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Meshless global radial point collocation method for three-dimensional partial differential equations
Engineering Analysis with Boundary Elements 35 (2011) 289–297
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Meshless global radial point collocation method for three-dimensional partial differential equations Song Xiang a,n, Yao-xing Jin b, Shao-xi Jiang c, Ze-yang Bi d, Ke-ming Wang a a
School of Engine & Energy Engineering, Shenyang Aerospace University, No. 37 Daoyi South Avenue, Shenyang, Liaoning 110136, People’s Republic of China Technology Department, Avic Shenyang Liming Aero-Engine (group) Corporation Ltd., Shenyang, Liaoning 110043, People’s Republic of China c Manufacturing Factory of Blade, Avic Shenyang Liming Aero-Engine (Group) Corporation Ltd., Shenyang, Liaoning 110043, People’s Republic of China d Steel_Making Plant, ShouGang Qian’an Iron&Steel Corporation Ltd., Qian’an, Hebei 064400, People’s Republic of China b
a r t i c l e in f o
abstract
Article history: Received 12 April 2010 Accepted 28 October 2010 Available online 24 November 2010
This study deals with the numerical solution of three-dimensional partial differential equations by the meshless global radial point collocation method based on various radial basis functions. First, second, third, and fourth-order three-dimensional partial differential equations are considered. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Meshless Global radial point collocation Three-dimensional partial differential equations Radial basis functions
1. Introduction The mathematic models have been developed for the physical phenomena in the areas of mechanics of solids, structures, and fluid flows. Different types of partial differential equations have also been derived for these phenomena. The method for solving partial differential equations includes the finite element method, the finite volume method, and the finite difference method. In recent years, a new method called the meshless method has been developed [1–3]. Meshless method is used to establish a system of algebraic equations for the whole problem domain without the use of a predefined mesh [4]. The meshless methods fall into three categories according to the formulation procedures: meshless methods based on weak-forms, meshless methods based on collocation techniques, meshless methods based on the combination of weakforms and collocation techniques [3]. The meshless collocation methods have the advantages of a simple algorithm, computational efficiency, and truly meshless. Many researchers have utilized the meshless collocation methods to solve partial differential equations. Hardy [5] solved the equations of topography by the meshless collocation methods based on the multiquadric radial basis function. Hardy [6] reviewed the development of multiquadric-biharmonic method from 1968 to 1988. Kansa [7] presented a powerful, enhanced multiquadrics (MQ) scheme for accurate interpolation and partial derivative estimates. The meshless collocation methods based on the
n
Corresponding author. E-mail address: [email protected] (S. Xiang).
0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2010.10.007
multiquadrics radial basis function was used as the spatial approximation scheme for parabolic, hyperbolic, and the elliptic Poisson’s equation by Kansa [8]. Golberg et al. [9] interpolated the forcing term of partial differential equations using multiquadric approximations, and then use them to approximate particular solutions. The technique of cross-validation was used to obtain a good shape parameter of the multiquadrics. Sharan et al. [10] applied the multiquadric (MQ) approximation scheme to solve two-dimensional Laplace, Poisson, and biharmonic equations with Dirichlet and/or Neumann boundary conditions. The method is also applied successfully to a problem with a curved boundary. Hon et al. [11–14] studied the numerical solution of a biphasic model, Burgers equation, shallow water equation, and options pricing model by the multiquadric method. Kansa and Hon [15] explored several techniques, each of which improves the conditioning of the coefficient matrix and the solution accuracy. Power and Barraco [16] presented a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations. Wong et al. [17] presented the application of the compactly supported radial basis functions (CSRBFs) in solving a system of shallow water hydrodynamics equations. The performances of domain-type meshless collocation methods and boundary-type meshless methods in solving partial differential equations were compared by Li et al. [18]. It was found from their studies that these two methods provide a similar optimal accuracy in solving both 2D Poisson’s and parabolic equations. Larsson and Fornberg [19] compared the RBF-baeed collocation methods against two standard techniques (a second-order finite difference method and a pseudospectral method), it was found that the former gave a much
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superior accuracy. Trahan and Wyatt [20] investigated multiquadric radial basis function (RBF) interpolation and its application in the quantum trajectory method (QTM) for wave packet propagation. In their studies, an algorithm similar to the leave-one-out method of cross-validation was utilized to obtain the optimized shape parameter at each time step. Lee et al. [21] solved 1D and 2D boundary value problems of partial differential equations based on the local multiquadric (LMQ) and the local inverse multiquadric (LIMQ) collocation procedures. Ferreira [22–31] solved the governing partial differential equations of laminated composite plates and functionally graded plates based on the meshless global radial basis function collocation methods. Ahmed [32] presented a meshless method based on a new combination between thin plate and multiquadric radial basis functions, and solved one-dimensional heat conduction and two-dimensional diffusion equations by the new combined radial basis functions. Dehghan and Tatari [33] solved an inverse semi-linear parabolic equation with a source parameter using radial basis functions. Wertz et al. [34] studied the shape parameters of the generalized multiquadrics in solving elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. It was found that convergence accelerations were obtained by permitting the shape parameters to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Zhang [35] solved partial differential equations by two or more radial basis functions. Vertnik and Sarler [36] developed a new local radial basis function collocation method (LRBFCM) for one-domain solving of the nonlinear convection-diffusion equation. Sarra [37] used integrated multiquadric radial basis function to solve two-dimensional Poisson equations. Dehghan and Shokri [38] solved the two-dimensional time-dependent Schrodinger equation using the collocation method based on the multiquadrics and thin plate splines radial basis functions. Bouhamidi and Jbilou [39] presented the meshless method based on thin plate splines radial basis function for solving modified Helmholtz equations. Numerical solution of the Korteweg-de Vries equation was obtained by using the meshless collocation method based on the five standard radial basis functions by Dag and Dereli [40]. The pressure–velocity formulation of the Navier–Stokes (N–S) equation was solved using the radial basis functions collocation method by Demirkaya et al. [41]. A local radial basic function based gridfree scheme had been developed to solve unsteady, incompressible Navier–Stokes equations in primitive variables by Sanyasiraju and Chandhini [42]. Islam et al. [43] solved regularized long wave (RLW) equation by the global collocation method using the radial basis functions. Tatari and Dehghan [44] solved the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions using the radial basis functions collocation method. Dereli et al. [45] presented a numerical study of the cubic non-linear ¨ Schrodinger equation using the meshless method based on collocation with radial basis functions. Xiang et al. [46–48] studied the static and vibration characteristics of the laminated composite plates by the meshless collocation methods based on various radial basis functions. Madych and Nelson [49] explained why the rates of convergence decrease with increase in orders of differentiation. Mai-Duy and Tran-Cong [50] used an integrated-RBF technique based on Galerkin formulation to solve elliptic differential equations. Huang et al. [51] found that error can be reduced by increasing the value of shape constant c in the MQ basis function, without refining the grid. It is believed that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting c-N. The global radial basis function method has some shortcomings, especially for real applications. To circumvent these shortcomings locally supported method has been proposed in the literatures. Liu and Gu [52] compared the local point interpolation method (LPIM) and the local radial point interpolation method (LR-PIM) for
structural analyses. It is found that LPIM and LR-PIM are easy to implement, and efficient obtaining numerical solutions to problems of computational mechanics. Gu [53] presented the categories of meshless methods, discussed the construction of meshless shape functions, studied interpolation qualities of them using the surface fitting, and addressed the future development of meshless methods. The studies in the previous literatures are mostly about the numerical solution of one-dimensional or two-dimensional partial differential equations using the meshless collocation methods based on the radial basis functions. The aim of the present paper is to study the numerical solution of three-dimensional partial differential equations by the meshless global radial point collocation method based on various radial basis functions. First, second, third, fourth-order three-dimensional partial differential equations are considered. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples.
2. Meshless global radial point collocation method Consider the linear boundary value problem of three-dimensional partial differential equations LuðsÞ ¼ f ðsÞ,s A O BuðsÞ ¼ gðsÞ,s A @O
ð1Þ
where @O is the boundary of the domain O, L is a linear elliptic partial differential operator, B is a linear boundary operator, f(s) and g(s) are the known functions. The solution of Eq. (1) can be approximated using multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function in the following form, respectively. ua ðsÞ ¼
N X
aj ½ðxxj Þ2 þ ðyyj Þ2 þ ðzzj Þ2 þ c2 0:5
ð2Þ
aj ½ðxxj Þ2 þ ðyyj Þ2 þ ðzzj Þ2 þ c2 0:5
ð3Þ
j¼1
ua ðsÞ ¼
N X j¼1
ua ðsÞ ¼
N X
2
aj ec½ðxxj Þ
þ ðyyj Þ2 þ ðzzj Þ2
ð4Þ
j¼1
ua ðsÞ ¼
N X
aj ½ðxxj Þ2 þ ðyyj Þ2 þ ðzzj Þ2 m
j¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log ðxxj Þ2 þ ðyyj Þ2 þ ðzzj Þ2
ð5Þ
where c and m are the shape parameters of radial basis functions; N is the total number of nodes; aj is unknown coefficients; xj, yj, and zj are coordinates of node j. Assume that there are NI nodes in the interior of the domain O, and NB nodes on the boundary @O. For the interior nodes, we have N X
aj Ljj ¼ f ðsi Þ, i ¼ 1,. . .,NI
ð6Þ
j¼1
For the boundary nodes, we have N X
aj Bjj ¼ gðsi Þ, i ¼ NI þ 1,. . .,N
j¼1
where jj is the radial basis function.
ð7Þ
S. Xiang et al. / Engineering Analysis with Boundary Elements 35 (2011) 289–297
"
Eqs. (6) and (7) can be expressed as # " # f ½a ¼ Bj g
Lj
Then " #1 " # Lj f ½a ¼ Bj g
291
ð8Þ
ð9Þ
3. Numerical examples The meshless global radial point collocation method based on various radial basis functions is utilized to solve the first, second, third, and fourth-order three-dimensional partial differential equations. In the multiquadric and inverse multiquadric radial basis functions, the pffiffiffiffiffiffi shape parameter c ¼ cc= Ns . In the Gaussian radial basis function, the shape parameter c¼cc Ns. Ns is the node number at each side. The total node number of the cubic domain N¼ Ns Ns Ns. The RMS errors were calculated using the following formulation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uP 2 u ui ¼ 1 ½ua ðiÞuðiÞ u RMS ¼ u ð10Þ N P t ½uðiÞ2
Fig. 1. RMS errors of the multiquadric radial basis function with the shape parameter pffiffiffiffiffiffi c ¼ cc= Ns solving the first-order three-dimensional partial differential equation.
i¼1
3.1. Effect of shape parameters on the numerical accuracy The cubic domain of 1 1 1 is discretized by the uniformly spaced nodes. The node distribution is increased from 6 6 6 to 10 10 10. 3.1.1. First-order three-dimensional partial differential equation Consider the boundary value problem of the first-order threedimensional partial differential equation @u @u @u þ þ ¼ pðcos px sin py sin pz þ sin px cos py sin pz @x @y @z þ sin px sin pycos pzÞ uð0,y,zÞ ¼ uð1,y,zÞ ¼ 0, uðx,0,zÞ ¼ uðx,1,zÞ ¼ 0, uðx,y,0Þ ¼ uðx,y,1Þ ¼ 0 x A ½0,1,y A ½0,1,z A ½0,1
ð11Þ
with the exact solution u ¼ sin pxsin pysin pz. Figs. 1–4 shows the RMS errors of the meshless global collocation method based on the multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function solving the first-order three-dimensional partial differential equation, respectively. As can be seen from Figs. 1–4 that the multiquadric with the shape parameter pffiffiffiffiffiffi c ¼ 1:6= Ns , the inverse multiquadric with shape parameter pffiffiffiffiffiffi c ¼ 1:95= Ns , the Gaussian with shape parameter c¼1.4 Ns, and the thin plate spline with shape parameter m ¼4 produce the highest accuracy, respectively. 3.1.2. Second-order three-dimensional partial differential equation Consider the boundary value problem of second-order threedimensional partial differential equation @2 u @2 u @2 u þ þ 2 ¼ 3p2 sin px sin py sin pz @x2 @y2 @z uð0,y,zÞ ¼ uð1,y,zÞ ¼ 0, uðx,0,zÞ ¼ uðx,1,zÞ ¼ 0, uðx,y,0Þ ¼ uðx,y,1Þ ¼ 0 x A ½0,1,y A ½0,1,z A ½0,1
ð12Þ
with the exact solution u ¼ sinpxsinpysinpz. Figs. 5–8 shows the RMS errors of the meshless global collocation method based on the
Fig. 2. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the first-order three-dimensional partial differential equation.
multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function solving the second-order three-dimensional partial differential equation, respectively. As can be seen from Figs. 5–8 that the multiquadric with the shape parameter pffiffiffiffiffiffi c ¼ 1:9= Ns , the inverse multiquadric with shape parameter pffiffiffiffiffiffi c ¼ 2:1= Ns , the Gaussian with shape parameter c¼1.2 Ns, and the thin plate spline with shape parameter m ¼4 produce the highest accuracy, respectively. 3.1.3. Third-order three-dimensional partial differential equation Consider the boundary value problem of third-order threedimensional partial differential equation @3 u @3 u @3 u þ þ 3 ¼ p3 ðcospx sinpy sinpz þsinpx cospy sinpz @x3 @y3 @z þsinpx sinpy cospzÞ
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Fig. 3. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns solving the first-order three-dimensional partial differential equation.
Fig. 4. RMS errors of the thin plate spline radial basis function with the shape parameter m solving the first-order three-dimensional partial differential equation.
uð0,y,zÞ ¼ uð1,y,zÞ ¼ 0, uðx,0,zÞ ¼ uðx,1,zÞ ¼ 0, uðx,y,0Þ ¼ uðx,y,1Þ ¼ 0 x A ½0,1,y A ½0,1,z A ½0,1
Fig. 5. RMS errors of the multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the second-order three-dimensional partial differential equation.
Fig. 6. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the second-order three-dimensional partial differential equation.
ð13Þ
with the exact solution u ¼ sinpxsinpysinpz. Figs. 9–12 shows the RMS errors of the meshless global collocation method based on the multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function solving the third-order three-dimensional partial differential equation, respectively. As can be seen from Figs. pffiffiffiffiffiffi 9–12 that the multiquadric with the shape parameter c ¼ 1:9= Ns , pffiffiffiffiffiffi the inverse multiquadric with shape parameter c ¼ 1:7= Ns , the Gaussian with shape parameter c ¼1.3 Ns, and the thin plate spline with shape parameter m¼4 produce the highest accuracy, respectively.
3.1.4. Fourth-order three-dimensional partial differential equation Consider the boundary value problem of fourth-order threedimensional partial differential equation @4 u @4 u @4 u þ þ 4 ¼ 3p4 sin px sin pysin pz @x4 @y4 @z uð0,y,zÞ ¼ uð1,y,zÞ ¼ 0, uðx,0,zÞ ¼ uðx,1,zÞ ¼ 0, uðx,y,0Þ ¼ uðx,y,1Þ ¼ 0 x A ½0,1,y A ½0,1,z A ½0,1
ð14Þ
with the exact solution u ¼ sin px sin pysin pz. Figs. 13–16 shows the RMS errors of the meshless global collocation method based on the
S. Xiang et al. / Engineering Analysis with Boundary Elements 35 (2011) 289–297
Fig. 7. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns solving the second-order three-dimensional partial differential equation.
Fig. 8. RMS errors of the thin plate spline radial basis function with the shape parameter m solving the second-order three-dimensional partial differential equation.
multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function solving the fourth-order three-dimensional partial differential equation, respectively. As can be seen from Figs. 13–16 that the multiquadric with the shape parameter pffiffiffiffiffiffi c ¼ 1:8= Ns , the inverse multiquadric with shape parameter pffiffiffiffiffiffi c ¼ 2:2= Ns , the Gaussian with shape parameter c¼1.3 Ns, and the thin plate spline with shape parameter m¼4 produce the highest accuracy, respectively. According to the Figs. 1–16, as the order number of partial differential equations increases, the numerical accuracy of meshless collocation method solving three-dimensional partial differential equations decreases. The numerical accuracy of meshless
293
Fig. 9. RMS errors of the multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the third-order three-dimensional partial differential equation.
Fig. 10. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the third-order three-dimensional partial differential equation.
collocation method solving third-order partial differential equations is the lowest. 3.2. Effect of grid pattern on the numerical accuracy In order to study the effect of grid pattern on the numerical accuracy, the uniform and cosine grid pattern are considered. The total node number of the cubic domain N ¼8 8 8. The coordinates of ith nodes for the cosine grid pattern are a cosði1Þp b cosði1Þp xðiÞ ¼ 1 1 ; yðiÞ ¼ ; 2 2 Nx 1 Ny 1 h cosði1Þp 1 zðiÞ ¼ ð15Þ 2 Nz 1
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Fig. 11. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns solving the third-order three-dimensional partial differential equation.
Fig. 12. RMS errors of the thin plate spline radial basis function with the shape parameter m solving the third-order three-dimensional partial differential equation.
where Nx, Ny and Nz are the number of nodes on the x, y and z side, respectively. a, b and h are the length, width, and height of the cubic domain, respectively. Figs. 17–20 shows the RMS errors of the meshless global collocation method based on various radial basis function with the uniform and cosine grid pattern solving the secondorder three-dimensional partial differential equation, respectively. According to Fig. 17, the RMS errors of multiquadric with cosine grid pattern are greater than that of multiquadric with uniform grid pffiffiffiffiffiffi pattern when the shape parameter c is smaller than 0:4= Ns . Converse condition emerges when the shape parameter c is greater pffiffiffiffiffiffi than 0:4= Ns . According to Fig. 18, the RMS errors of inverse multiquadric with cosine grid pattern are greater than that of inverse multiquadric with uniform grid pattern when the shape
Fig. 13. RMS errors of the multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the fourth -order three-dimensional partial differential equation.
Fig. 14. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the fourth-order three-dimensional partial differential equation.
pffiffiffiffiffiffi parameter c is smaller than 0:9= Ns . Converse condition emerges pffiffiffiffiffiffi when the shape parameter c is greater than 0:9= Ns . According to Fig. 19, the RMS errors of Gaussian with cosine grid pattern are greater than that of Gaussian with uniform grid pattern when the shape parameter c is greater than 2.4 Ns. Converse condition emerges when the shape parameter c is smaller than 2.4 Ns. According to Fig. 20, the RMS errors of thin plate spline with cosine grid pattern are greater than that of thin plate spline with uniform grid pattern when the shape parameter m is smaller than 2. Converse condition emerges when the shape parameter m is greater than 2. According to Figs. 17–20, the effect of grid pattern on the numerical accuracy is insignificant when the shape parameter is close to the optimal shape parameter.
S. Xiang et al. / Engineering Analysis with Boundary Elements 35 (2011) 289–297
Fig. 15. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns solving the fourth-order three-dimensional partial differential equation.
Fig. 16. RMS errors of the thin plate spline radial basis function with the shape parameter m solving the fourth-order three-dimensional partial differential equation.
4. Conclusions In this paper, first, second, third, and fourth-order threedimensional partial differential equations are solved by the meshless global radial point collocation method based on the multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis functions. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples. Through numerical examples, the study can draw following conclusions: (1) For the multiquadric radial basis functions, good shape parapffiffiffiffiffiffi pffiffiffiffiffiffi meter c is 1:6= Ns –1:9= Ns .
295
Fig. 17. RMS errors of the multiquadric radial basis function with the shape pffiffiffiffiffiffi parameterc ¼ cc= Ns in various grid pattern solving the second-order threedimensional partial differential equation.
Fig. 18. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns in various grid pattern solving the second-order threedimensional partial differential equation.
(2) For the inverse multiquadric radial basis functions, good shape pffiffiffiffiffiffi pffiffiffiffiffiffi parameter c is 1:7= Ns –2:2= Ns . (3) For the Gaussian radial basis functions, good shape parameter c is 1.2 Ns–1.4 Ns. (4) For the thin plate spline radial basis functions, optimal shape parameter m is 4. (5) As the order number of partial differential equations increases, the numerical accuracy of meshless collocation method solving three-dimensional partial differential equations decreases. The numerical accuracy of meshless collocation method solving third-order partial differential equations is the lowest.
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Fig. 19. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns in various grid pattern solving the second-order three-dimensional partial differential equation.
Fig. 20. RMS errors of the thin plate spline radial basis function with the shape parameter m in various grid pattern solving the second-order three-dimensional partial differential equation.
(6) The effect of grid pattern on the numerical accuracy is insignificant when the shape parameter is close to the optimal shape parameter.
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Contents lists available at ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Meshless global radial point collocation method for three-dimensional partial differential equations Song Xiang a,n, Yao-xing Jin b, Shao-xi Jiang c, Ze-yang Bi d, Ke-ming Wang a a
School of Engine & Energy Engineering, Shenyang Aerospace University, No. 37 Daoyi South Avenue, Shenyang, Liaoning 110136, People’s Republic of China Technology Department, Avic Shenyang Liming Aero-Engine (group) Corporation Ltd., Shenyang, Liaoning 110043, People’s Republic of China c Manufacturing Factory of Blade, Avic Shenyang Liming Aero-Engine (Group) Corporation Ltd., Shenyang, Liaoning 110043, People’s Republic of China d Steel_Making Plant, ShouGang Qian’an Iron&Steel Corporation Ltd., Qian’an, Hebei 064400, People’s Republic of China b
a r t i c l e in f o
abstract
Article history: Received 12 April 2010 Accepted 28 October 2010 Available online 24 November 2010
This study deals with the numerical solution of three-dimensional partial differential equations by the meshless global radial point collocation method based on various radial basis functions. First, second, third, and fourth-order three-dimensional partial differential equations are considered. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Meshless Global radial point collocation Three-dimensional partial differential equations Radial basis functions
1. Introduction The mathematic models have been developed for the physical phenomena in the areas of mechanics of solids, structures, and fluid flows. Different types of partial differential equations have also been derived for these phenomena. The method for solving partial differential equations includes the finite element method, the finite volume method, and the finite difference method. In recent years, a new method called the meshless method has been developed [1–3]. Meshless method is used to establish a system of algebraic equations for the whole problem domain without the use of a predefined mesh [4]. The meshless methods fall into three categories according to the formulation procedures: meshless methods based on weak-forms, meshless methods based on collocation techniques, meshless methods based on the combination of weakforms and collocation techniques [3]. The meshless collocation methods have the advantages of a simple algorithm, computational efficiency, and truly meshless. Many researchers have utilized the meshless collocation methods to solve partial differential equations. Hardy [5] solved the equations of topography by the meshless collocation methods based on the multiquadric radial basis function. Hardy [6] reviewed the development of multiquadric-biharmonic method from 1968 to 1988. Kansa [7] presented a powerful, enhanced multiquadrics (MQ) scheme for accurate interpolation and partial derivative estimates. The meshless collocation methods based on the
n
Corresponding author. E-mail address: [email protected] (S. Xiang).
0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2010.10.007
multiquadrics radial basis function was used as the spatial approximation scheme for parabolic, hyperbolic, and the elliptic Poisson’s equation by Kansa [8]. Golberg et al. [9] interpolated the forcing term of partial differential equations using multiquadric approximations, and then use them to approximate particular solutions. The technique of cross-validation was used to obtain a good shape parameter of the multiquadrics. Sharan et al. [10] applied the multiquadric (MQ) approximation scheme to solve two-dimensional Laplace, Poisson, and biharmonic equations with Dirichlet and/or Neumann boundary conditions. The method is also applied successfully to a problem with a curved boundary. Hon et al. [11–14] studied the numerical solution of a biphasic model, Burgers equation, shallow water equation, and options pricing model by the multiquadric method. Kansa and Hon [15] explored several techniques, each of which improves the conditioning of the coefficient matrix and the solution accuracy. Power and Barraco [16] presented a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations. Wong et al. [17] presented the application of the compactly supported radial basis functions (CSRBFs) in solving a system of shallow water hydrodynamics equations. The performances of domain-type meshless collocation methods and boundary-type meshless methods in solving partial differential equations were compared by Li et al. [18]. It was found from their studies that these two methods provide a similar optimal accuracy in solving both 2D Poisson’s and parabolic equations. Larsson and Fornberg [19] compared the RBF-baeed collocation methods against two standard techniques (a second-order finite difference method and a pseudospectral method), it was found that the former gave a much
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superior accuracy. Trahan and Wyatt [20] investigated multiquadric radial basis function (RBF) interpolation and its application in the quantum trajectory method (QTM) for wave packet propagation. In their studies, an algorithm similar to the leave-one-out method of cross-validation was utilized to obtain the optimized shape parameter at each time step. Lee et al. [21] solved 1D and 2D boundary value problems of partial differential equations based on the local multiquadric (LMQ) and the local inverse multiquadric (LIMQ) collocation procedures. Ferreira [22–31] solved the governing partial differential equations of laminated composite plates and functionally graded plates based on the meshless global radial basis function collocation methods. Ahmed [32] presented a meshless method based on a new combination between thin plate and multiquadric radial basis functions, and solved one-dimensional heat conduction and two-dimensional diffusion equations by the new combined radial basis functions. Dehghan and Tatari [33] solved an inverse semi-linear parabolic equation with a source parameter using radial basis functions. Wertz et al. [34] studied the shape parameters of the generalized multiquadrics in solving elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. It was found that convergence accelerations were obtained by permitting the shape parameters to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Zhang [35] solved partial differential equations by two or more radial basis functions. Vertnik and Sarler [36] developed a new local radial basis function collocation method (LRBFCM) for one-domain solving of the nonlinear convection-diffusion equation. Sarra [37] used integrated multiquadric radial basis function to solve two-dimensional Poisson equations. Dehghan and Shokri [38] solved the two-dimensional time-dependent Schrodinger equation using the collocation method based on the multiquadrics and thin plate splines radial basis functions. Bouhamidi and Jbilou [39] presented the meshless method based on thin plate splines radial basis function for solving modified Helmholtz equations. Numerical solution of the Korteweg-de Vries equation was obtained by using the meshless collocation method based on the five standard radial basis functions by Dag and Dereli [40]. The pressure–velocity formulation of the Navier–Stokes (N–S) equation was solved using the radial basis functions collocation method by Demirkaya et al. [41]. A local radial basic function based gridfree scheme had been developed to solve unsteady, incompressible Navier–Stokes equations in primitive variables by Sanyasiraju and Chandhini [42]. Islam et al. [43] solved regularized long wave (RLW) equation by the global collocation method using the radial basis functions. Tatari and Dehghan [44] solved the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions using the radial basis functions collocation method. Dereli et al. [45] presented a numerical study of the cubic non-linear ¨ Schrodinger equation using the meshless method based on collocation with radial basis functions. Xiang et al. [46–48] studied the static and vibration characteristics of the laminated composite plates by the meshless collocation methods based on various radial basis functions. Madych and Nelson [49] explained why the rates of convergence decrease with increase in orders of differentiation. Mai-Duy and Tran-Cong [50] used an integrated-RBF technique based on Galerkin formulation to solve elliptic differential equations. Huang et al. [51] found that error can be reduced by increasing the value of shape constant c in the MQ basis function, without refining the grid. It is believed that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting c-N. The global radial basis function method has some shortcomings, especially for real applications. To circumvent these shortcomings locally supported method has been proposed in the literatures. Liu and Gu [52] compared the local point interpolation method (LPIM) and the local radial point interpolation method (LR-PIM) for
structural analyses. It is found that LPIM and LR-PIM are easy to implement, and efficient obtaining numerical solutions to problems of computational mechanics. Gu [53] presented the categories of meshless methods, discussed the construction of meshless shape functions, studied interpolation qualities of them using the surface fitting, and addressed the future development of meshless methods. The studies in the previous literatures are mostly about the numerical solution of one-dimensional or two-dimensional partial differential equations using the meshless collocation methods based on the radial basis functions. The aim of the present paper is to study the numerical solution of three-dimensional partial differential equations by the meshless global radial point collocation method based on various radial basis functions. First, second, third, fourth-order three-dimensional partial differential equations are considered. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples.
2. Meshless global radial point collocation method Consider the linear boundary value problem of three-dimensional partial differential equations LuðsÞ ¼ f ðsÞ,s A O BuðsÞ ¼ gðsÞ,s A @O
ð1Þ
where @O is the boundary of the domain O, L is a linear elliptic partial differential operator, B is a linear boundary operator, f(s) and g(s) are the known functions. The solution of Eq. (1) can be approximated using multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function in the following form, respectively. ua ðsÞ ¼
N X
aj ½ðxxj Þ2 þ ðyyj Þ2 þ ðzzj Þ2 þ c2 0:5
ð2Þ
aj ½ðxxj Þ2 þ ðyyj Þ2 þ ðzzj Þ2 þ c2 0:5
ð3Þ
j¼1
ua ðsÞ ¼
N X j¼1
ua ðsÞ ¼
N X
2
aj ec½ðxxj Þ
þ ðyyj Þ2 þ ðzzj Þ2
ð4Þ
j¼1
ua ðsÞ ¼
N X
aj ½ðxxj Þ2 þ ðyyj Þ2 þ ðzzj Þ2 m
j¼1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log ðxxj Þ2 þ ðyyj Þ2 þ ðzzj Þ2
ð5Þ
where c and m are the shape parameters of radial basis functions; N is the total number of nodes; aj is unknown coefficients; xj, yj, and zj are coordinates of node j. Assume that there are NI nodes in the interior of the domain O, and NB nodes on the boundary @O. For the interior nodes, we have N X
aj Ljj ¼ f ðsi Þ, i ¼ 1,. . .,NI
ð6Þ
j¼1
For the boundary nodes, we have N X
aj Bjj ¼ gðsi Þ, i ¼ NI þ 1,. . .,N
j¼1
where jj is the radial basis function.
ð7Þ
S. Xiang et al. / Engineering Analysis with Boundary Elements 35 (2011) 289–297
"
Eqs. (6) and (7) can be expressed as # " # f ½a ¼ Bj g
Lj
Then " #1 " # Lj f ½a ¼ Bj g
291
ð8Þ
ð9Þ
3. Numerical examples The meshless global radial point collocation method based on various radial basis functions is utilized to solve the first, second, third, and fourth-order three-dimensional partial differential equations. In the multiquadric and inverse multiquadric radial basis functions, the pffiffiffiffiffiffi shape parameter c ¼ cc= Ns . In the Gaussian radial basis function, the shape parameter c¼cc Ns. Ns is the node number at each side. The total node number of the cubic domain N¼ Ns Ns Ns. The RMS errors were calculated using the following formulation: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uP 2 u ui ¼ 1 ½ua ðiÞuðiÞ u RMS ¼ u ð10Þ N P t ½uðiÞ2
Fig. 1. RMS errors of the multiquadric radial basis function with the shape parameter pffiffiffiffiffiffi c ¼ cc= Ns solving the first-order three-dimensional partial differential equation.
i¼1
3.1. Effect of shape parameters on the numerical accuracy The cubic domain of 1 1 1 is discretized by the uniformly spaced nodes. The node distribution is increased from 6 6 6 to 10 10 10. 3.1.1. First-order three-dimensional partial differential equation Consider the boundary value problem of the first-order threedimensional partial differential equation @u @u @u þ þ ¼ pðcos px sin py sin pz þ sin px cos py sin pz @x @y @z þ sin px sin pycos pzÞ uð0,y,zÞ ¼ uð1,y,zÞ ¼ 0, uðx,0,zÞ ¼ uðx,1,zÞ ¼ 0, uðx,y,0Þ ¼ uðx,y,1Þ ¼ 0 x A ½0,1,y A ½0,1,z A ½0,1
ð11Þ
with the exact solution u ¼ sin pxsin pysin pz. Figs. 1–4 shows the RMS errors of the meshless global collocation method based on the multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function solving the first-order three-dimensional partial differential equation, respectively. As can be seen from Figs. 1–4 that the multiquadric with the shape parameter pffiffiffiffiffiffi c ¼ 1:6= Ns , the inverse multiquadric with shape parameter pffiffiffiffiffiffi c ¼ 1:95= Ns , the Gaussian with shape parameter c¼1.4 Ns, and the thin plate spline with shape parameter m ¼4 produce the highest accuracy, respectively. 3.1.2. Second-order three-dimensional partial differential equation Consider the boundary value problem of second-order threedimensional partial differential equation @2 u @2 u @2 u þ þ 2 ¼ 3p2 sin px sin py sin pz @x2 @y2 @z uð0,y,zÞ ¼ uð1,y,zÞ ¼ 0, uðx,0,zÞ ¼ uðx,1,zÞ ¼ 0, uðx,y,0Þ ¼ uðx,y,1Þ ¼ 0 x A ½0,1,y A ½0,1,z A ½0,1
ð12Þ
with the exact solution u ¼ sinpxsinpysinpz. Figs. 5–8 shows the RMS errors of the meshless global collocation method based on the
Fig. 2. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the first-order three-dimensional partial differential equation.
multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function solving the second-order three-dimensional partial differential equation, respectively. As can be seen from Figs. 5–8 that the multiquadric with the shape parameter pffiffiffiffiffiffi c ¼ 1:9= Ns , the inverse multiquadric with shape parameter pffiffiffiffiffiffi c ¼ 2:1= Ns , the Gaussian with shape parameter c¼1.2 Ns, and the thin plate spline with shape parameter m ¼4 produce the highest accuracy, respectively. 3.1.3. Third-order three-dimensional partial differential equation Consider the boundary value problem of third-order threedimensional partial differential equation @3 u @3 u @3 u þ þ 3 ¼ p3 ðcospx sinpy sinpz þsinpx cospy sinpz @x3 @y3 @z þsinpx sinpy cospzÞ
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Fig. 3. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns solving the first-order three-dimensional partial differential equation.
Fig. 4. RMS errors of the thin plate spline radial basis function with the shape parameter m solving the first-order three-dimensional partial differential equation.
uð0,y,zÞ ¼ uð1,y,zÞ ¼ 0, uðx,0,zÞ ¼ uðx,1,zÞ ¼ 0, uðx,y,0Þ ¼ uðx,y,1Þ ¼ 0 x A ½0,1,y A ½0,1,z A ½0,1
Fig. 5. RMS errors of the multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the second-order three-dimensional partial differential equation.
Fig. 6. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the second-order three-dimensional partial differential equation.
ð13Þ
with the exact solution u ¼ sinpxsinpysinpz. Figs. 9–12 shows the RMS errors of the meshless global collocation method based on the multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function solving the third-order three-dimensional partial differential equation, respectively. As can be seen from Figs. pffiffiffiffiffiffi 9–12 that the multiquadric with the shape parameter c ¼ 1:9= Ns , pffiffiffiffiffiffi the inverse multiquadric with shape parameter c ¼ 1:7= Ns , the Gaussian with shape parameter c ¼1.3 Ns, and the thin plate spline with shape parameter m¼4 produce the highest accuracy, respectively.
3.1.4. Fourth-order three-dimensional partial differential equation Consider the boundary value problem of fourth-order threedimensional partial differential equation @4 u @4 u @4 u þ þ 4 ¼ 3p4 sin px sin pysin pz @x4 @y4 @z uð0,y,zÞ ¼ uð1,y,zÞ ¼ 0, uðx,0,zÞ ¼ uðx,1,zÞ ¼ 0, uðx,y,0Þ ¼ uðx,y,1Þ ¼ 0 x A ½0,1,y A ½0,1,z A ½0,1
ð14Þ
with the exact solution u ¼ sin px sin pysin pz. Figs. 13–16 shows the RMS errors of the meshless global collocation method based on the
S. Xiang et al. / Engineering Analysis with Boundary Elements 35 (2011) 289–297
Fig. 7. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns solving the second-order three-dimensional partial differential equation.
Fig. 8. RMS errors of the thin plate spline radial basis function with the shape parameter m solving the second-order three-dimensional partial differential equation.
multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis function solving the fourth-order three-dimensional partial differential equation, respectively. As can be seen from Figs. 13–16 that the multiquadric with the shape parameter pffiffiffiffiffiffi c ¼ 1:8= Ns , the inverse multiquadric with shape parameter pffiffiffiffiffiffi c ¼ 2:2= Ns , the Gaussian with shape parameter c¼1.3 Ns, and the thin plate spline with shape parameter m¼4 produce the highest accuracy, respectively. According to the Figs. 1–16, as the order number of partial differential equations increases, the numerical accuracy of meshless collocation method solving three-dimensional partial differential equations decreases. The numerical accuracy of meshless
293
Fig. 9. RMS errors of the multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the third-order three-dimensional partial differential equation.
Fig. 10. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the third-order three-dimensional partial differential equation.
collocation method solving third-order partial differential equations is the lowest. 3.2. Effect of grid pattern on the numerical accuracy In order to study the effect of grid pattern on the numerical accuracy, the uniform and cosine grid pattern are considered. The total node number of the cubic domain N ¼8 8 8. The coordinates of ith nodes for the cosine grid pattern are a cosði1Þp b cosði1Þp xðiÞ ¼ 1 1 ; yðiÞ ¼ ; 2 2 Nx 1 Ny 1 h cosði1Þp 1 zðiÞ ¼ ð15Þ 2 Nz 1
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Fig. 11. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns solving the third-order three-dimensional partial differential equation.
Fig. 12. RMS errors of the thin plate spline radial basis function with the shape parameter m solving the third-order three-dimensional partial differential equation.
where Nx, Ny and Nz are the number of nodes on the x, y and z side, respectively. a, b and h are the length, width, and height of the cubic domain, respectively. Figs. 17–20 shows the RMS errors of the meshless global collocation method based on various radial basis function with the uniform and cosine grid pattern solving the secondorder three-dimensional partial differential equation, respectively. According to Fig. 17, the RMS errors of multiquadric with cosine grid pattern are greater than that of multiquadric with uniform grid pffiffiffiffiffiffi pattern when the shape parameter c is smaller than 0:4= Ns . Converse condition emerges when the shape parameter c is greater pffiffiffiffiffiffi than 0:4= Ns . According to Fig. 18, the RMS errors of inverse multiquadric with cosine grid pattern are greater than that of inverse multiquadric with uniform grid pattern when the shape
Fig. 13. RMS errors of the multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the fourth -order three-dimensional partial differential equation.
Fig. 14. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns solving the fourth-order three-dimensional partial differential equation.
pffiffiffiffiffiffi parameter c is smaller than 0:9= Ns . Converse condition emerges pffiffiffiffiffiffi when the shape parameter c is greater than 0:9= Ns . According to Fig. 19, the RMS errors of Gaussian with cosine grid pattern are greater than that of Gaussian with uniform grid pattern when the shape parameter c is greater than 2.4 Ns. Converse condition emerges when the shape parameter c is smaller than 2.4 Ns. According to Fig. 20, the RMS errors of thin plate spline with cosine grid pattern are greater than that of thin plate spline with uniform grid pattern when the shape parameter m is smaller than 2. Converse condition emerges when the shape parameter m is greater than 2. According to Figs. 17–20, the effect of grid pattern on the numerical accuracy is insignificant when the shape parameter is close to the optimal shape parameter.
S. Xiang et al. / Engineering Analysis with Boundary Elements 35 (2011) 289–297
Fig. 15. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns solving the fourth-order three-dimensional partial differential equation.
Fig. 16. RMS errors of the thin plate spline radial basis function with the shape parameter m solving the fourth-order three-dimensional partial differential equation.
4. Conclusions In this paper, first, second, third, and fourth-order threedimensional partial differential equations are solved by the meshless global radial point collocation method based on the multiquadric, inverse multiquadric, Gaussian, and thin plate spline radial basis functions. The effect of shape parameters of various radial basis functions on the numerical accuracy is studied. The effect of grid pattern on accuracy is also studied by several numerical examples. Through numerical examples, the study can draw following conclusions: (1) For the multiquadric radial basis functions, good shape parapffiffiffiffiffiffi pffiffiffiffiffiffi meter c is 1:6= Ns –1:9= Ns .
295
Fig. 17. RMS errors of the multiquadric radial basis function with the shape pffiffiffiffiffiffi parameterc ¼ cc= Ns in various grid pattern solving the second-order threedimensional partial differential equation.
Fig. 18. RMS errors of the inverse multiquadric radial basis function with the shape pffiffiffiffiffiffi parameter c ¼ cc= Ns in various grid pattern solving the second-order threedimensional partial differential equation.
(2) For the inverse multiquadric radial basis functions, good shape pffiffiffiffiffiffi pffiffiffiffiffiffi parameter c is 1:7= Ns –2:2= Ns . (3) For the Gaussian radial basis functions, good shape parameter c is 1.2 Ns–1.4 Ns. (4) For the thin plate spline radial basis functions, optimal shape parameter m is 4. (5) As the order number of partial differential equations increases, the numerical accuracy of meshless collocation method solving three-dimensional partial differential equations decreases. The numerical accuracy of meshless collocation method solving third-order partial differential equations is the lowest.
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Fig. 19. RMS errors of the Gaussian radial basis function with the shape parameter c ¼cc Ns in various grid pattern solving the second-order three-dimensional partial differential equation.
Fig. 20. RMS errors of the thin plate spline radial basis function with the shape parameter m in various grid pattern solving the second-order three-dimensional partial differential equation.
(6) The effect of grid pattern on the numerical accuracy is insignificant when the shape parameter is close to the optimal shape parameter.
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