-
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.
-
4 Shock and Vibration
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
]]]] ,
-
Shock and Vibration 5
(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)
-
6 Shock and Vibration
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 120 × 8ICS-FEM 150 × 10ICS-FEM 30 × 2
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 60 × 4ICS-FEM 120 × 8ICS-FEM 150 × 10
ICS-FEM 30 × 2
ICS-FEM 30 × 2
= 0
= 0
= 0
= 0
= 1
ICS-FEM 120 × 8ICS-FEM 150 × 10ICS-FEM 30 × 2
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 60 × 4ICS-FEM 120 × 8ICS-FEM 150 × 10
ICS-FEM 30 × 2
ICS-FEM 30 × 2
= 0
= 0
= 0
= 0
= 1
ICS-FEM 120 × 8ICS-FEM 150 × 10ICS-FEM 30 × 2
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 60 × 4ICS-FEM 120 × 8ICS-FEM 150 × 10
ICS-FEM 30 × 2
ICS-FEM 30 × 2
= 0
= 0
= 0
= 0
= 1
ICS-FEM 120 × 8ICS-FEM 150 × 10ICS-FEM 30 × 2
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 60 × 4ICS-FEM 120 × 8ICS-FEM 150 × 10
ICS-FEM 30 × 2
ICS-FEM 30 × 2
= 0
= 0
= 0
= 0
= 1
ICS-FEM 120 × 8ICS-FEM 150 × 10ICS-FEM 30 × 2
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.
-
Shock and Vibration 7
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.
-
8 Shock and Vibration
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.
-
Shock and Vibration 9
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)
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
(b)
0300060009000
120001500018000210002400027000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(c)
1 2 3 4 5 6 7 8 9 10 110Mode
0300060009000
120001500018000210002400027000300003300036000
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
(d)
0100002000030000400005000060000700008000090000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(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:
-
10 Shock and Vibration
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.
-
Shock and Vibration 11
1 2 3 4 5 6 7 8 9 10 110Mode
02000400060008000
100001200014000160001800020000220002400026000
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)
02000400060008000
100001200014000160001800020000220002400026000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(b)
1 2 3 4 5 6 7 8 9 10 110Mode
040008000
120001600020000240002800032000
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
(c)
040008000
12000160002000024000280003200036000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(d)
0100002000030000400005000060000700008000090000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(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
-
12 Shock and Vibration
02000400060008000
100001200014000160001800020000220002400026000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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)
02000400060008000
100001200014000160001800020000220002400026000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(b)
02000400060008000
100001200014000160001800020000220002400026000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(c)
02000400060008000
100001200014000160001800020000220002400026000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(d)
02000400060008000
100001200014000160001800020000220002400026000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(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
-
Shock and Vibration 13
(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
-
14 Shock and Vibration
040008000
1200016000200002400028000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
040008000
120001600020000240002800032000
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
(b)
1 2 3 4 5 6 7 8 9 10 110Mode
040008000
120001600020000240002800032000
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
(c)
040008000
120001600020000240002800032000360004000044000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(d)
0100002000030000400005000060000700008000090000
100000110000120000
Freq
uenc
y (H
z)
1 2 3 4 5 6 7 8 9 10 110Mode
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
(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.
-
Shock and Vibration 15
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.
References
[1] W. Eerenstein, N. D. Mathur, and J. F. Scott, “Multiferroic
andmagnetoelectric materials,” Nature, vol. 442, no. 7104, pp.
759–765, 2006.
[2] J. F. Scott, “Applications of magnetoelectrics,” Journal of
Materi-als Chemistry, vol. 22, no. 11, pp. 4567–4574, 2012.
[3] A. Zhang and B. Wang, “Effects of crack surface
electrostatictractions on the fracture behaviour of magnetoelectric
compos-ite materials,”Mechanics of Materials, vol. 102, pp. 15–25,
2016.
[4] J. Zhai, Z. Xing, S. Dong, J. Li, and D. Viehland,
“Detectionof pico-Tesla magnetic fields using magneto-electric
sensors atroom temperature,”Applied Physics Letters, vol. 88, no.
6, ArticleID 062510, 2006.
[5] D. J. Huang, H. J. Ding, and W. Q. Chen, “Analytical
solutionfor functionally graded magneto-electro-elastic plane
beams,”International Journal of Engineering Science, vol. 45, no.
2-8, pp.467–485, 2007.
[6] A. Milazzo, C. Orlando, and A. Alaimo, “An analytical
solutionfor themagneto-electro-elastic bimorph beam forced
vibrationsproblem,” Smart Materials and Structures, vol. 18, no. 8,
ArticleID 085012, 2009.
http://www.ase.uc.edu/~liugr/software.htmlhttp://www.ase.uc.edu/~liugr/software.html
-
16 Shock and Vibration
[7] W. Q. Chen, K. Y. Lee, and H. J. Ding, “On free vibrationof
non-homogeneous transversely isotropic magneto-electro-elastic
plates,” Journal of Sound and Vibration, vol. 279, no. 1-2,pp.
237–251, 2005.
[8] C.-P. Wu and Y.-C. Lu, “A modified Pagano method for the3D
dynamic responses of functionally gradedmagneto-electro-elastic
plates,” Composite Structures, vol. 90, no. 3, pp.
363–372,2009.
[9] A. R. Annigeri, N. Ganesan, and S. Swarnamani, “Free
vibrationbehaviour of multiphase and layered
magneto-electro-elasticbeam,” Journal of Sound and Vibration, vol.
299, no. 1-2, pp. 44–63, 2007.
[10] R. K. Bhangale and N. Ganesan, “Static analysis of
simplysupported functionally graded and layered
magneto-electro-elastic plates,” International Journal of Solids
and Structures, vol.43, no. 10, pp. 3230–3253, 2006.
[11] R. K. Bhangale and N. Ganesan, “Free vibration studiesof
simply supported non-homogeneous functionally
gradedmagneto-electro-elastic finite cylindrical shells,” Journal
ofSound and Vibration, vol. 288, no. 1-2, pp. 412–422, 2005.
[12] R. K. Bhangale and N. Ganesan, “Free vibration of
simplysupported functionally graded and layered
magneto-electro-elastic plates by finite element method,” Journal
of Sound andVibration, vol. 294, no. 4, pp. 1016–1038, 2006.
[13] R. G. Lage, C. M. M. Soares, C. A. M. Soares, and J. N.
Reddy,“Layerwise partial mixed finite element analysis of
magneto-electro-elastic plates,”Computers & Structures, vol.
82, pp. 1293–1301, 2004.
[14] S. S. Phoenix, S. K. Satsangi, and B. N. Singh,
“Layer-wisemodelling of magneto-electro-elastic plates,” Journal of
Soundand Vibration, vol. 324, no. 3-5, pp. 798–815, 2009.
[15] G. R. Buchanan, “Free vibration of an infinite
magneto-electro-elastic cylinder,” Journal of Sound and Vibration,
vol. 268, no. 2,pp. 413–426, 2003.
[16] J. Sladek, V. Sladek, S. Krahulec, C. S. Chen, and D. L.
Young,“Analyses of circular magnetoelectroelastic plates with
func-tionally graded material properties,” Mechanics of
AdvancedMaterials and Structures, vol. 22, no. 6, pp. 479–489,
2015.
[17] J. Liu, P. Zhang, G. Lin, W. Wang, and S. Lu, “High order
solu-tions for the magneto-electro-elastic plate with
non-uniformmaterials,” International Journal of Mechanical
Sciences, vol. 115-116, pp. 532–551, 2016.
[18] L.-L. Ke and Y.-S. Wang, “Free vibration of
size-dependentmagneto-electro-elastic nanobeams based on the
nonlocal the-ory,” Physica E: Low-Dimensional Systems and
Nanostructures,vol. 63, pp. 52–61, 2014.
[19] G. R. Liu, H. Nguyen-Xuan, and T. Nguyen-Thoi, “A
theoreticalstudy on the smoothed FEM (S-FEM)models: properties,
accu-racy and convergence rates,” International Journal for
NumericalMethods in Engineering, vol. 84, no. 10, pp. 1222–1256,
2010.
[20] S. P. Bordas and S. Natarajan, “On the approximation in
thesmoothed finite elementmethod (SFEM),” International Journalfor
Numerical Methods in Engineering, vol. 81, no. 5, pp. 660–670,
2010.
[21] G. R. Liu and G. Y. Zhang, “A normed G space and weak-ened
weak (W-2) formulation of a cell-based smoothed pointinterpolation
method,” International Journal of ComputationalMethods, vol. 6, no.
1, pp. 147–179, 2009.
[22] G. R. Liu, T. T. Nguyen, K. Y. Dai, and K. Y. Lam,
“Theoreticalaspects of the smoothed finite element method (SFEM),”
Inter-national Journal for Numerical Methods in Engineering, vol.
71,no. 8, pp. 902–930, 2007.
[23] H. Feng, X. Cui, and G. Li, “A stable nodal integration
methodfor static and quasi-static electromagnetic field
computation,”Journal of Computational Physics, vol. 336, pp.
580–594, 2017.
[24] S. Li, X. Cui, H. Feng, and G.Wang, “An electromagnetic
form-ing analysis modelling using nodal integration
axisymmetricthin shell,” Journal of Materials Processing
Technology, vol. 244,pp. 62–72, 2017.
[25] S. Natarajan, A. J.M. Ferreira, S. P. A. Bordas, E.
Carrera, andM.Cinefra, “Analysis of composite plates by a unified
formulation-cell based smoothed finite element method and field
consistentelements,” Composite Structures, vol. 105, pp. 75–81,
2013.
[26] K. Nguyen-Quang, H. Dang-Trung, V. Ho-Huu, H. Luong-Van,and
T. Nguyen-Thoi, “Analysis and control of FGM plates inte-grated
with piezoelectric sensors and actuators using cell-basedsmoothed
discrete shear gap method (CS-DSG3),” CompositeStructures, vol.
165, pp. 115–129, 2017.
[27] T. Nguyen-Thoi, H. C. Vu-Do, T. Rabczuk, and H.
Nguyen-Xuan, “A node-based smoothed finite element method (NS-FEM)
for upper bound solution to visco-elastoplastic analysesof solids
using triangular and tetrahedral meshes,” ComputerMethods
AppliedMechanics and Engineering, vol. 199, no. 45-48,pp.
3005–3027, 2010.
[28] T. Nguyen-Thoi, G. R. Liu, H. Nguyen-Xuan, and C.
Nguyen-Tran, “Adaptive analysis using the node-based smoothed
finiteelement method (NS-FEM),” International Journal for
Numeri-calMethods in Biomedical Engineering, vol. 27, no. 2, pp.
198–218,2011.
[29] T. N. Tran, G. R. Liu, H. Nguyen-Xuan, and T.
Nguyen-Thoi,“An edge-based smoothed finite element method for
primal-dual shakedown analysis of structures,” International
Journal forNumerical Methods in Engineering, vol. 82, no. 7, pp.
917–938,2010.
[30] H. Nguyen-Xuan, L. V. Tran, T. Nguyen-Thoi, and H. C.
Vu-Do, “Analysis of functionally graded plates using an
edge-basedsmoothed finite element method,”Composite Structures,
vol. 93,no. 11, pp. 3019–3039, 2011.
[31] W. Li, Y. Chai, M. Lei, and G. R. Liu, “Analysis of
coupledstructural-acoustic problems based on the smoothed
finiteelement method (S-FEM),” Engineering Analysis with
BoundaryElements, vol. 42, pp. 84–91, 2014.
[32] T. Nguyen-Thoi, G. R. Liu, H. C. Vu-Do, and H. Nguyen-Xuan,
“A face-based smoothed finite elementmethod (FS-FEM)for
visco-elastoplastic analyses of 3D solids using tetrahedralmesh,”
Computer Methods Applied Mechanics and Engineering,vol. 198, no.
41-44, pp. 3479–3498, 2009.
[33] C. Jiang, Z.-Q. Zhang, G. R. Liu, X. Han, and W. Zeng,“An
edge-based/node-based selective smoothed finite elementmethod using
tetrahedrons for cardiovascular tissues,” Engi-neering Analysis
with Boundary Elements, vol. 59, pp. 62–77,2015.
[34] W. Zeng, G. R. Liu, D. Li, and X. W. Dong, “A
smoothingtechnique based beta finite element method (beta FEM)
forcrystal plasticity modeling,” Computers & Structures, vol.
162,pp. 48–67, 2016.
[35] H. Nguyen-Van, N. Mai-Duy, and T. Tran-Cong, “A
smoothedfour-node piezoelectric element for analysis of
two-dimension-al smart structures,” Computer Modeling in
Engineering andSciences, vol. 23, no. 3, pp. 209–222, 2008.
[36] Y. Onishi, R. Iida, and K. Amaya, “F-bar aided
edge-basedsmoothed finite element method using tetrahedral
elementsfor finite deformation analysis of nearly incompressible
solids,”
-
Shock and Vibration 17
International Journal for Numerical Methods in Engineering,
vol.109, no. 11, pp. 1582–1606, 2017.
[37] W. Zuo and K. Saitou, “Multi-material topology
optimizationusing ordered SIMP interpolation,” Structural and
Multidisci-plinary Optimization, vol. 55, no. 2, pp. 477–491,
2017.
[38] L. M. Zhou, G. W. Meng, F. Li, and H. Wang,
“Cell-basedsmoothed finite element method-virtual crack closure
tech-nique for a piezoelectric material of crack,”
MathematicalProblems in Engineering, vol. 2015, Article ID 371083,
10 pages,2015.
[39] C. V. Le, “Estimation of bearing capacity factors of
cohesive-frictional soil using the cell-based smoothed finite
elementmethod,” Computers & Geosciences, vol. 83, pp. 178–183,
2017.
[40] L. M. Zhou, G. W. Meng, F. Li, and S. Gu, “A
cell-basedsmoothed XFEM for fracture in piezoelectric
materials,”Advances inMaterials Science and Engineering, vol. 2016,
ArticleID 4125307, 14 pages, 2016.
[41] A. L. Pramod, S. Natarajan, A. J. Ferreira, E. Carrera,
andM. Cinefra, “Static and free vibration analysis of
cross-plylaminated plates using the Reissner-mixed variational
theoremand the cell based smoothed finite element method,”
EuropeanJournal of Mechanics-A/Solids, vol. 62, pp. 14–21,
2017.
[42] F.Wu, L. Y. Yao,M. Hu, and Z. C. He, “A stochastic
perturbationedge-based smoothed finite element method for the
analysisof uncertain structural-acoustics problems with random
vari-ables,” Engineering Analysis with Boundary Elements, vol. 80,
pp.116–126, 2017.
[43] J. L. Mantari, E. M. Bonilla, and C. Guedes Soares, “A
newtangential-exponential higher order shear deformation theoryfor
advanced composite plates,” Composites Part B: Engineering,vol. 60,
pp. 319–328, 2014.
[44] A. Daga, N. Ganesan, and K. Shankar, “Behaviour of
magneto-electro-elastic sensors under transient mechanical
loading,”Sensors andActuators A: Physical, vol. 150, no. 1, pp.
46–55, 2009.
-
International Journal of
AerospaceEngineeringHindawiwww.hindawi.com Volume 2018
RoboticsJournal of
Hindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com Volume 2018
Shock and Vibration
Hindawiwww.hindawi.com Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwww.hindawi.com
Volume 2018
Hindawi Publishing Corporation http://www.hindawi.com Volume
2013Hindawiwww.hindawi.com
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwww.hindawi.com Volume 2018
International Journal of
RotatingMachinery
Hindawiwww.hindawi.com Volume 2018
Modelling &Simulationin EngineeringHindawiwww.hindawi.com
Volume 2018
Hindawiwww.hindawi.com Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwww.hindawi.com Volume 2018
Hindawiwww.hindawi.com Volume 2018
Navigation and Observation
International Journal of
Hindawi
www.hindawi.com Volume 2018
Advances in
Multimedia
Submit your manuscripts atwww.hindawi.com
https://www.hindawi.com/journals/ijae/https://www.hindawi.com/journals/jr/https://www.hindawi.com/journals/apec/https://www.hindawi.com/journals/vlsi/https://www.hindawi.com/journals/sv/https://www.hindawi.com/journals/ace/https://www.hindawi.com/journals/aav/https://www.hindawi.com/journals/jece/https://www.hindawi.com/journals/aoe/https://www.hindawi.com/journals/tswj/https://www.hindawi.com/journals/jcse/https://www.hindawi.com/journals/je/https://www.hindawi.com/journals/js/https://www.hindawi.com/journals/ijrm/https://www.hindawi.com/journals/mse/https://www.hindawi.com/journals/ijce/https://www.hindawi.com/journals/ijap/https://www.hindawi.com/journals/ijno/https://www.hindawi.com/journals/am/https://www.hindawi.com/https://www.hindawi.com/