# Natural Frequencies of Shear Deformable Plates by Polyharmonic Splines

Authors: A. J. M. Ferreira; A. M. Zenkour
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We use the third-order shear deformation theory and a collocation technique with polyharmonic splines to predict natural frequencies of moderately thick isotropic plates. The natural frequencies of vibration are computed for various plates and compared with some available published results. Through numerical experiments, the capability and efficiency of the present method for eigenvalue problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.


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## Body

## 1. Introduction

Radial basis functions (RBFs) have recently proved to be an excellent technique for interpolating data and functions. A radial basis functionϕ(∥x-xj∥) can be considered a spline that depends on the Euclidean distance between distinct data centers xj,j=1,2,…,N∈ℝn, also called nodal or collocation points. Not only RBFs are adequate to scattered data approximation and in general to interpolation theory, as proposed by Hardy [1], but also excellent for solving partial differential equations (PDEs) as first introduced by Kansa [2].Kansa proposed an unsymmetric RBF collocation method based upon multiquadric interpolation functions, in which the shape parameter is considered to be variable across the problem domain. The distribution of the shape parameter is obtained by an optimization approach, in which the value of the shape parameter is assumed to be proportional to the curvature of the unknown solution of the original partial differential equation. In this way, it is possible to reduce the condition number of the matrix at the expense of implementing an additional iterative algorithm. In the present work, we will implement the unsymmetric global collocation method in a form that is independent of this shape parameter, based on unshifted polyharmonic splines. In some respect this can be seen as a more stable form than multiquadrics.Structures composed of laminated materials are among the most important structures used in modern engineering and, especially, in the aerospace industry. Such lightweight structures are also being increasingly used in civil, mechanical, and transportation engineering applications. The rapid increase of the industrial use of these structures has necessitated the development of new analytical and numerical tools that are suitable for the analysis and study of the mechanical behavior of such structures. The behavior of structures composed of advanced composite materials is considerably more complicated than for isotropic ones. The strong influences of anisotropy, the transverse stresses through the thickness of a laminate, and the stress distributions at interfaces are among the most important factors that affect the general performance of such structures. The use of shear deformation theories has been the topic of intensive research, as in [7–19], among many others.The analysis of laminated plates by finite element methods is now considerably established. The use of alternative methods such as the meshless methods based on radial basis functions is attractive due to the absence of a mesh and the ease of collocation methods. The use of radial basis function for the analysis of structures and materials has been previously studied by numerous authors [20–31]. More recently the authors have applied RBFs to the static deformations of composite beams and plates [32–35]. Recent meshless techniques were discussed in [36–42].In this paper the use of radial basis functions to isotropic and composite plates using a third-order shear deformation theory is investigated. The quality of the present method in predicting free vibrations of isotropic and laminated composite plates is compared and discussed with other methods in some numerical examples.


## 2. The Radial Basis Function Method

### 2.1. The Eigenproblem

Radial basis functions (RBF) approximations are grid-free numerical schemes that can exploit accurate representations of the boundary, are easy to implement, and can be spectrally accurate [43, 44].In this section the formulation of a global unsymmetrical collocation RBF-based method to compute eigenvalues of elliptic operators is presented.Consider a linear elliptic partial differential operatorL and a bounded region Ω in ℝn with some boundary ∂Ω. The eigenproblem looks for eigenvalues (λ) and eigenvectors (u) that satisfy
(1)Lu+λu=0inΩ,LBu=0on∂Ω,
where LB is a linear boundary operator. The eigenproblem of (1) is replaced by a finite-dimensional eigenvalue problem, based on RBF approximations.The operatorL is approximated by a matrix that incorporates the boundary conditions and then solve the eigenvalues and eigenvectors of this matrix by standard techniques.


### 2.2. Radial Basis Functions

The radial basis function(ϕ) approximation of a function (u) is given by
(2)u~(x)=∑i=1N‍αiϕ(∥x-yi∥2),x∈ℝn,
where yi,i=1,…,N is a finite set of distinct points (centers) in ℝn. The coefficients αi are calculated so that u~ satisfies some boundary conditions. Although many other functions could be used, as illustrated by several authors [1, 2, 45], here we use unshifted polyharmonic splines in the form
(3)ϕ(r)=r2m+1,m∈Z,
where the Euclidean distance r is real and nonnegative.ConsideringN distinct interpolations, and knowing u(xj),j=1,2,…,N, we find αi by the solution of a N×N linear system Aα_=u, where A=[ϕ(∥x-yi∥2)]N×N, α_=[α1,α2,…,αN]T and u = [u(x1),u(x2),…,u(xN)]T. The RBF interpolation matrix A is positive definite for some RBFs [46] but in general provides ill-conditioned systems. This ill-conditioning problem is due to full matrices produced by the collocation method and worsens with the increase of number of nodes. However it was shown by Schaback that the more ill-conditioned the problem the better the solution, until a flatness of the function occurs. In our numerical experiments this ill-conditioning did not produce any inconvenience to quality of solution.


### 2.3. Solution of the Eigenproblem

We follow a simple scheme for the solution of the eigenproblem (1). We consider NI nodes in the interior of the domain and NB nodes on the boundary, with N=NI+NB.We denote interpolation points byxi∈Ω,i=1,…,NI and xi∈∂Ω,i=NI+1,…,N. For the interior points we have that
(4)∑i=1N‍αiLϕ(∥x-yi∥2)=λu~(xj),j=1,2,…,NI
or
(5)LIα_=λu~I,
where
(6)LI=[Lϕ(∥x-yi∥2)]NI×N,For the boundary conditions we have(7)∑i=1N‍αiLBϕ(∥x-yi∥2)=0,j=NI+1,…,N
or
(8)Bα_=0.Therefore we can write a finite-dimensional problem as a generalized eigenvalue problem(9)[LIB]α_=λ[AI0]α_,
where
(10)AI=ϕ[(∥xNI-yj∥2)]NI×N,BI=LBϕ[(∥xNI+1-yj∥2)]NB×N.


## 2.1. The Eigenproblem

Radial basis functions (RBF) approximations are grid-free numerical schemes that can exploit accurate representations of the boundary, are easy to implement, and can be spectrally accurate [43, 44].In this section the formulation of a global unsymmetrical collocation RBF-based method to compute eigenvalues of elliptic operators is presented.Consider a linear elliptic partial differential operatorL and a bounded region Ω in ℝn with some boundary ∂Ω. The eigenproblem looks for eigenvalues (λ) and eigenvectors (u) that satisfy
(1)Lu+λu=0inΩ,LBu=0on∂Ω,
where LB is a linear boundary operator. The eigenproblem of (1) is replaced by a finite-dimensional eigenvalue problem, based on RBF approximations.The operatorL is approximated by a matrix that incorporates the boundary conditions and then solve the eigenvalues and eigenvectors of this matrix by standard techniques.


## 2.2. Radial Basis Functions

The radial basis function(ϕ) approximation of a function (u) is given by
(2)u~(x)=∑i=1N‍αiϕ(∥x-yi∥2),x∈ℝn,
where yi,i=1,…,N is a finite set of distinct points (centers) in ℝn. The coefficients αi are calculated so that u~ satisfies some boundary conditions. Although many other functions could be used, as illustrated by several authors [1, 2, 45], here we use unshifted polyharmonic splines in the form
(3)ϕ(r)=r2m+1,m∈Z,
where the Euclidean distance r is real and nonnegative.ConsideringN distinct interpolations, and knowing u(xj),j=1,2,…,N, we find αi by the solution of a N×N linear system Aα_=u, where A=[ϕ(∥x-yi∥2)]N×N, α_=[α1,α2,…,αN]T and u = [u(x1),u(x2),…,u(xN)]T. The RBF interpolation matrix A is positive definite for some RBFs [46] but in general provides ill-conditioned systems. This ill-conditioning problem is due to full matrices produced by the collocation method and worsens with the increase of number of nodes. However it was shown by Schaback that the more ill-conditioned the problem the better the solution, until a flatness of the function occurs. In our numerical experiments this ill-conditioning did not produce any inconvenience to quality of solution.


## 2.3. Solution of the Eigenproblem

We follow a simple scheme for the solution of the eigenproblem (1). We consider NI nodes in the interior of the domain and NB nodes on the boundary, with N=NI+NB.We denote interpolation points byxi∈Ω,i=1,…,NI and xi∈∂Ω,i=NI+1,…,N. For the interior points we have that
(4)∑i=1N‍αiLϕ(∥x-yi∥2)=λu~(xj),j=1,2,…,NI
or
(5)LIα_=λu~I,
where
(6)LI=[Lϕ(∥x-yi∥2)]NI×N,For the boundary conditions we have(7)∑i=1N‍αiLBϕ(∥x-yi∥2)=0,j=NI+1,…,N
or
(8)Bα_=0.Therefore we can write a finite-dimensional problem as a generalized eigenvalue problem(9)[LIB]α_=λ[AI0]α_,
where
(10)AI=ϕ[(∥xNI-yj∥2)]NI×N,BI=LBϕ[(∥xNI+1-yj∥2)]NB×N.


## 3. Third-Order Theory

The displacement field for the third-order shear deformation theory of Hardy [1] is obtained as
(11)u(x,y,z,t)=u0(x,y,t)+zθx(x,y,t)-43h2z3(θx+∂w∂x),v(x,y,z,t)=v0(x,y,t)+zθy(x,y,t)-43h2z3(θy+∂w∂y),w(x,y,z,t)=w0(x,y,t),
where u and v are the in-plane displacements at any point (x,y,z), u0,v0 denote the in-plane displacement of the point (x,y,0) on the midplane, w is the deflection, and θx and θy are the rotations of the normals to the midplane about the y- and x-axes, respectively.The strain-displacement relationships are given as(12){ϵxxϵyyγxyγxzγyz}={∂u∂x∂v∂y∂u∂y+∂v∂x∂u∂z+∂w∂x∂v∂z+∂w∂y}.Therefore strains can be expressed as(13){ϵxxϵyyγxy}={ϵxx(0)ϵyy(0)γxy(0)}+z{ϵxx(1)ϵyy(1)γxy(1)}+z3{ϵxx(3)ϵyy(3)γxy(3)},{γxzγyz}={γxz(0)γyz(0)}+z2{γxz(2)γyz(2)},
where
(14){ϵxx(0)ϵyy(0)γxy(0)}={∂u0∂x∂v0∂y∂u0∂y+∂v0∂x},{ϵxx(1)ϵyy(1)γxy(1)}={∂θx∂x∂θy∂y∂θx∂y+∂θy∂x},{ϵxx(3)ϵyy(3)γxy(3)}=-c1{∂θx∂x+∂2w0∂x2∂θy∂y+∂2w0∂y2∂θx∂y+∂θy∂x+2∂2w0∂x∂y},{γxz(0)γyz(0)}={∂w0∂x+θx∂w0∂y+θy};{γxz(2)γyz(2)}=-c2{∂w0∂x+θx∂w0∂y+θy}
and c1=4/3h2,c2=3c1.A laminate can be manufactured from orthotropic layers (or plies) of preimpregnated unidirectional fibrous composite materials. Neglectingσz for each layer, the stress-strain relations in the fiber local coordinate system can be expressed as
(15){σ1σ2τ12τ23τ31}=[Q11Q12000Q12Q2200000Q3300000Q4400000Q55]{ε1ε2γ12γ23γ31},
where subscripts 1 and 2 are, respectively, the fiber and the normal to fiber in-plane directions, 3 is the direction normal to the plate, and the reduced stiffness components, Qij, are given by
(16)Q11=E11-ν12ν21Q22=E21-ν12ν21Q12=ν21Q11Q33=G12Q44=G23Q55=G31ν21=ν12E2E1,
in which E1,E2,ν12,G12,G23, and G31 are materials properties of the lamina.By performing adequate coordinate transformation, the stress-strain relations in the globalx-y-z coordinate system can be obtained as
(17){σxxσyyτxyτyzτzx}=[Q¯11Q¯12Q¯1600Q¯12Q¯22Q¯2600Q¯16Q¯26Q¯6600000Q¯44Q¯45000Q¯45Q¯55]{εxxεyyγxyγyzγzx}.The third-order theory of Hardy [15, 16] satisfies zero transverse shear stresses on the bounding planes.The equations of motion of the third-order theory are derived from the principle of virtual displacements:(3)∂Nxx∂x+∂Nxy∂y=I0∂2u0∂t2+J1∂2θx∂t2-c1I3∂2∂t2(∂w0∂x),∂Nxy∂x+∂Nyy∂y=I0∂2v0∂t2+J1∂2θy∂t2-c1I3∂2∂t2(∂w0∂y),∂Q¯x∂x+∂Q¯y∂y+c1(∂2Pxx∂x2+2∂2Pxy∂x∂y+∂2Pyy∂y2)+q=I0∂2w0∂t2-c1I6∂2∂t2(∂2w0∂x2+∂2w0∂y2)+c1[I3∂2∂t2(∂u0∂x+∂v0∂y)+J4∂2∂t2(∂θx∂x+∂θy∂y)],∂M¯xx∂x+∂M¯xy∂y-Q¯x=∂2∂t2(J1u0+K2θx-c1J4∂w0∂x),∂M¯xy∂x+∂M¯yy∂y-Q¯y=∂2∂t2(J1v0+K2θy-c1J4∂w0∂y),
where q is the external distributed load and with
(19)M¯αβ=Mαβ-c1Pαβ;Q¯α=Qα-c2Rα,Ii=∑k=1N‍∫kk+1‍ρ(k)(z)idz,(i=0,1,2,…,6),==========Ji=Ii-c1Ii+2(i=1,4),K2=I2-2c1I4+c12I6,c1=43h2,c2=4h2=3c1,
where α,β take the symbols x,y. The resultants (Nxx,Nyy,Nxy) denote the in-plane force resultants, (Mxx,Myy,Mxy) the moment resultants, (Qx,Qy) the shear resultants and (Pxx,Pyy,Pxy) and (Rx,Ry) denote the higher-order stress resultants:
(20){NαβMαβPαβ}=∫-h/2h/2‍σαβ{1zz3}dz,{QαRα}=∫-h/2h/2‍σαz{1z2}dz,
where α,β take the symbols x,y.If needed, the first-order shear deformation theory equations are readily obtained from the third-order equations, just by puttingc1=0.


## 4. Numerical Examples

The eigenvalues presented in this problem are expressed in terms of the nondimensional frequency parameterλ=(ωb2/π2)(ρt/D)1/2. Unlike first-order shear deformation theories the present approach does not incorporate shear correction factors in the constitutive matrix. The Poisson ratio, ν, is taken to be 0.3 throughout the present problem. Figure 1 illustrates the grid scheme for 11×11 points.Figure 1
1
1
×
11 regular grid for discretization of the square laminated plate.In Tables1, 2, 3, 4, 5, and 6 we compare the present results with those of Liew et al. [3]. We consider here simply supported (SSSS) and clamped (CCCC) boundary conditions in all edges, thin and thick plates and square and rectangular plates. The results obtained in Tables 1 to 6 show that the present method presents very close results to those proposed by Liew et al. [3]. However, for thin plates we need more points than in thick plates in order to get convergent and accurate frequencies. The reason for this behaviour lies in the worse conditioning of thin plates. In Liew et al. [3] thin plates also showed slower convergence.Table 1
Convergence study of frequency parameters,λ=(ωb2/π2)(ρt/D)1/2 for rectangular isotropic plates with SSSS boundaries, h/a=0.001,b/a=1.0.
Mode
N
=
21
N
=
25
N
=
30
Liew et al. [3]
1
2.0146
2.0089
2.0087
2.0000
2
5.0630
5.0343
5.0218
5.0000
3
5.0644
5.0371
5.0255
5.0000
4
8.1648
8.0717
8.0508
7.9999
5
9.8773
9.9908
10.0138
9.9998
6
10.2520
10.2462
10.0472
9.9998
7
12.7965
12.9815
13.0339
12.9997
8
12.8308
13.1027
13.0579
12.9997Table 2
Convergence study of frequency parameters,λ=(ωb2/π2)(ρt/D)1/2 for rectangular isotropic plates with SSSS boundaries, h/a=0.2,b/a=1.0.
Mode
N
=
11
N
=
15
N
=
21
Liew et al. [3]
1
1.7725
1.7700
1.7688
1.7659
2
3.8769
3.8717
3.8698
3.8576
3
3.8770
3.8717
3.8699
3.8576
4
5.6201
5.6063
5.6009
5.5729
5
6.5939
6.6093
6.6149
6.5809
6
6.6098
6.6127
6.6153
6.5809
7
8.0235
8.0092
8.0044
7.9470
8
8.0250
8.0099
8.0046
7.9470Table 3
Convergence study of frequency parameters,λ=(ωb2/π2)(ρt/D)1/2 for rectangular isotropic plates with SSSS boundaries, h/a=0.001,b/a=2.5.
Mode
N
=
21
N
=
25
N
=
30
Liew et al. [3]
1
1.2528
1.1294
1.1511
1.1600
2
1.7969
1.5886
1.6694
1.6400
3
3.5791
2.4060
2.4977
2.4400
4
4.1586
2.8928
3.7214
3.5600
5
4.1586
4.5265
4.4835
4.1600
6
6.1938
5.4430
5.1762
4.6400
7
6.2026
5.7588
7.0559
5.0000
8
6.2026
6.4153
7.5293
5.4399Table 4
Convergence study of frequency parameters,λ=(ωb2/π2)(ρt/D)1/2 for rectangular isotropic plates with SSSS boundaries, h/a=0.2,b/a=2.5.
Mode
N
=
11
N
=
15
N
=
21
Liew et al. [3]
1
1.0699
1.0722
1.0732
1.0741
2
1.4836
1.4789
1.4702
1.4768
3
2.1267
2.1114
2.1041
2.1059
4
2.9901
2.9429
2.9328
2.9145
5
3.3120
3.3224
3.3288
3.3191
6
3.6449
3.6403
3.6375
3.6306
7
4.0418
3.9265
3.9239
3.8576
8
4.1796
4.1508
4.1451
4.1281Table 5
Convergence study of frequency parameters,λ=(ωb2/π2)(ρt/D)1/2 for rectangular isotropic plates with CCCC boundaries, h/a=0.001,b/a=1.0.
Mode
N
=
15
N
=
21
N
=
25
Liew et al. [3]
1
3.9078
3.7202
3.6892
3.6460
2
8.5057
7.7031
7.5817
7.4362
3
8.5057
7.7032
7.5818
7.4362
4
14.7680
11.9358
11.4953
10.9643
5
15.8127
13.7424
13.5536
13.3315
6
17.1194
14.0909
13.7236
13.3947
7
25.6747
18.6394
17.7457
16.7173
8
25.6747
18.6396
17.7458
16.7173Table 6
Convergence study of frequency parameters,λ=(ωb2/π2)(ρt/D)1/2 for rectangular isotropic plates with CCCC boundaries, h/a=0.2,b/a=1.0.
Mode
N
=
11
N
=
15
N
=
21
Liew et al. [3]
1
2.5935
2.5956
2.5976
2.6803
2
4.5153
4.5295
4.5366
4.6744
3
4.5153
4.5295
4.5366
4.6744
4
6.0888
6.1042
6.1166
6.2748
5
6.9553
6.9909
7.0045
7.1481
6
7.0281
7.0730
7.0833
7.2466
7
8.2813
8.3096
8.3290
8.4803
8
8.2813
8.3096
8.3290
8.4803In Tables7 and 8 we analyse thin and thick plates for SSSS and CCCC boundaries. Present results are compared with classical theory of Leissa [4], first-order theory of Dawe [5], and orthogonal polynomials of Liew et al. [3]. The present method shows very accurate results when compared to Liew et al. [3] and FSDT of Dawe [5], for both SSSS and CCCC boundary conditions.Table 7
Comparison study of frequency parameters,λ=(ωb2/π2)(ρt/D)1/2 for square isotropic plates with SSSS boundaries.
t
/
b
Method of solution
Mode
(1, 1)
(1, 2)
(2, 1)
(2, 2)
(1, 3)
(3, 1)
0.01
Classical theory [4]
2.000
5.000
5.000
8.000
10.000
10.000
FSDT, Ritz method [5]
1.999
4.995
4.995
7.998
9.981
9.981
Orthogonal polynomials [3]
1.999
4.995
4.995
7.998
9.981
9.981
Present, grid=11×11
1.9695
5.1285
5.1286
7.9350
10.4481
10.4689
Present, grid=15×15
1.9950
5.0563
5.0563
8.0279
10.1396
10.1719
Present, grid=21×21
1.9997
5.0189
5.0189
8.0180
10.0261
10.0464
0.1
Classical theory [4]
2.000
5.000
5.000
8.000
10.000
10.000
3D analysis [6]
1.934
4.662
4.622
7.103
8.662
8.662
FSDT, Ritz method [5]
1.931
4.605
4.605
7.064
8.605
8.605
Orthogonal polynomials [3]
1.931
4.605
4.605
7.064
8.605
8.605
Present, grid=11×11
1.9387
4.6608
4.6608
7.1929
8.7181
8.7134
Present, grid=15×15
1.9386
4.6448
4.6448
7.1589
8.7070
8.7134
Present, grid=21×21
1.9376
4.6394
4.6394
7.1449
8.7091
8.7110Table 8
Comparison study of frequency parameters,λ=(ωb2/π2)(ρt/D)1/2 for square isotropic plates with CCCC boundaries, h/a=0.1.
Mode
N
=
7
N
=
11
N
=
15
CPT [4]
FSDT (Ritz) [5]
Liew et al. [3]
1, 1
3.2900
3.2478
3.2461
3.646
3.297
3.297
3.292
1, 2
6.2129
6.1577
6.1648
7.436
6.290
6.290
6.276
2, 1
6.2129
6.1577
6.1648
7.436
6.290
6.290
6.276
2, 2
8.8256
8.6192
8.6169
10.964
8.837
8.842
8.792
1, 3
9.8350
10.0951
10.1473
13.333
10.376
10.376
10.356
3, 1
9.9634
10.2240
10.2712
13.395
10.465
10.461
10.455


## 5. Conclusions

Radial basis functions are increasingly popular in the analysis of science and engineering problems.In this paper, we use the third-order shear deformation theory and a collocation technique with polyharmonic splines to predict natural frequencies of moderately thick isotropic plates. The natural frequencies of vibration are computed for various plates and compared with some available published results. The formulation for higher-order plates and their interpolation with polyharmonic splines has been presented.Through numerical experiments, the capability and efficiency of the present method for eigenvalue problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.When compared to other RBFs, polyharmonic splines show good stability and accuracy. However, proper modelling of plates should consider both a good numerical technique and an adequate shear deformation theory.



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