Document Type : Research Article
Authors
Department of Mechanical Engineering, MMMUT, Gorakhpur273010, India
Abstract
Keywords
Assessment of RBFs Based Meshfree Method for the Vibration Response of FGM Rectangular Plate Using HSDT Model
M.C. Srivastva, J. Singh ^{*}
Department of Mechanical Engineering, MMMUT, Gorakhpur273010, India
KEYWORDS 

ABSTRACT 
Free vibration; FGM plate; Meshfree; RBF. 
Radial basis functions (RBFs) with modified radial distance are proposed for vibration analysis of functionally graded materials (FGM) rectangular plates. The displacement field with five variables higherorder shear deformation theory (HSDT) is considered. The governing differential equations (GDEs) and boundary conditions are obtained using Hamilton's principle. The governing differential equations formulations are solved via strongform solutions. The rectangular plates are analyzed in the framework of the RBFbased meshfree method. The novelty of the present modified method is to analyze the square and rectangular plates without changing the shape parameters. Here, the seventeen different RBFs are available in various literature to demonstrate the accuracy and efficiency of the present method in terms of the number of nodes and computational time. The results of several numerical examples have shown that the present modified RBFbased meshfree method can lead to much more accurate solutions. Computational times of different RBFs are also analyzed. 
Generally, models originating from practical applications in industry and engineering don't have an exact solution, or it is exorbitant to be implemented and actualized. Thus, depending upon the computational calculations, mainly numerical methods such intentions are unavoidable. The most significant achievements for solving governing differential equations in the simulation framework are the finite difference method, finite element method, and finite volume method, which relies on a mesh to construct the local approximation of functions. For more accuracy in these methods, we depend on quality mesh generation, and the time cost is very high. Recently, a novel numerical method known as the Meshfree method has drawn the attention of solving engineering problems by many researchers. The meshfree name method selfdefines the method in which no mesh is required and constructs a functional approximation based on scattered points without mesh connectivity. Recently detailed elucidation of various types of meshfree methods can be discussed by Chen et al. [1]. Meshfree methods formulations are developed under two categories which are known as strong formbased formulations, such as the radial basis collocation method Kansa [2],[3], and weak formbased formulations, such as the radial point interpolation method Wang and Liu [4]. The meshfree method based on the strong form formulation attracts many researchers due to its high accuracy and fast convergence rate, and it is also implemented easily. The time cost is also significantly reduced. In the previous 25 years, the strong form formulation of the meshfree method, which depends on the RBFs, has gotten an alluring response for solving partial differential equations (PDEs). The RBFbased meshfree method is truly meshfree in nature that can directly discretize GDEs of any order, along with their boundary conditions. The foundation thought of RBF interpolation was introduced by Hardy [5] to appraise the scattered data sets. After two decades, Kansa [3] pioneered the concept of solving PDEs utilized by multiquadric RBF. Franke [6] investigated the assessment of RBFs for scattered data interpolation in terms of time, cost, accuracy, and simplicity of usage. In the recent past, the RBFbased meshfree method on strong form formulation has been successfully adapted to problems of solid mechanics, especially in beams, plates, shells, and panels, and drew attention due to the absence of a mesh in the researcher's community. The meshfree method for the analysis of plates has been previously examined by numerous authors. The nonlinearity of plates on the von Kármán strain assumptions. Van Do and Lee [7] carried out a modified meshfree RPIM with a new RBF for the nonlinear bending response of the FGM plate using HSDT. Kumar et al [8] investigated the bending analysis of porous bidirectional FGM plates via meshfree methods. Liu and Gu [9] developed a local radial point interpolation method replacing the polynomial basis functions with radial basis functions. Liew et al. [10] introduced a review of the developments of elementfree or meshless methods and their applications for the analysis of laminated and FGM structures. Ferreira et al. [11] investigated the buckling and vibration response of laminated plates using the RBFbased meshfree method and FSDT. Rodrigues et al. [12] applied Murakami's zigzag theory with the RBFfinitedifference collocation method for the buckling, bending, and vibration analysis of composite plates. Zhang et al. [13] studied thermal and mechanical buckling responses of FGM plates using a local Kriging meshfree method. Liew et al. [14] studied the buckling and vibration response of plates by reproducing kernel particle approximate based meshfree method and firstorder shear deformation theory (FSDT).
The vibration response of FGM plates necessarily includes the solution of eigenvalue problems. The solution of eigenvalue problems utilizing 3D elasticity theories is challenging to get, especially when the material properties vary using the power law. Thus, 2D assumptionbased plate theories are developed and effectivelybeing utilized for the analysis of plates via analytical and numerical methods. Reddy and Cheng [15] studied 3D asymptotic theory in terms of the transfer matrix to investigate harmonic vibration analysis of the FGM plates. Vel and Batra [16] investigated the free and forced vibration of FGM plates using the power series expansion method. Zenkour [17] considered the effect of rotary inertia on the free vibration response of SS FGM thick plates. The free vibration analysis of the FGM plate using the collocation method was investigated by Ferreira et al. [18]. Uymaz and Aydogdu [19] utilized CPT for the 3D solution of the FGM plate using closedform formulation. Roque et al. [20] investigated the free vibration of the FGM plate using the RBFbased meshless method. Batra [21] applied the HSDT model for linearly elastic incompressible FGM plates. Fares et al. [22] introduced a refined ESL theory using the mixed variational approach for the deflection and free vibration analysis of FGM plates. Li et al. [23] studied a 3D elasticity solution for the vibration response of FGM plates using the Ritz method in uniformed, linear, and nonlinear types of temperature distribution. Malekzadeh [24] examined the 3D free vibration of the EFGM plate resting on the foundation, and the material properties vary exponentially through the thickness. Zhao et al. [25] implemented an elementfree kpRitz technique to investigate the free vibration of FGM plates. FSDT examines transverse shear strain and meshfree kernel functions used to approximate the 2D displacement fields. Atmane et al. [26] studied the free vibration of elastically supported FGM plate with new HSDT modeling the framework of Navier's method. Talha and Singh [27] examined the bending and vibration response of FGM plates via the FEM method. Giunta et al. [28] utilized the Chebyshev series to expand plate displacements, while the Ritz method determined the natural frequencies. Wu and Chiu [29] carried the RMVTbased meshless collocation and elementfree Galerkin technique for the quasi3D free vibration analysis of FGM plates. Zhu and Liew [30] examined the free vibration analysis of FGM plates using FSDT based on the local Kriging meshless technique. Neves et al. [31] studied the static and free vibration analysis of FGM plates via quasi3D hyperbolic shear deformation theory with an RBFbased meshless technique. Neves et al. [32] proposed to improve the accuracy by introducing a new sinusoidal HSDT for static bending, buckling, and free vibration response of the FGM. HosseiniHashemi e. al. [33] studied the free vibration of thick FGM plates based on exact 3D elasticity theory and closedform formulation. Xiang and Xing [34] introduced a new FSDT model with two independent variables for the free vibration of a rectangular plate. Chakravarty and Pradhan [35] used the RayleighRitz method to investigate the free vibration of FGM plates under various boundary conditions. Mahi et al. [36] introduced a new hyperbolic HSDT model to avoid utilizing the shear correction factor, which improved the efficiency of bending and free vibration response of isotropic, functionally graded, sandwich, and laminated composite. Su et al. [37] studied FSDT for the free vibration response of thick laminated FGM plate by using the modified Fourier–Ritz method. Chandra et al. [38] investigated vibration analysis of FGM plates using FSDT and a nearfield elemental radiator approach to find the radiated acoustic field. Sekkal et al. [39] examined buckling and free vibration analysis of the FGM plate utilizing a quasi3D solution. Shahsavari et al. [40] investigated the free vibration of three types of porosity distribution FGM plate by using quasi3D hyperbolic theory. Xing et al. [41] implemented multiquadric radial basis functions for the free vibration studies of thin FGM plates via the separationofvariable method. Zhao et al. [42] utilized a 3D exact solution for free vibration analysis of FGM plates under a porous medium with various boundary conditions. Parida and Mohanty [43] utilized HSDT for the free vibration of a skew FGM plate using a finite element approach.
The objective of this paper is to offer modified radial distance in different RBFs for the free vibration of FG rectangular plate by strong form formulation. To the best of the author's knowledge, the first time the RBFbased meshfree method has been used to analyze the free vibration response of the FG rectangular plate without changing the shape parameters. The influence of the grading index, foundation parameters, spantothickness ratio, and plate aspect ratio on the free vibration of rectangular FGM is discussed.
A porous FG rectangular plate of dimensions length a, breadth b, and thickness h in the Cartesian coordinate system (xyz) are shown in Figure 1. The modified powerlaw homogenization technique with porosity distribution effect for material properties is employed, which is formulated by Zhao et al., [42]

(1) 
where ‘n’ is the grading index and 'P' is the porosity volume fraction. E and ρ represent the effective material property Young's modulus and mass density respectively, subscripts m and c represent the metallic and ceramic constituents respectively, P is the porosity fraction (0 <P< 1) and P=0 means pure FG rectangular plate.
Fig. 1. The geometry of a rectangular FG plate with porosity in the rectangular coordinate system
The displacement field is a framework as an equivalent singlelayer approach with five variables higher order shear deformation theory which can be expressed by Kumar et al. [44].

(2) 
where , , , and are the five unknown displacement variables and is the algebraic transverse shear deformation function which is taken from Kumar et al. [44] and is the time derivative.
The stressstrain relation using generalized Hook's law with respect to the structural axis system (XYZ) can be expressed as Kumar et al. [44];

(3) 
is plane stressreduced stiffness and is given below [33].

(4) 
Hamilton's principle is utilized to formulate the GDEs of the FG plate, which is represented as;

(5) 
where KE = Kinetic energy, UE = Strain energy,
The kinetic energy of the FG plate can be expressed as.

(6) 
The strain energy of the FG plate can be expressed as.

(7) 

Hamilton's principle is used herein to derive the GDEs of the FG plate along with variationally admissible boundary conditions and collecting the coefficients of , the governing differential equations of the plate are obtained as:

(8) 


(9) 


(10) 



(11) 

(12) 

The axial force resultants, the bending moment resultants , the additional moment resultants related to the transverse shear function , and the transverse shear force resultants and are expressed as:

(13) 
And,

(14) 
The inertia terms of rectangular FG plate are expressed as:

(15) 
The GDEs are expressed in terms of displacement components represented as:

(16) 



(17) 



(18) 


(19) 


(20) 

The boundary conditions for an arbitrary edge with simply supported (SS) condition is as follows:

(21) 
The importance of RBFbased meshfree methods is that it discretizes the GDEs directly and produce a high rate of convergence with good accuracy. The firm establishment of RBF came to light in the early 1970s, when it was utilized for fitting scattered data [5]. And almost after two decades, Kansa [2][3] used RBF directly to solve partial differential equations that help to initiate the pillar of new methods by using different types of RBFs.
In the present analysis, we have considered nodes distribution uniformly for a 2D rectangular domain having IN interior nodes, BN boundary nodes, and N is the total nodes which are the sum of IN and BN which is shown in Singh and Shukla[45]. All computational calculations are carried out in MATLAB with a 2.7 GHz Corei7 processor.
Here, we have considered seventeen types of RBFs that are used in various kinds of computational engineering applications and are listed in Table 1. The considered GDEs with five unknown variables can be an interpolation in the form of the modified radial distance between nodes. In order to eliminate the singularity, an infinitesimally small value is added for zero radial distance. The radial distance between two nodes is denoted by ‘r’ and the modification of radial distance between the nodes for rectangular coordinates is done in such a way that the aspect ratio starts changing without changing the shape parameters. The expression used for square plate has been modified as for rectangular plate where a and b are the length and breadth of a rectangular plate.
Table 1 Various types of RBFs used in computation applications.
S.N. 
RBF 
1 
Polynomial, g1

2 
Gaussian quadratic, g2

3 
Thin Plate Spline, g3

4 
Wendland's C2, g4

5 
Wendland's C4, g5

6 
Wendland's C6, g6

7 
Hyperbolic secant, g7

8 
WuC2, g8

9 
WuC4, g9

10 
Hardy's Multiquadric, g10

S.N. 
RBF 
11 
Hardy's Inverse Quadric g11

12 
Multiquadratic, g12

13 
Inverse Multiquadratic, g13

14 
GeneralizedInverse Multiquadratic, g14

15 
Inverse quadratic, g15

16 
Multiquadratic Shu II, g16

17 
Inverse Multiquadratics, g17

'k' is the shape parameter that is responsible for the accurate numerical solution and stability of the method in the computational domain. It is additionally reported that stability and accuracy both simultaneously can't be ensured.
All the five variables of Equation (2) have been discretized from seventeen RBFs for nodes 1: N, as:

(22) 
where, is unknown coefficients of unknown variables.
In the eigenvalue problems, the objective is to obtain eigenvalues (λ) and corresponding eigenvectors . The eigenvalue problem can be expressed as:

(23) 
Equation (23) is solved by standard eigen solvers of computational software to obtain eigenvalues and eigenvectors.

(24) 

(25) 



(26) 
here, [K]_{I} represents the stiffness matrix for interior points resulting from LHS of Eq. (25).
The boundary conditions can be discretized in a similar fashion. For example, simply supported boundary condition at the edge x=0 is discretized and finally expressed as:

(27) 
where

(28) 

while discretizing the boundary, corner nodes are considered only once.
The unknown coefficients {δ} are calculated from equation (24) obtained, and finally, using equations (22), u_{0}, v_{0}, w_{0}, and at desired locations are obtained. Using equation (2), the displacement components, and using equation (3), the stress components are obtained.
In the following, we demonstrate the implementation of seventeen types of RBFbased meshfree methods for free vibration response of rectangular FG plate. The stability and performance of the proposed modified RBFsbased meshfree method have been checked by convergence and time study. A simply supported FG rectangular plate is considered throughout the study. Three types of FG rectangular plates, FG1, FG2, and FG3, are considered, and the material property is described in Table 2.
Convergence study
In this section, we examine the impact of the number of nodes on the normalized natural frequency of the rectangular FG1 plate. The plate geometry is defined by uniformly distributed nodes. The results related to the convergence study are explored in Table 3. It is noticed that the present solution obtained by the proposed modified RBFs converged well and is also in good agreement with the result presented in the literature by the 3D HSDT solution of Jin et al. [46]. It can also be noted that all the RBFs produced good results, and the convergence rate is less than 1% after 14x14 nodes. So, based on the convergence study, a 15×15 node is used throughout the study.
Fig. 2. Comparison study of the natural frequency with
the computational speed of different RBF
Table 2 Mechanical properties of metallic and ceramic materials considered.
Types of Functionally graded material 
Properties 


E 



FG1 
Metal (Al) 
70 
2702 
0.3 

(Al2O3) 
380 
3800 
0.3 

FG2 
(Al) 
70 
2702 
0.3 

(ZrO2) 
200 
5700 
0.3 

FG3 
(Ti6Al4V) 
105.7 
4429 
0.298 

(Aluminum oxide) 
320.2 
3750 
0.26 

Table 3. Convergence study of nondimensional frequency parameters of rectangular plates (a/h=10, n=1, b/a=2).
RBFs 
Number of nodes 
Jin et al., [46] 

11x11 
12x12 
13x13 
14x14 
15x15 
16x16 

g1 
0.05494 
0.05498 
0.05498 
0.05493 
0.05494 
0.05494 
0.055 
g2 
0.05493 
0.05482 
0.05475 
0.05470 
0.05471 
0.05469 
0.055 
g3 
0.05524 
0.05502 
0.05491 
0.05488 
0.05486 
0.05486 
0.055 
g4 
0.05494 
0.05498 
0.05498 
0.05493 
0.05494 
0.05494 
0.055 
g5 
0.0549 
0.0549 
0.0549 
0.0549 
0.0549 
0.0549 
0.055 
g6 
0.0550 
0.0547 
0.0548 
0.0548 
0.0549 
0.0549 
0.055 
g7 
0.05501 
0.05493 
0.05503 
0.05499 
0.05490 
0.05486 
0.055 
g8 
0.05455 
0.05468 
0.05476 
0.05479 
0.05482 
0.05483 
0.055 
g9 
0.05505 
0.05494 
0.05489 
0.05487 
0.05486 
0.05486 
0.055 
g10 
0.05476 
0.05478 
0.05474 
0.05474 
0.05473 
0.05472 
0.055 
g11 
0.05522 
0.05510 
0.05504 
0.05499 
0.05495 
0.05493 
0.055 
g12 
0.05554 
0.05532 
0.05512 
0.05503 
0.05496 
0.05492 
0.055 
g13 
0.05556 
0.05532 
0.05511 
0.05502 
0.05494 
0.05491 
0.055 
g14 
0.05447 
0.05547 
0.05515 
0.05508 
0.05496 
0.05494 
0.055 
g15 
0.05572 
0.05545 
0.05516 
0.05507 
0.05497 
0.05499 
0.055 
g16 
0.05481 
0.05480 
0.05478 
0.05478 
0.05476 
0.05479 
0.055 
g17 
0.05510 
0.05499 
0.05496 
0.05491 
0.05490 
0.05491 
0.055 
Table 4. Comparison of nondimensional frequency parameter with seventeen RBFs of SS FG2 plate with
different span to thickness ratio (a=b = 1, n=0.5).
RBFs 
Spantothickness ratio 
Aver diff % 

5 
10 
20 
50 
100 

Ref.[39] 
1.6149 
1.7504 
1.7902 
1.8017 
1.8034 
 
g1 
1.5832 
1.7393 
1.7837 
1.7965 
1.7983 
0.71 
g2 
1.5924 
1.7424 
1.7862 
1.7989 
1.8008 
0.47 
g3 
1.5826 
1.7395 
1.7834 
1.7959 
1.7977 
0.73 
g4 
1.5832 
1.7393 
1.7837 
1.7965 
1.7983 
0.71 
g5 
1.5804 
1.7389 
1.7842 
1.7974 
1.7993 
0.72 
g6 
1.5818 
1.7391 
1.7835 
1.7963 
1.7981 
0.73 
g7 
1.5772 
1.7390 
1.7849 
1.7975 
1.8152 
0.83 
g8 
1.5794 
1.7373 
1.7825 
1.7957 
1.7976 
0.81 
g9 
1.5818 
1.7390 
1.7835 
1.7963 
1.7963 
0.75 
g10 
1.5879 
1.7414 
1.7849 
1.7975 
1.7993 
0.59 
g11 
1.5842 
1.7414 
1.7861 
1.7992 
1.8011 
0.58 
g12 
1.5846 
1.7419 
1.7867 
1.7997 
1.8016 
0.55 
g13 
1.5837 
1.7411 
1.7859 
1.7990 
1.8008 
0.60 
g14 
1.5868 
1.7426 
1.7869 
1.7998 
1.8016 
0.51 
g15 
1.5853 
1.7420 
1.7867 
1.7997 
1.8015 
0.55 
g16 
1.5804 
1.7375 
1.7822 
1.7952 
1.7970 
0.81 
g17 
1.5945 
1.7397 
1.7847 
1.7977 
1.7996 
0.52 
Now compare all the RBFs in terms of effectiveness (CPU time) and accuracy, which is shown in Fig. 2. It can be noticed that RBFs g8 and g9 consume more time and RBFs g2, g10, g11, g12, g13, g14, g15, g16, and g17 take less time to execute the computational calculation. RBFs g4, g5, g11, g12, g13, g14 and g15 predict closer results with the 3D HSDT solution of Jin et al., [46]. It can be clearly noticed that RBFs g11, g12, g13, g14, and g15 give closer results to Jin et al. [46] with less time as compared to other RBFs. Table 4 represents the effect of the spantothickness ratio for seventeen RBFs. The grading index is taken as 0.5 with a=b. It can be observed that all the RBFs predict less than 1 % average difference and show good agreement with the 3D HSDT result by Sekkal et al. It can also be observed that by increasing spantothickness ratios, normalized frequency increases, and after a/h=50, the effect is negligible.
To further examine the effectiveness of the proposed modified RBFs for rectangular FG plate, various parametric study is carried out. Table 5 represents the comparison study of seventeen RBFs with various grading indexes. The spantothickness ratio is taken as 10 with an aspect ratio =0.5. It can be seen that all the RBFs predict good results and are in agreement with the existing 3D results by Jin et al. [46]. Results are also compared with 2D results of [47], and it is found that the present results are closer to 3D results, which shows the accuracy of the present solution methodology. It can also be observed that g14 and g15 results are more accurate. The normalized frequency decreases by increasing the grading index.
Figure 3 represents nondimensional frequency parameter with various porosity index for FG1 rectangular plate with a=20h, n=1 and a=0.5b. It can be noticed that RBFs g4, g6, g7, g8, g9, g11, g12, g13, g14, 15, and g17 predict closer to the HSDT result published by Rezaei et al. [48]. It can also be seen that normalized frequency parameters decrease with an increase in the porosity index. Figure 4 represents the comparison study of seventeen RBFs for rectangular FG plates with various aspect ratios. The grading index is taken as 1 with a spantothickness ratio =10. It can be seen that all the RBF results predict good agreement with the existing 3D HSDT solution of Jin et al. [46] and the 2D HSDT result by Thai and Choi [47]. It can be noticed that RBFs g1, g4, g11, g12, g13, g14, and g15 predict closer results as compared to other RBFs, and by increasing the aspect ratio, normalized natural frequency increases.
Fig. 3. Comparison study of nondimensional frequency parameter of FG1 rectangular plates
with various porosity index (a/h=20, n=1.a/b=0.5)
Fig. 4. Comparison study of nondimensional frequency parameters of FG1 rectangular
plates (a/h=10, n=1
Table 5. Comparison study of nondimensional frequency parameter of
FG1 rectangular plate. (a/h=10, a/b=0.5).
Methods 
Grading index 
Aver. diff% 
Aver. diff% 

0 
1 
2 
5 

Ref.[46] 3D 
0.0719 
0.0550 
0.0499 
0.0471 
 
 
Ref.[47] 2D 
0.0717 
0.0548 
0.0498 
0.0470 
 
 
g1 
0.0717 
0.0549 
0.0499 
0.0471 
0.18 
0.10 
g2 
0.0717 
0.0547 
0.0496 
0.0468 
0.24 
0.51 
g3 
0.0717 
0.0549 
0.0498 
0.0470 
0.04 
0.24 
g4 
0.0717 
0.0549 
0.0499 
0.0471 
0.18 
0.10 
g5 
0.0718 
0.0549 
0.0498 
0.0469 
0.04 
0.24 
g6 
0.0717 
0.0549 
0.0498 
0.0469 
0.00 
0.27 
g7 
0.0716 
0.0549 
0.0498 
0.0470 
0.02 
0.25 
g8 
0.0717 
0.0548 
0.0498 
0.0469 
0.03 
0.31 
g9 
0.0717 
0.0549 
0.0498 
0.0470 
0.04 
0.24 
g10 
0.0715 
0.0547 
0.0497 
0.0469 
0.19 
0.47 
g11 
0.0718 
0.0550 
0.0499 
0.0470 
0.16 
0.11 
g12 
0.0718 
0.0550 
0.0499 
0.0470 
0.17 
0.11 
g13 
0.0718 
0.0549 
0.0499 
0.0470 
0.16 
0.12 
g14 
0.0718 
0.0550 
0.0499 
0.0471 
0.22 
0.05 
g15 
0.0718 
0.0550 
0.0499 
0.0471 
0.23 
0.05 
g16 
0.0716 
0.0548 
0.0497 
0.0469 
0.15 
0.42 
g17 
0.0718 
0.0549 
0.0498 
0.0470 
0.09 
0.18 
Table 6 shows the results for all 10 modes of nondimensional frequency parameter of FGM3 plate (a=b = 1) for different values of the grading index n = 0, 0.5, 1, 2, 5, 8, 10 and metal with respect to span to thickness ratio. It can be observed that the value of increase by increasing the mode. The first three mode shapes and their normalized frequency response of SS FGM1 rectangular plate (a/b=0.5, a/h=10, g12, n=1, P=0) are examined and shown in Fig. 5. It seems that the figure with the higher modes plates buckles in different shapes.
Mode 1 =2.798
Mode 2= 4.422
Mode 3=7.035
Fig. 5. Mode shape of the normalized frequency response
of FGM1 rectangular plate.
Table 6. Effect of grading index 'n' with spantothickness ratio on nondimensional frequency parameter
of FGM3 square plates for mode 10 (g10)
a/h 
Modes 
Grading Index' n' 

0 
0.5 
1 
2 
4 
8 
10 
Metal 

4 
1 
4.858 
4.051 
3.673 
3.346 
3.115 
2.952 
2.909 
2.587 
2 
7.926 
6.784 
6.176 
5.537 
5.000 
4.628 
4.541 
4.133 

3 
7.926 
6.784 
6.176 
5.537 
5.000 
4.628 
4.541 
4.133 

4 
9.605 
8.107 
7.356 
6.644 
6.090 
5.721 
5.636 
5.086 

5 
9.614 
8.112 
7.361 
6.648 
6.093 
5.724 
5.639 
5.090 

6 
11.154 
9.568 
8.698 
7.690 
6.810 
6.279 
6.181 
5.835 

7 
11.154 
9.583 
8.699 
7.690 
6.810 
6.279 
6.181 
5.843 

8 
11.167 
9.583 
8.699 
7.692 
6.814 
6.284 
6.185 
5.843 

9 
11.167 
9.596 
8.708 
7.692 
6.814 
6.284 
6.185 
5.848 

10 
11.190 
9.596 
8.708 
7.740 
6.864 
6.332 
6.232 
5.848 

10 
1 
5.705 
4.722 
4.287 
3.944 
3.735 
3.568 
3.515 
3.056 
2 
13.575 
11.260 
10.217 
9.369 
8.822 
8.406 
8.282 
7.257 

3 
13.576 
11.260 
10.218 
9.370 
8.822 
8.407 
8.282 
7.257 

4 
19.814 
16.978 
15.476 
13.890 
12.542 
11.591 
11.368 
10.334 

5 
19.814 
16.978 
15.476 
13.890 
12.542 
11.591 
11.368 
10.334 

6 
20.734 
17.235 
15.633 
14.294 
13.392 
12.732 
12.546 
11.065 

7 
25.136 
20.923 
18.974 
17.317 
16.176 
15.357 
15.133 
13.399 

8 
25.192 
20.969 
19.016 
17.356 
16.215 
15.394 
15.169 
13.429 

9 
27.975 
23.967 
21.842 
19.599 
17.697 
16.359 
16.045 
14.588 

10 
31.293 
26.104 
23.670 
21.552 
20.051 
18.995 
18.719 
16.657 

30 
1 
5.878 
4.858 
4.413 
4.070 
3.868 
3.701 
3.645 
3.153 
2 
14.606 
12.075 
10.966 
10.108 
9.600 
9.184 
9.045 
7.832 

3 
14.607 
12.075 
10.966 
10.108 
9.600 
9.184 
9.045 
7.832 

4 
23.239 
19.216 
17.449 
16.078 
15.258 
14.593 
14.372 
12.457 

5 
28.933 
23.928 
21.725 
20.011 
18.981 
18.150 
17.876 
15.506 

6 
28.971 
23.960 
21.755 
20.039 
19.009 
18.176 
17.902 
15.527 

7 
37.386 
30.928 
28.080 
25.855 
24.508 
23.427 
23.074 
20.032 

8 
37.386 
30.928 
28.080 
25.855 
24.508 
23.427 
23.074 
20.032 

9 
48.730 
40.324 
36.606 
33.686 
31.902 
30.484 
30.026 
26.102 

10 
48.730 
40.324 
36.606 
33.686 
31.902 
30.484 
30.026 
26.102 

50 
1 
5.893 
4.870 
4.423 
4.080 
3.878 
3.712 
3.656 
3.161 
2 
14.697 
12.146 
11.031 
10.174 
9.669 
9.254 
9.113 
7.882 

3 
14.697 
12.146 
11.031 
10.174 
9.669 
9.254 
9.113 
7.882 

4 
23.468 
19.397 
17.615 
16.243 
15.434 
14.769 
14.545 
12.585 

5 
29.291 
24.210 
21.985 
20.270 
19.256 
18.426 
18.147 
15.706 

6 
29.326 
24.239 
22.012 
20.295 
19.281 
18.449 
18.169 
15.725 

7 
37.982 
31.398 
28.512 
26.284 
24.963 
23.884 
23.521 
20.365 

8 
37.982 
31.398 
28.512 
26.284 
24.963 
23.884 
23.521 
20.365 

9 
49.737 
41.118 
37.338 
34.411 
32.672 
31.255 
30.781 
26.664 

10 
49.737 
41.118 
37.338 
34.411 
32.672 
31.255 
30.781 
26.664 

100 
1 
5.898 
4.874 
4.427 
4.084 
3.883 
3.717 
3.660 
3.164 
2 
14.735 
12.176 
11.059 
10.201 
9.699 
9.283 
9.142 
7.904 

3 
14.735 
12.176 
11.059 
10.201 
9.699 
9.283 
9.142 
7.904 

4 
23.566 
19.474 
17.687 
16.314 
15.510 
14.845 
14.619 
12.640 

5 
29.446 
24.332 
22.097 
20.381 
19.376 
18.546 
18.263 
15.793 

6 
29.478 
24.360 
22.123 
20.406 
19.399 
18.567 
18.285 
15.810 

7 
38.239 
31.600 
28.699 
26.470 
25.162 
24.083 
23.717 
20.509 

8 
38.239 
31.600 
28.699 
26.470 
25.162 
24.083 
23.717 
20.509 

9 
50.176 
41.464 
37.657 
34.729 
33.010 
31.595 
31.114 
26.909 

10 
50.176 
41.465 
37.657 
34.729 
33.010 
31.595 
31.114 
26.909 
In the present analysis, seventeen radial basis functions have been compared for normalized frequency parameters obtained from five variables HSDT of rectangular FG plate. Hamilton’s principle is used to derive a strong form governing differential equations based on the displacement fields. The governing differential equations and boundary conditions are discretized in simultaneous equations using the generalized radial basis function. Modified radial distancebased RBF methods results are in good agreement with various examples available in the literature. The computational speed of RBFs g2, g10, g11, g12, g13, g14, g15, g16, and g17 is good as compared to other RBFs taken here. It also concluded that results produced by g14 g15 are closer to 3D results.
Through the implementation of numerical model results, it is concluded that
Finally, it is up to the research society to sensibly select the RBF that offers more accurate results with less computational cost.
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript
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