Generalized Finite Difference Method for Plate Bending Analysis Of

mathematics

Article Generalized Finite Diﬀerence Method for Plate Bending Analysis of Functionally Graded Materials

Yu-Dong Li, Zhuo-Chao Tang * and Zhuo-Jia Fu Center for Numerical Simulation Software in Engineering & Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China; [email protected] (Y.-D.L.); [email protected] (Z.-J.F.) * Correspondence: [email protected]

Received: 4 September 2020; Accepted: 22 October 2020; Published: 3 November 2020

Abstract: In this paper, an easy-to-implement domain-type meshless method—the generalized finite difference method (GFDM)—is applied to simulate the bending behavior of functionally graded (FG) plates. Based on the first-order shear deformation theory (FSDT) and Hamilton’s principle, the governing equations and constrained boundary conditions of functionally graded plates are derived. Based on the multivariate Taylor series and the weighted moving least-squares technique, the partial derivative of the underdetermined displacement at a certain node can be represented by a linear combination of the displacements at its adjacent nodes in the GFDM implementation. A certain node of the local support domain is formed according to the rule of “the shortest distance”. The proposed GFDM provides the sparse resultant matrix, which overcomes the highly ill-conditioned resultant matrix issue encountered in most of the meshless collocation methods. In addition, the studies show that irregular distribution of structural nodes has hardly any impact on the numerical performance of the generalized finite difference method for FG plate bending behavior. The method is a truly meshless approach. The numerical accuracy and efficiency of the GFDM are firstly verified through some benchmark examples, with different shapes and constrained boundary conditions. Then, the effects of material parameters and thickness on FG plate bending behavior are numerically investigated.

Keywords: meshless method; functionally graded material; ﬁrst-order shear deformation theory; bending of plates; numerical simulation

1. Introduction Functionally gradient materials (FGMs) [1] are attractive to various fields such as aerospace, machinery, civil, nuclear power, chemistry, biological medicine, and electronic information due to its good mechanical performance. The FGMs often consist of two or more materials whose volume fractions are derived from a function of position through their thickness, namely, it changes continuously along certain dimensions of the structure. Recently, Bodaghi et al. [2] applied FGMs to the design of porous femoral prostheses under normal walking load conditions and discovered that using FGMs could be better than the conventional solutions. Furthermore, the development of 4D printing technology makes FGMs more popular when fabricating metamaterials [3–5]. Therefore, in this study, we are committed to numerically studying the FGMs’ bending behavior. Rapid developments of numerical methods have included the application of both meshed-based methods and meshless methods to the structural analysis of mechanical problems, which have been extensively investigated in research works. The objective of meshless methods is to eliminate, at least in part, the structure of elements, as in the finite element method (FEM), by constructing the approximation entirely in terms of nodes [6,7]. The presentation of a number of meshless methods is aimed at either improving the process in the calculated performance of existing methods or developing new ones. The bending analysis of plates using different versions of meshless methods has been studied in the past few decades [8–14]. In 1996,

Mathematics 2020, 8, 1940; doi:10.3390/math8111940 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1940 2 of 19 the static bending problem of elastic thin plates and shells was analyzed by Belytschko using the element-free Galerkin method [15]. Subsequently, Liu et al. [16] analyzed laminated plates using the element-free Galerkin method. Batra et al. [17] applied the Petrov–Galerkin meshless method, with radial basis functions to simulate the bending behavior of FGM thick plates based on higher-order shear and the normal deformable plate theory. The axisymmetric bending of functionally graded circular plates has also been studied based on the fourth-order shear deformation theory [18]. Fu et al. studied the bending analysis of the Kirchhoff and Winkler plate using the boundary particle method [19,20]. The generalized finite difference method (GFDM) [21,22] is a relatively new localized meshless method that was developed from the classical finite difference method (FDM) [23]. A point of local support domain is formed according to the rule of “the shortest distance”. Based on the multivariate Taylor series expansion and weighted least-squares fitting, the partial derivative of the underdetermined displacement at a certain node can be represented by a linear combination of the displacements at its adjacent nodes in the GFDM implementation. Unlike the finite difference method, the GFDM eliminates the need for orthogonal grid generation and extends the FDM to a more flexible scheme. While ensuring computational accuracy, the efficiency in computations is also improved under the advantage of localization in numerical algorithms. Belytschko et al. [24] developed an alternative implementation using MLS approximation. They called their approach the element-free Galerkin (EFG) method. The use of a constrained variational principle, with a penalty function to alleviate the treatment of Dirichlet boundary conditions, has been proposed in the EFG method. Otherwise, the convenience of having no transformation to boundary conditions in mathematics is highlighted in the GFDM. For the Laplace equation in the case of different domains with essential boundary conditions and irregular clouds of points, Gavete et al. [22] summarized that the GFDM appeared to be more accurate compared to the EFG method with linear approximation. As an easy-to-implement domain-type meshless method with simple mathematics and easy programming, the GFDM has the advantage of describing physical equations and mathematics without any transformation, unlike the technology in most meshless collocation methods. Similarly, the GFDM avoids the ill-conditional dense matrix issue commonly encountered in the boundary-type meshless collocation methods. Recently, this method has been widely used to solve various scientific and engineering computing problems. In 2012, Urena et al. [25] proposed the generalized finite difference method for higher-order partial differential equations. Gu et al. [26,27] successfully applied the GFDM to engineering inverse problems. The double-diffusive natural convection in fluid-saturated porous media was studied using the generalized finite difference method [28]. The application of the generalized finite difference method to the wave and diffusion problem was also studied by Fu et al. [29–31]. In this paper, the generalized finite difference method is applied to analyze the plate bending behavior of functionally graded materials. Based on the first-order shear deformation theory, the generalized finite difference method was developed. Firstly, a numerical discrete model of FGM plate bending, based on the first-order shear deformation theory, is presented. The effectiveness and convergence of the proposed method are verified by solving the benchmark examples with various boundaries and shapes. The effect of functionally graded material plate bending analysis, with various gradient distributions, is numerically studied.

2. Numerical Model Consider the FGM plate, as shown in Figure1. The length and width of the plate are shown as a and b with the thickness h. The components of functionally graded materials (FGMs) have a continuous gradient range, changing with spatial coordinates from one side to the other along a certain direction. The material properties Q, (i.e., Young’s modulus of elasticity E, Poisson’s ratio µ, density ρ), changing along the thickness direction of the FGM plate and obeying the law of power function, are assumed [32]. Q(z) = QtV + Q (1 V) b − (1) V = ( 1 + z )n( h z h , 0 n ) 2 h − 2 ≤ ≤ 2 ≤ ≤ ∞ Mathematics 2020, 8, x FOR PEER REVIEW 5 of 20

Mathematics 2020, 8, x FOR PEER REVIEW 5 of 20 =+θθ uun cos v sin uu=+=−sinθθθθ + v cos uuns cos v sin φφ=−=+cosθθ θφθ + sin uusnxsin v cosy φφφφθφθ=+=− θφθ + nxsxcossin ysin y cos φφθφθMM=−=++sincos +22θθθ cos M sin M sin 2 sxnxx y yy xy (12) MMMM=++=−+−cos(cos2222θθθθθ M sin sin ) ( M M sinM 2 )sin() 2 θ / 2 nxxsxy yy yyxx xy (12) =−+−=++2222θθθθθ() θ MMNNsxynxxcos(cos N sin yy sin ) ( M N yyxx xy sin M 2 )sin 2 / 2 =++=−+−2222θθθθθ() θ Mathematics 2020, 8, 1940 NNNNnxxsxycos(cos N sin yy sin ) ( N N yyxx xy sin N 2 )sin 2 / 2 3 of 19 ==−+−θθ22+θθ() θ NNVRnxsxycos(cosRy sinsin ) ( N yyxx N )sin 2 / 2 = θθ+ where V is the volume fractionVRnx showncos inR Figurey sin 2, n is the gradient index of FGMs.

Figure 1. Geometry of the functionally gradient material (FGM) plate. Figure 1. Geometry of the functionally gradient material (FGM) plate. Figure 1. Geometry of the functionally gradient material (FGM) plate.

Figure 2. Volume ratio with thickness of the FGM plate. Figure 2. Volume ratio with thickness of the FGM plate. The first-order shearFigure deformation 2. Volume theory ratio iswith governed thickness by of the FGM plate. 3. Generalized Finite Difference Method  ( ) ( ) ( ) 3. GeneralizedFirst of all, Finite an irre Differencegular cloud Methodu ofx ,points,y, z = uNo ,x ,isy generated+ zφx x, y in a domain. A set of NS points is  expressed as local collocation pointsv( surroundingx, y, z) = vo(x a, ycentral) + zφ ynode(x, y ) ()x , y , shown as Figure 3. (2) First of all, an irregular cloud of points, N , is generated in aii domain. A set of NS points is  w(x, y, z) = wo(x, y) expressed as local collocation points surrounding a central node ()x , y , shown as Figure 3. A differential function F ()xy, as a fourth-order Taylor expansionii is governed by where (u, v, w) is the displacement of the FGM plate, (uo, vo, wo) is a projection point on the middle A differential function F ()xy, as a fourth-order Taylor expansion is governed by surface of a board along the coordinates’ direction22(22x, y, z), φ , φ is 2the corresponding rotation of the ∂∂FFhk ∂ F ∂ Fx y ∂ F ()=+ii + + i + i + i original middle surfaceFxjj, y after F ideformation, h k and z is the22 distancehk to the middle plane of the FGM plate. ∂∂22∂∂22 ∂∂ 2 ∂∂FFxyhk22 ∂xy F ∂ xy F ∂ F Based on theFx() effect, y of=+ first-order F hii + shear k +deformation, i + hk the istrain + model i [33] is governed by jj32∂∂ i33FF∂∂∂∂ 222 ∂∂ 33 FF ∂∂ 3 ++hhkhkkiixy +22xy ii + xy 32n o 23n o (13) 32∂∂∂∂∂∂33 2 33 3 62∂∂xxyxyyFFε = 2ε0 ∂∂+ FFz ε1 6 ++hhkhkkii + ii + 4344{ } 223n o 4 44 4 (13)(3) ∂∂∂∂∂∂∂∂32FF 0 ∂ 23 F ∂∂ FF ++62hhkhkhkkxxyxyyiiγ += 2γ i +6 ii ++5 43 2234O()r 43∂∂∂∂∂∂∂∂44 223 4 44 4 hhkhkhkk24∂∂xxyxyxyyFF 6 4 ∂ F 6 ∂∂ FF 24 where ++ii + i + ii ++O()r5 ∂∂∂∂∂∂∂∂43 2234T 24xxyxyxyyn 0 6o n 4 6o 24 ε = uo,x vo,y uo,y + vo,x T n 1o n o ε = φx,x φy,y φx,y + φy,x (4) T n 0o n o γ = φx + wo,x φy + wo,y The constitutive relation for functionally graded materials is governed by

    n 0o   N   AB 0  ε   { }    n o   M  =  BD  1     0  ε  (5)  { }    n o  R 0 0 A1  γ0  { } Mathematics 2020, 8, 1940 4 of 19 where        A11 A12 0   B11 B12 0   D11 D12 0  " #       A 0 [A] =  A A [B] =  B B [D] =  D D [A ] = 44  12 11 0   12 11 0   12 11 0  1 (6)       0 A55 0 0 A66 0 0 B66 0 0 D66

Z h/2 − 2 Aij, Bij, Dij = Sij 1, z, z dz (7) +h/2 S = E/ 1 µ2 , S = µS , S = (1 µ)S /2 11 − 12 11 66 − 11 In Equation (7), Aij, Bij, Dij(i, j = 1, 2, 6) are the tensile stiffness, coupling stiffness, and bending stiffness, respectively. A44, A55 are the shear stiffness for various orders. It assumed that the shear strain is constant along the thickness of the plate, using the first-order shear deformation theory. The shear correctional factor is considered due to the quadratic functional form in reality, along the thickness. Considering the equal principle of energy between shear strain energy and shear residual strain energy, the shear correctional factor χ is derived by " # " #" #" # " #" # R χ 0 A 0 φ + w ϑ 0 φ + w x = x 44 x o,x = 44 x o,x (8) Ry 0 χy 0 A55 φy + wo,y 0 ϑ55 φy + wo,y

R h/2 2 1 = 9 1 4 z2 /S dz − ϑ44 4h2 h/2 h2 44 − − (9) R h/2 2 1 = 9 1 4 z2 /S dz − ϑ55 4h2 h/2 h2 55 − − R h/2 2 1 R h/2 χ = Rx 9 1 4 z2 /S dz − / S dz x R0 4h2 h/2 h2 44 h/2 44 x − − − (10) R R h/2 2 1 R h/2 χ = y = 9 1 4 z2 /S dz − / S dz y R0 4h2 h/2 h2 55 h/2 55 y − − − where ϑ44, ϑ55 are the shear stiffness with shear correctional factor; the equal shear modulus is shown as S44 = S55 = S66. The static differential governing equations of the FGM plate, based on the first-order shear deformation theory, are processed [34,35]

Nij,i= 0 Mij,i = Rj (11) R = q i,i − where N is the internal force tensor of two-dimensional elastic mechanics, M is the bending moment tensor, R is the shear vector of two-dimensional elastic mechanics, and q is the transverse load in a domain. In order to obtain the unique deﬂection, certain constraints are imposed on the boundary of the FGM plate. Three boundary conditions are common, as follows, where θ is the angle between the outer normal vector and the axis vector. Clamped boundary conditions (CBC): un = 0, us = 0, w = 0, φn = 0, φs = 0 Simply boundary conditions (SBC): un = 0, us = 0, w = 0, φs = 0, Mn = 0 Free boundary conditions (FBC): Nn = 0, Ns = 0, Mn = 0, Ms = 0, Vn = 0 where Mathematics 2020, 8, 1940 5 of 19

Mathematics 2020, 8, x FOR PEERun REVIEW= u cos θ + v sin θ 8 of 20 us = u sin θ + v cos θ − φ uu= cos v+ sin φφ y  ++φn ()φx θ φy θ xx + () + A11GGxxyy A 66 AA 12 66 G xy 0 B 11 GG xxyy B 66 BB 12 66 G xy  φs = φx sin θ + φy cos θ φ φφ  ()AA++GGGuvv A− A 0 () BB + Gx B GGyy + B 12 66xyxxyy= 66 112 + 2 + 12 66 xy 66 xxyy 11  Mn Mxx cos θ Myy sin θ Mxy sin 2θφ φ (12) T = 00GGG ϑϑww+ ϑ x ϑ Gy  = 2 44xx2 55+ yy 44 x( ) 55 y Ms Mxy cos θ sin θ Myy Mxx sin 2θ /2 φ  uu v w φφφ y BBGG++() BB G2 − −ϑϑ G2 DD G−xx +− GGx ()DD+ G  11xx 66N yyn = 12Nxx 66cos xyθ + Nyy 44sin xθ + N 11xy xxsin 2 66θ yy 44 12 66 xy uvv w φ φφ  ()BB++−+GGG B B2 ϑϑ2 G ( DD ) Gx D GGyy +− D  12 66xyNs = 66N xxxy cos 11 yyθ sin 55θ y+ Nyy Nxx 12sin 66(2θ xy)/2 66 xx 11 yy 55 − − Vn = Rx cos θ + Ry sin θ

3. Generalized Finite Diﬀerence Method T Φ =U×× V W ×φφ 11NN 1 Nxy11××NN First of all, an irregular cloud of points, N, is generated in a domain. A set of NS points is expressed as local collocation points surrounding[] a− central node (x T, y ), shown as Figure3. L= 011××NN 0 q 111 ××× NNN 0 0 i i

FigureFigure 3. Schematic 3. Schematic diagram diagram of collocation of collocation nodes nodes in the in generalized the generalized finite finite difference difference method method (GFDM). (GFDM). A differential function F(x, y) as a fourth-order Taylor expansion is governed by

4. Numerical Results and Discussion 2 2 F x , y = F + h ∂Fi + k ∂Fi + h ∂Fi + hk ∂Fi + k ∂Fi j j i ∂x ∂y 2 ∂x2 ∂x∂y 2 ∂y2 3 3 To illustrate the accuracy hof3 ∂ theFi presenth2k ∂ F i approachhk2 ∂ F, idescribedk3 ∂Fi in the previous sections, the static + 3 + 2 + 2 + (13) bending of homogeneous and 6FGM∂x plates2 ∂x with∂y diff2 ∂erentx∂y shapes6 ∂y3 is considered and analyzed. More 4 3 4 2 2 4 3 4 4 + h ∂Fi + h k ∂ Fi + h k ∂ Fi + hk ∂ Fi + k ∂Fi + (r5) types of the FGM plates consisting24 ∂x of4 AlAlO/6 ∂23x3∂,yAlZrO/ 4 2 ,∂ andx2∂y2 AlFeO/ 6 23∂x∂ y were3 24 used∂y4 throughout this work. The materials of the FGM plates are listed in Table 1. Young’s modulus and mass density are analyzed 2 2 whereusing Voigt’sFi = F( xrulei, yi )ofh =mixturesxj xik in= Equationyj yir = (1).√ hFor+ convenience,k the boundary conditions of the FGM − − platesIn are any specified case, the as extension follows: to free other boundary problems condition is obvious. (F), Wesimply consider supported norm Θ(S), or fully clamped (C). Both thin and thick plates were studied through the specified thickness-span aspect ratios ah/ 2 2 2  *3 h 2 k 2   wwEhq=∂F0 / ∂F0 j ∂F0 ∂ F0 j ∂ F0i in Figure 1. Normalized deflectionFi Fj + hj Al+ kj is +implemented+ hjk jto ignore+ the effect of the transverse  ∂x ∂y 2 ∂x2 ∂x∂y 2 ∂y2   N  − 3 2 2 3   load. Unless otherwiseP  stated, a regularhj 3 set hofj k j123 × 12 scatteredhjkj 3 nodeskj 3was used in a square plate, Θ(F) =  + ∂ F0 + ∂ F0 + ∂ F0 + ∂ F0 ωj (14)  6 x3 2 x2 y 2 x y2 6 ∂y   shown as Figure 4b. j=1 ∂ ∂ ∂ ∂ ∂ 3    h 4 4 h 3k 4 h 2k 2 4 h k 3 4 k 4 4    + j ∂ F0 + j j ∂ F0 + j j ∂ F0 + j j ∂ F0 + j ∂ F0   24 ∂x4 6 ∂x3∂y 4 ∂x2∂y2 6 ∂x∂y3 24 ∂y4 where ωj is the denominated weighting function [22]. The solution to the linear system may be obtained by minimizing norm Θ using the calculus of variations. HDF = M (15)

(a) (b)

Figure 4. Clouds (a) 8 × 8 in a domain and (b) 12 × 12 in a domain.

Mathematics 2020, 8, 1940 6 of 19

P P P P P  ω2h2 ω2h k ω 2h3/2 ω2h2k3/6 ω2h2k4/24   j j j j j j j ··· j j j j j j   P 2 2 P 2 2 P 2 4 P 2 5   ω k ω h k /2 ω hjk /6 ω k /24   j j j j j ··· j j j j   P 2 3 P 2 2 3 P 2 2 4   ωj h /4 ω h k /12 ω h k /48   j ··· j j j j j j  H14 14 =  ......  ×  ......   ......   P P   ω2h2k6/36 ω2h2k7/144   j j j j j j   ··· P   SYM ω2h2k8/576  ······ j j j  P P  F ω2h + F ω2h  − i j j j j j   P 2 P 2   Fi ω kj + Fj ω kj   − j j   P 2 2 P 2 2   Fi ω h /2 + Fj ω h /2   − j j j j  M14 1 =  .  ×  .   .   P P   F ω2h k3/6 + F ω2h k3/6   i j j j j j j j   − P P   F ω2k4/24 + F ω2k4/24  − i j j j j j 2 4 4 D = ∂Fi ∂Fi ∂ Fi ∂ Fi ∂ Fi F14 1 ∂x ∂y x2 x y3 4 × ∂ ··· ∂ ∂ ∂y where H14 14 is the coeﬃcient matrix formed by Euclidean distance in the appointed domain. M14 1 is × × the matrix including explicit function value Fi. Hence DF14 1 is expressed by ×  NS   x,i + P x,i i   α0 Fi αj Fj   =   j 1     NS  ∂Fi  y,i y,i     + P i   ∂x   α0 Fi αj Fj   ∂Fi       j=1   ∂y   NS   2   xx,i P xx,i   ∂ Fi   α F + α Fi   2   0 i j j  D =  ∂x  =  j=1  F14 1     (16) ×    .   4   .   ∂ Fi   .       ∂x∂y3   NS   4   xyyy,i P xyyy,i i   ∂ Fi   +     α0 Fi αj Fj  ∂y4  j=1     NS   yyyy,i + P yyyy,i i   α0 Fi αj Fj  j=1

NS NS NS NS x,i + P x,i = y,i + P y,i = xx,i + P xx,i = yyyy,i + P yyyy,i = where α0 αj 1,α0 αj 1,α0 αj 1,...,α0 αj 1. j=1 j=1 j=1 j=1 Then, the process of using the generalized finite difference method will be shown at each node, according to Equations (11) and (16), and is expressed in a differential form, shown as Equation (17).

2 2 2 2 2 2 ∂ φ ∂ φ ∂ φy A ∂ uo + A ∂ uo + (A + A ) ∂ vo + B x + B x + (B + B ) = 0 11 ∂x2 66 ∂y2 12 66 ∂x∂y 11 ∂x2 66 ∂y2 12 66 ∂x∂y 2 2 2 2 2 2 ∂ φ ∂ φy ∂ φy (A + A ) ∂ uo + A ∂ vo + A ∂ vo + (B + B ) x + B + B = 0 12 66 ∂x∂y 66 ∂x2 11 ∂y2 12 66 ∂x∂y 66 ∂x2 11 ∂y2 2 2 2 2 2 2 ∂ uo ∂ uo ∂ vo ∂wo ∂ φx ∂ φx ∂ φy B + B + (B + B ) ϑ ϑ φx + D + D + (D + D ) = 0 (17) 11 ∂x2 66 ∂y2 12 66 ∂x∂y − 44 ∂x − 44 11 ∂x2 66 ∂y2 12 66 ∂x∂y 2 2 2 2 2 2 ∂ uo ∂ vo ∂ vo ∂wo ∂ φx ∂ φy ∂ φy (B + B ) + B + B ϑ ϑ φy + (D + D ) + D + D = 0 12 66 ∂x∂y 66 ∂x2 11 ∂y2 − 55 ∂y − 55 12 66 ∂x∂y 66 ∂x2 11 ∂y2 2 2 ∂φ ∂φy ϑ ∂ wo + ϑ ∂ wo + ϑ x + ϑ = q 44 ∂x2 55 ∂y2 44 ∂x 55 ∂y − For better understanding and less writing-space, consider the ﬁrst governing equation of Equation (17) as an example to describe the process. Herein, the partial derivative of the underdetermined displacement can be written in matrix forms according to Equation (16). Mathematics 2020, 8, 1940 7 of 19

  αxx,1 αxx,1 αxx,1 0 0 0 αxx,1 αxx,1    1 2 3 N 1 N   u1   xx,2 xx,2 xx,2 xx···,2 ··· ··· − xx,2     α α α α 0 0 0 α   u   1 2 3 4 ··· ··· ··· N   2   xx,3 xx,3 xx,3 xx,3 xx,3     α1 α2 α3 α4 α5 0 0 0   u3   xx,4 xx,4 xx,4 xx,4 xx···,4 ··· ···     0 α α α α α 0 0   u  2  2 3 4 5 6   4  ∂ uo u  ··· ··· ···    2 = GxxU =  ......   .  ∂x  ......   .       ···   .   ......   .   ......   .   ···     xx,N 1 xx,N 1 xx,N 1 xx,N 1 xx,N 1   u   α1 − 0 0 0 αN 3− αN 2− αN 1− αN −   N 1   xx,N xx···,N ··· ··· − xx−,N xx−,N xx,N   −    uN α1 α2 0 0 0 αN 2 αN 1 αN N N N 1 ··· ··· ··· − − × ×  xy,1 xy,1 xy,1 xy,1 xy,1   α α α 0 0 0 α α     1 2 3 N 1 N   v1   xy,2 xy,2 xy,2 xy···,2 ··· ··· − xy,2     α α α α 0 0 0 α   v   1 2 3 4 ··· ··· ··· N   2   xy,3 xy,3 xy,3 xy,3 xy,3     α1 α2 α3 α4 α5 0 0 0   v3   xy,4 xy,4 xy,4 xy,4 xy···,4 ··· ···     0 α α α α α 0 0   v4  2  2 3 4 5 6    ∂ vo = Gv V =  ··· ··· ···   .  ∂x∂y xy  ......   .   ......   .   ···     ......   .   ......   .       xy···,N 1 xy,N 1 xy,N 1 xy,N 1 xy,N 1     − − − − −   vN 1   α1 0 0 0 αN 3 αN 2 αN 1 αN   −   xy,N xy···,N ··· ··· − xy− ,N xy− ,N xy,N   v   α α 0 0 0 α α α  N N 1 1 2 N 2 N 1 N N N × ··· ··· ··· − − × xx,i Based on the analyzed Euclidean distance, the coefficient αj at each node can be solved by the previously obtained matrix in Equation (15). In Equation (16), we shall select NS(nodes of star in a local region), considering a fixed radius (the same radius for all the stars of the domain) or a variable radius (the radius for each of the stars depends on their nodal distribution). That is why the sparse resultant matrix is provided in the GFDM. In the same way, all of the partial derivatives of the underdetermined displacements can be expressed as matrix forms in the GFDM. Then, the first governing equation of Equation (17) can be derived by

u u v φx φx φy A11GxxU + A66GyyU + (A12 + A66)GxyV + B11Gxx ϕx + B66Gyyϕx + (B12 + B66)Gxy ϕy = 0 (18)

According to the details of the first governing equation derived by the GFDM, the other governing equations of Equation (17) are similarly derived, like Equation (18). Lastly, the complete linear equation systems of functionally graded material plates using the generalized finite difference method (GFDM) are derived, shown as Equation (19), and the displacements Φ can be solved by the sparse linear system.

TΦ = L (19)

 φ φ φy   A Gu + A Gu (A + A )Gv 0 B G x + B G x (B + B )G   11 xx 66 yy 12 66 xy 11 xx 66 yy 12 66 xy   u v v φx φy φy   (A + A66)G A66G + A G 0 (B + B66)G B66G + B G   12 xy xx 11 yy 12 xy xx 11 yy   w w φx φy  T =  0 0 ϑ44G + ϑ55G ϑ44G ϑ55G   xx yy x y   u u v w φx φx φx φy   B11G + B66G (B12 + B66)G ϑ44G D11G + D66G ϑ44G (D12 + D66)G   xx yy xy x xx yy xy   − −φ φy φy   (B + B )Gu B Gv + B Gv ϑ Gw (D + D )G x D G + D G ϑ Gφy  12 66 xy 66 xx 11 yy 55 y 12 66 xy 66 xx 11 yy 55 5N 5N − − × h iT = U V W Φ 1 N 1 N 1 N ϕx1 N ϕy1 N × × × × × h iT L= 01 N 01 N q1 N 01 N 01 N × × − × × × 4. Numerical Results and Discussion To illustrate the accuracy of the present approach, described in the previous sections, the static bending of homogeneous and FGM plates with different shapes is considered and analyzed. More types of the FGM plates consisting of Al/Al2O3,Al/ZrO2, and Al/Fe2O3 were used throughout this work. The materials of the FGM plates are listed in Table1. Young’s modulus and mass density are analyzed using Voigt’s rule of mixtures in Equation (1). For convenience, the boundary conditions of the FGM plates are specified as follows: free boundary condition (F), simply supported (S), or fully Mathematics 2020, 8, 1940 8 of 19 clamped (C). Both thin and thick plates were studied through the specified thickness-span aspect 3 ratios a/h in Figure1. Normalized deflection w∗ = wEAlh /q is implemented to ignore the effect of the transverse load. Unless otherwise stated, a regular set of 12 12 scattered nodes was used in a square × plate, shown as Figure4b.

Table 1. Material properties of the FGM plates used for the analysis [32].

Ceramic Properties Aluminum (Al) Alumina (Al2O3) Zirconia (ZrO2) Castiron (Fe2O3) E(GP a) 70 380 200 150 µ 0.3 0.3 0.3 0.3 ρ kg/m3 2707 3800 5700 5240

Consider an Al/ZrO2 FGM square plate with different thickness-span aspect ratios, a/h, subjected to a transverse uniform load q that is fully simply supported (SSSS) or completely clamped supported (CCCC). Four different values of gradient index are considered for calculations in this study. Previous methods to analyze the bending of FGM plates are quoted. Table2 reports the normalized center 3 deflection w∗ = wEAlh /q, where the obtained results are compared with first-order shear deformation theory (FSDT)-based MK [32], FSDT-based IGA [36], and FSDT-based ES-DSG (edge-based smoothed discrete shear gap) [37]. Obviously, the numerical results solved by the present method, GFDM, agreed well with the results of the other mentioned methods.

3 Table 2. Normalized center deﬂection w∗ = wEAlh /q of an Al/ZrO2 square plate with diﬀerent length-thickness ratio a/h and gradient index n under the transverse uniform load q.

Boundary Condition a/h Method n = 0 n = 0.5 n = 1 n = 2 SSSS 5 IGA [36] 0.0187 0.0253 0.0295 0.0340 ES-DSG [37] 0.0186 0.0243 0.0275 0.0308 MK [32] 0.0188 0.0254 0.0297 0.0340 present 0.0187 0.0248 0.0286 0.0328 20 IGA [36] 0.0157 0.0214 0.0252 0.0285 MK [32] 0.0159 0.0217 0.0253 0.0286 present 0.0157 0.0213 0.0242 0.0270 SFSS 5 IGA [36] 0.0345 0.0469 0.0551 0.0630 MK [32] 0.0348 0.0472 0.0550 0.0627 present 0.0347 0.0460 0.0546 0.0621 20 IGA [36] 0.0306 0.0419 0.0551 0.0631 MK [32] 0.0308 0.0422 0.0549 0.0627 present 0.0303 0.0415 0.0538 0.0618 CCCC 5 ES-DSG [38] 0.0083 0.0110 0.0128 0.0149 MITC4 [38] 0.0082 0.0110 0.0127 0.0149 IGA [39] 0.0082 0.0110 0.0129 0.0149 MLPG [17] 0.0079 0.0117 0.0137 0.0157 present 0.0083 0.0110 0.0130 0.0156

A further study on the effect of the nodes of star was also performed. Figure5 depicts the convergence curves of normalized central deflection of the homogeneous plate, obtained with various values of NS (nodes of star) and with different sets of nodes. A wide range of nodes of star, from 28 to 45, was considered. As the number of nodes increases, it is observed that the situation has the same convergence to the numerical solution given by the FSDT approach [40]. Here, again, it is concluded that when NS 30, the process of convergence occurs, namely, the phenomenon of instability, because ≤ the nearly singular matrix in Equation (15) is formed with relatively fewer nodes. While 30 < NS 45, ≤ it is obviously stable enough to gain the convergence curve. For appropriate computational efficiency and convergence of stability, the best result is commonly obtained with NS = 35, verified by abundant numerical calculations. Mathematics 2020, 8, x FOR PEER REVIEW 8 of 20

φφ φ  uu++() v xx + () +y A11GGxxyy A 66 AA 12 66 G xy 0 B 11 GG xxyy B 66 BB 12 66 G xy  φ φφ ()++uvv () +x yy +  AA12 66GGGxyxxyy A 66 A 11 0 BB 12 66 G xy B 66 GG xxyy B 11  φ φ = ϑϑww+ ϑx ϑy T  00GGG44xx 55 yy 44 x 55 G y  φφφ φ uu++() v −ϑϑ w xx +−x ()+ y BB11GGxx 66 yy BB 12 66 G xy 44 G x DD 11 G xx 66 GG yy 44 DD12 66 Gxy φ φφ  ()++−+uvvϑϑ w ( )x yy +−  BB12 66GGGxy B 66 xx B 11 yy 55 G y DD 12 66 G xy D 66 GG xx D 11 yy 55

T Φ =U×× V W ×φφ 11NN 1 Nxy11××NN

[]− T L= 011××NN 0 q 111 ××× NNN 0 0

Figure 3. Schematic diagram of collocation nodes in the generalized finite difference method (GFDM).

4. Numerical Results and Discussion To illustrate the accuracy of the present approach, described in the previous sections, the static bending of homogeneous and FGM plates with different shapes is considered and analyzed. More

types of the FGM plates consisting of AlAlO/ 23, AlZrO/ 2 , and AlFeO/ 23 were used throughout this work. The materials of the FGM plates are listed in Table 1. Young’s modulus and mass density are analyzed using Voigt’s rule of mixtures in Equation (1). For convenience, the boundary conditions of the FGM plates are specified as follows: free boundary condition (F), simply supported (S), or fully clamped (C). Both thin and thick plates were studied through the specified thickness-span aspect ratios ah/ *3= in Figure 1. Normalized deflection wwEhqAl / is implemented to ignore the effect of the transverse load. Unless otherwise stated, a regular set of 12 × 12 scattered nodes was used in a square plate, Mathematics 2020, 8, 1940 9 of 19 shown as Figure 4b.

Mathematics 2020, 8, x FOR PEER REVIEW 10 of 20 various values of NS (nodes of star) and with different sets of nodes. A wide range of nodes of star, from 28 to 45, was considered. As the number of nodes increases, it is observed that the situation has the same convergence to the numerical solution given by the FSDT approach [40]. Here, again, it is concluded that when NS ≤ 30, the process of convergence occurs, namely, the phenomenon of instability, because the nearly singular matrix in Equation (15) is formed with relatively fewer nodes. < ≤ While 30 NS 45, it is obviously stable enough to gain the convergence curve. For appropriate computational efficiency (anda) convergence of stability, the best result( bis) commonly obtained with NS = 35, verified by abundant numerical calculations. FigureFigure 4. Clouds 4. Clouds (a) ( 8a) 88 × in 8 ain domain a domain and and (b )(b 12) 1212 × 12 in in a domain.a domain. × ×

3 Figure 5. Convergence curves of normalized central deflection w∗ = wEAlh /q with different values of wwEhq*3= / Figurenodes 5. of Convergence star in a local domaincurves of in annormalized SSSS FG plate, central with deflectiona/h = 5. Al with different values of nodes of star in a local domain in an SSSS FG plate, with a/h = 5. The effect of the computational cost (CPU-time) using the proposed method is investigated herein. As shown in Figure4, seven regular sets of 5 5, 8 8, 10 10, 12 12, 16 16, 20 20 and The effect of the computational cost (CPU-time)× × using× the proposed× ×method ×is investigated 32 32 nodes are used for the analysis of an SSSS homogeneous plate and a CCCC FG plate (n = 1). herein.× As shown in Figure 4, seven regular sets of 5 × 5, 8 × 8, 10 × 10, 12 × 12, 16 × 16, 20 × 20 and 32 The intensity of scattered nodes is characterized by the distance of two neighboring nodes (d), shown × 32 nodes are used for the analysis of an SSSS homogeneous plate and a CCCC FG plate (n = 1). The as Figure3. The computational time (CPU-time) required for preprocessing and the direct full-matrix intensity of scattered nodes is characterized by the distance of two neighboring nodes ( d ), shown as solver is evaluated, involving the definition of parameters, the partition of sets of nodes, and the Figureimplementation 3. The computational of the proposed time method. (CPU-time) All the tasks requ areired performed for preprocessing using a personal and computerthe direct with full-matrix a solverCore-i7 is CPUevaluated, @ 2.5 GHz involving and 8 GB the of RAM,definition based of on parameters, MATLAB software. the partition Figure6 showsof sets the of measurednodes, and the implementationCPU-time versus of the the nodal proposed spacing method. (d) for diAllfferent the ta valuessks are of performed the nodes of using star on a apersonal logarithmic computer scale. with a Core-i7It is obvious CPU that @ the2.5 CPU-timeGHz and increases 8 GB of as RAM, the nodes based of star on increaseMATLAB because software. of an increasingFigure 6 localshows the measureddomain at CPU-time each node. versus Similarly, the Figure nodal7 depicts spacing the ( measuredd ) for different CPU-time values versus of the the nodal nodes spacing of (stard) on a logarithmicfor a CCCC scale. FG plate It is ( nobvious= 1). It isthat concluded the CPU-time that diff increaseserent boundary as the conditions nodes of andstar various increase values because of anof increasing gradient index local havedomain hardly at each any impact node. onSimilarly, the computational Figure 7 depicts time. It the is undisputed measured thatCPU-time the sets versus theof nodal nodes spacing have a strong ( d ) for impact a CCCC on the FG computational plate (n = 1). It time, is concluded with 0.32 sthat to generate different the boundary displacement conditions vector at 12 12 nodes and 2.08 s to generate the displacement vector at 32 32 nodes (NS = 35). and various values× of gradient index have hardly any impact on the ×computational time. It is Key aspects attributed to CPU-time are the efficient configuration of the point rigid body without undisputed that the sets of nodes have a strong impact on the computational time, with 0.32 s to meshing, the formation of the local support domain at a certain node, and the solver comprised of the generatesparse resultantthe displacement matrix in the vector GFDM at implementation.12 × 12 nodes and Compared 2.08 s to to generate the finite the element displacement method with vector at 32node-based × 32 nodes strain(NS = smoothing 35). Key aspects [38], accuracy attributed and e toffectiveness CPU-time are are exemplified the efficient by usingconfiguration the generalized of the point rigidfinite body difference without method meshing, to analyze the theformation bending of athe CCCC local FG support plate. Based domain on the at numerical a certainsolutions node, and the solver comprised of the sparse resultant matrix in the GFDM implementation. Compared to the finite element method with node-based strain smoothing [38], accuracy and effectiveness are exemplified by using the generalized finite difference method to analyze the bending of a CCCC FG plate. Based on the numerical solutions of the IGA [39] in Table 2, Figure 8 depicts the relative deviation versus increasing nodes (grids), including the finite element method with node-based strain smoothing and the proposed method. The same setting on sets of nodes and grids is used. It is clear that the relative deviation is decreasing as the sets of nodes (grids) increase, respectively generated by the two methods. It is observed that the GFDM has higher precision and faster convergence than the ES-DSG, with different values of gradient index.

Mathematics 2020, 8, 1940 10 of 19

of the IGA [39] in Table2, Figure8 depicts the relative deviation versus increasing nodes (grids), including the ﬁnite element method with node-based strain smoothing and the proposed method. The Mathematicssame setting 2020,on 8, x sets FOR of PEER nodes REVIEW and grids is used. It is clear that the relative deviation is decreasing as the 11 of 20 sets of nodes (grids) increase, respectively generated by the two methods. It is observed that the GFDM Mathematics 2020, 8, x FOR PEER REVIEW 11 of 20 has higher precision and faster convergence than the ES-DSG, with diﬀerent values of gradient index.

Figure 6. The computational time (CPU-time) for solving the bending deflection of an SSSS Figure 6. The computational time (CPU-time) for solving the bending deﬂection of an SSSS homogeneous homogeneous plate subjected to a uniform load using the present method, with different values of Figureplate subjected6. The tocomputational a uniform load time using the(CPU-time) present method, for solving with di ﬀtheerent bending values of deflection nodes of star of in an a SSSS nodes of star in a local domain. homogeneouslocal domain. plate subjected to a uniform load using the present method, with different values of nodes of star in a local domain.

Figure 7. The computational time (CPU-time) for solving the bending deflection of a CCCC FG plate Figure(n = 1 7.) subjected The computational to a uniform loadtime using(CPU-time) the present for method,solving withthe bending different valuesdeflection of nodes of a of CCCC star in FG a plate (n local= 1) subjected domain. to a uniform load using the present method, with different values of nodes of star in Figure 7. The computational time (CPU-time) for solving the bending deflection of a CCCC FG plate a local domain. (n = 1) subjected to a uniform load using the present method, with different values of nodes of star in An Al/Al2O3 FGM square thick plate a/h= 10 that is fully simply supported is considered to examinea local domain. the eff ect of different transverse loads and different gradient indices. The normalized 3 center deflection w∗ = wEAlh /q solved by the present method is also used to compare with the results of Singha [41] and Zenkour [42]. A transverse uniform load q and transverse sinusoidal load q sin(πx/a) sin(πy/b) are implemented to verify the accuracy of the GFDM and analyze the bending of the FGM plate, shown in Table3.

Figure 8. The relative deviation, based on the IGA [39] in Table 2 versus 8 × 8, 12 × 12, 16 × 16, 20 × 20, and 32 × 32 nodes (grids), including the finite element method, with node-based strain smoothing [38] Figure 8. The relative deviation, based on the IGA [39] in Table 2 versus 8 × 8, 12 × 12, 16 × 16, 20 × 20, and the present method. and 32 × 32 nodes (grids), including the finite element method, with node-based strain smoothing [38] and the present method.

Mathematics 2020, 8, x FOR PEER REVIEW 11 of 20

Figure 6. The computational time (CPU-time) for solving the bending deflection of an SSSS homogeneous plate subjected to a uniform load using the present method, with different values of nodes of star in a local domain.

Figure 7. The computational time (CPU-time) for solving the bending deflection of a CCCC FG plate (n = 1) subjected to a uniform load using the present method, with different values of nodes of star in Mathematicsa local2020 domain., 8, 1940 11 of 19

Figure 8. The relative deviation, based on the IGA [39] in Table2 versus 8 8, 12 12, 16 16, 20 20, Figure 8. The relative deviation, based on the IGA [39] in Table 2 versus× 8 × 8, ×12 × 12, 16× × 16, 20× × 20, and 32 32 nodes (grids), including the ﬁnite element method, with node-based strain smoothing [38] and 32× × 32 nodes (grids), including the finite element method, with node-based strain smoothing [38] and the present method. and the present method. 3 Table 3. Normalized center deﬂection w∗ = wEAlh /q of an Al/Al2O3 square thick plate a/h = 10 with fully simply supported boundary conditions and gradient index n under the transverse uniform load q and the transverse sinusoidal load q sin(πx/a) sin(πy/b).

Method n = 0 (Ceramic) n = 1 n = 2 n = 4 n = (Metal) ∞ Uniformly distributed load Singha [41] 0.0086 0.0172 0.0221 0.0257 - Zenkour [42] 0.0085 0.0170 0.0218 0.0253 - present 0.0086 0.0168 0.0215 0.0248 0.0466 Lateral sinusoidal load Singha [41] 0.0055 0.0109 0.0141 0.0164 0.0296 Zenkour [42] 0.0054 0.0107 0.0138 0.0161 0.0296 present 0.0054 0.0106 0.0135 0.0158 0.0296

Based on the previous example and validation, the skew angle of the FGM plate was processed, shown as Table4. An Al/ZrO2 square thick plate a/h = 5 with a fully clamped boundary condition was implemented by the present method to analyze the effect of different skew angles. As the skew 3 angle of the FGM square plate increases, the normalized maximum deflection w∗ = wEAlh /q reduces in the figure. Obviously, the stiffness of the structure increases as the span of the structure decreases. The normalized maximum deflection of plate composed with pure ceramic (n = 0) is slightly less than that result of pure metal (n = ) because of the super-stiff ceramic. ∞ The effect on various levels of irregular clouds of points in a domain is also investigated herein. The level of irregular clouds of points is characterized by a series of numbers, shown in Figure9. Figure 10 depicts the relative deviation versus various irregular levels of sketching nodes, comparing with the original numerical solution of the present method with a regular cloud of points. In Figure 10, the results with different skew angles and gradient indices are evaluated. It is observed that the distribution of the irregular cloud of points has a slight effect on the numerical solution using the proposed method. All of the relative deviations are slightly below 1.0%. The process of numerical calculation using the generalized finite difference method has favorable stability, with various irregular clouds of points in a domain. As for the numerical analysis of structural, mechanical properties, it is absolutely essential to produce a configured model with a relatively uniform collocation of nodes (grids). The classical finite difference method needs orthogonal grids to gain equal results. The Jacobian ratio is also considered as the evaluation index of uniform grid quality in the conventional finite element method. In contrast, based on the multivariate Taylor series and the weighted moving least-squares technique, the GFDM has the advantage of obtaining stable numerical solutions from a configured model with irregular clouds of points. MathematicsMathematicsMathematicsMathematics 2020 2020 2020 2020, 8, ,8, x,8, x,8FOR x,FOR xFOR FOR PEER PEER PEER PEER REVIEW REVIEW REVIEW REVIEW 1313 13of 13of of21 21of 21 21

MathematicsMathematicsMathematicsMathematics 2020 2020 2020 2020, 8, ,,8 x8,, ,x8FOR x ,FOR xFOR FOR PEER PEER PEER PEER REVIEW REVIEW REVIEW REVIEW *3wwEhq *3=*3=*3== / AlZrO/ ahah/5/5==== 1313 13of 13of of21 of21 21 21 TableTableTableTable 4. 4. 4.Normalized 4.Normalized Normalized Normalized deflection deflection deflection deflection wwEhq wwEhq wwEhqAlAlAl/Al/ / ofof of an ofan an an A AlZrO /lZrOA/ lZrO/ 2 2 square2 square2 square square thick thick thick thick plate plate plate plate ah /5 ah/5 withwith with with fully fully fully fully clampedclampedclamped boundary boundary boundary conditions conditions conditions and and and different different different skew skew skew angles angles angles θ θ θ underθ under under the the the transverse transverse transverse uniform uniform uniform load load load q q. q. q. MathematicsMathematicsMathematicsMathematicsclamped 2020 2020 2020 2020, 8, 8, boundaryx,8, x ,8FOR x,FOR xFOR FOR PEER PEER PEER PEER conditions REVIEW REVIEW REVIEW REVIEW and *3= *3*3== *3different= skew angles under the transverse uniform==== load 1313 13.of 13of of21 21of 21 21 TableTableTableTable 4. 4. 4.Normalized 4.Normalized Normalized Normalized deflection deflection deflection deflection wwEhq wwEhq wwEhqwwEhqAl AlAl /Al/ / /of of of an ofan an an A AlZrO A/lZrOlZrOA/ /lZrO/ 2 2 square2 square2 square square thick thick thick thick plate plate plate plate ah ah/5 ah/5ah/5/5 with withwith with fully fully fully fully clamped boundary conditionsw* w* * * andθ θdifferent=θ=θ= =   skew anglesθθ=θ=θ= θ=θθ θ under the transverseθθ=θ=θ==    uniform loadθθ=θ=θ= q=qq. q MathematicsMathematicsMathematicsMaximumMathematicsMaximumMaximumclampedMaximumMathematicsclampedclamped 2020 2020 2020 2020boundary ,Deflection 8 Deflection,boundary ,,82020Deflection boundaryx8,,Deflection ,x8FOR x ,FOR ,xFOR8 FOR, PEER 1940 conditionsPEER PEER conditions PEER conditions w REVIEW REVIEW REVIEWw REVIEW and and and *3different *3= *3=different 0=*3different0= 0 0 skew skew skew angles angles angles 1515 15 15 under under under the the thetransverse transverse transverse303030 30 uniform uniform uniform==== load load load 4545 45. 4513 13. 13.of 13of of21 of2112 21 21 of 19 TableTableTableTable 4. 4. 4.Normalized 4.Normalized Normalized Normalized deflection deflection deflection deflection wwEhq wwEhq wwEhqwwEhqAlAlAl Al/ / / / of of of an ofan an an A AlZrO AlZrO/ A/lZrO/lZrO/ 2 2 2 square square2 square square thick thick thick thick plate plate plate plate ah ah/5 ah/5ah/5/5 with withwith with fully fully fully fully * * * * θθ=θ= =   θθ=θ=θ=θθθ θθ=θ==    θθ=θ=q=qq q MaximumMaximumMaximumclampedMaximumclampedclampedclamped boundaryDeflection boundary Deflection boundaryDeflection boundaryDeflection conditions conditions conditions conditions w w w w and and and andθ *3different *3=different= *3=different0*3 =different0= 0 0 skew skew skew skew angles angles angles θangles= 15 15 15 15 under under under under the the the thetransverse transverse θtransverse =transverse303030 30 uniform uniform uniform uniform==== load loadθ load =load 45 45 45 . 45 . . . TableTableTableTable 4. 4. 4.Normalized 4.Normalized Normalized Normalized deflection deflection deflection deflection wwEhq wwEhq wwEhqwwEhqAlAlAlAl/ / / /of of of an ofan an an A AlZrO A/lZrOAlZrO/ /lZrO/ 2 2 2 square 2 square square square thick thick thick thick plate plate plate plate ah ah/5 ah/5ah/5/5 with withwith with fully fully fully fully Table 4. Normalized deflection w = wE h3/q of an Al/ZrO square thick plate a/h = 5 with fully * * * *  ∗  Al θθθ  2         MaximumMaximumMaximumclampedMaximumclampedclampedclamped boundaryDeflection boundary Deflection boundaryDeflection boundaryDeflection conditions conditions conditions conditions w w w w and and and and θdifferentθ different= θdifferent= 0θdifferent=0= 0 0 skew skew skew skew angles angles angles anglesθθ=θ= 15θ= 15 =15 15 under θ under under under the the the thetransverse transverse θtransverse θtransverse=θ=30θ=30=30 30 uniform uniform uniform uniform load loadθ loadθ =loadθ= 45θ= 45q= 45q. 45 q . q. . clamped boundary conditions and different skew angles θ under the transverse uniform load q. nn=n=0n=0(ceramic)=0(ceramic)0(ceramic)(ceramic) * * * *                 MaximumMaximumMaximumMaximum Deflection Deflection Deflection Deflection w w w w θθ=θ=θ0=0= 0 0 θθ=θ=15θ=15=15 15 θθ=θ=30θ=30=30 30 θθ=θ=θ45=45=45 45 Maximum Deflection w∗ θ = 0◦ θ = 15◦ θ = 30◦ θ = 45◦ nn=n=0n=0(ceramic)=0(ceramic)0(ceramic)(ceramic) * * * *= * * * *= * * * *= * * * *= www=w=0.00830.0083=0.00830.0083 www=w=0.00750.0075=0.00750.0075 www=w=0.00560.0056=0.00560.0056 www=w=0.00330.0033=0.00330.0033 nn=n=0n=0(ceramic)=0(ceramic)0(ceramic)(ceramic) w* w=* *==*0.0083=0.0083 w* w=* *==*0.0075=0.0075 w* w=* *==*0.0056=0.0056 w* w=* *==*0.0033=0.0033 n = 0 (ceramic) w w0.00830.0083 w w0.00750.0075 w w0.00560.0056 w w0.00330.0033 nn=n=0n=0(ceramic)=0(ceramic)0(ceramic)(ceramic) ww* w*=w*=0.0083=*0.0083=0.00830.0083 ww* w*=w*=0.0075=*0.0075=0.00750.0075 ww* w*=w*=0.0056=*0.0056=0.00560.0056 ww* w*=w*=0.0033=*0.0033=0.00330.0033 == w = = = nn=n0.5n0.5=0.5 0.5 w∗ = 0.0083 ∗ 0.0075 w∗ 0.0056 w∗ 0.0003 * * * * * * * * * * * * * * * * www=w=0.0083=0.0083=0.00830.0083 www=w=0.0075=0.0075=0.00750.0075 www=w=0.0056=0.0056=0.00560.0056 www=w=0.0033=0.0033=0.00330.0033 nn=n=0.5n=0.5=0.5 0.5 ww* w*=w*=0.0110*=0.0110=0.01100.0110 ww* w*=w*=0.0100*=0.0100=0.01000.0100 ww* w*=w*=0.0075*=0.0075=0.00750.0075 ww* w*=w*=0.0044*=0.0044=0.00440.0044 n = 0.5 nn=n=0.5n=0.5=0.5 0.5 * * * * ww* w=w*=0.0110=*0.0110=0.01100.0110 ww* w=w*=0.0100=*0.0100=0.01000.0100 ww* w=w*=0.0075=*0.0075=0.00750.0075 ww* w=w*=0.0044=*0.0044=0.00440.0044 nn=n=0.5n=0.5=0.5 0.5 w∗ = 0.0110 w∗ = 0.0100 w∗ = 0.0075 w∗ = 0.0044 ww* w*=w*=0.0110=*0.0110=0.01100.0110 ww* w*=w*=0.0100=*0.0100=0.01000.0100 ww* w*=w*=0.0075=*0.0075=0.00750.0075 ww* w*=w*=0.0044=*0.0044=0.00440.0044 nn=n=1n=1 =1 1 ww* w*=w*=0.0110*=0.0110=0.01100.0110 ww* w*=w*=0.0100*=0.0100=0.01000.0100 ww* w*=w*=0.0075*=0.0075=0.00750.0075 ww* w*=w*=0.0044*=0.0044=0.00440.0044 n = 1 nn=n=1n=1 =1 1 * * * *= * * * *= * * * *= * * * *= www=w=0.01300.0130=0.01300.0130 www=w=0.01190.0119=0.01190.0119 www=w=0.00880.0088=0.00880.0088 www=w=0.00520.0052=0.00520.0052 = nn=n=1n=1 =1 1 w∗ = 0.0130 w∗ = 0.0119 w∗ 0.0088 w∗ = 0.0052 * * * * ww* w=w*=0.0130=*0.0130=0.01300.0130 ww* w=w*=0.0119=*0.0119=0.01190.0119 ww* w=w*=0.0088=*0.0088=0.00880.0088 ww* w=w*=0.0052=*0.0052=0.00520.0052 nn=n=1n=1 =1 1 ww* w*=w*=0.0130=*0.0130=0.01300.0130 ww* w*=w*=0.0119=*0.0119=0.01190.0119 ww* w*=w*=0.0088=*0.0088=0.00880.0088 ww* w*=w*=0.0052=*0.0052=0.00520.0052 n = (metal) nn=∞n=∞n=∞=∞(metal)(metal)(metal)∞(metal) * * * * * * * * * * * * * * * * www=w=0.0130=0.0130=0.01300.0130 www=w=0.0119=0.0119=0.01190.0119 www=w=0.0088=0.0088=0.00880.0088 www=w=0.0052=0.0052=0.00520.0052 = nn=∞n=∞n=∞=∞(metal)(metal)(metal)(metal) w∗ = 0.0237 w∗ = 0.0215 w∗ = 0.0160 w∗ 0.0095 * * * *= * * * *= * * * *= * * * *= www=w=0.02370.0237=0.02370.0237 www=w=0.02150.0215=0.02150.0215 www=w=0.01600.0160=0.01600.0160 www=w=0.00950.0095=0.00950.0095 nn=∞n=∞n=∞=∞(metal)(metal)(metal)(metal) * * * * * * * * * * * * * * * * A further study for staticwww= bendingw=0.0237=0.0237=0.02370.0237 of an FGMwww=w=0.0215=0.0215= (0.0215Al0.0215/ Fe 2 O 3 ) circlewww=w=0.0160=0.0160= thick0.01600.0160 plate Rw/ww=hw=0.0095=0.0095=0.0095100.0095 with torus TheThe effect =∞effect=∞=∞ on on various various levels levels of of irregular irregular clouds clouds of of points points in in a a domain domain is is also also investigated investigated herein. herein. ThewasThe effectn performednn effectn =∞(metal) on(metal)(metal)(metal) onvarious various using levels the levels proposedof irregularof irregular method. clouds clouds of The pointsof dimensionalpoints in ain domain a domain parameters is also is also investigated (the investigated diameter herein. herein.R = 1 of * *=*==* = * *=*==* = * *=*==* = * *=*==* = TheTheTheThe level level level level of of of irregular ofirregular irregular irregular clouds clouds clouds clouds of of of points ofpointsw pointsw pointswhw0.02370.0237 0.0237is 0.0237is is characte is characte characte characte rizedwrizedwrizedwrizedw0.0215 0.0215by 0.0215by 0.0215by bya a seriesa seriesa series series of wof w of wnumbers, ofwnumbers, 0.0160numbers, 0.0160numbers,0.01600.0160 shown shown shown shownww win win 0.0095 in0.0095 Figure in0.0095Figure 0.0095Figure Figure 9. 9. 9. 9. ThetheTheTheThe effect circleeffect effect effect on plate,on on variouson various various various the diameterlevels levels levels levels of of of irregular R ofirregular irregular irregular= 0.7 cloudsof clouds clouds theclouds hole,of of of points ofpoints points and points thein in in a ina gradient domaina domain adomain domain is indexis isalso isalso also also investigatedn investigated investigated= investigated1) were provided,herein. herein. herein. herein. as FigureFigureFigureFigure 10 10 10 depicts10 depicts depicts depicts the the the the relative relative relative relative deviation deviation deviation deviation* *=*=*= =versus versus versus versus vari vari vari variousousous* ous*= irregular*= irregular*= irregular= irregular levels levels levels levels of* of*= of*=sketching of*=sketching =sketching sketching nodes,3 nodes, nodes, nodes,* *= *comparing= *=comparing =comparing comparing TheTheTheThe level levelshown level level of of of irregular ofinirregular irregular Figuresirregular clouds clouds11clouds clouds–13 of. of Theof points ofpoints wpointsw pointsw normalizedw0.02370.0237 0.0237is 0.0237is is characte is charactecharacte characte maximumrizedwrizedwrizedwrizedw0.0215 0.0215by 0.0215by 0.0215by bya deflection a series a series a series series of wof wof wnumbers, ofwnumbers, 0.0160∗numbers, 0.0160numbers,=0.01600.0160wE Al shown hshown shown /shownqwofw win win FGM0.0095 in 0.0095Figure in0.0095Figure 0.0095Figure Figure plate 9. 9. 9. with9. withwithwithwith Thethe Thethe Thethe Thethe originaleffect originaleffect originaleffect originaleffect on on onnumerical onvariousnumerical variousnumerical variousnumerical various levels levels solutionlevels solutionlevels solution solution of of of irregular ofirregular ofirregular ofirregular of the ofthe the the present presentclouds presentclouds presentclouds clouds method of methodof methodof pointsmethod ofpoints points points with with in within within a in a a domaina a domain regulara a domain regular a domainregular regular is isclou iscloualso clouisalso cloualso alsod d investigated ofd investigated ofd investigated of investigatedpoints. ofpoints. points. points. In In In herein. Figure Inherein. Figure herein.Figure herein.Figure FigureFigureFigureFiguretorus 10 10 10 depicts10 depicts anddepicts depicts di theff the erentthe the relative relative relative relative boundary deviation deviation deviation deviation conditions versus versus versus versus vari under vari vari variousousous theous irregular irregular irregular transverse irregular levels levels levels levels uniform of of of sketching ofsketching sketching sketching load was nodes, nodes, nodes, nodes, studied, comparing comparing comparing comparing as shown in 10,TheThe10,The10, The10,the thelevel thelevel levelthe resultslevel results results of resultsof of irregular ofirregular withirregular withirregular with with different different different clouds differentclouds clouds clouds skew of skew of skew of skewpoints ofpoints points angles pointsangles angles angles is is is andcharacte ischaracteand characteand andcharacte gradie gradie gradie gradierizedrizedrizedntrizedntnt indices ntby indices by indices by indices bya a aseries seriesa areseries areseries are are evaluated. ofevaluated. of evaluated. of evaluated. numbers, ofnumbers, numbers, numbers, It It Itis shown Itis shown isobserved shown isobservedshown observed observed in in in Figure in Figure that Figurethat Figurethat that the the 9. the9. the 9. 9. withwithwithwith Thethe FiguresThethe Thethe Thethe originaleffect originaleffect originaleffect originaleffect 11 on– on 13onnumerical onvariousnumerical .numericalvarious various Thenumerical various next levels levels solutionlevels examplesolutionlevels solution solution of of of irregularof irregular ofirregular ofirregular of the of staticthe the thepresent present clouds presentclouds bending, presentclouds clouds method of method of methodof pointsmethodof withpoints points points with the with inwith in within gradient ain a domaina a domainregulara domain regular adomainregular regular index is clouis iscloualso clouisalso cloualson dalso= d investigateddof investigated1 ofd investigatedof andinvestigatedpoints. ofpoints. points. points. diff erentIn In In herein. Figure In herein. Figure herein.Figure diametersherein.Figure distributionFigureFiguredistributionFiguredistributionFiguredistribution 10 10 10 depicts10 depicts depicts ofdepicts of of the ofthe the the irregularthe irregularthe relativeirregular relativeirregular relative relative cloud deviation clouddeviation clouddeviation clouddeviation of of of points of pointsversus versuspoints versuspoints versus has hasvari varihas varihas varia ousa ousslighta ousslighta ousslight irregularslight irregular irregular irregulareffect effect effect effect levelson levelson levelson levelsonthe the the ofthe ofnumerical of numerical sketching ofnumericalsketching numericalsketching sketching solution nodes, solution nodes, solution nodes, solutionnodes, comparing comparingusing comparingusing comparingusing using the the the the 10,The10,The10,The The10,the thelevel thelevelof levelthe resultslevel results theresults ofresults of of hole,irregularof irregularwithirregular with irregularwith with was different different different clouds calculated.differentclouds clouds clouds skew of skew of skew of skewpointsof points points The angles pointsangles angles angles boundaryis is is andcharacte isandcharacte andcharacte characteand gradie gradie gradie gradie conditionrizedrizedrizedntrizedntnt indices nt by indices by indices by indices bya ofa seriesa seriesa clampedareseries areseries are areevaluated. ofevaluated. ofevaluated. of evaluated. numbers,of numbers, supportnumbers, numbers, It It It is shown aroundItis shown isobserved shown isobserved shownobserved observed in thein in Figurein Figure that holeFigure that Figurethat that the the was9. the 9. the 9. 9. set. proposedwithwithproposedwithproposedwithproposed the the the the original originalmethod. originalmethod. originalmethod. method. numerical numerical numericalAll Allnumerical All All of of of the ofthe thesolution thesolution relativesolution relative solutionrelative relative of ofdeviations of deviations the ofdeviationsthe deviationsthe the present present present present are are aremethod methodare slightlymethod slightlymethod slightly slightly with with belowwith belowwith below belowa a regulara regulara 1.0%. regular 1.0%. regular1.0%. 1.0%. Theclou Theclou Theclou Theclou dprocess dprocess dofprocess ofdprocess of points. ofpoints. points. points.of of of numerical ofInnumerical Innumerical In numerical Figure InFigure Figure Figure distributionFiguredistributionFiguredistributionFigureFiguredistributionThe 10 10 10 depicts e10 depictsff depicts ect ofdepicts of of the ofthe the thethe irregularthe irregularthe relativeirregular FGMrelativeirregular relative relative circle clouddeviation cloud deviation clouddeviation clouddeviation plate of of of points withof versuspoints pointsversus versuspoints versus di hasff varihas erenthasvari varihas varia ousa slightousa ousslight diametersaslightous irregularslight irregular irregular irregulareffect effect effect effect of levelson levelson thelevelson levelsonthe the holethe ofthe ofnumerical of numerical sketchingof numerical sketchingwas numericalsketching sketching studied solution nodes,solution nodes,solution nodes, solutionnodes, under comparing comparingusing comparingusing thecomparingusing using transverse the the the the calculation10,10,calculation10,calculation 10,calculationthe the the the results results results results using using usingwith usingwith with with the thedifferent thedifferent differentthe differentgeneralized generalized generalizedgeneralized skew skew skew skew angles finiteangles finiteangles finiteanglesfinite anddifference anddifference and difference anddifference gradie gradie gradie gradie nt methodnt methodnt methodindices ntmethodindices indices indices has hasare hasare hasare are favorableevaluated. favorableevaluated. favorableevaluated. favorableevaluated. stability, stability,It It stability, Itstability,is Itis isobserved isobserved observed observed with with withwith that variousthat variousthat various thatvarious the the the 3the proposedwithproposedwithproposedwithwithproposed the uniformthe the the original originalmethod. originalmethod. originalmethod. method. load, numerical numerical Allnumerical numericalAll asAll Allof shown of of the ofthe thesolution thesolution relativesolution inrelative solutionrelative relative Figures of ofdeviations of deviations theof deviations the deviationsthe13 the present –present 16present present. Theare are aremethod aremethod slightlymethod normalizedslightlymethod slightly slightly with with belowwith below withbelow belowa maximuma regulara regulara 1.0%.regular 1.0%. regular1.0%. 1.0%. Theclou The clou Theclou deflectionTheclou dprocess dprocess dofprocess dofprocess of points.of points. points. points.of wof of ∗ numerical ofInnumerical= numericalIn In numerical FigurewEIn Figure Figure FigureAlh / q of irregulardistributiondistributionirregulardistributionirregulardistributionirregular clouds clouds cloudsof cloudsof of the ofthe the ofthe ofirregular ofirregular pointsofirregular pointsirregularpoints points in incloud cloud in cloud ain a cloud a domain.a domain. domain. of domain.of of points ofpoints points points As As As hasAshas forhasfor forhas fora thea theslighta the slighta slightthe slightnumerical numericalnumerical numericaleffect effect effect effect on on analysison analysisonthe analysis the analysisthe the numerical numerical numerical numerical of of of structural,ofstructural, structural, structural, solution solution solution solution mechanical mechanicalusingmechanical using mechanicalusing using the the the the calculation10,calculation10,calculation10, 10,calculationthe the thethe the results results results FGMresults using usingusing withusing platewith with with the thedifferent the withdifferent thedifferent different generalized generalizedgeneralized torusgeneralized skew skew skew andskew angles finite angles di finiteangles finite anglesfffiniteerent differenceand anddifference anddifference anddifference gradient gradie gradie gradie gradie ntmethod nt method indicesnt methodindicesnt methodindices indices indices has( hasnare has are =hasare are favorableevaluated. 0,favorableevaluated. favorableevaluated. 0.5,favorableevaluated. 1, 2, stability, Itstability, stability, It ) Itstability,is wasItis isobserved isobserved observed observed calculated with withwith with that various thatvarious thatvarious thatvarious usingthe the the the the properties,proposedproposedproperties,proposedproperties,proposedproperties, method. method. itmethod. itmethod. it isit is is absolutelyis absolutely absolutelyAll Allabsolutely All All of of of the ofthe the essentialthe essentialrelative relativeessential relativeessential relative deviations todeviations todeviations todeviations toproduce produce produceproduce are are area aare slightlyaconfiguredslightlya configuredslightly configuredslightlyconfigured below below below below model model model1.0%. model1.0%. 1.0%. 1.0%. with withThe withThe withThe ∞The processa processa process a relativelyprocessa relatively relativelyrelatively of of of numerical of numerical uniformnumerical uniformnumerical uniformuniform irregulardistributionirregulardistributionirregulardistributiondistributionirregularpresent clouds cloudsclouds of clouds method,ofof theof thetheof the ofirregularof irregular pointsofirregular pointsirregularpoints changingpoints in incloud incloud acloud in acloud alonga domain. adomain. domain. of domain.of of pointsof withpoints points points As As theAs has Asfor has hasfor diameterforhas fora the a theslighta the slighta the slight numericalslight numericalnumerical ofnumericaleffect effect effectthe effect structuralon on analysis on analysisontheanalysis the analysisthe the numerical numerical numerical hole,numericalof ofof structural, of structural, asstructural, structural, shownsolution solution solution solution mechanical in mechanical usingmechanical Figureusing mechanicalusing using the the 17the the . It is collocationcalculationcalculationcollocationcalculationcollocationcalculationcollocation of usingof using ofusing nodes ofusingnodes nodes nodes the the the(grids). the (grids). generalized(grids). generalized(grids). generalizedgeneralized The The The The classical classical finiteclassical finiteclassicalfinite finite difference differencefinite differencefinite finitedifference finite difference difference difference difference method methodmethod method method method method has methodhas hashas favorable favorableneedsfavorable needsfavorable needs needs orthogonal orthogonal orthogonalstability, stability,orthogonalstability, stability, withgrids withgrids withgrids withgrids variousto variousto various to variousgain togain gain gain properties,proposedproperties,proposedproperties,proposedproposedproperties,observed method. method. itmethod. itmethod. it isit thatis is absolutelyis absolutely absolutelyAll theabsolutelyAll All All of rangeof of theof the the essentialthe ofrelativeessential essentialrelative relativeessential sizerelative of deviations to thedeviations todeviations todeviations toproduce structuralproduce produceproduce are are aare a are holeaslightly configuredslightly aconfiguredslightly configuredslightlyconfigured has below abelow below nonlinear below model model model1.0%. model1.0%. 1.0%. 1.0%. with e withTheff withThe ectwithThe The aprocess on processa processa relativelyprocess arelatively structural relativelyrelatively of of of numericalof numerical uniformnumerical stiuniformnumerical uniformuniformffness with equalirregularirregularequalirregularequalirregularequal results. results. results. results. clouds cloudsclouds clouds The The The The of Jacobian of Jacobianof Jacobian ofpoints Jacobian pointspoints points ratio inratio inratioin ratio ain a isa domain.isa domain.isalsodomain. isalsodomain. also also considered considered considered consideredAs AsAs As for forfor for the as the asthe as the theas numericalthe numericalthe numerical the evaluationnumerical evaluation evaluation evaluation analysis analysisanalysis indexanalysis index index index of of of uniform ofstructural,uniform structural,uniformstructural, uniform3structural, grid grid grid mechanicalgrid mechanical qualitymechanical qualitymechanical quality quality in in in in collocationcalculationcollocationcalculationcollocationcalculationcalculationcollocationdifferent of usingof using of usingnodes gradientofusingnodes nodes nodes the the the(grids). the (grids). generalized(grids). indices. generalized(grids). generalizedgeneralized The The The TheThe classical classical finiteclassical normalized finiteclassical finitefinite difference finitedifference differencefinitefinitedifference finite maximum difference difference difference difference method method methodmethod deflection method method method has methodhas hashas favorable favorableneeds favorablewneeds favorable∗needs needs= wE orthogonal orthogonalorthogonal stability,Al orthogonalstability,h stability,stability,/q of thewithgrids withgrids withgrids withgrids FGM variousto variousto various to variousgain platetogain gain gain with theproperties,properties,theproperties,theproperties,the conventional conventional conventional conventional it ith it isit is is finiteabsolutelyis absolutelyfinite absolutelyfinite absolutelyfinite element element element element essential essential essential method.essential method. method. method. to to toIn toInproduce produceIn contrastproduceIn contrastproduce contrast contrast a, a ,based a ,configuredbased a configured,based configuredbasedconfigured on on on onthe the the the modelmultivariate modelmultivariate modelmultivariate modelmultivariate with with withwith aTaylor aTaylor aTaylorrelatively a Taylorrelatively relativelyrelatively series series series series uniformanduniform and uniformand uniformand the the the the equalirregularequalirregularequalirregularirregularequal torusresults. results. results. results.clouds cloudsRclouds clouds The =The The The0.7of Jacobian of Jacobianof Jacobianofpoints andJacobian pointspoints points di ratioff inratio erentinratio inratioina a isa domain. isa domain.isalso boundarydomain. isalsodomain. also also considered considered considered consideredAs AsAs conditionsAsfor forfor forthe as the asthe as the theas numerical the numericalthenumerical the wasevaluationnumerical evaluation evaluation evaluation analyzed, analysis analysisanalysis indexanalysis index index index changing of of of uniform ofstructural, uniform structural,uniformstructural, uniformstructural, along grid grid grid withmechanicalgrid mechanicalqualitymechanical qualitymechanical quality thequality gradient in in in in weightedcollocationcollocationweightedcollocationweightedcollocationweighted moving ofmoving ofmoving of moving nodes ofnodes nodes nodes least-squares least-squares(grids). least-squares(grids). least-squares(grids). (grids). The The The The classical technique, classicaltechnique,classical technique,classicaltechnique, finite finite finite finitethe the the differencethe difference GFDMdifference GFDMdifference GFDMGFDM has hasmethod methodhas methodhas method the the thethe advantageneeds advantageneeds advantageneeds advantageneeds orthogonal orthogonal orthogonal orthogonal of of of ofobtaining obtaining obtaining obtaininggrids grids grids grids toto stableto stablegain tostablegain stablegain gain theproperties,theproperties,theproperties,properties,the conventional conventional indexconventional conventional underit it it isit is is finiteabsolutelyis thefiniteabsolutely finiteabsolutely absolutelyfinite transverse element element element element essential essential essential method.essential uniformmethod. method. method. to to toIn load,to produceIn In produce contrast produceIn contrastproduce contrast contrast as shown a, a ,based a, configuredbased a ,basedconfigured configuredbased inconfigured Figureon on on onthe the the the model18multivariate modelmultivariate model.multivariate modelmultivariate The with e withff withectwith Tayloraof aTaylor aTaylor relatively deformationa Taylorrelatively relativelyrelatively series series series series uniformand uniformand uniformand withuniform and the the the athe fully numericalequalequalnumericalequalnumericalequalnumerical results. results. results. results. solutions solutions solutions solutionsThe The The The Jacobian Jacobian Jacobianfrom Jacobianfrom from from a a ratio configured a ratio configured a ratioconfigured ratioconfigured is is isalso isalso also also modelconsidered modelconsidered modelconsidered modelconsidered with with with with as irregular as irregular as irregular the asirregularthe the the evaluation evaluation evaluation evaluationclouds clouds clouds clouds of index of index ofindex points. ofpoints.index points. points. of of of uniform ofuniform uniform uniform grid grid grid grid quality quality quality quality in in in in weightedcollocationweightedcollocationweightedcollocationcollocationweightedclamped moving ofmoving movingof of movingnodesof porednodes nodes nodes least-squares least-squares(grids). plateleast-squares(grids). least-squares(grids). (grids). is relativelyThe The The The classicaltechnique, classical technique,classical technique,classicaltechnique, minor finite finite finite thefinite in the the spitedifferencethe difference GFDMdifference GFDMdifference GFDMGFDM of the has hasmethod changinghasmethod methodhas method the the thethe advantageneeds advantageneeds advantageneeds gradientadvantageneeds orthogonal orthogonal orthogonal orthogonal index,of of of ofobtaining obtaining obtaining obtaining comparedgrids grids grids grids to tostable to stablegainto stablegain stable withgain gain the thethethethe conventional conventional conventional conventional finite finite finite finite element element element element method. method. method. method. In In In contrast Incontrast contrast contrast, ,based ,based ,based based on on on onthe the the the multivariate multivariate multivariate multivariate Taylor Taylor Taylor Taylor series series series series and and and and the the the the numericalequalnumericalequalnumericalequalequalnumerical otherresults. results. results. results. solutions boundarysolutions solutions solutionsThe The The The Jacobian Jacobian Jacobianfrom Jacobianfrom conditions.from from a a ratioconfigureda ratio configured aratioconfigured ratioconfigured is is It isalso is followsalso also also modelconsidered modelconsidered modelconsidered modelconsidered that with with with the with asirregular FGMsas irregular as irregular the as irregularthe the the evaluation evaluation with evaluation evaluationclouds clouds clouds variousclouds of indexof indexof index points. ofindexpoints. gradient points. points. of of of uniformof uniform uniform uniform indices grid grid grid have grid quality quality quality aquality substantial in in in in weightedweightedweightedweighted moving moving movingmoving least-squares least-squares least-squaresleast-squares technique, technique, technique,technique, the the thethe GFDM GFDM GFDMGFDM has has hashas the the thethe advantage advantage advantageadvantage of of of ofobtaining obtaining obtainingobtaining stable stable stablestable thethethethe conventional conventional nonlinearconventional conventional e fffinite ectfinite finite finite on element element mechanicalelement element method. method. method. method. behavior. In In In contrastIn contrast contrast contrast Some, ,based engineering ,based ,based based on on on onthe the the the significancemultivariate multivariate multivariate multivariate and Taylor Taylor Taylor Taylor reference series series series series valuesand and and and the the shouldthe the numericalnumericalnumericalnumerical solutions solutions solutions solutions from from from from a a configureda configureda configured configured model model model model with with with with irregular irregular irregular irregular clouds clouds clouds clouds of of of points. ofpoints. points. points. weightedweightedweightedweightedbe provided moving moving movingmoving by least-squares least-squares theleast-squaresleast-squares proposed technique, method,technique, technique,technique, with the the thethe some GFDM GFDM GFDMGFDM operating has has hashas the the conditions.thethe advantage advantage advantageadvantage of of of of obtaining obtaining obtainingobtaining stable stable stablestable numericalnumericalnumericalnumerical solutions solutions solutions solutions from from from from a a configureda configureda configured configured model model model model with with with with irregular irregular irregular irregular clouds clouds clouds clouds of of of points.of points. points. points.

Mathematics 2020, 8, x FOR PEER REVIEW 13 of 20

n =1

w* = 0.0130 w* = 0.0119 w* = 0.0088 w* = 0.0052

n =∞(metal)

w* = 0.0237 w* = 0.0215 w* = 0.0160 w* = 0.0095

The effect on various levels of irregular clouds of points in a domain is also investigated herein. The level of irregular clouds of points is characterized by a series of numbers, shown in Figure 9. Figure 10 depicts the relative deviation versus various irregular levels of sketching nodes, comparing with the original numerical solution of the present method with a regular cloud of points. In Figure 10, the results with different skew angles and gradient indices are evaluated. It is observed that the distribution of the irregular cloud of points has a slight effect on the numerical solution using the proposed method. All of the relative deviations are slightly below 1.0%. The process of numerical calculation using the generalized finite difference method has favorable stability, with various irregular clouds of points in a domain. As for the numerical analysis of structural, mechanical properties, it is absolutely essential to produce a configured model with a relatively uniform collocation of nodes (grids). The classical finite difference method needs orthogonal grids to gain equal results. The Jacobian ratio is also considered as the evaluation index of uniform grid quality in the conventional finite element method. In contrast, based on the multivariate Taylor series and the Mathematicsweighted 2020moving, 8, 1940 least-squares technique, the GFDM has the advantage of obtaining 13stable of 19 numerical solutions from a configured model with irregular clouds of points.

(a)

Mathematics 2020, 8, x FOR PEER REVIEW 14 of 20 Mathematics 2020, 8, x FOR PEER REVIEW 14 of 20 (b) (c)

(d) (e) (d) (e) Figure 9. The level of irregular clouds of points in a domain: (a) 0 level, (b) 1 level, (c) 2 level, (d) 3 Figure 9. The level of irregular clouds of points in a domain: (a) 0 level; (b) 1 level; (c) 2 level; (d) 3 level; Figurelevel, 9. and The (e level) 4 level. of irregular clouds of points in a domain: (a) 0 level, (b) 1 level, (c) 2 level, (d) 3 and (e) 4 level. level, and (e) 4 level.

Figure 10. The relative deviation, based on the numerical solution of the present method with a regular Figure 10. The relative deviation, based on the numerical solution of the present method with a cloud of points versus various irregular levels of sketching nodes. Figureregular 10. cloud The relativeof points deviation, versus various based irregular on the numeri levels ofcal sketching solution nodes.of the present method with a regular cloud of points versus various irregular levels of sketching nodes. = A further study for static bending of an FGM ( AlFeO/ 23) circle thick plate Rh/10 with torus = wasA performedfurther study using for the static proposed bending method. of an FGM The (dimensionalAlFeO/ 23) circle parameters thick plate (the diameterRh/10 withR =1 torus of the h = wascircle performed plate, the using diameter the proposed R = 0.7 method. of the hole, The and dimensional the gradient parameters index n = (the1 ) were diameter provided, R 1 as of shown the h = circle plate, the diameter R = 0.7 of the hole, and the gradient*3 =index n 1 were provided, as shown in Figures 11–13. The normalized maximum deflection wwEhqAl / of) FGM plate with torus and wwEhq*3= / in differentFigures 11–13.boundary The conditionsnormalized under maximum the transverse deflection uniform loadAl was of FGMstudied, plate as shownwith torus in Figures and different11–13. Theboundary next example conditions of static under bending, the transverse with the uniform gradient load index was n studied, = 1 and asdifferent shown diametersin Figures of 11–13.the hole, The nextwas calculated.example of The static boundary bending, condition with the ofgradient clamped index support n = 1 aroundand different the hole diameters was set. ofThe theeffect hole, of was the calculated. FGM circle The plate boundary with different condition diam ofeters clamped of the support hole was around studied the under hole wasthe transverseset. The *3= effectuniform of the load, FGM as circle shown plate in Figureswith different 13–16. diamThe normalizedeters of the maximumhole was studied deflection under wwEhq the transverseAl / of the wwEhq*3= / uniformFGM plate load, with as shown torus andin Figures different 13–1 gradient6. The normalizedindices ()n =∞ maximum0, 0.51,, 2, wasdeflection calculated usingAl the ofpresent the ()=∞ FGMmethod, plate changingwith torus along and differentwith the diametergradient indicesof the struct n ural0, 0.51, hole,, 2, aswas shown calculated in Figure using 17. theIt is presentobserved method,that the changing range of alongsize of with the structuralthe diameter hole of has the astruct nonlinearural hole, effect as on shown structural in Figure stiffness 17. It withis observed different *3= thatgradient the range indices. of size The of the normalized structural maximum hole has a nodeflectionnlinear effect wwEhq onAl structural/ of the stiffness FGM plate with differentwith torus *3 h wwEhq= / gradientR = 0.7 indices. and different The normalized boundary maximumconditions deflectionwas analyzed, changingAl of along the FGMwith theplate gradient with torus index Rhunder= 0.7 andthe transversedifferent boundary uniform load,conditions as shown was analyzed,in Figure 18.changing The effect along of with deformation the gradient with index a fully underclamped the transversepored plate uniform is relatively load, minor as shown in spite in Fi ofgure the changing18. The effect gradient of deformation index, compared with awith fully the clampedother boundary pored plate conditions. is relatively It follows minor that in spite the FGMs of the withchanging various gradient gradient index, indices compared have a substantialwith the othernonlinear boundary effect conditions. on mechanical It follows behavior. that the Some FGMs engineering with various significance gradient andindices reference have a values substantial should nonlinearbe provided effect by on the mechanical proposed behavior. method, withSome some engineering operating significance conditions. and reference values should be provided by the proposed method, with some operating conditions.

Mathematics 2020, 8, x FOR PEER REVIEW 15 of 20

Mathematics 2020, 8, 1940 14 of 19 Mathematics 2020, 8, x FOR PEER REVIEW 15 of 20

Figure 11. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and fully clamped Al , boundary conditions under a transverse uniform load q . Figure 11. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and fully clamped Al , 3 h boundaryFigure conditions 11. Normalized under maximum a transverse deﬂection uniformw∗ = loadwEAl hq ./ q, with torus R = 0.7 and fully clamped Figure 11. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and fully clamped boundary conditions under a transverse uniform load Alq. , boundary conditions under a transverse uniform load q .

Figure 12. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and external Al , 3 h clampedFigure boundary 12. Normalized conditions maximum under deﬂection a transversew∗ = wE uniformAlh /q, withload torusq . R = 0.7 and external clamped Figure 12. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and external boundary conditions under a transverse uniform load q. Al , clamped boundary conditions under a transverse uniform load q . Figure 12. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and external Al , clamped boundary conditions under a transverse uniform load q .

3 h Figure 13. Normalized maximum deﬂection w∗ = wEAlh /q, with torus R = 0.7 and internal clamped Figure 13. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and internal boundary conditions under a transverse uniform load q. Al , clamped boundary conditions under a transverse uniform load q . Figure 13. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and internal Al , clamped boundary conditions under a transverse uniform load q . Figure 13. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.7 and internal Al , clamped boundary conditions under a transverse uniform load q .

Mathematics 2020, 8, x FOR PEER REVIEW 16 of 20

MathematicsMathematics 20202020, 8, x, 8FOR, 1940 PEER REVIEW 15 of 1916 of 20

Figure 14. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.3 and internal Al , clamped boundary conditions under a transverse uniform load q . Figure 14. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.3 and internal Al , 3 h clampedFigure boundary 14. Normalized conditions maximum under deﬂection a transversew∗ = wE uniformAlh /q, load with torusq . R = 0.3 and internal clamped Figure 14. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.3 and internal boundary conditions under a transverse uniform load q. Al , clamped boundary conditions under a transverse uniform load q .

Figure 15. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.5 and internal Al , 3 h clampedFigure boundary 15. Normalized conditions maximum under deﬂection a transversew∗ = wE uniformAlh /q, load with torusq . R = 0.5 and internal clamped Figure 15. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.5 and internal boundary conditions under a transverse uniform load q. Al , clamped boundary conditions under a transverse uniform load q . Figure 15. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.5 and internal Al , clamped boundary conditions under a transverse uniform load q .

3 h Figure 16. Normalized maximum deﬂection w∗ = wEAlh /q, with torus R = 0.8 and internal clamped Figure 16. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.8 and internal boundary conditions under a transverse uniform load q. Al , clamped boundary conditions under a transverse uniform load q . Figure 16. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.8 and internal Al , clamped boundary conditions under a transverse uniform load q . Figure 16. Normalized maximum deflection wwEhq*3= / with torus Rh = 0.8 and internal Al , clamped boundary conditions under a transverse uniform load q .

Mathematics 2020, 8, x FOR PEER REVIEW 17 of 20

MathematicsMathematics 20202020, 8,, x8 ,FOR 1940 PEER REVIEW 16 of 1917 of 20

Figure 17. Normalized maximum deflection wwEhq*3= / versus diameter Rh of the hole with Al 3 h internalFigure clamped 17. Normalized boundary maximum conditions deﬂection under aw transverse∗ = wEAlh uniform/q versus load diameter q . R of the hole with *3= h Figureinternal 17. clampedNormalized boundary maximum conditions deflection under a transversewwEhqAl uniform/ versus load qdiameter. R of the hole with internal clamped boundary conditions under a transverse uniform load q .

3 h Figure 18. Normalized maximum deflection w∗ = wEAlh /q with a hole R = 0.7, ranging with the Figure 18. Normalized maximum deflection wwEhq*3= / with a hole Rh = 0.7 ranging with the gradient index under a transverse uniform load q. Al , gradient index under a transverse uniform load q . 5. Conclusions wwEhq*3= / h = Figure 18. Normalized maximum deflection Al with a hole R 0.7 , ranging with the 5. ConclusionsgradientIn this index paper, under a simple a transverse and easy-to-implement uniform load q domain-type. meshless method (GFDM) is applied to simulate the bending behavior of functionally graded material plates based on the first-order shearIn this deformation paper, a simple theory. and The easy-to-implement process of using the GFDMdomain-type to analyze meshless the governing method equations (GFDM) of is theapplied 5. Conclusions to simulateFGM plate the highlights bending thebehavior features of of functionally simple mathematics graded withoutmaterial meshing plates based generation on the and first-order numerical shear deformationintegrationIn this paper, theory. in the a GFDM.simpleThe process Someand easy-to-implement benchmarkof using the examples GFDM domain-type areto analyze used to verify the meshless governing the accuracy method equations and (GFDM) availability of isthe applied FGM plateto simulateof highlights the method. the bending the Numerical features behavior results of simpleof calculated functionally mathem by the gratics generalizedaded without material finite meshing plates difference based generation method on the in first-orderand this papernumerical shear integrationdeformationagree well in theory. withthe GFDM. those The studied processSome by benchmark otherof using mentioned the examples GFDM methods. toare analyze used The convergenceto the verify governing the and accuracy computational equations and availabilityof costthe FGM of the GFDM are investigated with different boundary conditions and various values of the gradient ofplate the method.highlights Numerical the features results of calculatedsimple mathem by theatics generalized without finite meshing difference generation method and in thisnumerical paper index. A further study shows that the GFDM has the advantage of obtaining stable numerical solutions agreeintegration well with in the those GFDM. studied Some by benchmarkother mentioned examples methods. are used The to convergence verify the accuracy and computational and availability cost from a configured model with irregular clouds of points. Then, the effects of material constants, of the method. Numerical results calculated by the generalized finite difference method in this paper of thegeometry, GFDM andare skewinvestigated angles of with the FGM different plate boundary are also studied conditions by the proposedand various method. values It is of observed the gradient index.agreethat, wellA di furtherff witherent those from study the studied isotropicshows by that materials,other the mentioned GFDM the FGMs has methods. with the various advantage The gradientconvergence of indicesobtaining and have computational stable a substantial numerical cost solutionsof thenonlinear GFDM from e ffare ecta investigatedconfigured on mechanical model with behavior. different with The irregular applicationboundary clouds conditions of the of meshless points. and generalized variousThen, the values finiteeffects of diff theoference materialgradient constants,index.method A furthergeometry, has been study preliminarily and showsskew angles verifiedthat the of to the analyzeGFDM FGM FGMhas plate platethe are advantage bending also studied behavior, of byobtaining andthe theproposed applicationstable method.numerical to It issolutions observedmore complex from that, a different structural configured from analysis model the will isotropic with be further irregular materials, studied cl oudsthe in the FGMs of future. points. with variousThen, the gradient effects indices of material have aconstants, substantial geometry, nonlinear and effect skew on angles mechanical of the behavior.FGM plate The are applicationalso studied of by the the meshless proposed generalized method. It finiteis observed difference that, method different has from been the preliminarily isotropic materials, verified theto analyze FGMs with FGM various plate bending gradient behavior, indices haveand thea substantial application nonlinear to more complexeffect on structural mechanical analysis behavior. will Thebe further application studied of inthe the meshless future. generalized finite difference method has been preliminarily verified to analyze FGM plate bending behavior, and the application to more complex structural analysis will be further studied in the future.

Mathematics 2020, 8, 1940 17 of 19

Author Contributions: The study was carried out in collaboration between all authors. Y.-D.L. completed the main part of this paper and gave examples to simulate the bending behavior of functionally graded material plates using a novel domain-type meshless method (GFDM); Z.-C.T. and Z.-J.F. corrected the main theorems and refined the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: The work described in this paper was supported by the National Science Fund of China (Grant No. 11772119, 11572111), the Fundamental Research Funds for the Central Universities (Grant No. B200202124), the Foundation for Open Project of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University Of Aeronautics And Astronautics) (Grant No. MCMS-E-0519G01) and the Six Talent Peaks Project in Jiangsu Province of China (Grant No. 2019-KTHY-009). Acknowledgments: The authors express their gratitude to the referees for their helpful suggestions. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

References

1. Zenkour, A.M.; Hafed, Z.S.; Radwan, A.F. Bending analysis of functionally graded nanoscale plates by using nonlocal mixed variational formula. Mathematics 2020, 8, 1162. [CrossRef] 2. Chashmi, M.J.; Fathi, A.; Shirzad, M.; Jafari-Talookolaei, R.A.; Bodaghi, M.; Rabiee, S.M. Design and analysis of porous functionally graded femoral prostheses with improved stress shielding. Designs 2020, 4, 12. [CrossRef] 3. Scheithauer, U.; Weingarten, S.; Johne, R.; Schwarzer, E.; Abel, J.; Richter, H.-J.; Moritz, T.; Michaelis, A. Ceramic-based 4D components: Additive manufacturing (AM) of ceramic-based functionally graded materials (FGM) by thermoplastic 3D printing (T3DP). Materials 2017, 10, 1368. [CrossRef][PubMed] 4. Bodaghi, M.; Noroozi, R.; Zolfagharian, A.; Fotouhi, M.; Norouzi, S. 4D printing self–morphing structures. Materials 2019, 12, 1353. [CrossRef][PubMed] 5. Bodaghi, M.; Damanpack, A.R.; Liao, W.-H. Adaptive metamaterials by functionally graded 4D printing. Mater. Des. 2017, 135, 26–36. [CrossRef] 6. Reddy, B.; Reddy, A.; Kumar, J.; Reddy, K. Bending analysis of laminated composite plates using finite element method. Int. J. Eng. Sci. Technol. 2012, 4, 177–190. [CrossRef] 7. Pham, Q.H.; Pham, T.D.; Trinh, Q.V.; Phan, D.H. Geometrically nonlinear analysis of functionally graded shells using an edge-based smoothed MITC3 (ES-MITC3) finite elements. Eng. Comput. 2020, 36, 1069–1082. [CrossRef] 8. Bui, T.Q.; Nguyen, M.N. A moving Kriging interpolation-based meshfree method for free vibration analysis of Kirchhoff plates. Comput. Struct. 2011, 89, 380–394. [CrossRef] 9. Bui, T.Q.; Nguyen, M.N. Eigenvalue Analysis of Thin Plate with Complicated Shapes by A Novel Mesh-free Method. Int. J. Appl. Mech. 2011, 3, 21–46. [CrossRef] 10. Bui, T.Q.; Nguyen, M.N.; Zhang, C. An efficient meshfree method for vibration analysis of laminated composite plates. Comput. Mech. 2011, 48, 175–193. [CrossRef] 11. Dinis, L.M.J.S.; Jorge, R.M.N.; Belinha, J. An Unconstrained Third-Order Plate Theory Applied to Functionally Graded Plates Using a Meshless Method. Mech. Adv. Mater. Struct. 2010, 17, 108–133. [CrossRef] 12. Wu, C.-P.; Yang, S.-W.; Wang, Y.-M.; Hu, H.-T. A meshless collocation method for the plane problems of functionally graded material beams and plates using the DRK interpolation. Mech. Res. Commun. 2011, 38, 471–476. [CrossRef] 13. Zhu, P.; Liew, K.M. Free vibration analysis of moderately thick functionally graded plates by local Kriging meshless method. Compos. Struct. 2011, 93, 2925–2944. [CrossRef] 14. Zhou, H.; Qin, G.; Wang, Z. Heat conduction analysis for irregular functionally graded material geometries using the meshless weighted least-square method with temperature-dependent material properties. Numer. Heat Transf. Part B Fundam. 2019, 75, 1–13. [CrossRef] 15. Krysl, P.; Belytschko, T. Analysis of thin plates by the element-free Galerkin method. Comput. Mech. 1995, 17, 26–35. [CrossRef] 16. Liu, G.R.; Chen, X.L.; Reddy, J.N. Buckling of Symmetrically Laminated Composite Plates Using the Element-free Galerkin Method. Int. J. Struct. Stab. Dyn. 2002, 2, 281–294. [CrossRef] Mathematics 2020, 8, 1940 18 of 19

17. Gilhooley, D.F.; Batra, R.C.; Xiao, J.R. Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radius basis functions. Compos. Part B 2008, 39, 414–427. [CrossRef] 18. Sahraee, S.; Saidi, A.R. Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory. Eur. J. Mech. A/Solids 2009, 28, 974–984. [CrossRef] 19. Fu, Z.-J.; Chen, W. A truly boundary-only meshfree method applied to Kirchhoff plate bending problem. Adv. Appl. Math. Mech. 2009, 1, 341–352. 20. Fu, Z.-J.; Chen, W.; Yang, W. Winkler plate bending problems by a truly boundary-only boundary particle method. Comput. Mech. 2009, 44, 757–763. [CrossRef] 21. Benito, J.J.; Urena, F.; Gavete, L. Influence of several factors in the generalized finite difference method. Appl. Math. Model. 2001, 25, 1039–1053. [CrossRef] 22. Gavete, L.; Gavete, M.L.; Benito, J.J. Improvements of generalized finite difference method and comparison with other meshless method. Appl. Math. Model. 2003, 27, 831–847. [CrossRef] 23. Mitchell, A.R.; Griffiths, D.F. The Finite Difference Method in Partial Differential Equations; John Wiley: Hoboken, NJ, USA, 1980. 24. Belytschko, T.; Lu, Y.-Y.; Gu, L. Element-free Galerkin methods. Int. J. Numer. Meth. Eng. 1994, 37, 229–256. [CrossRef] 25. Ureña, F.; Salete, E.; Benito, J.J. Solving third- and fourth-order partial differential equations using GFDM: Application to solve problems of plates. Comput. Mech. 2012, 89, 366–376. [CrossRef] 26. Gu, Y.; Wang, L.; Chen, W. Application of the meshless generalized finite difference method to inverse heat source problems. Int. J. Heat Mass Transf. 2017, 108, 721–729. [CrossRef] 27. Li, P.-W.; Fu, Z.-J.; Gu, Y. The generalized finite difference method for the inverse Cauchy problem in two-dimensional isotropic linear elasticity. Int. J. Solids Struct. 2019, 174, 69–84. [CrossRef] 28. Li, P.-W.; Chen, W.; Fu, Z.-J. Generalized finite difference method for solving the double-diffusive natural convection in fluid-saturated porous media. Eng. Anal. Bound. Elem. 2018, 95, 175–186. [CrossRef] 29. Fu, Z.-J.; Xie, Z.-Y.; Ji, S.-Y. Meshless generalized finite difference method for water wave interactions with multiple-bottom-seated-cylinder-array structures. Ocean Eng. 2020, 195, 106736. [CrossRef] 30. Fu, Z.-J.; Tang, Z.-C.; Zhao, H.-T. Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method. Eur. Phys. J. Plus 2019, 134, 134. [CrossRef] 31. Li, P.-W.; Fan, C.-M. Generalized finite difference method for two-dimensional shallow water equations. Eng. Anal. Bound. Elem. 2017, 80, 58–71. [CrossRef] 32. Vua, T.V.; Nguyenb, N.H.; Khosravifardc, A. A simple FSDT-based meshfree method for analysis of functionally graded plates. Eng. Anal. Bound. Elem. 2017, 79, 1–12. [CrossRef] 33. Yu, T.T.; Yin, S.H.; Tinh, Q.B. A simple FSDT-based isogeometric analysis for geometrically nonlinear analysis of functionally graded plates. Finite Elem. Anal. Des. 2015, 96, 1–10. [CrossRef] 34. Mantari, J.L.; Granados, E.V. An original FSDT to study advanced composites on elastic foundation. Thin-Walled Struct. 2016, 107, 80–89. [CrossRef] 35. Mantari, J.L.; Granados, E.V. A refined FSDT for the static analysis of functionally graded sandwich plates. Thin-Walled Struct. 2015, 90, 150–158. [CrossRef] 36. Yin, S.H.; Jack, S.H.; Yu, T.T. Isogeometric locking-free plate element: A simple first order shear deformation theory for functionally graded plates. Compos. Struct. 2014, 118, 121–138. [CrossRef] 37. Nguyen, X.H.; Tran, L.V.; Nguyen, T.T. Analysis of functionally graded plates using an edge-based smoothed finite element method. Compos. Struct. 2011, 93, 3019–3039. [CrossRef] 38. Nguyen, X.H.; Tran, L.V.; Thai, H. Analysis of functionally graded plates by an efficient finite element method with node-based strain smoothing. Thin Walled Struct. 2012, 54, 1–18. [CrossRef] 39. Valizadeh, N.; Natarajan, S.; Gonzalez-Estrada, O.A.; Rabczuk, T.; Bui, T.Q.; Bordas, S.P.A.NURBS-based finite element analysis of functionally graded plates: Static bending, vibration, buckling and flutter. Compos. Struct. 2013, 99, 309–326. [CrossRef] 40. Reddy, J.N. Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 2000, 47, 663–684. [CrossRef] 41. Singha, M.K.; Prakash, T.; Ganapathi, M. Finite element analysis of functionally graded plates under transverse load. Finite Elem. Anal. Des. 2011, 47, 453–460. [CrossRef] Mathematics 2020, 8, 1940 19 of 19

42. Zenkour, A.M. Generalized shear deformation theory for bending analysis of functionally graded plates. Appl. Math. Model. 2006, 30, 67–84. [CrossRef]

Sample Availability: The datasets analyzed during the current study are available from the corresponding author and the ﬁrst author upon reasonable request.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).