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, Gorakhpur-273010, India

1.     Introduction

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 self-defines 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 form-based formulations, such as the radial basis collocation method Kansa [2],[3], and weak form-based 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 RBF-based 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 RBF-based 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 mesh-free 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 element-free 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 RBF-based meshfree method and FSDT. Rodrigues et al. [12] applied Murakami's zig-zag theory with the RBF-finite-difference 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 first-order 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 assumption-based plate theories are developed and effectively-being 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 closed-form formulation. Roque et al. [20] investigated the free vibration of the FGM plate using the RBF-based 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 element-free kp-Ritz technique to investigate the free vibration of FGM plates. FSDT examines transverse shear strain and mesh-free 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 RMVT-based meshless collocation and element-free Galerkin technique for the quasi-3D 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 quasi-3D hyperbolic shear deformation theory with an RBF-based 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. Hosseini-Hashemi e. al. [33] studied the free vibration of thick FGM plates based on exact 3D elasticity theory and closed-form 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 Rayleigh-Ritz 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 near-field 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 quasi-3D solution. Shahsavari et al. [40] investigated the free vibration of three types of porosity distribution FGM plate by using quasi-3D hyperbolic theory. Xing et al. [41] implemented multiquadric radial basis functions for the free vibration studies of thin FGM plates via the separation-of-variable 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 RBF-based 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, span-to-thickness ratio, and plate aspect ratio on the free vibration of rectangular FGM is discussed.

2.     Theoretical Formulations

2.1. Material properties and constitutive equations

A porous FG rectangular plate of dimensions length a, breadth b, and thickness h in the Cartesian coordinate system (x-y-z) are shown in Figure 1. The modified power-law homogenization technique with porosity distribution effect for material properties is employed, which is formulated by Zhao et al., [42]

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

2.2. Displacement field

The displacement field is a framework as an equivalent single-layer approach with five variables higher order shear deformation theory which can be expressed by Kumar et al. [44].

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 stress-strain relation using generalized Hook's law with respect to the structural axis system (X-Y-Z) can be expressed as Kumar et al. [44];

is plane stress-reduced stiffness and is given below [33].

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

where KE = Kinetic energy, UE = Strain energy,

The kinetic energy of the FG plate can be expressed as.

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

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:

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:

And,

The inertia terms of rectangular FG plate are expressed as:

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

2.3. Boundary Conditions

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

3.     Solution Methodology

The importance of RBF-based 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 2-D 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.

'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:

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:

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

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:

where

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), u₀, v₀, w₀,  and  at desired locations are obtained. Using equation (2), the displacement components, and using equation (3), the stress components are obtained.

4.     Result and Discussions

In the following, we demonstrate the implementation of seventeen types of RBF-based meshfree methods for free vibration response of rectangular FG plate. The stability and performance of the proposed modified RBFs-based 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.

Table 3. Convergence study of non-dimensional frequency parameters  of rectangular plates (a/h=10, n=1, b/a=2).

Table 4. Comparison of non-dimensional frequency parameter with seventeen RBFs of SS FG2 plate with
different span to thickness ratio (a=b = 1, n=0.5).

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 span-to-thickness 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 span-to-thickness 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 span-to-thickness 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 non-dimensional 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 span-to-thickness 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 non-dimensional frequency parameter  of FG1 rectangular plates
with various porosity index (a/h=20, n=1.a/b=0.5)

Fig. 4. Comparison study of non-dimensional frequency parameters of FG1 rectangular
plates (a/h=10, n=1

Table 5. Comparison study of non-dimensional frequency parameter  of
FG1 rectangular plate. (a/h=10, a/b=0.5).

Table 6 shows the results for all 10 modes of non-dimensional frequency parameter  of FGM-3 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 FGM-1 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 FGM-1 rectangular plate.

Table 6. Effect of grading index 'n' with span-to-thickness ratio on non-dimensional frequency parameter
of FGM-3 square plates for mode 10 (g10)

5.     Conclusions

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 distance-based 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

• The results obtained by the RBFs have a fast convergence rate (15x15 nodes) with high precision.
• By increasing the grading index, the normalized natural frequency starts decreasing.
• There is an increment in the normalized natural frequency for thick to thin plates, and after a/h=50, the effect of the span-to-thickness ratio is constant.
• The normalized frequency decrease with an increase in the porosity index.

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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