# Frequency Domain Spectral Element Model for the Vibration Analysis of a Thin Plate with Arbitrary Boundary Conditions

Authors: Ilwook Park; Taehyun Kim; Usik Lee
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

We propose a new spectral element model for finite rectangular plate elements with arbitrary boundary conditions. The new spectral element model is developed by modifying the boundary splitting method used in our previous study so that the four corner nodes of a finite rectangular plate element become active. Thus, the new spectral element model can be applied to any finite rectangular plate element with arbitrary boundary conditions, while the spectral element model introduced in the our previous study is valid only for finite rectangular plate elements with four fixed corner nodes. The new spectral element model can be used as a generic finite element model because it can be assembled in any plate direction. The accuracy and computational efficiency of the new spectral element model are validated by a comparison with exact solutions, solutions obtained by the standard finite element method, and solutions from the commercial finite element analysis package ANSYS.


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## Body

## 1. Introduction

The plate is a representative structural element that is widely used in many engineering fields such as mechanical, civil, aerospace, shipbuilding, and structural engineering. Severe or unwanted vibration of a plate is a very important engineering problem. Thus, it is required to accurately predict the vibration characteristics of a plate during the design phase. Exact solutions are available only for Levy-type plates [1, 2]. Thus, numerous computational methods have been developed for the vibrations of plates during the last two centuries.The finite element method (FEM) is one of the most widely used computational methods that can be applied to various complex structures including the plates. The FEM in general provides reliable solutions in the low frequency range, but poor solutions in the high frequency range. Thus, to improve the solution accuracy in the high frequency range, a finite element must be divided into many smaller finite elements so that their sizes are smaller than the wavelengths of the highest vibration mode of interest. However, this will result in a significant increase in computation cost. Thus, as an alternative to FEM, we can consider the spectral element method (SEM) for the vibration analysis of plates.The SEM considered in this study is the fast Fourier transform- (FFT-) based frequency domain analysis method [3, 4]. The spectral element matrix (or exact dynamic stiffness matrix) used in the SEM is formulated from free wave solutions that satisfy governing differential equations of motion in the frequency domain. Thus, compared with FEM, the SEM can provide exact solutions by representing a uniform structure as a single finite element, regardless of the size of the uniform structure. Accordingly the SEM is known as an exact solution method that has the flexibility of FEM and the exactness of continuum elements [3].Despite the outstanding features of the SEM, it is mostly used in one-dimensional (1D) structures [3, 4]. The SEM application to two-dimensional (2D) structures such as plates has been limited to very specific geometries and boundary conditions, for example, Levy-type plates [6–10] and infinite or semi-infinite plates [11–15]. Some researchers [16, 17] have introduced the spectral super element method (SSEM) for rectangular plates with prespecified boundary conditions on two parallel edges in one direction (say, the y-direction). However, as their spectral element models can be assembled only in another direction (the x-direction), their applications must be limited to very specific boundary conditions. Recently, Park et al. [5, 18, 19] developed spectral element models for rectangular membrane, isotropic plate, and orthotropic laminated composite plate elements by using two methods in combination: the boundary splitting method [20] and the spectral super element method (SSEM) [16]. They derived frequency-dependent shape functions by applying a Kantorovich method-based finite strip element method in one direction and the SEM in another direction in combination, andvice versa. Accordingly, their spectral element models can be assembled in both the x- and y-directions. However, the spectral element models by Park et al. [5, 18, 19] still have some limitations because they are valid only for finite rectangular membranes or plate elements whose four corner nodes are fixed. To the authors’ best knowledge, there have been no reports on a generic type of spectral element model that can be assembled in any direction of a plate subjected to arbitrary boundary conditions.Thus, the purpose of this study is to develop a new spectral element model for finite rectangular plate elements that can be applied to any plate subjected to arbitrary boundary conditions. The new spectral element model is developed by modifying the boundary splitting technique used in our previous study [5] so that the four corner nodes of a finite rectangular plate element become active. The performance of the new spectral element model is evaluated by comparison with exact solutions, FEM solutions, and solutions using the commercial finite element analysis package ANSYS [21].


## 2. Spectral Element Model for a Finite Plate Element

### 2.1. Governing Equations in the Frequency Domain

The time domain equation of motion and the boundary conditions of plate structures with transverse vibrations are described in [2]. The time domain equation of motion of a plate can be transformed into a frequency domain equation of motion of the plate by using the FFT as follows [5]:(1)D∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4-ρω2w=fx,y,where w(x,y) is the transverse displacement in spectral form, f(x,y) is the external force in spectral form, ρ is the mass per unit area of the plate, and D=Eh3/[12(1-ν2)] is the flexural bending rigidity of the plate where E is the modulus of elasticity,ν is Poisson’s ratio, andh is the plate thickness. Similarly, the time domain boundary conditions can be transformed into frequency domain boundary conditions as follows:(2)Myx,-12Ly=-My1xorθyx,-12Ly=θy1x,Vyx,-12Ly=-Vy1xorwx,-12Ly=w1x,Myx,12Ly=My2xorθyx,12Ly=θy2x,Vyx,12Ly=Vy2xorwx,12Ly=w2x,Mx-12Lx,y=-Mx1yorθx-12Lx,y=θx1y,Vx-12Lx,y=-Vx1yorw-12Lx,y=w3y,Mx12Lx,y=Mx2yorθx12Lx,y=θx2y,Vx12Lx,y=Vx2yorw12Lx,y=w4y,where Lx and Ly are the dimensions of a finite plate in the x- and y-directions, respectively; Mx and My are the resultant moments; and Vx and Vy are the resultant transverse shear forces defined by(3)Mx=-D∂2w∂x2+ν∂2w∂y2,Vx=-D∂∂x∂2w∂x2+ν∂2w∂y2-2D1-ν∂3w∂x∂y2My=-D∂2w∂y2+ν∂2w∂x2,Vy=-D∂∂y∂2w∂y2+ν∂2w∂x2-2D1-ν∂3w∂x2∂y.And θx and θy are the slopes defined by(4)θx=∂w∂x,θy=∂w∂y.We need to obtain frequency domain free wave solutions for a homogeneous equation of motion in order to formulate the spectral element model for a finite plate element. To realize this, the homogeneous equation of motion is considered by removing external forcef(x,y) in (1) as follows:(5)D∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4-ρω2w=0.The weak form of (5) can be obtained in the following form:(6)∫x∫yD∂2w∂x2+ν∂2w∂y2δ∂2w∂x2+21-νD∂2w∂x∂yδ∂2w∂x∂y+D∂2w∂y2+ν∂2w∂x2δ∂2w∂y2-ρω2wδwdxdy=0.A free vibration solution satisfying the weak form given in (6) can be obtained approximately by using two combined methods: the boundary splitting method [20] and the spectral super element method (SSEM) [16]. The SSEM uses a combination of the Kantorovich method (based on the finite strip element method) and the frequency domain 1D spectral element method.The concept of the boundary splitting method is illustrated in Figures1 and 2. Figure 1 indicates the concept used in our previous study [5]. Figure 2 indicates the concept used in the present study. The original problems, shown in Figures 1(a) and 2(a), are represented by the sum of two partial problems, Problem A and Problem B. In Figures 1 and 2, the geometric boundary conditions of the original problems are presented in simple forms by using the following definitions:(7)wxx,y=wx,y,θxx,y=∂w∂xTwyx,y=wx,y,θyx,y=∂w∂yTwxAx,y=wAx,y,θxAx,y=∂wA∂xTwyAx,y=wAx,y,θyAx,y=∂wA∂yTwxBx,y=wBx,y,θxBx,y=∂wB∂xTwyBx,y=wBx,y,θyBx,y=∂wB∂yT.Figure 1
Boundary splitting method used in the previous study [5] to derive w(x,y) for a rectangular plate element subjected to arbitrary boundary conditions (●: active nodes; O: fixed nodes).Figure 2
Boundary splitting method used in this study to derivew(x,y) for a rectangular plate element subjected to arbitrary boundary conditions (●: active nodes).In our previous study [5], Problem A, shown in Figure 1(b), has fixed (null) boundary conditions on the parallel edges at y=-Ly/2 and Ly/2. Problem B, shown in Figure 1(c), has fixed (null) boundary conditions on the parallel edges at x=-Lx/2 and Lx/2. As a result, the spectral element model developed in [5] is valid only for finite rectangular plate elements whose four corner nodes are fixed, as shown in Figure 1(d). Accordingly, an application of this approach should be limited to very specific problems as considered in [5].We propose a new boundary splitting method by modifying the boundary splitting method used in [5] such that the four corner nodes of a finite plate element become active. Problem A, shown in Figure 2(b), has arbitrary boundary conditions rather than fixed boundary conditions on the parallel edges at y=-Ly/2 and Ly/2, and its solution is represented by wA(x,y). Problem B, shown in Figure 2(c), has fixed (null) boundary conditions on the parallel edges at x=-Lx/2 and Lx/2. However, the boundary conditions at y=-Ly/2 and Ly/2 in Problem B must be specified such that the sum of the boundary conditions at y=-Ly/2 and Ly/2 in Problem A and those in Problem B is identical to the boundary conditions at y=-Ly/2 and Ly/2 in the original problem. The solution of Problem B is represented by wB(x,y). Then, the solution w(x,y) of the original problem can be obtained by summing the solutions to Problem A and Problem B as follows:(8)wx,y=wAx,y+wBx,y.Accordingly, compared to the spectral element model developed in our previous study [5] based on the boundary splitting shown in Figure 1, the present spectral element model that was developed based on the boundary splitting shown in Figure 2 has four active corner nodes. Thus, it can be used as a generic finite element model that can be assembled in both the x- and y-directions of a plate with arbitrary boundary conditions.


### 2.2. Derivation ofwA(x,y)

To obtain the solutionwA(x,y) for Problem A by using the SSEM, a rectangular finite plate element is divided into Ny finite strip elements in the y-direction, as shown in Figure 3(a). The jth finite strip element, which has a width of lyj=yj-yj-1 in the y-direction, is shown in Figure 3(b).Figure 3
Finite strip element representation of a rectangular finite plate element subjected to arbitrary boundary conditions aty=-Ly/2 and Ly/2 (●: nodes; grey circles: nodal values).
(a) (b)The displacement field in thejth finite strip element can be represented by(9)wAjx,y=YAjywAjxyj-1≤y≤yj,where YA(j)(y) is a one-by-four interpolation function matrix and wA(j)(x) are the nodal line degree of freedom (DOF) functions defined by(10)YAjy=YA1jy,YA2jywAjx=wyAj-1xwyAjx,where(11)YA1jy=lyj-3y-yj22y+yj-3yj-1,lyj-2y-yj2y-yj-1YA2jy=-lyj-3y-yj-122y+yj-1-3yj,lyj-2y-yj-12y-yj,(12)wyAj-1x=wyAx,yj-1=wAx,yj-1,θyAx,yj-1TwyAjx=wyAx,yj=wAx,yj,θyAx,yjT.By using (9), the displacement field wA(x,y) over the entire domain of the finite plate element can be represented as(13)wAx,y=YAywAx-12Ly≤y≤12Ly,where(14)wAx=wyA0Tx,wyA1Tx,…,wyAjTx,…,wyANy-1Tx,wyANyTxT,(15)YAy=LA0y,LA1y,…,LAjy,…,LANy-1y,LANyywith(16)LA0y=hA1YA11yLAjy=hAjyYA2jy+hAj+1yYA1j+1yj=1,2,…,Ny-1LANyy=hANyYA2Nyy.In (16), hA(j)(y) are functions defined by(17)hAjy=Hy-yj-1-Hy-yj,where H(y) is the Heaviside unit step function.Substituting (13) into (5) yields(18)A4∂4wA∂x4+A2∂2wA∂x2+A0-ω-2MAwA=0,where(19)ω-=ωρD,A0=ΛA4,A2=νΛA3+ΛA3T-21-νΛA2A4=ΛA1,MA=ΛA1with the following definitions:(20)ΛA1=∫-1/2Ly+1/2LyYATYAdy,ΛA2=∫-1/2Ly+1/2Ly∂YAT∂y∂YA∂ydyΛA3=∫-1/2Ly+1/2LyYAT∂2YA∂y2dy,ΛA4=∫-1/2Ly+1/2Ly∂2YAT∂y2∂2YA∂y2dy.The constant matrices ΛA1, ΛA2, ΛA3, and ΛA4 are provided in Appendix A.Next, we assume solutions of (18) to be in the following form:(21)wAx=1rA2⋮rA2Ny+1aAe+kxx-1/2k-xLx=rAaAe+kxx-1/2k-xLx,where aA is a constant, kx is the wavenumber in the x-direction, and(22)k-x=+kxifRekx>0-kxifRekx<00ifRekx=0.Substituting (21) into (18) gives the following eigenvalue problem:(23)A4kx4+A2kx2+A0-ω-2MArA=0or(24)AAλx2+A2λx+A0-ω-2MArA=0λx=kx2.The dispersion relation (i.e., the frequency-wavenumber relationship) can be obtained from (24) as follows:(25)detA4λx2+A2λx+A0-ω-2MA=0.From (25), the wavenumbers can be computed as(26)kxj=+λxjkx4Ny+1+j=-λxj=-kxjj=1,2,…,4Ny+1.By using the wavenumberskxj given by (26), we can write the general solution of (18) in the following form:(27)wAx=RAEAx;ωaA,where(28)aA=aA1aA2aA3⋯aAi⋯aA8Ny+1T,RA=R-AR-AEAx;ω=EA1x;ω00EA2x;ωwith(29)R-A=rA1,rA2,rA3,…,rAj,…,rA4×Ny-1EA1x;ω=diagonale+kxjx-1/2k-xjLxj=1,2,…,4Ny+1EA2x;ω=diagonale-kxjx-1/2k-xjLxj=1,2,…,4Ny+1.In (29), rAj is the jth eigenvector, which can be readily computed from (23) using kx=kxj.The nodal DOFs at the nodes defined on the edges atx=-Lx/2 and Lx/2 can be written in vector form as(30)d-A=d-Ld-R,where(31)d-L=wA-12Lx∂wA-1/2Lx∂x,d-R=wA12Lx∂wA1/2Lx∂x∂x.Here, the superscriptsL andR denote the nodal values on the left edge (i.e., at x=-Lx/2) and the right edge (i.e., at x=Lx/2) of the plate, respectively. SuperscriptA denotes the quantities related to or contributed by wA(x,y).By substituting (27) into (31), the nodal DOF vector d-A can be written in terms of the constant vector aA as follows:(32)d-A=RAEA-12LxRAKAEA-12LxRAEA+12LxRAKAEA+12LxaA≡HAωaA,where(33)KA=κA00-κAκA=diagonalkxjj=1,2,…,4Ny+1.The constant vectoraA can be removed from (27) by using (32) to obtain the following expression:(34)wAx=X-Ax;ωd-A,where(35)X-Ax;ω=RAEAx;ωHA-1.From (12), by using the nodal DOFs defined in Figure 4, we obtain the following expressions:(36)wyAj-12Lx=wA-12Lx,yjθyA-12Lx,yj=wj+1Lθyj+1L∂wyAj-1/2Lx∂x=θxA-12Lx,yjθxyA-12Lx,yj=θxj+1Lθxyj+1L,wyAj12Lx=wA12Lx,yjθyA12Lx,yj=wj+1Rθyj+1R∂wyAj1/2Lx∂x=θxA12Lx,yjθxyA12Lx,yj=θxj+1Rθxyj+1R.By applying (36) to (14), we obtain the following expressions:(37)wA-12Lx=w1L,θy1L,w2L,θy2L,…,wj+1L,θyj+1L,…,wNyL,θyNyL,wNy+1L,θyNy+1LT∂wA-1/2Lx∂x=θx1L,θxy1L,θx2L,θxy2L,…,θxj+1L,θxyj+1L,…,θxNyL,θxyNyL,θxNy+1L,θxyNy+1LTwA12Lx=w1R,θy1R,w2R,θy2R,…,wj+1R,θyj+1R,…,wNyR,θyNyR,wNy+1R,θyNy+1RT∂wA1/2Lx∂x=θx1R,θxy1R,θx2R,θxy2R,…,θxj+1R,θxyj+1R,…,θxNyR,θxyNyR,θxNy+1R,θxyNy+1RT.Applying (37) to (31) gives(38)d-L=w1L,θy1L,w2L,θy2L,…,wj+1L,θyj+1L,…,wNyL,θyNyL,wNy+1L,θyNy+1Lθx1L,θxy1L,θx2L,θxy2L,…,θxj+1L,θxyj+1L,…,θxNyL,θxyNyL,θxNy+1L,θxyNy+1LTd-R=w1R,θy1R,w2R,θy2R,…,wj+1R,θyj+1R,…,wNyR,θyNyR,wNy+1R,θyNy+1Rθx1R,θxy1R,θx2R,θxy2R,…,θxj+1R,θxyj+1R,…,θxNyR,θxyNyR,θxNy+1R,θxyNy+1RT.Figure 4
Spectral nodal degrees of freedom (DOFs) of a rectangular finite plate element (●: nodes).We rearrange the order of nodal DOFs in (38) to define a new nodal DOF vector as follows:(39)dA=dLdRdL=w1L,θy1L,θx1L,θxy1L,w2L,θy2L,θx2L,θxy2L,…,wj+1L,θyj+1L,θxj+1L,θxyj+1L,…,wNyL,θyNyL,θxNyL,θxyNyL,wNy+1L,θyNy+1L,θxNy+1L,θxyNy+1LTdR=w1R,θy1R,θx1R,θxy1R,w2R,θy2R,θx2R,θxy2R,…,wj+1R,θyj+1R,θxj+1R,θxyj+1R,…,wNyR,θyNyR,θxNyR,θxyNyR,wNy+1R,θyNy+1R,θxNy+1R,θxyNy+1RT.The nodal DOF vectord-A can be related to the new nodal DOF vector dA as follows:(40)d-A=d-Ld-R=TA00TAdLdR≡T1dA,where  T1 is the transformation matrix defined by(41)T1=TA00TA,where(42)TA=TA1TA2.The matricesTA1 and TA2 in (42) are (Ny+1)-by-(Ny+1) block diagonal matrices defined by(43)TA1=diagonaltA1,TA2=diagonaltA2,where(44)tA1=10000100,tA2=00100001.Applying (40) to (34) gives (45)wAx=XAx;ωdA,where(46)XAx;ω=RAEAx;ωHA-1T1.Finally, substituting (45) into (13) gives(47)wAx,y=NAx,y;ωdA,where (48)NAx,y;ω=YAyXAx;ω.


### 2.3. Derivation ofwB(x,y)

We can find the solutionwB(x,y) to Problem B by using a procedure similar to that used to obtain the solution for wA(x,y) in Problem A. Problem B can be obtained from Problem A by rotating the coordinate system (x,y) 90° clockwise. However, the differences for Problem B are as follows: (1) the fixed (null) boundary conditions are placed at x=-Lx/2 and Lx/2; (2) the finite plate element is divided into Nx finite strip elements in the x-direction, as shown in Figure 5; and (3) wB(x,y) should be determined to satisfy the boundary conditions at y=-Ly/2 and Ly/2 in combination with the boundary values contributed by wA(x,y).Figure 5
Finite strip element representation of a rectangular finite plate element subjected to null boundary conditions aty=-Lx/2 and Lx/2 (●: nodes).By following the solution procedure used for ProblemB, the displacement field in theith finite strip element, which has a width of lx(i)=xi-xi-1, can be written in the following form:(49)wBix,y=XBixwBiyxi-1≤x≤xi,where XB(i)(x) is a one-by-four interpolation function matrix and wB(i)(y) are the nodal line DOF functions defined by(50)XBix=XB1ix,XB2ixwBiy=wxBi-1ywxBiy,where(51)XB1ix=lxi-3x-xi22x-3xi-1+xi,lxi-2x-xi2x-xi-1XB2ix=-lxi-3x-xi-122x-3xi+xi-1,lxi-2x-xi-12x-xi,wxBi-1y=wxBxi-1,y=wBxi-1,y,θxBxi-1,yTwxBiy=wxBxi,y=wBxi,y,θxBxi,yT.By using (49), the displacement field wB(x,y) in the whole domain of the finite plate element can be represented by(52)wBx,y=XBxwBy-12Lx≤x≤12Lx,where(53)wBy=wxB1Ty,wxB2Ty,…,wxBiTy,…,wxBNx-2Ty,wxBNx-1TyT,(54)XBx=LB1x,LB2x,…,LBix,…,LBNx-2x,LBNx-1x.In (54), the following definition is used:(55)LBix=hBiXB2ix+hBi+1XB1i+1xi=1,2,…,Nx-1,where (56)hBix=Hx-xi-1-Hx-xiand H(x) is the Heaviside unit step function. Note that the null boundary conditions at x=-Lx/2 and x=Lx/2 have been applied to obtain (52).Substituting (52) into (5) gives(57)B4∂4wB∂y4+B2∂2wB∂y2+B0-ω-2MBwB=0,where(58)B4=ΛB1,B2=νΛB3+ΛB3T-21-νΛB2B0=ΛB4,MB=ΛB1with the following definitions:(59)ΛB1=∫-1/2Lx+1/2LxXBTXBdx,ΛB2=∫-1/2Lx+1/2Lx∂XBT∂x∂XB∂xdxΛB3=∫-1/2Lx+1/2LxXBT∂2XB∂x2dx,ΛB4=∫-1/2Lx+1/2Lx∂2XBT∂x2∂2XB∂x2dxThe constant matrices ΛB1, ΛB2, ΛB3, and ΛB4 are provided in Appendix B.Now we assume solutions to (57) in the following form:(60)wBy=1rB2⋮rB2Nx-1aBe+kyy-1/2k-yLy=rBaBe+kyy-1/2k-yLy,where aB is a constant, ky is the wavenumber in the y-direction, and(61)k-y=+kyifReky>0-kyifReky<00ifReky=0.Substituting (60) into (57) gives the following eigenvalue problem:(62)B4ky4+B2ky2+B0-ω-2MBrB=0or(63)B4λy2+B2λy+B0-ω-2MBrB=0λy=ky2.The dispersion relation can be obtained from (63) as follows:(64)detB4λy2+B2λy+B0-ω-2MB=0.From (64), the wavenumbers can be readily computed in the following forms:(65)kyj=+λyjky4Nx-1+j=-λyj=-kyjj=1,2,…,4Nx-1.By using the8×(Nx-1) wavenumbers computed from (64), we can write the general solution of (57) in the following form:(66)wBy=RBEBy;ωaB,where(67)aB=aB1aB2aB3⋯aBj⋯aB8Nx-1T,RB=R-BR-BEBx;ω=EB1x;ω00EB2x;ωwith(68)R-B=rB1,rB2,rB3,…,rBj,…,rB4×Nx-1EB1y;ω=diagonale+kyjy-1/2k-yjLyj=1,2,…,4Nx-1EB2y;ω=diagonale-kyjy-1/2k-yjLyj=1,2,…,4Nx-1.In (61), rB(j) is the jth eigenvector that can be readily computed from (63) using ky=ky(j).By using (47), the nodal values contributed by wA(x,y) at the ith nodes on the bottom edge at y=-Ly/2 and the upper edge at y=Ly/2 can be related to the nodal DOF vector dA as follows:(69)dAiB=wAiBθyAiBθxAiBθxyAiB=NAxi,-12LyNA,yxi,-12LyNA,xxi,-12LyNA,xyxi,-12LydA=YA-12LyXAxiYA′-12LyXAxiYA-12LyXA′xiYA′-12LyXA′xidAdAiU=wAiUθyAiUθxAiUθxyAiU=NAxi,+12LyNA,yxi,+12LyNA,xxi,+12LyNA,xyxi,+12LydA=YA+12LyXAxiYA′+12LyXAxiYA+12LyXA′xiYA′+12LyXA′xidAor(70)dAiB=QiBdA,dAiU=QiUdA,where the primes (′) denote the derivatives with respect to x or y, and(71)QiB=YA-12LyXAxiYA′-12LyXAxiYA-12LyXA′xiYA′-12LyXA′xi,QiU=YA+12LyXAxiYA′+12LyXAxiYA+12LyXA′xiYA′+12LyXA′xi.The superscriptsB andU denote the quantities at the bottom edge (i.e., at y=-Ly/2) and the upper edge (i.e., at y=Ly/2) of the plate, respectively. The superscriptsA andB denote the quantities related to or contributed by wA(x,y) and wB(x,y), respectively.By using (70), the nodal values contributed by wA(x,y) at all nodes on the bottom and upper edges, except for four corner nodes, can be written in vector form as(72)d^A=dABdAU,where(73)dAB=dA1BdA2B⋮dAiB⋮dANx-1B,dAU=dA1UdA2U⋮dAiU⋮dANx-1U.By using (70), (72) can be written in terms of dA as follows:(74)d^A=QdA,where(75)Q=QBQUwith(76)QB=Q1BQ2B⋮QNx-1B,QU=Q1UQ2U⋮QNx-1U.Similarly, the nodal values contributed bywB(x,y) at all nodes on the bottom and upper edges can be computed from (66), and they can be written in the vector form as(77)d-^B=d-BBd-BU,where(78)d-BB=wB-12Ly∂wB-1/2Ly∂y,d-BU=wB+12Ly∂wB+1/2Ly∂y,(79)wB-12Ly=wB1B,θxB1B,wB2B,θxB2B,…,wBiB,θxBiB,…,wBNx-1B,θxBNx-1BT∂wB-1/2Ly∂y=θyB1B,θxyB1B,θyB2B,θxyB2B,…,θyBiB,θxyBiB,…,θyBNx-1B,θxyBNx-1BTwB12Ly=wB1U,θxB1U,wB2U,θxB2U,…,wBiU,θxBiU,…,wBNx-1U,θxBNx-1UT∂wB1/2Ly∂y=θyB1U,θxyB1U,θyB2U,θxyB2U,…,θyBiU,θxyBiU,…,θyNx-1U,θxyNx-1UT.By substituting (66) into (78), the nodal values d-^B can be written in terms of the constant vector aB as follows:(80)d-^B=RBEB-12LyRBKBEB-12LyRBEB+12LyRBKBEB+12LyaB≡HBωaB,where(81)KB=κB00-κBκB=diagonalkyjj=1,2,…,4Nx-1.We rearrange the order of nodal values in (77) to define a new vector as (82)d^B=dBBdBU,where(83)dBB=wB1B,θyB1B,θxB1B,θxyB1B,wB2B,θyB2B,θxB2B,θxyB2B,…,wBiB,θyBiB,θxBiB,θxyBiB,…,wBNx-2B,θyBNx-2B,θxBNx-2B,θxyBNx-2B,wBNx-1B,θyBNx-1B,θxBNx-1B,θxyBNx-1BTdBU=wB1U,θyB1U,θxB1U,θxyB1U,wB2U,θyB2U,θxB2U,θxyB2U,…,wBiU,θyBiU,θxBiU,θxyBiU,…,wBNx-2U,θyBNx-2U,θxBNx-2U,θxyBNx-2U,wBNx-1U,θyBNx-1U,θxBNx-1U,θxyBNx-1UT.By introducing a proper transformation matrix, the nodal DOF vector d-^B can be related to the new vector d^B in the following form: (84)d-^B=d-BBd-BU=TB00TBdBBdBU≡T2d^B,where T2 is the transformation matrix defined by(85)T2=TB00TB,where(86)TB=TB1TB2and TB1 and TB2 are (Nx-1)-by-(Nx-1) block diagonal matrices defined by(87)TB1=diagonaltB1,TB2=diagonaltB2,where(88)tB1=10000010,tB2=01000001.By substituting (80) into (84), we obtain(89)d^B=T2-1HBωaB.The nodal DOFs defined at the nodes on the bottom and upper edges of the finite plate element (see Figure4) can be written in vector form as(90)dB=dBdU,where(91)dB=w1B,θy1B,θx1B,θxy1B,w2B,θy2B,θx2B,θxy2B,…,wiB,θyiB,θxiB,θxyiB,…,wNx-2B,θyNx-2B,θxNx-2B,θxyNx-2B,wNx-1B,θyNx-1B,θxNx-1B,θxyNx-1BTdU=w1U,θy1U,θx1U,θxy1U,w2U,θy2U,θx2U,θxy2U,…,wiU,θyiU,θxiU,θxyiU,…,wNx-2U,θyNx-2U,θxNx-2U,θxyNx-2U,wNx-1U,θyNx-1U,θxNx-1U,θxyNx-1UT.The nodal DOF vectordB must be identical to the sum of the nodal values contributed by wA(x,y) and wB(x,y). Thus, (92)dB=d^A+d^B.Substituting (74) and (89) into (92) gives(93)dB=QdA+T2-1HBωaB.From (93), we obtain the constant vector aB as (94)aB=HB-1ωT2dB-QdA.The constant vector aB can be removed from (66) by using (94) to obtain(95)wBy=YBydB-YABydA,where(96)YABy;ω=RBEBy;ωHB-1ωT2QYBy;ω=RBEBy;ωHB-1ωT2.Finally, substituting (95) into (52) yields(97)wBx,y=-NABx,y;ωdA+NBx,y;ωdB,where(98)NABx,y;ω=XBxYABy;ωNBx,y;ω=XBxYBy;ω.


### 2.4. Derivation ofw(x,y)

The solutionw(x,y) of the original problem shown in Figure 2(a) can be obtained by substituting (47) and (97) into (8) as follows:(99)wx,y;ω=Nx,y;ωdω,where d(ω) is the 8(Nx+Ny)-by-one spectral nodal DOF vector defined by (100)d=dAdB=dLdRdBdUand N(x,y;ω) is the one-by-8(Nx+Ny) dynamic shape function matrix defined by(101)Nx,y;ω=NAx,y;ω-NABx,y;ωNBx,y;ω.


### 2.5. Formulation of Spectral Element Equation

To formulate the spectral element model for the finite plate element, we first derive the weak form of governing equation (5) in the following form:(102)∫x∫yD∂2w∂x2+ν∂2w∂y2δ∂2w∂x2+21-νD∂2w∂x∂yδ∂2w∂x∂y+D∂2w∂y2+ν∂2w∂x2δ∂2w∂y2-ρω2wδwdxdy=∫x∫yfx,yδwdxdy+∫yVx1δw-12Lx,ydy+∫yVx2δw12Lx,ydy+∫xVy1δwx,-12Lydx+∫xVy2δwx,12Lydx+∫yMx1δ∂w-1/2Lx,y∂xdy+∫yMx2δ∂w1/2Lx,y∂xdy+∫xMy1δ∂wx,-1/2Ly∂ydx+∫xMy2δ∂wx,1/2Ly∂ydx,where Vx1(y), Vx2(y), Vy1(x), and Vy2(x) are the resultant transverse shear forces acting on the four boundary edges. Similarly Mx1(y), Mx2(y), My1(x), and My2(x) are the resultant bending moments acting on the four boundary edges.By substituting (101) into (102), we obtain the spectral element equation in the following form:(103)Sωdω=f1ω+f2ω,where(104)Sω=ΦTωHTωDωHωΦω,f1ω=ΦT∫x∫yfx,yNTx,ydxdyf2ω=ΦT∫yVx1NT-12Lx,ydy+∫yVx2NT12Lx,ydy+∫xVy1NTx,-12Lydx+∫xVy2NTx,12Lydx+∫yMx1N,xT-12Lx,ydy+∫yMx2N,xT12Lx,ydy+∫xMy1N,yTx,-12Lydx+∫xMy2N,yTx,12Lydx.In the preceding equations, the following definitions are used:(105)Φω=IA0-QIB,Hω=HA-1ωT100HB-1ωT2,where IA is the 8(Ny+1)-by-8(Ny+1) identity matrix and IB is the 8(Nx-1)-by-8(Nx-1) identity matrix.The matrixS(ω) in (103) is the 8(Nx+Ny)-by-8(Nx+Ny) symmetric dynamic stiffness matrix (often called the “spectral element matrix” in the literature), and matrix D(ω) is defined by(106)Dω=DD1ω+D2ω+21-νD3ω+νD4ω+D4Tω-ρω2Dρω,where(107)D1ω=K1Z1K1,D2ω=K2Z2K2D3ω=K3Z3K3,D4ω=K1Z4K2,Dρω=Z0,where(108)K1=KA200IB,K2=IA00KB2,K3=KA00KB,Z0=ZAA0ZAB0ZAB0TZBB0,Z1=ZAA1ZAB1ZAB1TZBB1,Z2=ZAA2ZAB2ZAB2TZBB2Z3=ZAA3ZAB3ZAB3TZBB3,Z4=ZAA4ZAB4ZBA4ZBB4with the use of the following definitions:(109)ZAA0=RATΛA1RA.∗Γ1X,ZAB0=Γ2XRB.∗RATΓ2YTZBB0=RBTΛB1RB.∗Γ1Y,ZAA1=RATΛA1RA.∗Γ1XZAB1=Γ4XRB.∗RATΓ2YT,ZBB1=RBTΛA4RB.∗Γ1YZAA2=RATΛA4RA.∗Γ1X,ZAB2=Γ2XRB.∗RATΓ4YTZBB2=RBTΛB1RB.∗Γ1Y,ZAA3=RATΛA2RA.∗Γ1XZAB3=Γ3XRB.∗RATΓ3YT,ZBB3=RBTΛB2RB.∗Γ1YZAA4=RATΛA3RA.∗Γ1X,ZAB4=Γ2XRB.∗RATΓ2YTZBB4=RBTΛB3TRB.∗Γ1Y,ZBA4=Γ4YRA.∗RBTΓ4XTΓ1X=∫xvectEAvectEATdx,Γ2X=∫xvectEAXBdxΓ3X=∫xvectEA∂XB∂xdx,Γ4X=∫xvectEA∂2XB∂x2dxΓ1Y=∫yvectEBvectEBTdy,Γ2Y=∫yvectEBYAdyΓ3Y=∫yvectEB∂YA∂ydy,Γ4Y=∫yvectEB∂2YA∂y2dy.In (109), the symbol vect() denotes a vector-form representation of the components of a diagonal matrix, and the symbol (.∗) denotes the elementwise matrix multiplication defined in MATLAB® [23] as follows:(110)A.∗B=C,where the components of matrix C are defined by(111)Cij=Aij×Bij


## 2.1. Governing Equations in the Frequency Domain

The time domain equation of motion and the boundary conditions of plate structures with transverse vibrations are described in [2]. The time domain equation of motion of a plate can be transformed into a frequency domain equation of motion of the plate by using the FFT as follows [5]:(1)D∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4-ρω2w=fx,y,where w(x,y) is the transverse displacement in spectral form, f(x,y) is the external force in spectral form, ρ is the mass per unit area of the plate, and D=Eh3/[12(1-ν2)] is the flexural bending rigidity of the plate where E is the modulus of elasticity,ν is Poisson’s ratio, andh is the plate thickness. Similarly, the time domain boundary conditions can be transformed into frequency domain boundary conditions as follows:(2)Myx,-12Ly=-My1xorθyx,-12Ly=θy1x,Vyx,-12Ly=-Vy1xorwx,-12Ly=w1x,Myx,12Ly=My2xorθyx,12Ly=θy2x,Vyx,12Ly=Vy2xorwx,12Ly=w2x,Mx-12Lx,y=-Mx1yorθx-12Lx,y=θx1y,Vx-12Lx,y=-Vx1yorw-12Lx,y=w3y,Mx12Lx,y=Mx2yorθx12Lx,y=θx2y,Vx12Lx,y=Vx2yorw12Lx,y=w4y,where Lx and Ly are the dimensions of a finite plate in the x- and y-directions, respectively; Mx and My are the resultant moments; and Vx and Vy are the resultant transverse shear forces defined by(3)Mx=-D∂2w∂x2+ν∂2w∂y2,Vx=-D∂∂x∂2w∂x2+ν∂2w∂y2-2D1-ν∂3w∂x∂y2My=-D∂2w∂y2+ν∂2w∂x2,Vy=-D∂∂y∂2w∂y2+ν∂2w∂x2-2D1-ν∂3w∂x2∂y.And θx and θy are the slopes defined by(4)θx=∂w∂x,θy=∂w∂y.We need to obtain frequency domain free wave solutions for a homogeneous equation of motion in order to formulate the spectral element model for a finite plate element. To realize this, the homogeneous equation of motion is considered by removing external forcef(x,y) in (1) as follows:(5)D∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4-ρω2w=0.The weak form of (5) can be obtained in the following form:(6)∫x∫yD∂2w∂x2+ν∂2w∂y2δ∂2w∂x2+21-νD∂2w∂x∂yδ∂2w∂x∂y+D∂2w∂y2+ν∂2w∂x2δ∂2w∂y2-ρω2wδwdxdy=0.A free vibration solution satisfying the weak form given in (6) can be obtained approximately by using two combined methods: the boundary splitting method [20] and the spectral super element method (SSEM) [16]. The SSEM uses a combination of the Kantorovich method (based on the finite strip element method) and the frequency domain 1D spectral element method.The concept of the boundary splitting method is illustrated in Figures1 and 2. Figure 1 indicates the concept used in our previous study [5]. Figure 2 indicates the concept used in the present study. The original problems, shown in Figures 1(a) and 2(a), are represented by the sum of two partial problems, Problem A and Problem B. In Figures 1 and 2, the geometric boundary conditions of the original problems are presented in simple forms by using the following definitions:(7)wxx,y=wx,y,θxx,y=∂w∂xTwyx,y=wx,y,θyx,y=∂w∂yTwxAx,y=wAx,y,θxAx,y=∂wA∂xTwyAx,y=wAx,y,θyAx,y=∂wA∂yTwxBx,y=wBx,y,θxBx,y=∂wB∂xTwyBx,y=wBx,y,θyBx,y=∂wB∂yT.Figure 1
Boundary splitting method used in the previous study [5] to derive w(x,y) for a rectangular plate element subjected to arbitrary boundary conditions (●: active nodes; O: fixed nodes).Figure 2
Boundary splitting method used in this study to derivew(x,y) for a rectangular plate element subjected to arbitrary boundary conditions (●: active nodes).In our previous study [5], Problem A, shown in Figure 1(b), has fixed (null) boundary conditions on the parallel edges at y=-Ly/2 and Ly/2. Problem B, shown in Figure 1(c), has fixed (null) boundary conditions on the parallel edges at x=-Lx/2 and Lx/2. As a result, the spectral element model developed in [5] is valid only for finite rectangular plate elements whose four corner nodes are fixed, as shown in Figure 1(d). Accordingly, an application of this approach should be limited to very specific problems as considered in [5].We propose a new boundary splitting method by modifying the boundary splitting method used in [5] such that the four corner nodes of a finite plate element become active. Problem A, shown in Figure 2(b), has arbitrary boundary conditions rather than fixed boundary conditions on the parallel edges at y=-Ly/2 and Ly/2, and its solution is represented by wA(x,y). Problem B, shown in Figure 2(c), has fixed (null) boundary conditions on the parallel edges at x=-Lx/2 and Lx/2. However, the boundary conditions at y=-Ly/2 and Ly/2 in Problem B must be specified such that the sum of the boundary conditions at y=-Ly/2 and Ly/2 in Problem A and those in Problem B is identical to the boundary conditions at y=-Ly/2 and Ly/2 in the original problem. The solution of Problem B is represented by wB(x,y). Then, the solution w(x,y) of the original problem can be obtained by summing the solutions to Problem A and Problem B as follows:(8)wx,y=wAx,y+wBx,y.Accordingly, compared to the spectral element model developed in our previous study [5] based on the boundary splitting shown in Figure 1, the present spectral element model that was developed based on the boundary splitting shown in Figure 2 has four active corner nodes. Thus, it can be used as a generic finite element model that can be assembled in both the x- and y-directions of a plate with arbitrary boundary conditions.


## 2.2. Derivation ofwA(x,y)

To obtain the solutionwA(x,y) for Problem A by using the SSEM, a rectangular finite plate element is divided into Ny finite strip elements in the y-direction, as shown in Figure 3(a). The jth finite strip element, which has a width of lyj=yj-yj-1 in the y-direction, is shown in Figure 3(b).Figure 3
Finite strip element representation of a rectangular finite plate element subjected to arbitrary boundary conditions aty=-Ly/2 and Ly/2 (●: nodes; grey circles: nodal values).
(a) (b)The displacement field in thejth finite strip element can be represented by(9)wAjx,y=YAjywAjxyj-1≤y≤yj,where YA(j)(y) is a one-by-four interpolation function matrix and wA(j)(x) are the nodal line degree of freedom (DOF) functions defined by(10)YAjy=YA1jy,YA2jywAjx=wyAj-1xwyAjx,where(11)YA1jy=lyj-3y-yj22y+yj-3yj-1,lyj-2y-yj2y-yj-1YA2jy=-lyj-3y-yj-122y+yj-1-3yj,lyj-2y-yj-12y-yj,(12)wyAj-1x=wyAx,yj-1=wAx,yj-1,θyAx,yj-1TwyAjx=wyAx,yj=wAx,yj,θyAx,yjT.By using (9), the displacement field wA(x,y) over the entire domain of the finite plate element can be represented as(13)wAx,y=YAywAx-12Ly≤y≤12Ly,where(14)wAx=wyA0Tx,wyA1Tx,…,wyAjTx,…,wyANy-1Tx,wyANyTxT,(15)YAy=LA0y,LA1y,…,LAjy,…,LANy-1y,LANyywith(16)LA0y=hA1YA11yLAjy=hAjyYA2jy+hAj+1yYA1j+1yj=1,2,…,Ny-1LANyy=hANyYA2Nyy.In (16), hA(j)(y) are functions defined by(17)hAjy=Hy-yj-1-Hy-yj,where H(y) is the Heaviside unit step function.Substituting (13) into (5) yields(18)A4∂4wA∂x4+A2∂2wA∂x2+A0-ω-2MAwA=0,where(19)ω-=ωρD,A0=ΛA4,A2=νΛA3+ΛA3T-21-νΛA2A4=ΛA1,MA=ΛA1with the following definitions:(20)ΛA1=∫-1/2Ly+1/2LyYATYAdy,ΛA2=∫-1/2Ly+1/2Ly∂YAT∂y∂YA∂ydyΛA3=∫-1/2Ly+1/2LyYAT∂2YA∂y2dy,ΛA4=∫-1/2Ly+1/2Ly∂2YAT∂y2∂2YA∂y2dy.The constant matrices ΛA1, ΛA2, ΛA3, and ΛA4 are provided in Appendix A.Next, we assume solutions of (18) to be in the following form:(21)wAx=1rA2⋮rA2Ny+1aAe+kxx-1/2k-xLx=rAaAe+kxx-1/2k-xLx,where aA is a constant, kx is the wavenumber in the x-direction, and(22)k-x=+kxifRekx>0-kxifRekx<00ifRekx=0.Substituting (21) into (18) gives the following eigenvalue problem:(23)A4kx4+A2kx2+A0-ω-2MArA=0or(24)AAλx2+A2λx+A0-ω-2MArA=0λx=kx2.The dispersion relation (i.e., the frequency-wavenumber relationship) can be obtained from (24) as follows:(25)detA4λx2+A2λx+A0-ω-2MA=0.From (25), the wavenumbers can be computed as(26)kxj=+λxjkx4Ny+1+j=-λxj=-kxjj=1,2,…,4Ny+1.By using the wavenumberskxj given by (26), we can write the general solution of (18) in the following form:(27)wAx=RAEAx;ωaA,where(28)aA=aA1aA2aA3⋯aAi⋯aA8Ny+1T,RA=R-AR-AEAx;ω=EA1x;ω00EA2x;ωwith(29)R-A=rA1,rA2,rA3,…,rAj,…,rA4×Ny-1EA1x;ω=diagonale+kxjx-1/2k-xjLxj=1,2,…,4Ny+1EA2x;ω=diagonale-kxjx-1/2k-xjLxj=1,2,…,4Ny+1.In (29), rAj is the jth eigenvector, which can be readily computed from (23) using kx=kxj.The nodal DOFs at the nodes defined on the edges atx=-Lx/2 and Lx/2 can be written in vector form as(30)d-A=d-Ld-R,where(31)d-L=wA-12Lx∂wA-1/2Lx∂x,d-R=wA12Lx∂wA1/2Lx∂x∂x.Here, the superscriptsL andR denote the nodal values on the left edge (i.e., at x=-Lx/2) and the right edge (i.e., at x=Lx/2) of the plate, respectively. SuperscriptA denotes the quantities related to or contributed by wA(x,y).By substituting (27) into (31), the nodal DOF vector d-A can be written in terms of the constant vector aA as follows:(32)d-A=RAEA-12LxRAKAEA-12LxRAEA+12LxRAKAEA+12LxaA≡HAωaA,where(33)KA=κA00-κAκA=diagonalkxjj=1,2,…,4Ny+1.The constant vectoraA can be removed from (27) by using (32) to obtain the following expression:(34)wAx=X-Ax;ωd-A,where(35)X-Ax;ω=RAEAx;ωHA-1.From (12), by using the nodal DOFs defined in Figure 4, we obtain the following expressions:(36)wyAj-12Lx=wA-12Lx,yjθyA-12Lx,yj=wj+1Lθyj+1L∂wyAj-1/2Lx∂x=θxA-12Lx,yjθxyA-12Lx,yj=θxj+1Lθxyj+1L,wyAj12Lx=wA12Lx,yjθyA12Lx,yj=wj+1Rθyj+1R∂wyAj1/2Lx∂x=θxA12Lx,yjθxyA12Lx,yj=θxj+1Rθxyj+1R.By applying (36) to (14), we obtain the following expressions:(37)wA-12Lx=w1L,θy1L,w2L,θy2L,…,wj+1L,θyj+1L,…,wNyL,θyNyL,wNy+1L,θyNy+1LT∂wA-1/2Lx∂x=θx1L,θxy1L,θx2L,θxy2L,…,θxj+1L,θxyj+1L,…,θxNyL,θxyNyL,θxNy+1L,θxyNy+1LTwA12Lx=w1R,θy1R,w2R,θy2R,…,wj+1R,θyj+1R,…,wNyR,θyNyR,wNy+1R,θyNy+1RT∂wA1/2Lx∂x=θx1R,θxy1R,θx2R,θxy2R,…,θxj+1R,θxyj+1R,…,θxNyR,θxyNyR,θxNy+1R,θxyNy+1RT.Applying (37) to (31) gives(38)d-L=w1L,θy1L,w2L,θy2L,…,wj+1L,θyj+1L,…,wNyL,θyNyL,wNy+1L,θyNy+1Lθx1L,θxy1L,θx2L,θxy2L,…,θxj+1L,θxyj+1L,…,θxNyL,θxyNyL,θxNy+1L,θxyNy+1LTd-R=w1R,θy1R,w2R,θy2R,…,wj+1R,θyj+1R,…,wNyR,θyNyR,wNy+1R,θyNy+1Rθx1R,θxy1R,θx2R,θxy2R,…,θxj+1R,θxyj+1R,…,θxNyR,θxyNyR,θxNy+1R,θxyNy+1RT.Figure 4
Spectral nodal degrees of freedom (DOFs) of a rectangular finite plate element (●: nodes).We rearrange the order of nodal DOFs in (38) to define a new nodal DOF vector as follows:(39)dA=dLdRdL=w1L,θy1L,θx1L,θxy1L,w2L,θy2L,θx2L,θxy2L,…,wj+1L,θyj+1L,θxj+1L,θxyj+1L,…,wNyL,θyNyL,θxNyL,θxyNyL,wNy+1L,θyNy+1L,θxNy+1L,θxyNy+1LTdR=w1R,θy1R,θx1R,θxy1R,w2R,θy2R,θx2R,θxy2R,…,wj+1R,θyj+1R,θxj+1R,θxyj+1R,…,wNyR,θyNyR,θxNyR,θxyNyR,wNy+1R,θyNy+1R,θxNy+1R,θxyNy+1RT.The nodal DOF vectord-A can be related to the new nodal DOF vector dA as follows:(40)d-A=d-Ld-R=TA00TAdLdR≡T1dA,where  T1 is the transformation matrix defined by(41)T1=TA00TA,where(42)TA=TA1TA2.The matricesTA1 and TA2 in (42) are (Ny+1)-by-(Ny+1) block diagonal matrices defined by(43)TA1=diagonaltA1,TA2=diagonaltA2,where(44)tA1=10000100,tA2=00100001.Applying (40) to (34) gives (45)wAx=XAx;ωdA,where(46)XAx;ω=RAEAx;ωHA-1T1.Finally, substituting (45) into (13) gives(47)wAx,y=NAx,y;ωdA,where (48)NAx,y;ω=YAyXAx;ω.


## 2.3. Derivation ofwB(x,y)

We can find the solutionwB(x,y) to Problem B by using a procedure similar to that used to obtain the solution for wA(x,y) in Problem A. Problem B can be obtained from Problem A by rotating the coordinate system (x,y) 90° clockwise. However, the differences for Problem B are as follows: (1) the fixed (null) boundary conditions are placed at x=-Lx/2 and Lx/2; (2) the finite plate element is divided into Nx finite strip elements in the x-direction, as shown in Figure 5; and (3) wB(x,y) should be determined to satisfy the boundary conditions at y=-Ly/2 and Ly/2 in combination with the boundary values contributed by wA(x,y).Figure 5
Finite strip element representation of a rectangular finite plate element subjected to null boundary conditions aty=-Lx/2 and Lx/2 (●: nodes).By following the solution procedure used for ProblemB, the displacement field in theith finite strip element, which has a width of lx(i)=xi-xi-1, can be written in the following form:(49)wBix,y=XBixwBiyxi-1≤x≤xi,where XB(i)(x) is a one-by-four interpolation function matrix and wB(i)(y) are the nodal line DOF functions defined by(50)XBix=XB1ix,XB2ixwBiy=wxBi-1ywxBiy,where(51)XB1ix=lxi-3x-xi22x-3xi-1+xi,lxi-2x-xi2x-xi-1XB2ix=-lxi-3x-xi-122x-3xi+xi-1,lxi-2x-xi-12x-xi,wxBi-1y=wxBxi-1,y=wBxi-1,y,θxBxi-1,yTwxBiy=wxBxi,y=wBxi,y,θxBxi,yT.By using (49), the displacement field wB(x,y) in the whole domain of the finite plate element can be represented by(52)wBx,y=XBxwBy-12Lx≤x≤12Lx,where(53)wBy=wxB1Ty,wxB2Ty,…,wxBiTy,…,wxBNx-2Ty,wxBNx-1TyT,(54)XBx=LB1x,LB2x,…,LBix,…,LBNx-2x,LBNx-1x.In (54), the following definition is used:(55)LBix=hBiXB2ix+hBi+1XB1i+1xi=1,2,…,Nx-1,where (56)hBix=Hx-xi-1-Hx-xiand H(x) is the Heaviside unit step function. Note that the null boundary conditions at x=-Lx/2 and x=Lx/2 have been applied to obtain (52).Substituting (52) into (5) gives(57)B4∂4wB∂y4+B2∂2wB∂y2+B0-ω-2MBwB=0,where(58)B4=ΛB1,B2=νΛB3+ΛB3T-21-νΛB2B0=ΛB4,MB=ΛB1with the following definitions:(59)ΛB1=∫-1/2Lx+1/2LxXBTXBdx,ΛB2=∫-1/2Lx+1/2Lx∂XBT∂x∂XB∂xdxΛB3=∫-1/2Lx+1/2LxXBT∂2XB∂x2dx,ΛB4=∫-1/2Lx+1/2Lx∂2XBT∂x2∂2XB∂x2dxThe constant matrices ΛB1, ΛB2, ΛB3, and ΛB4 are provided in Appendix B.Now we assume solutions to (57) in the following form:(60)wBy=1rB2⋮rB2Nx-1aBe+kyy-1/2k-yLy=rBaBe+kyy-1/2k-yLy,where aB is a constant, ky is the wavenumber in the y-direction, and(61)k-y=+kyifReky>0-kyifReky<00ifReky=0.Substituting (60) into (57) gives the following eigenvalue problem:(62)B4ky4+B2ky2+B0-ω-2MBrB=0or(63)B4λy2+B2λy+B0-ω-2MBrB=0λy=ky2.The dispersion relation can be obtained from (63) as follows:(64)detB4λy2+B2λy+B0-ω-2MB=0.From (64), the wavenumbers can be readily computed in the following forms:(65)kyj=+λyjky4Nx-1+j=-λyj=-kyjj=1,2,…,4Nx-1.By using the8×(Nx-1) wavenumbers computed from (64), we can write the general solution of (57) in the following form:(66)wBy=RBEBy;ωaB,where(67)aB=aB1aB2aB3⋯aBj⋯aB8Nx-1T,RB=R-BR-BEBx;ω=EB1x;ω00EB2x;ωwith(68)R-B=rB1,rB2,rB3,…,rBj,…,rB4×Nx-1EB1y;ω=diagonale+kyjy-1/2k-yjLyj=1,2,…,4Nx-1EB2y;ω=diagonale-kyjy-1/2k-yjLyj=1,2,…,4Nx-1.In (61), rB(j) is the jth eigenvector that can be readily computed from (63) using ky=ky(j).By using (47), the nodal values contributed by wA(x,y) at the ith nodes on the bottom edge at y=-Ly/2 and the upper edge at y=Ly/2 can be related to the nodal DOF vector dA as follows:(69)dAiB=wAiBθyAiBθxAiBθxyAiB=NAxi,-12LyNA,yxi,-12LyNA,xxi,-12LyNA,xyxi,-12LydA=YA-12LyXAxiYA′-12LyXAxiYA-12LyXA′xiYA′-12LyXA′xidAdAiU=wAiUθyAiUθxAiUθxyAiU=NAxi,+12LyNA,yxi,+12LyNA,xxi,+12LyNA,xyxi,+12LydA=YA+12LyXAxiYA′+12LyXAxiYA+12LyXA′xiYA′+12LyXA′xidAor(70)dAiB=QiBdA,dAiU=QiUdA,where the primes (′) denote the derivatives with respect to x or y, and(71)QiB=YA-12LyXAxiYA′-12LyXAxiYA-12LyXA′xiYA′-12LyXA′xi,QiU=YA+12LyXAxiYA′+12LyXAxiYA+12LyXA′xiYA′+12LyXA′xi.The superscriptsB andU denote the quantities at the bottom edge (i.e., at y=-Ly/2) and the upper edge (i.e., at y=Ly/2) of the plate, respectively. The superscriptsA andB denote the quantities related to or contributed by wA(x,y) and wB(x,y), respectively.By using (70), the nodal values contributed by wA(x,y) at all nodes on the bottom and upper edges, except for four corner nodes, can be written in vector form as(72)d^A=dABdAU,where(73)dAB=dA1BdA2B⋮dAiB⋮dANx-1B,dAU=dA1UdA2U⋮dAiU⋮dANx-1U.By using (70), (72) can be written in terms of dA as follows:(74)d^A=QdA,where(75)Q=QBQUwith(76)QB=Q1BQ2B⋮QNx-1B,QU=Q1UQ2U⋮QNx-1U.Similarly, the nodal values contributed bywB(x,y) at all nodes on the bottom and upper edges can be computed from (66), and they can be written in the vector form as(77)d-^B=d-BBd-BU,where(78)d-BB=wB-12Ly∂wB-1/2Ly∂y,d-BU=wB+12Ly∂wB+1/2Ly∂y,(79)wB-12Ly=wB1B,θxB1B,wB2B,θxB2B,…,wBiB,θxBiB,…,wBNx-1B,θxBNx-1BT∂wB-1/2Ly∂y=θyB1B,θxyB1B,θyB2B,θxyB2B,…,θyBiB,θxyBiB,…,θyBNx-1B,θxyBNx-1BTwB12Ly=wB1U,θxB1U,wB2U,θxB2U,…,wBiU,θxBiU,…,wBNx-1U,θxBNx-1UT∂wB1/2Ly∂y=θyB1U,θxyB1U,θyB2U,θxyB2U,…,θyBiU,θxyBiU,…,θyNx-1U,θxyNx-1UT.By substituting (66) into (78), the nodal values d-^B can be written in terms of the constant vector aB as follows:(80)d-^B=RBEB-12LyRBKBEB-12LyRBEB+12LyRBKBEB+12LyaB≡HBωaB,where(81)KB=κB00-κBκB=diagonalkyjj=1,2,…,4Nx-1.We rearrange the order of nodal values in (77) to define a new vector as (82)d^B=dBBdBU,where(83)dBB=wB1B,θyB1B,θxB1B,θxyB1B,wB2B,θyB2B,θxB2B,θxyB2B,…,wBiB,θyBiB,θxBiB,θxyBiB,…,wBNx-2B,θyBNx-2B,θxBNx-2B,θxyBNx-2B,wBNx-1B,θyBNx-1B,θxBNx-1B,θxyBNx-1BTdBU=wB1U,θyB1U,θxB1U,θxyB1U,wB2U,θyB2U,θxB2U,θxyB2U,…,wBiU,θyBiU,θxBiU,θxyBiU,…,wBNx-2U,θyBNx-2U,θxBNx-2U,θxyBNx-2U,wBNx-1U,θyBNx-1U,θxBNx-1U,θxyBNx-1UT.By introducing a proper transformation matrix, the nodal DOF vector d-^B can be related to the new vector d^B in the following form: (84)d-^B=d-BBd-BU=TB00TBdBBdBU≡T2d^B,where T2 is the transformation matrix defined by(85)T2=TB00TB,where(86)TB=TB1TB2and TB1 and TB2 are (Nx-1)-by-(Nx-1) block diagonal matrices defined by(87)TB1=diagonaltB1,TB2=diagonaltB2,where(88)tB1=10000010,tB2=01000001.By substituting (80) into (84), we obtain(89)d^B=T2-1HBωaB.The nodal DOFs defined at the nodes on the bottom and upper edges of the finite plate element (see Figure4) can be written in vector form as(90)dB=dBdU,where(91)dB=w1B,θy1B,θx1B,θxy1B,w2B,θy2B,θx2B,θxy2B,…,wiB,θyiB,θxiB,θxyiB,…,wNx-2B,θyNx-2B,θxNx-2B,θxyNx-2B,wNx-1B,θyNx-1B,θxNx-1B,θxyNx-1BTdU=w1U,θy1U,θx1U,θxy1U,w2U,θy2U,θx2U,θxy2U,…,wiU,θyiU,θxiU,θxyiU,…,wNx-2U,θyNx-2U,θxNx-2U,θxyNx-2U,wNx-1U,θyNx-1U,θxNx-1U,θxyNx-1UT.The nodal DOF vectordB must be identical to the sum of the nodal values contributed by wA(x,y) and wB(x,y). Thus, (92)dB=d^A+d^B.Substituting (74) and (89) into (92) gives(93)dB=QdA+T2-1HBωaB.From (93), we obtain the constant vector aB as (94)aB=HB-1ωT2dB-QdA.The constant vector aB can be removed from (66) by using (94) to obtain(95)wBy=YBydB-YABydA,where(96)YABy;ω=RBEBy;ωHB-1ωT2QYBy;ω=RBEBy;ωHB-1ωT2.Finally, substituting (95) into (52) yields(97)wBx,y=-NABx,y;ωdA+NBx,y;ωdB,where(98)NABx,y;ω=XBxYABy;ωNBx,y;ω=XBxYBy;ω.


## 2.4. Derivation ofw(x,y)

The solutionw(x,y) of the original problem shown in Figure 2(a) can be obtained by substituting (47) and (97) into (8) as follows:(99)wx,y;ω=Nx,y;ωdω,where d(ω) is the 8(Nx+Ny)-by-one spectral nodal DOF vector defined by (100)d=dAdB=dLdRdBdUand N(x,y;ω) is the one-by-8(Nx+Ny) dynamic shape function matrix defined by(101)Nx,y;ω=NAx,y;ω-NABx,y;ωNBx,y;ω.


## 2.5. Formulation of Spectral Element Equation

To formulate the spectral element model for the finite plate element, we first derive the weak form of governing equation (5) in the following form:(102)∫x∫yD∂2w∂x2+ν∂2w∂y2δ∂2w∂x2+21-νD∂2w∂x∂yδ∂2w∂x∂y+D∂2w∂y2+ν∂2w∂x2δ∂2w∂y2-ρω2wδwdxdy=∫x∫yfx,yδwdxdy+∫yVx1δw-12Lx,ydy+∫yVx2δw12Lx,ydy+∫xVy1δwx,-12Lydx+∫xVy2δwx,12Lydx+∫yMx1δ∂w-1/2Lx,y∂xdy+∫yMx2δ∂w1/2Lx,y∂xdy+∫xMy1δ∂wx,-1/2Ly∂ydx+∫xMy2δ∂wx,1/2Ly∂ydx,where Vx1(y), Vx2(y), Vy1(x), and Vy2(x) are the resultant transverse shear forces acting on the four boundary edges. Similarly Mx1(y), Mx2(y), My1(x), and My2(x) are the resultant bending moments acting on the four boundary edges.By substituting (101) into (102), we obtain the spectral element equation in the following form:(103)Sωdω=f1ω+f2ω,where(104)Sω=ΦTωHTωDωHωΦω,f1ω=ΦT∫x∫yfx,yNTx,ydxdyf2ω=ΦT∫yVx1NT-12Lx,ydy+∫yVx2NT12Lx,ydy+∫xVy1NTx,-12Lydx+∫xVy2NTx,12Lydx+∫yMx1N,xT-12Lx,ydy+∫yMx2N,xT12Lx,ydy+∫xMy1N,yTx,-12Lydx+∫xMy2N,yTx,12Lydx.In the preceding equations, the following definitions are used:(105)Φω=IA0-QIB,Hω=HA-1ωT100HB-1ωT2,where IA is the 8(Ny+1)-by-8(Ny+1) identity matrix and IB is the 8(Nx-1)-by-8(Nx-1) identity matrix.The matrixS(ω) in (103) is the 8(Nx+Ny)-by-8(Nx+Ny) symmetric dynamic stiffness matrix (often called the “spectral element matrix” in the literature), and matrix D(ω) is defined by(106)Dω=DD1ω+D2ω+21-νD3ω+νD4ω+D4Tω-ρω2Dρω,where(107)D1ω=K1Z1K1,D2ω=K2Z2K2D3ω=K3Z3K3,D4ω=K1Z4K2,Dρω=Z0,where(108)K1=KA200IB,K2=IA00KB2,K3=KA00KB,Z0=ZAA0ZAB0ZAB0TZBB0,Z1=ZAA1ZAB1ZAB1TZBB1,Z2=ZAA2ZAB2ZAB2TZBB2Z3=ZAA3ZAB3ZAB3TZBB3,Z4=ZAA4ZAB4ZBA4ZBB4with the use of the following definitions:(109)ZAA0=RATΛA1RA.∗Γ1X,ZAB0=Γ2XRB.∗RATΓ2YTZBB0=RBTΛB1RB.∗Γ1Y,ZAA1=RATΛA1RA.∗Γ1XZAB1=Γ4XRB.∗RATΓ2YT,ZBB1=RBTΛA4RB.∗Γ1YZAA2=RATΛA4RA.∗Γ1X,ZAB2=Γ2XRB.∗RATΓ4YTZBB2=RBTΛB1RB.∗Γ1Y,ZAA3=RATΛA2RA.∗Γ1XZAB3=Γ3XRB.∗RATΓ3YT,ZBB3=RBTΛB2RB.∗Γ1YZAA4=RATΛA3RA.∗Γ1X,ZAB4=Γ2XRB.∗RATΓ2YTZBB4=RBTΛB3TRB.∗Γ1Y,ZBA4=Γ4YRA.∗RBTΓ4XTΓ1X=∫xvectEAvectEATdx,Γ2X=∫xvectEAXBdxΓ3X=∫xvectEA∂XB∂xdx,Γ4X=∫xvectEA∂2XB∂x2dxΓ1Y=∫yvectEBvectEBTdy,Γ2Y=∫yvectEBYAdyΓ3Y=∫yvectEB∂YA∂ydy,Γ4Y=∫yvectEB∂2YA∂y2dy.In (109), the symbol vect() denotes a vector-form representation of the components of a diagonal matrix, and the symbol (.∗) denotes the elementwise matrix multiplication defined in MATLAB® [23] as follows:(110)A.∗B=C,where the components of matrix C are defined by(111)Cij=Aij×Bij


## 3. Numerical Results and Discussion

To evaluate the performance of the present spectral element model, a square plate is considered as a numerical example. The plate is made of aluminum, and its material properties are as follows: Young’s modulusE = 69 GPa, Poisson’s ratioν = 0.33, and mass density ρ = 2700 kg/m3. The size of the square plate isL = 1 m, and its thickness ish = 0.001 m. For numerical studies, three types of boundary conditions are considered: (1) Example 1: a square plate with simple-simple-simple-simple (S-S-S-S) edge support, as shown in Figure 6; (2) Example 2: a square plate with free-clamped-free-clamped (F-C-F-C) edge support, as shown in Figure 7; and (3) Example 3: a square plate with free-free-free-clamped (F-F-F-C) edge support, as shown in Figure 8.Figure 6
Square plate with simple-simple-simple-simple (S-S-S-S) edge support, Example1 (●: active nodes).
(a)
Example1 (b)
FEM model (c)
SEM modelFigure 7
Square plate with free-clamped-free-clamped (F-C-F-C) edge support, Example2 (●: active nodes; O: fixed nodes).
(a)
Example2 (b)
FEM model (c)
SEM modelFigure 8
Square plate with free-free-free-clamped (F-F-F-C) edge support, Example3 (●: active nodes; O: fixed nodes).
(a)
Example3 (b)
FEM model (c)
SEM modelAn evaluation of the present spectral element model (denoted by “SEM”) was conducted by comparing the natural frequencies obtained from the SEM with those obtained by using exact theory, the standard finite element method (FEM), and the commercial finite element analysis package ANSYS [21]. For the FEM results, a four-node 12-DOF conforming rectangular finite element model [22] is used.It is well known that exact solutions are available in analytical forms only for Levy-type plates. This is why we considered S-S-S-S square plate (Example1) as the first example problem to evaluate the accuracy of the present spectral element model. The exact natural frequencies of S-S-S-S square plate are given by [24](112)ωmn=π2m2+n2L2Dρm,n=1,2,3,…,where L is the dimension of the square plate and (m,n) indicates the mode number.Table1 compares the natural frequencies of the S-S-S-S square plate (Example 1) obtained by exact theory, FEM [22], and the present SEM. The SEM results obtained by using a single finite element are found to be identical to the exact solutions given by (112). The FEM results are found to converge to the exact solutions or the SEM results as the number of finite elements is increased to more than 100×100. From Table 1, we can compare the range of natural frequencies that can be calculated with reasonable accuracy by the present SEM and FEM by comparison with exact solutions. The mesh size of the FEM when 50×50 elements are used (see the 4th column of Table 1) is the same as the inner mesh size of the SEM when a single element is used (see the last column of Table 1). Table 1 shows that the natural frequencies by the FEM are accurate only up to the mode (1,2), whereas the natural frequencies by the SEM are accurate up to the higher mode (10,10). This clearly verifies the high accuracy of the present SEM when compared with the FEM.Table 1
Comparison of the natural frequencies (Hz) of a square plate with simple-simple-simple-simple (S-S-S-S) edge support (Example1) obtained by exact theory, FEM [22], and SEM.
Mode Exact FEM SEM n E = 10 × 10 50 × 50 100 × 100 1 n D = 283 7403 29803 800 ( 1,1 ) 4.857 4.831 4.856 4.856 4.857 ( 1,2 ) 12.14 12.04 12.14 12.14 12.14 ( 2,2 ) 19.43 19.03 19.41 19.42 19.43 ( 1,3 ) 24.28 24.07 24.27 24.28 24.28 ( 2,3 ) 31.57 30.72 31.53 31.56 31.57 ( 1,4 ) 41.28 40.97 41.26 41.28 41.28 ( 3,3 ) 43.71 41.87 43.62 43.70 43.71 ( 5,5 ) 121.4 115.7 121.1 121.3 121.4 ( 10,10 ) 485.7 N/A 480.4 485.3 485.7 ( 20,20 ) 1943 N/A 1879 1922 1949 Note:nE: total number of finite elements; nD: total number of degrees of freedom.Table2 compares the natural frequencies of the F-C-F-C square plate (Example 2) obtained by using ANSYS [21], the FEM [22], and the present SEM. The ANSYS results were obtained by using a sufficient number of finite elements (100×100) and are very close to the SEM results obtained by using a single finite element. The FEM results converged to the SEM results as the number of finite elements is increased to more than 100×100.Table 2
Comparison of the natural frequencies (Hz) of a square plate with free-clamped-free-clamped (F-C-F-C) edge support obtained by ANSYS [21], FEM [22], and SEM.
Mode ANSYS FEM SEM n E = 100 × 100 10 × 10 50 × 50 100 × 100 1 n D = 39996 297 7497 29997 792 1 5.441 5.454 5.442 5.441 5.441 2 6.449 6.453 6.449 6.448 6.448 3 10.66 10.58 10.65 10.66 10.66 4 15.02 15.09 15.02 15.02 15.02 5 16.45 16.48 16.44 16.44 16.44 6 19.57 19.36 19.56 19.57 19.57 7 21.43 21.19 21.42 21.43 21.43 8 29.50 29.69 29.50 29.50 29.50 9 30.47 29.78 30.44 30.47 30.47 10 31.06 31.20 31.06 31.06 31.06 Note:nE: total number of finite elements; nD: total number of degrees of freedom.Table3 compares the natural frequencies of the F-F-F-C square plate (Example 3) obtained by ANSYS [21], FEM [22], and the present SEM. Similar to Example 2, the ANSYS results, which were obtained by using a sufficient number of finite elements (100×100), are also found to be very close to the SEM results obtained by using a single finite element. Table 3 also shows that the FEM results converge to the SEM results as the number of finite elements is increased to more than 100×100. Figure 9 shows the lowest six mode shapes of the F-F-F-C square plate (Example 3) obtained by the present SEM.Table 3
Comparison of the natural frequencies (Hz) of a square plate with free-free-free-clamped (F-F-F-C) edge support obtained by ANSYS [21], FEM [22], and SEM.
Modes ANSYS FEM SEM n E = 100 × 100 10 × 10 50 × 50 100 × 100 1 n D = 40400 330 7650 30300 1204 1 0.851 0.851 0.851 0.851 0.851 2 2.059 2.060 2.059 2.059 2.059 3 5.194 5.206 5.194 5.193 5.193 4 6.662 6.655 6.662 6.662 6.663 5 7.527 7.530 7.526 7.526 7.524 6 13.19 13.13 13.18 13.18 13.18 7 15.04 15.09 15.04 15.04 15.04 8 15.67 15.68 15.67 15.66 15.67 9 17.39 17.41 17.39 17.39 17.38 10 22.70 22.47 22.69 22.70 22.70 Note:nE: total number of finite elements; nD: total number of degrees of freedom.Figure 9
Lowest six mode shapes of a square plate with free-free-free-clamped (F-F-F-C) edge support obtained by SEM, Example3.Figure10 compares the vibration responses of the S-S-S-S square plate (Example 1) for the first 0.3 s obtained by the modal analysis method (denoted as “MAM”), the FEM [22], and the SEM. An impulse is applied to the middle of the plate, and the dynamic responses are predicted at x = 0.1 m and 0.2 m on the x-axis. The analytical solution by the MAM is given by [24](113)wx,y,t=∑m=1∞∑n=1∞Wmnx,yηmnt,where(114)Wmnx,y=2Lρsin⁡mπxLsin⁡nπxLηmnt=2ωmnLρsin⁡mπ2sin⁡nπ2sin⁡ωmnt,where ωmn are the natural frequencies provided by (112). Figure 10 shows that the analytical solution by the MAM is very close to that by the SEM and that the FEM results approach the result by the SEM as the number of finite elements is increased to more than 50×50.Figure 10
Comparison of the impulse-induced vibration responses atx = 0.1 m and 0.2 m on the x-axis of a simply supported square plate predicted by the present SEM, the modal analysis method (MAM), and the FEM(n), wheren is the number of finite elements used in the analysis.
(a)
Atx=0.1 m (b)
Atx=0.2 mFigure11 compares the transient dynamic responses at x = 0.1 m on the x-axis obtained by the MAM [24], the FEM [22], and the present SEM. Similar to Figure 10, the transient dynamic response by the MAM is very close to the SEM results, and the FEM results approach the SEM results as the number of finite elements is increased to more than 100×100.Figure 11
Comparison of the impulse-induced transient dynamic responses atx = 0.1 m on the x-axis of a simply supported square plate obtained by the present SEM, the modal analysis method (MAM), and the FEM(n), wheren is the number of finite elements used in the analysis.The computation times (CPU times) for the SEM and FEM are compared in Table4. To measure the CPU times, we used a standard desktop PC equipped with two sockets of Intel Xeon E5-2630 v3 processors and 320 GB of DDR4 RAM memory clocked at 2133 MHz. Table 4 shows that the CPU time for the present SEM is much smaller than that required to obtain the same accuracy level using the FEM.Table 4
Comparison of the computation times (CPU times) required to obtain the dynamic responses shown in Figure11.
Methods SEM FEM n E = 1 10 × 10 100 × 100 CPU time (hr) 1.363 0.002 42.57 Note:nE: total number of finite elements used in the analysis.Based on aforementioned observations, we conclude that the present spectral element model has the capability to provide extremely accurate natural frequencies and dynamic responses very efficiently.


## 4. Conclusion

We propose a new frequency domain spectral element model for finite rectangular plate elements with arbitrary boundary conditions. The new spectral element model is developed by using the boundary splitting method and the spectral super element method in combination. The high solution accuracy and computational efficiency of the proposed new spectral element model are validated by comparison with exact theory, the standard FEM, and the ANSYS commercial finite element analysis package. The conclusions drawn from our results are as follows:(1)
The proposed new spectral element model can be applied to any finite rectangular plate element with arbitrary boundary conditions, whereas the spectral element model introduced in the authors’ previous work [5] can be applied only to finite rectangular plate elements whose four corner nodes are fixed. Furthermore, the proposed new spectral element model can be assembled in the x- and y-directions to represent a plate structure, whereas most existing spectral element models are valid only for plates with very specific boundary conditions such as Levy-type plates, infinite or semi-infinite plates, and plates in which assembly is allowed only in one direction.(2)
Through numerical studies, we show that the proposed spectral element model provides highly accurate solutions by using a relatively small number of finite elements.(3)
The proposed spectral element model is found to be computationally efficient compared to the standard FEM and ANSYS, because the proposed spectral element model has nodal DOFs only on four edges of a rectangular finite element to significantly reduce the total number of DOFs used in the spectral element analysis.



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*Source: 9475397-2016-12-26.xml*