3 times10minus9m 77799 77682 10154 10146 6165 6123 3727 3725 0514 0516φ (V) 1434 1406 0129 0125 00756 00740 00414 00401 000224 000221







00025 u3 times10minus8m 16122 16006 2017 2003 1118 1117 0741 0736 0107 0106φ (V) 2522 2584 0219 0219 0122 0123 00691 00689 000392 000393

0004 u3 times10minus9m 76019 75052 10339 10264 6264 6261 3795 3768 0501 0496φ (V) 1402 1437 0113 0113 00649 00653 00351 00350 000179 000178

100 1000

1E ndash 3

001

01

Err

t (s)

FEMISFEM











(θ14)















000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)



a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls












(θ14)















000 001 002 003 004

ndash04

ndash02

00

02

04

06

F = 0 5cos (2πt001)

F (N

)

t (s)



a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




a0 6

(θΔt)2

a1 3

(θΔt)

a2 2a1

a3 θΔt2

a4 a0

θ

a5 minusa2

θ1113874 1113875

a6 1minus3θ

a7 Δt2

a8 Δt2

6

(22)




+ M a0qt + a2 _qt + 2euroqt( 1113857

+ C a1qt + 2 _qt + a3 euroqt( 1113857

(23)







(25)



ckl x3( 1113857 c0klf x3( 1113857

ekl x3( 1113857 e0klf x3( 1113857

χkl x3( 1113857 χ0klf x3( 1113857

(26)



000 001 002 003 00400

01

02

03

04

05

06

F = 05

F (N

)

t (s)







000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls








000 001 002 003 004 005 006

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 006t (s)

00

04

08

12

16

20

24

28

32

φ (V

)



(b)


000 001 002 003 004 005 006t (s)

u 3 (m

)

28 times 10ndash7

24 times 10ndash7

20 times 10ndash7

16 times 10ndash7

12 times 10ndash7

80 times 10ndash8

40 times 10ndash8

00



(a)



t (s)000 001 002 003 004 005 006

00

05

10

15

20

25

30

35

φ (V

)

(b)








000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls









000 001 002 003 00400

01

02

03

04

05


50(t ndash 002)002 lt t le 003


50t0 le t le 001

F (N

)

t (s)


000 001 002 003 004 005 00600

20 times 10ndash8

40 times 10ndash8

60 times 10ndash8

80 times 10ndash8

10 times 10ndash7

12 times 10ndash7

14 times 10ndash7

u 3 (m

)

t (s)



(a)

000 001 002 003 004 005 00600

04

08

12

16

20

24

28

t (s)

φ (V

)



(b)




6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





6 Conclusions






Data Availability






Acknowledgments


References





u 3 (m

)

000 001 002 003 004 005 00600

30 times 10ndash8

60 times 10ndash8

90 times 10ndash8

12 times 10ndash7

15 times 10ndash7

18 times 10ndash7

t (s)



(a)

000 001 002 003 004 005 00600

03

06

09

12

15

18

21

24

27

t (s)

φ (V

)



(b)











































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls












































































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls










































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls










Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




Documents

ACouplingElectromechanicalInhomogeneousCell-Based ...downloads.hindawi.com/journals/amse/2019/2812748.pdf · in FGPMs were adopted directly to replace the homogenization parameters