Liming ZHOU, Shuhui REN, Bin NIE, Guiki GUO, Xingyng CUI,b,*

a School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China

b State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China

KEYWORDS

Abstract A Coupling Magneto-Electro-Elastic (MEE) Node-based Smoothed Radial Point Interpolation Method(CM-NS-RPIM)was proposed to solve the free vibration and transient responses of Functionally Graded Magneto-Electro-Elastic (FGMEE) structures. By introducing the modified Newmark method, the displacement, electrical potential and magnetic potential of the structures under transient mechanical loading were obtained. Based on G space theory and the weakened weak (W2) formulation, the equations of the multi-physics coupling problems were derived. Using triangular background elements, the free vibration and transient responses of three numerical examples were studied. Results proved that CM-NS-RPIM performed better than the standard FEM by reducing the overly-stiff of structures. Moreover, CM-NS-RPIM could reduce the number of nodes while guaranteeing the accuracy.Besides, triangular elements could be generated automatically even for complex geometries. Therefore, the effectiveness and validity of CMNS-RPIM were demonstrated, which were valuable for the design of intelligence devices, such as energy harvesters and sensors.

1. Introduction

Nowadays, a new type of intelligence material, which is called Functionally Graded Magneto-Electro-Elastic (FGMEE)material, is composed of piezo-electric phase and piezomagnetic phase.This material has been studied to manufacture smart structures for sensing, actuating, etc.1–5Material properties gradually vary along the thickness direction and no obvious interfaces inside the material,which could integrate with other materials to constitute smart devices.6These smart devices were widely used in aircraft and spacecraft.7Therefore,the comprehensive and thorough studies for FGMEE materials are urgently needed as they are of great value for the design and development of intelligent devices.

Fig.1 Functionally Graded Magneto-Electro-Elastic material.

Hitherto,to obtain the characteristics of FGMEE materials including static and dynamic behavior, researchers have performed numerous studies on structures made of such materials. Two different classes of vibrations of simply supported plates were obtained by employing appropriate functions of displacement and stress, the factors influenced frequencies were also studied by Chen et al.8Later,solution for the simply supported plates composed of FGMEE materials was proposed by Pan and Han,9with the assumption of exponential in the thickness direction.Numerical results that could be used as benchmark showed the influence of payload types,exponential factor and material properties.A sequence of solutions for FGMEE beams were obtained by Huang et al.10,and the plane stress condition was considered. These solutions were appropriate for the beams under different loads and boundary conditions, which were very versatile. Two different methods for determining natural frequencies of FGMEE composite plates were acquired by Ramirez et al.11, different boundary conditions,aspect ratios of the plates and different material properties were considered. A solution for the three-dimensional FGMEE circular plate subjected to uniformly distributed load was presented by Li et al.12,this solution was suitable for arbitrary material properties.Three-dimensional static characteristics of doubly curved FGMEE shells and plates were brought up by Wu et al.13,14, different methods were applied to the analysis and parametric studies were performed to investigate the factors that influenced the structural responses. Buckling analysis of FGMEE nanostructures was performed by Ebrahimi et al.15,16in recent years, several numerical results were provided to prove the effect of various parameters on the critical buckling load. The solution could offer a basis for the design of FGMEE structures.

In order to improve the performance of such intelligent devices, studies of dynamic characteristics of FGMEE structures are also necessary. The dynamic anti-plane problem of FGMEE structures with an internal crack was investigated by Feng and Su,17results indicated that the loading was the key influencing factor in the dynamic fracture behavior. In addition, the measure of retarding the crack extension was proposed. The harmonic wave and circumferential harmonic wave propagation in FGMEE plates were studied and the dynamic solution was presented by Wu et al.18,19By introducing the Legendre orthogonal polynomial series expansion approach,the characteristics of wave propagation were identified.Dynamic behavior of simply supported three-dimensional FGMEE plate was investigated by Wu and Tsai20by using the modified Pagano method, which can markedly improve the accuracy of solutions combining with a successive approximation method.

Analytical solutions could accurately describe the responses of the structures,whereas these solutions are only applicable to simple boundary conditions and loadings, which cannot deal with complex engineering practical problems. To eliminate the limitation of analytical solutions and better solve complex questions, researchers started to develop approximate solutions for multi-physics coupling problems. With the development of computers, more and more numerical methods were developed for the analysis of Functionally Graded Materials(FGMs). The surface-waves and SH surface-waves in the FGM plates were investigated using hybrid numerical method by Liu et al.21–23, the responses in the frequency domain, the general solution for the motion equation were derived.Besides,the stress waves in FGM plates were also studied,and the relationships between the surface displacement and the material properties of the plate were obtained.24,25A neural network with a modified hybrid numerical method and a computational inverse procedure were proposed for simulating the transient waves and elastic waves in FGM structures. These methods were proved robust for characterizing the material properties.26–30Then,the transient responses of FGM structures were obtained by Liu et al.31,32, which indicated the efficiency and accuracy of the numerical method.

Among the numerical methods, Finite Element Method(FEM) did a remarkable job. An increasing number of researchers selected FEM as a tool for studies. Bhangale and Ganesan33–35have investigated free vibrations of linear FGMEE plates and cylindrical shells using FEM.Considering coupling effects between elasticity, electric and magnetic,FEMs for FGMEE structures were derived and the factors effecting MEE field have been studied. Milazzo36,37put forward a new model for solving the dynamic problems of onedimensional FGMEE beams using the first-order shear beam theory, with the assumption of quasi-static electric and magnetic fields. Before long, he extended the solution to twodimensional models of FGMEE plates. Such models were appropriate for analyzing behavior of plates made of intelligence materials. Later, Kattimani and Ray38have studied the geometrically nonlinear problems of FGMEE structures with damping, results indicated that the influence of electroelastic and magneto-elastic couplings was negligible.Recently,Cui et al.39put forward an original triangular prism solid and the performance was validated by comparing the solutions obtained by this method with those of traditional method. A shell interactive mapping element was proposed simultaneously.

Though FEM has been adopted to simulate various practical engineering problems and many commercial software,some insurmountable disadvantages can’t be ignored. Shear locking, volumetric locking and overly-stiff of FEM make it impossible to provide accurate results, and the values of the solution are lower than the true values. Furthermore, FEM is sensitive to the mesh distortion and not efficient in some cases.40To overcome the shortcomings of this method,several methods were developed including cell-based smoothed FEM,a meshfree radial point interpolation method so on.41–43Recently, Liu44,45developed a gradient smoothing technique and the weakened weak (W2) formulation that can be used to establish numerical methods superior to FEM.Some studies have obtained satisfying results that were more accurate by introducing such technique.46–49By introducing the gradient smoothing technique into the displacement field, a range of Smoothed Point Interpolation Methods (S-PIMs) and Smoothed Radial Point Interpolation Methods (S-RPIMs)were presented, including Node-based Smoothed PIM (NSPIM)50,51and Node-based Smoothed RPIM (NS-RPIM),52edge-based smoothed PIM53,54and cell-based smoothed PIM.55–58Shape functions established by PIM and RPIM own Kronecker delta function property, thus it was easier to apply boundary conditions.However,using RPIM shape functions possesses several merits. The singularity of moment matrix in PIM can be eliminated using Radial Basis Function(RBF). Moreover, the RPIM shape functions are more stable and easier to form than the PIM shape functions,because RBF is the function of distance.57By applying the gradient smoothing technique and W2formulation, the overly-stiff of FEM is overcome. Therefore, a softer model and more precise results are obtained. Moreover, in the progress of establishing RPIM shape functions, Jacobian matrix is not involved because of no mapping and coordinate transformation. Thus,CM-NS-RPIM is insensitive to mesh distortion. Because of these advantages, CM-NS-RPIM is developed for solving multi-physics coupling problems including free vibration and transient responses of FGMEE structures, which could be a robust and valid tool.

In recent years, researchers have concerned the damping analysis. Among these analyses, Rayleigh damping is usually applied to the analysis of transient responses with damping of various devices, which could accurately simulate the dynamic performance.59Hall60studied the latent problems in damped FEM models and proposed several remedies to ensure the correctness of the results. Roy and Chakraborty61have presented a system identification algorithm based on the free vibration responses of devices. The robustness of such algorithm was proved by the error sensitivity analysis. To better design a composite propeller blade with damping,Hong et al.59developed a hybrid method to calculate the damping parameters of high-frequency damping, material damping and structural damping, which was validated by experiments. Min et al.62performed parametric studies by altering two damping coefficients of shear-flexible and damped Beck’s columns and established stability maps under three damping cases. An et al.63have investigated the effects of Rayleigh damping in the analysis of FRP-concrete bonded joints and proposed a proper method to determine the damping coefficients, the results showed the accuracy of the simulation could be improved. Though researchers have been performing investigations of Rayleigh damping in numerous cases,no researches on free vibrations nor transient responses analysis of FGMEE structures with Rayleigh damping have been done. Using an effective algorithm to do these studies is very meaningful for the development and application of FGMEE materials.

In this paper, the CM-NS-RPIM equations for solving the free vibration and the transient responses problems of FGMEE material are deduced. The modified Newmark method is applied to the calculation of the mechanicalelectro-magnetic coupling dynamic equilibrium equation.Smoothed strain fields are formed by employing the gradient smoothing technique. The W2formulation is adopted to construct the stiffness matrix, which can reduce the overstiffness of FEM markedly. Three numerical examples robustly demonstrate the accuracy,efficiency and convergence of CM-NS-RPIM, including the free vibration of a cantilever beam, the transient responses without damping of a clamped-clamped beam and the transient responses of an FGMEE sensor with damping. Results were compared with those of FEM, which shows that CM-NS-RPIM outperforms the standard FEM.

The layout of this paper is given below: Section 2 presents the basic formulations and the material properties of FGMEE.Section 3 introduces the equations of CM-NS-RPIM and cellbased T6-scheme.Sections 4 and 5 contain the Rayleigh damping and the modified Newmark method,respectively.Section 6 lists investigations of three numerical examples consisting of free vibrations, dynamic characteristics with damping and without damping. Finally, Section 7 lists several conclusions.

2. Computation scheme

2.1. Basic formulations

A two-dimensional FGMEE structure defined in domain Ω bounded by the global boundary Γ, which is governed by the equations below.

Equilibrium equations of FGMEE materials are given as

whereLdandLbare matrices of differential operator, whose expressions are as follow

The geometric equations are given as

S（x，t）= [Sx（x，t）Sz（x，t）Sxz（x，t）]T，E（x，t）= [Ex（x，t）Ez（x，t）]T, andH（x，t）= [Hx（x，t）Hz（x，t）]Tare vectors of strain components,electric field components and magnetic field components, respectively.u（x，t）= [ux（x，t）uz（x，t）]Tis the vector of displacement components. Φ(x,t) and Ψ(x,t) denote the electrical potential and the magnetic potential,respectively.

The constitutive equations are

whereC(x),e(x) andq(x) represent the matrices of elastic,piezo-electric and piezo-magnetic coefficients, respectively. ε(x),m(x) and μ(x) are the matrices of dielectric coefficients,the magneto-electric material coefficients and the magnetic permeability coefficients, respectively. The expressions are as follows

The natural boundary conditions of mechanical, electrical and magnetic fields on Γu, Γpand Γaare

The essential boundary conditions of mechanical,electrical and magnetic fields on Γβ, Γqand Γbare

2.2. Functionally Graded Magneto-Electro-Elastic materials

FGMEE material,whose mechanical properties have a certain relationship with the coordinates, is a new type of functional material composed of at least two materials. Obvious interfaces are eliminated by making the component content change continuously in space. FGMEE is composed of piezo-electric and piezo-magnetic phase and shows a superior performance than the laminated MEE structures. Therefore,FGMEE materials are widely used in engineering practice.

The FGMEE material is shown in Fig.1,the material properties vary gradient along the thickness direction. During the manufacture of the material, continuous gradient change can be considered to follow a particular continuous function.Two appropriate functions for FGMEE materials are the exponential and the power law function. In this paper, the exponential function is applied.

For exponentially varying FGMEE materials, the coefficients can be defined as64

whereCij,ekiandqkiare the elastic constants,the piezo-electric coefficients and the piezo-magnetic coefficients, respectively,εlk,μlkandmlkare the dielectric coefficients,the magnetic permeability coefficients and the magneto-electric material coefficients, respectively.his the width of the whole model. ξ denotes the exponential factor,superscript 0 denotes the corresponding material coefficients at the lower surface of the structure. When ξ=0, the homogeneous MEE material is used.

3. Coupling Magneto-Electro-Elastic node-based smoothed radial point interpolation method (CM-NS-RPIM)

3.1. T-schemes for node selection

In this paper, triangular background elements are used to discretize the structures. The quadrature points are situated in cells, and some field nodes are needed as local support nodes to establish the shape function. Triangular/tetrahedral-meshbased node selection schemes (T-schemes) are very effective in selecting the support nodes. For a two-dimensional CM-NS-RPIM model, there are several alternative schemes including T3-scheme, Tr2L/3-scheme and Tr6-scheme.Because cell-based T6-scheme is appropriate to create RPIM shape functions, such method is chosen in the present work.

Totally six field nodes are chosen by cell-based T6-scheme for the establishment of shape functions.When the quadrature point situated in an interior cell, we select three field nodes of the home cell and three nodes of three neighboring cells.When the quadrature point situated in a boundary cell, we choose three nodes of the home cell, one (or two) field node of one(or two) neighboring cells and two (or one) field nodes that are nearest to the center of the cell. Fig.2 shows cell-based T6-schemes in detail.

Fig.2 Cell-based T6-scheme.

3.2. Establishment of RPIM shape functions

Consider a local support domain containingnnodes, the displacement fieldu(x) is given as

whereRp(x) denotes the RBFs,pr(x) is the polynomial basis functions;apandbrrepresent the corresponding parameters to be determined;nandmare the number of local support nodes and the number of polynomial basis functions, which could be selected according to the requirements. In CM-NSRPIM,RBFs are supplemented withmpolynomial basis functions. Whenm=0, pure RBFs are used.

Four alternative types of RBFs including Multi-Quadrics(MQ) function, Logarithmic RBF, Thin Plate Spline (TPS)function and Gaussian radial function can be used in the establishment of the RPIM shape functions. In this article, MQ function is adopted.

wheredcdenotes the average dimension of the background cells and generally equals to the characteristic length. αcandqare two parameters used in the calculation. According to the investigations in literatures,57αc=4.0 andq=1.03 can acquire a close-to-exact result. The expression ofrp(x) is

The polynomial basis functions are chosen from the Pascal triangle in a top-down mean, which is as follow

The undetermined parametersapandbrcan be calculated using Eq. (25). However,m+nunknowns are in equation while onlynequations are inclusive.

To determine the values of the parametersapandbr, equations should be satisfied at every nodes of the local support domain.nequations can be expressed in a matrix form,which are

whereUsis the vector of function values whose expression isUs= [u1u2···un]T.Rqis the moment matrix of RBFs,Pmdenotes the polynomial moment matrix, which are defined as

wherexs= [x1x2···xn]Tis the vector of coordinates ofnlocal support nodes.

adenotes the RBFs coefficients vector,bdenotes the polynomial basis functions vector.

To determine the parametersapandbr,mconstraint conditions are augmented to Eq. (29).

Combining Eqs.(29)and(34),an equation set includingm+nlinear equations are obtained and written as

Coefficients vectorsaandbcan be expressed as

Rewrite Eq. (25) as follows

Substituting Eq. (36) into Eq. (37), we obtain

Atnlocal support nodes, the RPIM shape functions are obtained.

The displacement is rewritten as

3.3. Node-based smoothed strains

Using the neighboring cells, the node-based smoothing domain of nodekis established, which is shown in Fig.3.The figure also illustrates the field nodes, the mid-edge point,the center nodes, Gauss points and the node smoothing domain of different cells.

Fig.3 Node-based smoothing domain of node k bounded by Γ.

Substituting Eqs. (42)–(44) into Eqs. (45)–(47), we obtain

wherel=x,z.

The integrands in Eqs.(56)–(58)are performed easily using the standard Gauss quadrature, because no singular terms are included. Eqs. (56)–(58) are rewritten as

Fig.4 Flowchart of modified Newmark method.

The global smoothing strain, smoothing electric field and smoothing magnetic field are expressed as

3.4. Discretized equations for CM-NS-RPIM

Assuming the surface forces, surface charge and surface normal to be absent, based on the generalized Hamilton’s principle, we obtain

wherePf, ρcandJare the body force, free charge density and free magnetic current, respectively.

Substituting Eqs. (42)–(44) and (62)–(64) into Eq. (65), the discretized system equations for free vibration for twodimensional FGMEE solids can be expressed as

The discretized system equations of damped dynamic equations under mechanical loadings are

whereF(t) is the time varying mechanical loading.

The components in the Eqs. (66) and (67) are given below.

where

Fig.5 Geometry of FGMEE cantilever beam.

Because of the coupling ofu, Φ and Ψ, the degree of freedom associated with Φ and Ψ can be condensed when performing eigenvalue and transient analysis.66,67Eqs. (66) and(67) can be rewritten as

Electrical potential Φ and magnetic potential Ψ are obtain using equations as follows

4. Rayleigh damping

In an actual motion system,an energy dissipation source is provided by damping in the transient responses analysis that cannot be ignored in most cases.However,because of the difficulty of the measurement and calculation of actual damping matrix, it is usually abstracted into a mathematical model whose parameters are determined based on the principle of equivalence with physical quantities of structural reactions.The most widely used method is Rayleigh damping which can simulate dissipative forces of structures under transient loads accurately.

Consider a mechanical-electro-magnetic coupling dynamic system with damping effects, the vibration equation can be expressed as Eq.(79),and the second term of it is the damping matrix which is written as a linear combination of mass matrixMand the equivalent stiffness matrixKeq.63

where γ1denotes the mass-proportional damping coefficients,γ2denotes the stiffness-proportional damping coefficients.Under the assumption thatVis proportional toMandKeq,γ1and γ2can be easily determined using the following equations.68

where ω1and ω2are the first and second natural frequencies of the FGMEE structures. ζ1and ζ2are the damping ratios. In this paper,the values of ζ1and ζ2are set as 0.05 with the consideration of MEE coupling effects.Using Eqs.(88)and(89)to calculate the values of γ1and γ2, then substitute the obtained damping matrix into the modified Newmark method to analyze the transient responses of FGMEE structures.

5. Modified Newmark method

Two methods are applied to solve the mechanical-electromagnetic coupling dynamic problems,which are direct numerical integration method and mode superposition method. The direct numerical integration method refers to integrate the mechanical-electro-magnetic coupling dynamic equilibrium equation stepwise without transforming it,which usually based on two concepts. The first one is the dynamic equilibrium equation which can be approximately satisfied under certain conditions in the solution domain. The other is to assume the function of generalized displacements (displacement, electrical potential and magnetic potential), velocities and accelerations in a certain number of temporal interval Δt.

The modified Newmark method is a method of the direct numerical integration methods which is fit for solving themechanical-electro-magnetic coupling dynamic problems, and obtain the generalized displacement, velocities and accelerations under mechanical load. This method is an implicit integration algorithm which is stable, accurate and efficient in analyzing transient responses of multi-physics systems. The algorithm is unconditionally stable when δ ≥0.5 and α ≥0.25(0.5+δ)2. In the present work, δ=0.5, α=0.25 are chosen. The flowchart of the modified Newmark method is shown in Fig.4.

Table 1 Material properties coefficients of BaTiO3-CoFe2O469.

Fig.6 Frequencies for FGMEE cantilever beam under different exponential factor ξ.

5.1. Initial calculation

(1) FormingKeq,Mand damping matrixV;

(3) Choosing a time step Δtand the parameters α and δ,calculating the integral constant

Fig.7 Frequencies for FGMEE cantilever beam under different number of nodes and exponential factor ξ.

5.2. For each time step (t=0, Δt, 2Δt...)

(2) Calculating the displacement at timet+Δt;

(3) Calculating the acceleration and velocity at timet+Δt;

Fig.8 Comparison of convergence rates for an FGMEE clamped-clamped beam under different ξ.

Fig.9 The first three order modal figures under different numbers of nodes.

Fig.10 Geometry of an FGMEE clamped–clamped beam.

(4) Using Eqs.(85)and(86),Φ and Ψ of the FGMEE structures are acquired.

6. Numerical examples

6.1. Free vibration analysis of an FGMEE cantilever beam

Fig.11 Waveform of F.

An FGMEE cantilever beam was shown in Fig.5 with geometrical parameters of lengthL=30 mm, widthh=2 mm and the plane stress condition was considered. We studied the free vibrations of the beam by changing the exponential factor ξ of FGMEE materials. The boundary condition of the beam isux=uz=Φ=Ψ=0 at the fixed end. Table 1 specifies the material properties coefficients of BaTiO3-CoFe2O4.

Fig.12 Variation of uz, Φ and Ψ at point B with respect to time when ξ=0, 0.05, 0.10, 0.20 and 0.40.

Fig.13 Comparison of convergence rates for an FGMEE C–C beam under different ξ.

The free vibration frequencies for the exponential factor ξ=0, 0.5, 1.0, 2.0, 4.0 were calculated using CM-NS-RPIM(272 nodes) and FEM (1089 nodes) as shown in Fig.6. FEM using 2353 nodes was adopted as the reference solution. CMNS-RPIM-Tr6 in the figures represents the model performed with cell-based T6-scheme and triangular elements. Figures indicate that the inherent frequencies of the former 4 orders of CM-NS-RPIM and FEM were roughly identical with the reference solution,which validates CM-NS-RPIM can acquire a close-to-exact results when using less nodes than the standard FEM.The requirements of the shape functions continuity were lowered. Besides, the boundary integral, which was easy to perform using the Gauss integration, was employed instead of the area integral. Meanwhile the derivation of the shape functions is not demanded.Moreover,the gradient smoothing technique is introduced to alleviate the system stiffness.Therefore,CM-NS-RPIM is a more reliable and accurate numerical method than FEM.

Frequencies under different number of nodes and exponential factor are shown in Fig.7. Clearly, the results of CM-NS-RPIM under different element division modes are basically the same, which indicates that this method has good convergence.

To further prove the convergence of CM-NS-RPIM, the error in total energy norm was computed, whose expression is given as70

The components in the equation above are

Fig.14 Geometry of FGMEE energy harvester.

The error in total energy norm of cantilever beam was computed with a force of 100 N applied at pointA,meanwhile the exponential factor was ξ=0 and ξ=4.0, respectively. The total energy norm error versus the number of nodes are shown in Fig.8.Moreover,the first third order modal figures with different numbers of nodes are shown in Fig.9.Modal figures are basic unanimously. The results validated the convergence of CM-NS-RPIM under static load.

6.2. Transient responses of an FGMEE clamped–clamped beam

For the sake of validating the convergence of CM-NSRPIM, different node numbers (159 nodes, 272 nodes, 441 nodes and 938 nodes)were used in the analysis of the behavior under dynamic mechanical load of the FGMEE C–C beam.The generalized displacement (displacementuz, electrical potential Φ and magnetic potential Ψ) with the varying exponential factor ξ at pointBis shown in Fig.12. The figures showed that the results tended to be a stable value as the number of nodes increased,which robustly demonstrated the excellent convergence of CM-NS-RPIM.

The error in total energy norm was computed att=0.05 s,with the exponential factor ξ=0 and ξ=0.40, the total energy norm error versus the number of nodes are shown in Fig.13. Values of the results are very low, which indicates CM-NS-RPIM possesses an excellent convergence under different values of ξ. Linear independent is one of the properties of the CM-NS-RPIM shape functions,a proper shape function can guarantee the convergence of the method though the displacement functions, electric field functions and magnetic functions are discontinuous.

Fig.15 Variation of uz, Φ and Ψ at point A with respect to time when ξ=0, 0.05, 0.10, 0.20 and 0.40.

6.3. Transient responses of an FGMEE sensor with Rayleigh damping

CM-NS-RPIM (903 nodes) and FEM (2186 nodes) were applied to analyze the transient characteristics of the FGMEE sensor under different ξ,respectively.The displacementuz,the electrical potential Φ and the magnetic potential Ψ at pointAare shown in Fig.15.Two curves are consistent,and results of CM-NS-RPIM are higher than the values of FEM,which indicates that by employing the gradient smoothing technique,the overly-stiff of the standard FEM is alleviated.Moreover,CMNS-RPIM using less nodes returns more accurate results than FEM. Therefore, CM-NS-RPIM can significantly reduce errors and provide reliable and accurate solutions, and it increases the computational efficiency by using less nodes.

7. Conclusions

In this work,a Coupling Magneto-Electro-Elastic Node-based Smoothed Radial Point Interpolation Method (CM-NSRPIM) for analyzing the static and dynamic characteristics of Functionally Graded Magneto-Electro-Elastic (FGMEE)structures is proposed. By introducing the gradient smoothing technique, equations for the multi-physics coupling problems of FGMEE materials are derived and the overly-stiff of the standard Finite Element Method (FEM) is deduced markedly. By combining with the modified Newmark method, the transient responses of the structures are calculated.Finally,three numerical examples are calculated by CM-NS-RPIM in comparison with FEM.

(1) CM-NS-RPIM works well with calculating the free vibration, transient responses of FGMEE structures,which is robust and stable.

(2) CM-NS-RPIM possesses high accuracy and convergence.

(3) CM-NS-RPIM could reduce the number of nodes used in the calculating while guaranteeing the accuracy.

(4) CM-NS-RPIM with the modified Newmark method is proved valid to analyze the transient responses of complex FGMEE structures with damping.

Acknowledgements

This study was co-supported by the National Key R&D Program of China (Nos. 2018YFF01012401-05) the National Natural Science Foundation of China (No. 51975243); Jilin Provincial Department of Education (No.JJKH20180084KJ),China; the Fundamental Research Funds for the Central Universities and Jilin Provincial Department of Science &Technology Fund Project, China (Nos. 20170101043JC and 20180520072JH) and Graduate Innovation Fund of Jilin University, China (No. 101832018C184).