Research Article  Open Access
A Coupling Electromechanical Inhomogeneous CellBased Smoothed Finite Element Method for Dynamic Analysis of Functionally Graded Piezoelectric Beams
Abstract
To accurately simulate the continuous property change of functionally graded piezoelectric materials (FGPMs) and overcome the overstiffness of the finite element method (FEM), we present an electromechanical inhomogeneous cellbased smoothed FEM (ISFEM) of FGPMs. Firstly, ISFEM formulations were derived to calculate the transient response of FGPMs, and then, a modified Wilsonθ method was deduced to solve the integration of the FGPM system. The true parameters at the Gaussian integration point in FGPMs were adopted directly to replace the homogenization parameters in an element. ISFEM provides a closetoexact stiffness of the continuous system, which could automatically and more easily generate for complicated domains and thus significantly decrease numerical errors. The accuracy and trustworthiness of ISFEM were verified as higher than the standard FEM by several numerical examples.
1. Introduction
Because of their outstanding electromechanical properties, easy fabricability, and preparation flexibility, piezoelectric materials are extensively applied as sensors and actuators to monitor and modulate the response of structures [1, 2]. Piezoelectric actuators and sensors are innovative for microscopic electromechanical systems and intelligent material systems, particularly in aerospace and medical fields [3]. Conventional piezoelectric sensors and actuators comprise multiple layers of various piezoelectric materials [4–7]. Moreover, piezoelectric layers with uniform material properties are limited by large bending displacement, stress concentration, creeping at high temperature, and failure from interfacial unbounding. All these phenomena are induced by mechanical or electric loading at layer interfaces [8].
To overcome the above limitations, Zhu et al. introduced and fabricated functionally graded piezoelectric material (FGPM) sensors and actuators [9, 10]. FGPMs change nonstop in one or more directions without generating internal stress concentration despite the production of large displacements. Takagi et al. fabricated FGPM bimorph actuators by using a mixed system of leadzirconatetitanate (PZT) and Pt [11]. Nowadays, FGPMs are widely used intelligent materials for sensors and actuators in microstructural engineering. Many efforts have been made to analyze the behaviors and static/dynamic responses of FGPMs (e.g., shells, beams, and plates), such as the wave propagation study of FGPM plates based on the laminate theory [12]. Moreover, exact 3D analysis of FGPM rectangular plates was conducted by using a statespace approach [13] and to investigate the natural frequencies and mode shapes after being poled perpendicular to the middle plane [14]. The above method was also applied to explore the free vibration of rectangular FGPM plates [15]. The semianalytical finite element method (FEM) was used to investigate the static response of anisotropic and linear functionally graded magnetoelectroelastic plates [16]. LezgyNazargah et al. [17] carried out static and dynamic analyses on piezoelectric beams by using a refined sinus model. The results have been found in good agreement with the reference solutions for various electrical and mechanical constrained conditions. Meanwhile, the 3D exact statespace solution [18] and Peano series solution [19] were developed for the cylindrical bending vibration of the FGPM laminates, respectively. Layerwise FEM was adopted to investigate the displacement and stress responses of an FGPM bimorph actuator [20]. Qiu et al. inhibited the vibration of a smart flexible clamped plate by using piezoelectric ceramic patch sensors and actuators [21]. The Timoshenko beam theory was used to analyze the static and dynamic responses of FGPM actuators to thermoelectromechanical loading [22]. The firstorder shear deformation theory was used to study the static bending, free vibration, and dynamic responses of FGPM plates under electromechanical loading [23]. The free and induced vibrations of FGPM beams under thermoelectromechanical loading were characterized using the 3order shear deformation beam theory [24]. A highorder theory for FGPM shells was proposed based on the generalized Hamilton’s principle [8]. Although the FEM (hversion) is adequate for lowfrequency vibration analysis, it is not well suited to the vibration analysis of medium or highfrequency regimes [25]. The spectral finite element method (SFEM) [26, 27] and the weak form quadrature method (QEM) [28, 29] are developed for the dynamic analysis of FGPM beams and structures.
Though FEM is the most widely used and effective numerical approach in practical issues in research and engineering (including mechanics of vibration), it is not necessarily fully perfect or cannot be further improved. For example, the probable overestimation of stiffness in solid structures may lead to locking behavior and inaccurate stresssolving [30]. By adding strain smoothing into FEM[31], Liu et al. established a series of cellbased [32–36], nodebased [37, 38], edgebased [39–41], or facebased [42, 43] smoothed FEMs (SFEMs) and their combinations [44–47]. These SFEMs with different properties can be used to get desired solutions for a variety of benchmarks and practical mechanic issues [48–50]. The strainsmoothing operations can reduce or alleviate the overstiffness of standard FEM, significantly improving the accuracy of both primal and dual quantities [51]. Moreover, owing to absence of parametric mapping, the shape function derivatives and SFEM models established in elasticity are not required to be insensitive to mesh distortion [52]. SFEMs have been successfully extended to analyze the dynamic control of piezoelectric sensors and actuators, topological optimization of linear piezoelectric micromotors, statics, frequency, or defects of smart materials [53–63]. Zheng et al. [64] utilized the cellbased smoothed finite element method with the asymptotic homogenization method to analyze the dynamic issues on micromechanics of piezoelectric composite materials. Zhou et al. [65, 66] deduced the linear and nonlinear cellbased smoothed finite element method of functionally graded magnetoelectroelastic (MEE) structures and further examined the transient responses of MEE sensors or energy harvest structures considering the damping factors. However, there is little literature reported concerning the dynamic response of FGPMs using the electromechanical inhomogeneous cellbased smoothed finite element method. Because of versatility, SFEMs become convenient and efficient numerical approaches to address different physical issues.
Given the continuous change of the gradient of material properties along the thickness x_{3} direction and with cellbased gradient smoothing, we deduced the basic formula of ISFEM and a modified Wilsonθ method to solve the integral solution of the FGPM system. The displacements and potentials of FGPM cantilever beams under sine wave load, cosine wave load, step wave load, and triangular wave load were analyzed in comparison with FEM.
2. Basic Equations for Piezoelectric Materials
2.1. Geometry and Coordinate System
Each beam has a rectangular uniform cross section and is made of N_{i} layers either completely or partially composed of FGPM beams. The Cartesian coordinate system (x_{1}, x_{2}, x_{3}) and geometric parameters are illustrated in Figure 1.
2.2. Constitutive Equations
At the kth layer, 3D linear constitutive equations are polarized along its global coordinates as follows:where and are the stress tensor and infinitesimal strain tensor, respectively, and are electric field and electric displacement vector components, respectively; , , and are the piezoelectric, elastic, and dielectric material constants, respectively. Different from homogeneous piezoelectric materials, the three constants are dependent on coordinate . We assume that the material properties along the thickness direction are arbitrarily distributed as follows:where is an arbitrary function and , , and are values at the plane x_{3} = 0.
In an FGPM beam, equations (1) and (2) are reduced towhere
2.3. Weak Formulation
The principle of virtual work for a piezoelectric medium of volume Ω and regular boundary surface Γ can be written aswhere F_{s}, F_{v}, u, and φ are the vectors of surface force, mechanical body force, node displacements, and node electrical potentials, respectively; , , and are the electrical body charge, surface charge, and mass density, respectively; and is the virtual quantity.
3. Electromechanical ISFEM
The solving domain Ω is discretized into n_{p} elements, which contain N_{n} nodes; the approximation displacement and the approximation electrical potential for the FGPM problem can be expressed aswhere and are the ISFEM displacement shape function and electrical potential shape function, respectively.
Fournode element is divided into four smoothing subdomains. Field nodes, edge smoothing nodes, center smoothing nodes and edge Gaussian points, the outer normal vector distribution, and the shape function values are shown in Figure 2.
At any point in the smoothing subdomains , the smoothed strain and the smoothed electric field arewhere and are the strain and electric field in FEM, respectively, and is the constant function:where
Substituting equation (10) into equations (8) and (9), then we havewhere is the boundary of and and are the outer normal vector matrixes of the smoothing domain boundary
Equations (12) and (13) can be rewritten aswhere is the number of smoothing elements.
At the Gaussian point , equations (16) and (17) are where and are the Gaussian point and the length of the smoothing boundary, respectively, and is the total number of boundaries for each smoothing subdomain. As the shape function is linearly changed along each side of the smoothing subdomain, one Gauss point is sufficient for accurate boundary integration [30].
The essential distinction between ISFEM and FEM is that FEM needs to construct the shape function matrix of the element, while ISFEM only needs to use the shape function at the Gaussian point of the smoothing element boundary and does not require to involve the shape function derivatives. The above can reduce the continuity requirement of the shape function, and therefore, the accuracy and convergence of the method are improved.
The dynamic model of the FGPM electromechanical system can be derived from the Hamilton principle in the following form:wherewhere , (i = 1, 2, 3, 4) is the mass of the ith smoothing element corresponding to node i, T is the smoothing element thickness, and is the density of Gaussian integration point of the ith smoothing subdomain:where .
The application of the inhomogeneous smoothing element is to calculate stiffness matrix of the element. The parameters of four smoothing subdomains (i = 1, 2, 3, 4) are various in the elements, so the actual parameters at the Gaussian integration point are taken directly in order to reflect the changes of material property in each element.
4. Modified Wilsonθ Method
The modified Wilsonθ method is an important scheme and an implicit integral way to solve the dynamic system equations [63]. If θ > 1.37, the solution is unconditionally stable. The detailed procedures are showed as follows:
4.1. Initial Calculation
(1)Formulate generalized stiffness matrix , mass matrix , and damping matrix (2)Calculate initial values of , , (3)Select the time step Δt and the integral constant θ (θ = 1.4)(4)Formulate an effective generalized stiffness matrix :
4.2. For Each Time Step
(1)Calculate the payload at time t + θΔt:(2)Calculate the generalized displacement at time t + θΔt:(3)Calculate the generalized acceleration, generalized speed, and generalized displacement at time t + Δt:
5. Numerical Examples
Four numerical examples were conducted under sine wave load, cosine wave load, step wave load and triangular wave load, respectively. FGPM cantilever beams of the same dimensions (length L = 40 mm, width h = 5 mm and thickness b = 1 mm) were subjected to forced vibration (Figure 3). The material constants are shown in Table 1. And initial conditions were and at t = 0 moment. The FGPM beams were made of PZT4 or PZT5H on basis of exponentially graded piezoelectric materials with the following material properties:where and n is the gradient parameter.

5.1. Sine Wave Load
The load F applied to the free end and the load waveform is demonstrated in Figure 4. A convergence investigation with respect to meshes was first carried out. Four smoothing subcells were used for electromechanical ISFEM with ∆t = 1 × 10^{−3} s. The variations of displacement u_{3} and electrical potential φ at the loading point of the PZT4based FGPM beam combined with respect to time are shown in Figure 5. The results at n = −5 with the element number of 480, 800, 1200, or 1680 were compared with the reference solution [67]. The variations of u_{3} and φ at the loading point combined with respect to time at n = −1, 0, 1, and 5 in comparison with the reference solution are shown in Figure 6 [67]. Figure 7 illustrates the total energy norm Err versus the mesh density at t = 0.002 s and t = 0.01 s. The simulation results are well consistent among different numbers of meshes, which demonstrate the high convergence of ISFEM.
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(b)
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Figure 8 shows a comparison of calculation time between ISFEM and FEM at Intel ® Xeon ® CPU E31220 v3 @ 3.10 GHz, 16GB RAM. The bandwidth of the system matrices for the FEM and ISFEM is identical. However, in the ISFEM, only the values of shape functions (not the derivatives) at the quadrature points are needed and the requirement of traditional coordinate transform procedure is not necessary to perform the numerical integration. Therefore, the ISFEM generally needs less computational cost than the FEM for handling the dynamic analysis problems.
The variations of u_{3} and φ at the loading point combined with t = 0.0001 s, 0.0004 s, 0.0025 s, and 0.004 s in the PZT4based FGPM cantilever beam are shown in Table 2. The 80 × 10 meshes of ISFEM at n = −5, −1, 0, 1, and 5 are shown in Figure 9, and FEM is considered 160 × 20 elements. Clearly, the results of ISFEM with 80 × 10 elements are the same as the calculated results of FEM using 160 × 20 elements, suggesting ISFEM has higher accuracy.

The variations of u_{3} and φ at the loading point combined with time of the PZT5Hbased FGPM cantilever beam are listed in Table 3. Clearly, when n changes from −5 to 5, the maximum u_{3} and φ decrease, which is consistent with the PZT4based FGPM cantilever beam. Furthermore, the results of ISFEM with 80 × 10 elements are the same as the calculated results of FEM using 160 × 20 elements, suggesting ISFEM has higher accuracy.

5.2. Cosine Wave Load
The cosine load F applied to the free end and load waveform is shown in Figure 10. The variations of u_{3} and φ at the loading point combined with time of PZT4 and PZT5Hbased FGPM cantilever beams are listed in Tables 4 and 5, respectively. The calculated results of ISFEM with 80 × 10 elements are the same as those of FEM using 160 × 20 elements, implying that ISFEM possesses higher accuracy.


5.3. Step Wave Load
The step load F applied to the free end and load waveform is indicated in Figure 11. The variations of u_{3} and φ at the loading point combined with time of PZT4 and PZT5Hbased FGPM cantilever beams are illustrated in Figures 12 and 13, respectively. It is clearly shown that ISFEM possesses higher accuracy than FEM for the calculated results of ISFEM using 80 × 10 elements are the same as FEM using 160 × 20 elements.
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5.4. Triangular Wave Load
The triangular load F applied to the free end and load waveform is shown in Figure 14. The variations of u_{3} and φ at the loading point combined with time of PZT4 and PZT5Hbased FGPM cantilever beams are shown in Figures 15 and 16, respectively. It shows that the solutions of CSFEM with less elements are the same as the solutions of FEM using more elements.
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6. Conclusions
An electromechanical ISFEM was proposed given the continuous changes of the gradient of material properties along the thickness x_{3} direction and with cellbased gradient smoothing. The modified Wilsonθ method was deduced to solve the integral solution of the FGPM system. The displacements and potentials of cantilever beams combining with sine load, cosine load, step load, and triangular load were analyzed by ISFEM in comparison with FEM.(1)ISFEM is correct and effective in solving the dynamic response of FGPM structures(2)ISFEM can reduce the systematic stiffness of FEM and provides calculations closer to the true values(3)ISFEM is more efficient than FEM and takes less computation time at the same accuracy
This study indicates a possibility to select suitable grading controlled by the power law index according to the application.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
Liming Zhou and Bin Cai performed the simulations. Bin Cai contributed to the writing of the manuscript.
Acknowledgments
Special thanks are due to Professor Guirong Liu for the SFEM Source Code in http://www.ase.uc.edu/∼liugr/software.html. This work was financially supported by the National Key R&D Program of China (grant no. 2018YFF0101240106), Jilin Provincial Department of Science and Technology Fund Project (grant no. 20170101043JC), Jilin Provincial Department of Education (grant nos. JJKH20180084KJ and JJKH20170788KJ), Fundamental Research Funds for the Central Universities, Science and Technological Planning Project of Ministry of Housing and Urban–Rural Development of the People's Republic of China (2017K9047) and Graduate Innovation Fund of Jilin University (grant no. 101832018C184).
References
 K. M. Liew, X. Q. He, and S. Kitipornchai, “Finite element method for the feedback control of FGM shells in the frequency domain via piezoelectric sensors and actuators,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 3–5, pp. 257–273, 2004. View at: Publisher Site  Google Scholar
 G. Song, V. Sethi, and H.N. Li, “Vibration control of civil structures using piezoceramic smart materials: a review,” Engineering Structures, vol. 28, no. 11, pp. 1513–1524, 2006. View at: Publisher Site  Google Scholar
 B. Legrand, J.P. Salvetat, B. Walter, M. Faucher, D. Théron, and J.P. Aimé, “MultiMHz microelectromechanical sensors for atomic force microscopy,” Ultramicroscopy, vol. 175, pp. 46–57, 2017. View at: Publisher Site  Google Scholar
 F. Pablo, I. Bruant, and O. Polit, “Use of classical plate finite elements for the analysis of electroactive composite plates. Numerical validations,” Journal of Intelligent Material Systems and Structures, vol. 20, no. 15, pp. 1861–1873, 2009. View at: Publisher Site  Google Scholar
 M. D’Ottavio and O. Polit, “Sensitivity analysis of thickness assumptions for piezoelectric plate models,” Journal of Intelligent Material Systems and Structures, vol. 20, no. 15, pp. 1815–1834, 2009. View at: Publisher Site  Google Scholar
 P. Vidal, M. D’Ottavio, M. Ben Thaïer, and O. Polit, “An efficient finite shell element for the static response of piezoelectric laminates,” Journal of Intelligent Material Systems and Structures, vol. 22, no. 7, pp. 671–690, 2011. View at: Publisher Site  Google Scholar
 S. B. BeheshtiAval, M. LezgyNazargah, P. Vidal, and O. Polit, “A refined sinus finite element model for the analysis of piezoelectriclaminated beams,” Journal of Intelligent Material Systems and Structures, vol. 22, no. 3, pp. 203–219, 2011. View at: Publisher Site  Google Scholar
 X.H. Wu, C. Chen, Y.P. Shen, and X.G. Tian, “A high order theory for functionally graded piezoelectric shells,” International Journal of Solids and Structures, vol. 39, no. 20, pp. 5325–5344, 2002. View at: Publisher Site  Google Scholar
 X. H. Zhu and Z. Y. Meng, “Operational principle, fabrication and displacement characteristic of a functionally gradient piezoelectric ceramic actuator,” Sensors and Actuators A. Physical, vol. 48, no. 3, pp. 169–176, 1995. View at: Publisher Site  Google Scholar
 C. C. M. Wu, M. Kahn, and W. Moy, “Piezoelectric ceramics with functional gradients: a new application in material design,” Journal of the American Ceramic Society, vol. 79, no. 3, pp. 809–812, 2005. View at: Publisher Site  Google Scholar
 K. Takagi, J.F. Li, S. Yokoyama, R. Watanabe, A. Almajid, and M. Taya, “Design and fabrication of functionally graded PZT/Pt piezoelectric bimorph actuator,” Science and Technology of Advanced Materials, vol. 3, no. 2, pp. 217–224, 2002. View at: Publisher Site  Google Scholar
 G. R. Liu and J. Tani, “Surface waves in functionally gradient piezoelectric plates,” Journal of Vibration and Acoustics, vol. 116, no. 4, pp. 440–448, 1994. View at: Publisher Site  Google Scholar
 Z. Zhong and E. T. Shang, “Threedimensional exact analysis of a simply supported functionally gradient piezoelectric plate,” International Journal of Solids and Structures, vol. 40, no. 20, pp. 5335–5352, 2003. View at: Publisher Site  Google Scholar
 C. Piotr, “Threedimensional natural vibration analysis and energy considerations for a piezoelectric rectangular plate,” Journal of Sound and Vibration, vol. 283, no. 3–5, pp. 1093–1113, 2005. View at: Publisher Site  Google Scholar
 W. Q. Chen and H. J. Ding, “On free vibration of a functionally graded piezoelectric rectangular plate,” Acta Mechanica, vol. 153, no. 34, pp. 207–216, 2002. View at: Publisher Site  Google Scholar
 R. K. Bhangale and N. Ganesan, “Static analysis of simply supported functionally graded and layered magnetoelectroelastic plates,” International Journal of Solids and Structures, vol. 43, no. 10, pp. 3230–3253, 2006. View at: Publisher Site  Google Scholar
 M. LezgyNazargah, P. Vidal, and O. Polit, “An efficient finite element model for static and dynamic analyses of functionally graded piezoelectric beams,” Composite Structures, vol. 104, pp. 71–84, 2013. View at: Publisher Site  Google Scholar
 M. LezgyNazargah, “A threedimensional exact statespace solution for cylindrical bending of continuously nonhomogenous piezoelectric laminated plates with arbitrary gradient composition,” Archive of Mechanics, vol. 67, pp. 25–51, 2015. View at: Google Scholar
 M. LezgyNazargah, “A threedimensional Peano series solution for the vibration of functionally graded piezoelectric laminates in cylindrical bending,” Scientia Iranica, vol. 23, no. 3, pp. 788–801, 2016. View at: Publisher Site  Google Scholar
 H.J. Lee, “Layerwise laminate analysis of functionally graded piezoelectric bimorph beams,” Journal of Intelligent Material Systems and Structures, vol. 16, no. 4, pp. 365–371, 2005. View at: Publisher Site  Google Scholar
 Z.C. Qiu, X.M. Zhang, H.X. Wu, and H.H. Zhang, “Optimal placement and active vibration control for piezoelectric smart flexible cantilever plate,” Journal of Sound and Vibration, vol. 301, no. 3–5, pp. 521–543, 2007. View at: Publisher Site  Google Scholar
 J. Yang and H. J. Xiang, “Thermoelectromechanical characteristics of functionally graded piezoelectric actuators,” Smart Materials and Structures, vol. 16, no. 3, pp. 784–797, 2007. View at: Publisher Site  Google Scholar
 B. Behjat, M. Salehi, A. Armin, M. Sadighi, and M. Abbasi, “Static and dynamic analysis of functionally graded piezoelectric plates under mechanical and electrical loading,” Scientia Iranica, vol. 18, no. 4, pp. 986–994, 2011. View at: Publisher Site  Google Scholar
 A. Doroushi, M. R. Eslami, and A. Komeili, “Vibration analysis and transient response of an FGPM beam under thermoelectromechanical loads using higherorder shear deformation theory,” Journal of Intelligent Material Systems and Structures, vol. 22, no. 3, pp. 231–243, 2011. View at: Publisher Site  Google Scholar
 G. W. Wei, Y. B. Zhao, and Y. Xiang, “A novel approach for the analysis of highfrequency vibrations,” Journal of Sound and Vibration, vol. 257, no. 2, pp. 207–246, 2002. View at: Publisher Site  Google Scholar
 P. Kudela, M. Krawczuk, and W. Ostachowicz, “Wave propagation modelling in 1D structures using spectral finite elements,” Journal of Sound and Vibration, vol. 300, no. 12, pp. 88–100, 2007. View at: Publisher Site  Google Scholar
 Y. Kim, S. Ha, and F.K. Chang, “Timedomain spectral element method for builtin piezoelectricactuatorinduced lamb wave propagation analysis,” AIAA Journal, vol. 46, no. 3, pp. 591–600, 2008. View at: Publisher Site  Google Scholar
 H. Zhong and Z. Yue, “Analysis of thin plates by the weak form quadrature element method,” Science China Physics, Mechanics and Astronomy, vol. 55, no. 5, pp. 861–871, 2012. View at: Publisher Site  Google Scholar
 X. Wang, Z. Yuan, and C. Jin, “Weak form quadrature element method and its applications in science and engineering: a stateoftheart review,” Applied Mechanics Reviews, vol. 69, no. 3, Article ID 030801, 2017. View at: Publisher Site  Google Scholar
 G. R. Liu and T. NguyenThoi, Smoothed Finite Element Methods, CRC Press, Taylor and Francis Group, Boca Raton, FL, USA, 2010.
 J.S. Chen, C.T. Wu, S. Yoon, and Y. You, “A stabilized conforming nodal integration for Galerkin meshfree methods,” International Journal for Numerical Methods in Engineering, vol. 50, no. 2, pp. 435–466, 2001. View at: Publisher Site  Google Scholar
 K. Y. Dai and G. R. Liu, “Free and forced vibration analysis using the smoothed finite element method (SFEM),” Journal of Sound and Vibration, vol. 301, no. 3–5, pp. 803–820, 2007. View at: Publisher Site  Google Scholar
 S. P. A. Bordas and S. Natarajan, “On the approximation in the smoothed finite element method (SFEM),” International Journal for Numerical Methods in Engineering, vol. 81, no. 5, pp. 660–670, 2010. View at: Publisher Site  Google Scholar
 C. V. Le, H. NguyenXuan, H. Askes, S. P. A. Bordas, T. Rabczuk, and H. NguyenVinh, “A cellbased smoothed finite element method for kinematic limit analysis,” International Journal for Numerical Methods in Engineering, vol. 83, no. 12, pp. 1651–1674, 2010. View at: Publisher Site  Google Scholar
 G. R. Liu, W. Zeng, and H. NguyenXuan, “Generalized stochastic cellbased smoothed finite element method (GS_CSFEM) for solid mechanics,” Finite Elements in Analysis and Design, vol. 63, pp. 51–61, 2013. View at: Publisher Site  Google Scholar
 K. NguyenQuang, H. DangTrung, V. HoHuu, H. LuongVan, and T. NguyenThoi, “Analysis and control of FGM plates integrated with piezoelectric sensors and actuators using cellbased smoothed discrete shear gap method (CSDSG3),” Composite Structures, vol. 165, pp. 115–129, 2017. View at: Publisher Site  Google Scholar
 Y. H. Bie, X. Y. Cui, and Z. C. Li, “A coupling approach of statebased peridynamics with nodebased smoothed finite element method,” Computer Methods in Applied Mechanics and Engineering, vol. 331, pp. 675–700, 2018. View at: Publisher Site  Google Scholar
 G. Liu, M. Chen, and M. Li, “Lower bound of vibration modes using the nodebased smoothed finite element method (NSFEM),” International Journal of Computational Methods, vol. 14, no. 4, Article ID 1750036, 2016. View at: Publisher Site  Google Scholar
 Z. C. He, G. Y. Li, Z. H. Zhong et al., “An ESFEM for accurate analysis of 3D midfrequency acoustics using tetrahedron mesh,” Computers & Structures, vol. 106107, pp. 125–134, 2012. View at: Publisher Site  Google Scholar
 X. Y. Cui, G. Wang, and G. Y. Li, “A nodal integration axisymmetric thin shell model using linear interpolation,” Applied Mathematical Modelling, vol. 40, no. 4, pp. 2720–2742, 2016. View at: Publisher Site  Google Scholar
 X. Y. Cui, X. B. Hu, and Y. Zeng, “A copulabased perturbation stochastic method for fiberreinforced composite structures with correlations,” Computer Methods in Applied Mechanics and Engineering, vol. 322, pp. 351–372, 2017. View at: Publisher Site  Google Scholar
 T. NguyenThoi, G. R. Liu, K. Y. Lam, and G. Y. Zhang, “A facebased smoothed finite element method (FSFEM) for 3D linear and geometrically nonlinear solid mechanics problems using 4node tetrahedral elements,” International Journal for Numerical Methods in Engineering, vol. 78, no. 3, pp. 324–353, 2009. View at: Publisher Site  Google Scholar
 Z. C. He, G. Y. Li, Z. H. Zhong et al., “An edgebased smoothed tetrahedron finite element method (ESTFEM) for 3D static and dynamic problems,” Computational Mechanics, vol. 52, no. 1, pp. 221–236, 2013. View at: Publisher Site  Google Scholar
 E. Li, Z. C. He, X. Xu, and G. R. Liu, “Hybrid smoothed finite element method for acoustic problems,” Computer Methods in Applied Mechanics and Engineering, vol. 283, pp. 664–688, 2015. View at: Publisher Site  Google Scholar
 C. Jiang, Z.Q. Zhang, G. R. Liu, X. Han, and W. Zeng, “An edgebased/nodebased selective smoothed finite element method using tetrahedrons for cardiovascular tissues,” Engineering Analysis with Boundary Elements, vol. 59, pp. 62–77, 2015. View at: Publisher Site  Google Scholar
 X. B. Hu, X. Y. Cui, Z. M. Liang, and G. Y. Li, “The performance prediction and optimization of the fiberreinforced composite structure with uncertain parameters,” Composite Structures, vol. 164, pp. 207–218, 2017. View at: Publisher Site  Google Scholar
 X. Cui, S. Li, H. Feng, and G. Li, “A triangular prism solid and shell interactive mapping element for electromagnetic sheet metal forming process,” Journal of Computational Physics, vol. 336, pp. 192–211, 2017. View at: Publisher Site  Google Scholar
 G. R. Liu, H. NguyenXuan, and T. NguyenThoi, “A theoretical study on the smoothed FEM (SFEM) models: properties, accuracy and convergence rates,” International Journal for Numerical Methods in Engineering, vol. 84, no. 10, pp. 1222–1256, 2010. View at: Publisher Site  Google Scholar
 E. Li, Z. C. He, G. Wang, and G. R. Liu, “An efficient algorithm to analyze wave propagation in fluid/solid and solid/fluid phononic crystals,” Computer Methods in Applied Mechanics and Engineering, vol. 333, pp. 421–442, 2018. View at: Publisher Site  Google Scholar
 W. Zeng, G. R. Liu, D. Li, and X. W. Dong, “A smoothing technique based beta finite element method (βFEM) for crystal plasticity modeling,” Computers & Structures, vol. 162, pp. 48–67, 2016. View at: Publisher Site  Google Scholar
 H. NguyenXuan, G. R. Liu, T. NguyenThoi, and C. NguyenTran, “An edgebased smoothed finite element method for analysis of twodimensional piezoelectric structures,” Smart Materials and Structures, vol. 18, no. 6, Article ID 065015, 2009. View at: Publisher Site  Google Scholar
 H. NguyenVan, N. MaiDuy, and T. TranCong, “A smoothed fournode piezoelectric element for analysis of twodimensional smart structures,” Computer Modeling in Engineering and Sciences, vol. 23, no. 3, pp. 209–222, 2008. View at: Publisher Site  Google Scholar
 P. PhungVan, T. NguyenThoi, T. LeDinh, and H. NguyenXuan, “Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cellbased smoothed discrete shear gap method (CSFEMDSG3),” Smart Materials and Structures, vol. 22, no. 9, Article ID 095026, 2013. View at: Publisher Site  Google Scholar
 Z. C. He, E. Li, G. R. Liu, G. Y. Li, and A. G. Cheng, “A massredistributed finite element method (MRFEM) for acoustic problems using triangular mesh,” Journal of Computational Physics, vol. 323, pp. 149–170, 2016. View at: Publisher Site  Google Scholar
 W. Zuo and K. Saitou, “Multimaterial topology optimization using ordered SIMP interpolation,” Structural and Multidisciplinary Optimization, vol. 55, no. 2, pp. 477–491, 2017. View at: Publisher Site  Google Scholar
 E. Li, Z. C. He, J. Y. Hu, and X. Y. Long, “Volumetric locking issue with uncertainty in the design of locally resonant acoustic metamaterials,” Computer Methods in Applied Mechanics and Engineering, vol. 324, pp. 128–148, 2017. View at: Publisher Site  Google Scholar
 L. Chen, Y. W. Zhang, G. R. Liu, H. NguyenXuan, and Z. Q. Zhang, “A stabilized finite element method for certified solution with bounds in static and frequency analyses of piezoelectric structures,” Computer Methods in Applied Mechanics and Engineering, vol. 241–244, pp. 65–81, 2012. View at: Publisher Site  Google Scholar
 L. Zhou, M. Li, Z. Ma et al., “Steadystate characteristics of the coupled magnetoelectrothermoelastic multiphysical system based on cellbased smoothed finite element method,” Composite Structures, vol. 219, pp. 111–128, 2019. View at: Publisher Site  Google Scholar
 L. Zhou, S. Ren, G. Meng, X. Li, and F. Cheng, “A multiphysics nodebased smoothed radial point interpolation method for transient responses of magnetoelectroelastic structures,” Engineering Analysis with Boundary Elements, vol. 101, pp. 371–384, 2019. View at: Publisher Site  Google Scholar
 H. NguyenVan, N. MaiDuy, and T. TranCong, “A nodebased element for analysis of planar piezoelectric structures,” CMES: Computer Modeling in Engineering & Sciences, vol. 36, no. 1, pp. 65–96, 2008. View at: Google Scholar
 E. Li, Z. C. He, Y. Jiang, and B. Li, “3D massredistributed finite element method in structuralacoustic interaction problems,” Acta Mechanica, vol. 227, no. 3, pp. 857–879, 2016. View at: Publisher Site  Google Scholar
 L. Zhou, M. Li, G. Meng, and H. Zhao, “An effective cellbased smoothed finite element model for the transient responses of magnetoelectroelastic structures,” Journal of Intelligent Material Systems and Structures, vol. 29, no. 14, pp. 3006–3022, 2018. View at: Publisher Site  Google Scholar
 L. Zhou, M. Li, H. Zhao, and W. Tian, “Cellbased smoothed finite element method for the intensity factors of piezoelectric bimaterials with interfacial crack,” International Journal of Computational Methods, vol. 16, no. 7, Article ID 1850107, 2019. View at: Google Scholar
 J. Zheng, Z. Duan, and L. Zhou, “A coupling electromechanical cellbased smoothed finite element method based on micromechanics for dynamic characteristics of piezoelectric composite materials,” Advances in Materials Science and Engineering, vol. 2019, Article ID 4913784, 16 pages, 2019. View at: Publisher Site  Google Scholar
 L. Zhou, S. Ren, C. Liu, and Z. Ma, “A valid inhomogeneous cellbased smoothed finite element model for the transient characteristics of functionally graded magnetoelectroelastic structures,” Composite Structures, vol. 208, pp. 298–313, 2019. View at: Publisher Site  Google Scholar
 L. Zhou, M. Li, B. Chen, F. Li, and X. Li, “An inhomogeneous cellbased smoothed finite element method for the nonlinear transient response of functionally graded magnetoelectroelastic structures with damping factors,” Journal of Intelligent Material Systems and Structures, vol. 30, no. 3, pp. 416–437, 2019. View at: Publisher Site  Google Scholar
 R. X. Yao and Z. F. Shi, “Steadystate forced vibration of functionally graded piezoelectric beams,” Journal of Intelligent Material Systems and Structures, vol. 22, no. 8, pp. 769–779, 2011. View at: Publisher Site  Google Scholar
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Copyright © 2019 Bin Cai and Liming Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.