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Advances in Mathematical Physics
Volume 2015, Article ID 916029, 38 pages
http://dx.doi.org/10.1155/2015/916029
Review Article

Hybrid Fundamental Solution Based Finite Element Method: Theory and Applications

Research School of Engineering, The Australian National University, Acton, ACT 2601, Australia

Received 13 October 2014; Revised 23 December 2014; Accepted 24 December 2014

Academic Editor: Luigi C. Berselli

Copyright © 2015 Changyong Cao and Qing-Hua Qin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An overview on the development of hybrid fundamental solution based finite element method (HFS-FEM) and its application in engineering problems is presented in this paper. The framework and formulations of HFS-FEM for potential problem, plane elasticity, three-dimensional elasticity, thermoelasticity, anisotropic elasticity, and plane piezoelectricity are presented. In this method, two independent assumed fields (intraelement filed and auxiliary frame field) are employed. The formulations for all cases are derived from the modified variational functionals and the fundamental solutions to a given problem. Generation of elemental stiffness equations from the modified variational principle is also described. Typical numerical examples are given to demonstrate the validity and performance of the HFS-FEM. Finally, a brief summary of the approach is provided and future trends in this field are identified.

1. Introduction

A novel hybrid finite element formulation, called the hybrid fundamental solution based FEM (HFS-FEM), was recently developed based on the framework of hybrid Trefftz finite element method (HT-FEM) and the idea of the method of fundamental solution (MFS) [15]. In this method, two independent assumed fields (intraelement filed and auxiliary frame field) are employed and the domain integrals in the variational functional can be directly converted to boundary integrals without any appreciable increase in computational effort as in HT-FEM [68]. It should be mentioned that the intraelement field of HFS-FEM is approximated by the linear combination of fundamental solutions analytically satisfying the related governing equation, instead of -complete functions as in HT-FEM. The resulting system of equations from the modified variational functional is expressed in terms of symmetric stiffness matrix and nodal displacements only, which is easy to implement into the standard FEM. It is noted that no singular integrals are involved in the HFS-FEM by locating the source point outside the element of interest and do not overlap with field point during the computation [9].

The HFS-FEM mentioned above inherits all the advantages of HT-FEM over the traditional FEM and the boundary element method (BEM), namely, domain decomposition and boundary integral expressions, while avoiding the major weaknesses of BEM [1012], that is, singular element boundary integral and loss of symmetry and sparsity [13]. The employment of two independent fields also makes the HFS-FEM easier to generate arbitrary polygonal or even curve-sided elements. It also obviates the difficulties that occur in HT-FEM [14, 15] in deriving -complete functions for certain complex or new physical problems [16]. The HFS-FEM has simpler expressions of interpolation functions for intraelement fields (fundamental solutions) and avoids the coordinate transformation procedure required in the HT-FEM to keep the matrix inversion stable. Moreover, this approach also has the potential to achieve high accuracy using coarse meshes of high-degree elements, to enhance insensitivity to mesh distortion, to give great liberty in element shape, and to accurately represent various local effects (such as hole, crack, and inclusions) without troublesome mesh adjustment [1720]. Additionally, HFS-FEM makes it possible for a more flexible element material definition which is important in dealing with multimaterial problems, rather than the material definition being the same in the entire domain in BEM. However, we noticed that there are also some limitations of HFS-FEM, for example, determining the positions of source points used for approximation interpolations. It is also known that fundamental solution based approximations can perform remarkably well in smooth problems but tend to deteriorate when high-gradient stress fields are presented.

This paper is organized as follows: in Section 2, the basic idea and formulations of the HFS-FEM are presented through a simple potential problem. Then, plane elasticity problems are described in Section 3. Section 4 extends the 2D formulations of the HFS-FEM to general three-dimensional (3D) elasticity problems. The method of particular solution and radial basis function approximation are shown to deal with body force in this Section. In Section 5 we extend the HFS-FEM to thermoelastic problems with arbitrary body force and temperature change. In Section 6, the HFS-FEM for 2D anisotropic elastic materials is described based on the powerful Stroh formalism. Plane piezoelectric problem is discussed in Section 7. Finally, typical numerical examples are presented in Section 8 to illustrate applications and performance of the HFS-FEM. Concluding remarks and future development are discussed at the end of this paper.

2. Potential Problems

2.1. Basic Equations of Potential Problems

The Laplace equation of a well-posed potential problem (e.g., heat conduction) in a general plane domain can be expressed as [21, 22]with the boundary conditionswhere is the unknown field variable and represents the boundary flux, is the component of outward normal vector to the boundary , and and are specified functions on the related boundaries, respectively. The space derivatives are indicated by a subscript comma, that is, , and the subscript index takes values for two-dimensional and for three-dimensional problems. Additionally, the repeated subscript indices imply summation convention.

For convenience, (3) can be rewritten in matrix form aswith .

2.2. Assumed Independent Fields

In this section, the procedure for developing a hybrid finite element model with fundamental solution as interior trial function is described based on the boundary value problem defined by (1)–(3). Similar to the conventional FEM and HT-FEM, the domain under consideration is divided into a series of elements [15, 16, 21, 2330]. In each element, two independent fields are assumed in the way as described in [31] and are given in Section 2.2.

2.2.1. Intraelement Field

Similar to the method of fundamental solution (MFS) in removing singularities of fundamental solution, for a particular element occupying subdomain , we assume that the field variable defined in the element domain is extracted from a linear combination of fundamental solutions centered at different source points located outside the element (see Figure 1):where is undetermined coefficients, is the number of virtual sources, and is the fundamental solution to the partial differential equation:asEvidently, (5) analytically satisfies (1) due to the solution property of .

Figure 1: Intraelement field and frame field of a HFS-FEM element for 2D potential problems.

In implementation, the number of source points is taken to be the same as the number of element nodes, which is free of spurious energy modes and can keep the stiffness equations in full rank, as indicated in [21]. The source point can be generated by means of the method employed in the MFS [3235]where is a dimensionless coefficient, is the point on the element boundary (the nodal point in this work), and is the geometrical centroid of the element (see Figure 1). Determination of was discussed in [31, 36] and is usually used in practice.

The corresponding outward normal derivative of on iswherewith

2.2.2. Auxiliary Frame Field

In order to enforce the conformity on the field variable , for instance, on of any two neighboring elements and , an auxiliary interelement frame field is used and expressed in terms of the same degrees of freedom (DOF) as those used in the conventional finite elements. In this case, is confined to the whole element boundary aswhich is independently assumed along the element boundary in terms of nodal DOF , where represents the conventional finite element interpolating functions. For example, a simple interpolation of the frame field on a side with three nodes of a particular element can be given in the formwhere stands for shape functions in terms of natural coordinate defined in Figure 2.

Figure 2: Typical quadratic interpolation for frame field.
2.3. Modified Variational Principle

For the boundary value problem defined in (1)–(3) and (5), since the stationary conditions of the traditional potential or complementary variational functional cannot guarantee the interelement continuity condition required in the proposed HFS FE model, as in the HT FEM [21, 26], a variational functional corresponding to the new trial functions should be constructed to assure the additional continuity across the common boundaries between intraelement fields of element “” and element “” (see Figure 3) [36, 37]:

Figure 3: Illustration of continuity between two adjacent elements “” and “.”

A modified variational functional is developed as follows:wherein which the governing equation (1) is assumed to be satisfied, a priori, for deriving the HFS FE model. The boundary of a particular element consists of the following parts:where represents the interelement boundary of the element “” shown in Figure 3.

To show that the stationary condition of the functional (15) leads to the governing equation (Euler equation), boundary conditions, and continuity conditions, invoking (16) and (15) gives the following functional for the problem domain:from which the first-order variational yieldsUsing divergence theoremwe can obtain For the displacement-based method, the potential conformity should be satisfied in advancethen (21) can be rewritten asThe Euler equation and boundary conditions can be obtained asusing the stationary condition .

As for the continuous requirement between two adjacent elements “” and “” given in (14), we can obtain it in the following way. When assembling elements “” and “,” we havefrom which the vanishing variation of leads to the reciprocity condition on the interelement boundary .

If the following expression: is uniformly positive (or negative) in the neighborhood of , where the displacement has such a value that and where stands for the stationary value of , we have in which the relation that is identical on has been used. This is due to the definition in (14) in Section 2.3.

Proof. For the proof of the theorem on the existence of extremum, we may complete it by the so-called “second variational approach” [38]. In doing this, performing variation of and using the constrained conditions (26), we findTherefore the theorem has been proved from the sufficient condition of the existence of a local extreme of a functional [38].

2.4. Element Stiffness Equation

With the intraelement field and frame field independently defined in a particular element (see Figure 1), we can generate element stiffness equation by the variational functional derived in Section 2.3. Following the approach described in [21], the variational functional corresponding to a particular element of the present problem can be written asAppling the divergence theorem (20) to the functional (29), we have the final functional for the HFS-FE modelThen, substituting (5), (9), and (12) into the functional (30) producesin whichThe symmetry of is obvious from the scalar definition (31) of variational functional .

To enforce interelement continuity on the common element boundary, the unknown vector should be expressed in terms of nodal DOF . The minimization of the functional with respect to and , respectively, yieldsfrom which the optional relationship between and and the stiffness equation can be producedwhere stands for the element stiffness matrix.

2.5. Numerical Integral for and Matrix

Generally, it is difficult to obtain the analytical expression of the integral in (32) and numerical integration along the element boundary is required. Herein the widely used Gaussian integration is employed [22].

For the matrix, one can express it asby introducing the matrix function Equation (36) can be further rewritten aswhereand is the Jacobean expressed as whereThus, the Gaussian numerical integration for matrix can be calculated bywhere is the number of edges of the element and is the Gaussian sampling points employed in the Gaussian numerical integration. Similarly, we can calculate the matrix using

The calculation of vector in (32) is the same as that in the conventional FEM, so it is convenient to incorporate the proposed HFS-FEM into the standard FEM program. Besides, the flux is directly computed from (9). The boundary DOF can be directly computed from (12) while the unknown variable at interior points of the element can be determined from (5) plus the recovered rigid-body modes in each element, which is discussed in the following section.

2.6. Recovery of Rigid-Body Motion

Considering the physical definition of the fundamental solution, it is necessary to recover the missing rigid-body motion modes from the above results. Following the method presented in [21], the missing rigid-body motion can be recovered by writing the internal potential field of a particular element aswhere the undetermined rigid-body motion parameter can be calculated using the least square matching of and at element nodeswhich finally givesin which and is the number of element nodes. Once the nodal field is determined by solving the final stiffness equation, the coefficient vector can be evaluated from (34), and then is evaluated from (45). Finally, the potential field at any internal point in an element can be obtained by means of (43).

3. Plane Elasticity Problems

3.1. Linear Theory of Plane Elasticity

In linear elastic theory, the strain displacement relations can be used and equilibrium equations refer to the undeformed geometry [39]. In the rectangular Cartesian coordinates (), the governing equations of a plane elastic body can be expressed asIf written as matrix form, it can be presented aswhere is a stress vector, is a body force vector, and the differential operator matrix is given aswhere is a strain vector and is a displacement vector.

The constitutive equations for the linear elasticity are given in matrix form aswhere is the material coefficient matrix with constant components for isotropic materials, which can be expressed as follows: whereThe two different kinds of boundary conditions can be expressed aswhere denotes the traction vector and is a transformation matrix related to the direction cosine of the outward normal Substituting (49) and (50) into (47) yields the well-known Navier partial differential equations in terms of displacements:

3.2. Assumed Independent Fields

For elasticity problem, two different assumed fields are employed as in potential problems: intraelement and frame field [1, 31, 36]. The intraelement continuity is enforced on nonconforming internal displacement field chosen as the fundamental solution of the problem [36]. The intraelement displacement fields are approximated in terms of a linear combination of fundamental solutions of the problem of interest:where is again the number of source points outside the element domain, which is equal to the number of nodes of an element in the present research based on the generation approach of the source points [31]. The vector and the fundamental solution matrix are now in the form in which and are, respectively, the field point and source point in the local coordinate system (). The components are the fundamental solution, that is, induced displacement component in -direction at the field point due to a unit point load applied in -direction at the source point , which are given by [40, 41]where . The virtual source points for elasticity problems are generated in the same manner as that in potential problems described in Section 2.

With the assumption of intraelement field in (56), the corresponding stress fields can be obtained by the constitutive equation (50):where As a consequence, the traction is written as in which The components for plane strain problems are given as

The unknown in (56) is calculated by a hybrid technique [31], in which the elements are linked through an auxiliary conforming displacement frame which has the same form as that in conventional FEM (see Figure 1). This means that, in the HFS-FEM, a conforming displacement field should be independently defined on the element boundary to enforce the field continuity between elements and also to link the unknown with nodal displacement . Thus, the frame is defined as where the symbol “” is used to specify that the field is defined on the element boundary only, is the matrix of shape functions, and is the nodal displacements of elements. Taking the side 3-4-5 of a particular 8-node quadrilateral element (see Figure 1) as an example, and can be expressed as where , and can be expressed by natural coordinate as

3.3. Modified Functional for the Hybrid FEM

As in Section 2, HFS-FE formulation for a plane elastic problem can also be established by the variational approach [36]. In the absence of body forces, the hybrid functional used for deriving the present HFS-FEM can be constructed as [22]where and are the intraelement displacement field defined within the element and the frame displacement field defined on the element boundary, respectively. and are the element domain and element boundary, respectively.  , and stand, respectively, for the specified traction boundary, specified displacement boundary, and interelement boundary (). Compared to the functional employed in the conventional FEM, the present variational functional is constructed by adding a hybrid integral term related to the intraelement and element frame displacement fields to guarantee the satisfaction of displacement and traction continuity conditions on the common boundary of two adjacent elements.

By applying the Gaussian theorem, (67) can be simplified toDue to the satisfaction of the equilibrium equation with the constructed intraelement field, we have the following expression for HFS finite element model:The variational functional in (69) contains boundary integrals only and will be used to derive HFS-FEM formulation for the plane isotropic elastic problem.

3.4. Element Stiffness Matrix

As in Section 2, the element stiffness equation can be generated by setting . Substituting (56), (64), and (61) into the functional of (69), we havewhere To enforce interelement continuity on the common element boundary, the unknown vector should be expressed in terms of nodal DOF . The stationary condition of the functional with respect to and yields, respectively, (33) and (34).

3.5. Recovery of Rigid-Body Motion

For the same reason stated in Section 2.6, it is necessary to reintroduce the discarded rigid-body motion terms after we have obtained the internal field of an element. The least squares method can be employed for this purpose and the missing terms can be recovered easily by setting for the augmented internal field [22]where the undetermined rigid-body motion parameter can be calculated using the least square matching of and at element nodeswhich finally giveswherein which and is the number of element nodes. Once the nodal field is determined by solving the final stiffness equation, the coefficient vector can be evaluated from (56), and then is evaluated from (74). Finally, the displacement field at any internal point in an element can be obtained by (72).

4. Three-Dimensional Elastic Problems

In this section, the HFS-FEM approach is extended to three-dimensional (3D) elastic problem with/without body force. The detailed 3D formulations of HFS-FEM are firstly derived for elastic problems by ignoring body forces, and then a procedure based on the method of particular solution and radial basis function approximation are introduced to deal with the body force [42]. As a consequence, the homogeneous solution is obtained by using the HFS-FEM and the particular solution associated with body force is approximated by using the strong form of basis function interpolation.

4.1. Governing Equations and Boundary Conditions

Let () denote the coordinates in a Cartesian coordinate system and consider a finite isotropic body occupying the domain , as shown in Figure 4. The equilibrium equation for this finite body with body force can be expressed as

Figure 4: Geometrical definitions and boundary conditions for a general 3D solid.

The constitutive equations for linear elasticity and the kinematical relation are given as where is the stress tensor, is the strain tensor, and is the Kronecker delta. Substituting (77) into (76), the equilibrium equation is rewritten as

For a well-posed boundary value problem, the boundary conditions are prescribed as follows: where is the boundary of the solution domain and are the prescribed boundary values. In the following parts, we will present the procedure for handling the body force appearing in (78).

4.2. The Method of Particular Solution

The inhomogeneous term associated with the body force in (78) can be effectively handled by means of the method of particular solution presented in [22]. In this approach, the displacement is decomposed into two parts, a homogeneous solution and a particular solution where the particular solution should satisfy the governing equation: without any restriction of boundary condition. However, the homogeneous solution should satisfywith the modified boundary conditions

From the above equations it can be seen that once the particular solution is known, the homogeneous solution in (82) and (83) can be obtained using HFS-FEM. The final solution can then be given by (80). In the next section, radial basis function approximation is introduced to obtain the particular solution, and the HFS-FEM is presented for solving (82) and (83).

4.3. Radial Basis Function Approximation

For body force , it is generally impossible to find an analytical solution which enables us to convert the domain integral into a boundary one. So we must approximate it by a combination of basis (trial) functions or other methods with the HFS-FEM. Here, we use radial basis function (RBF) [33, 43] to interpolate the body force. We assumewhere is the number of interpolation points, are the RBFs, and are the coefficients to be determined. Subsequently, the particular solution can be approximated bywhere is the approximated particular solution kernel of displacement satisfying (86) below. Once the RBFs are selected, the problem of finding a particular solution is reduced to solve the following equation:

To solve this equation, the displacement is expressed in terms of the Galerkin-Papkovich vectorsSubstituting (87) into (86), we can obtain the following biharmonic equation:Taking the Spline Type RBF as an example, we get the following solutions:whereand represents the Euclidean distance between a field point and a given point in the domain of interest. The corresponding particular solution of stresses can be obtained aswhere . Substituting (90) into (92), we havewhere

4.4. HFS-FEM for Homogeneous Solution

After obtaining the particular solution in Sections 4.2 and 4.3, we can determine the modified boundary conditions (83). Finally, we can treat the 3D problem as a homogeneous problem governed by (82) and (83) by using the HFS-FEM presented below. It is clear that once the particular and homogeneous solutions for displacement and stress components at nodal points are determined, the distribution of displacement and stress fields at any point in the domain can be calculated using the element interpolation function. However, for 3D elasticity problems in the absent of body force, that is, , the procedures in Sections 4.2 and 4.3 will become unnecessary and we can employ the following procedures to find the solution directly.

4.4.1. Assumed Intraelement and Auxiliary Frame Fields

The intraelement displacement fields are approximated in terms of a linear combination of fundamental solutions of the problem as where the matrix and unknown vector can be further written asin which and are, respectively, the field point and source point in the local coordinate system (). The fundamental solution is given by [40]where , is the number of source points. The source point can also be generated by means of the following method [36] as in two-dimensional cases where is a dimensionless coefficient, is the point on the element boundary (the nodal point in this work), and is the geometrical centroid of the element (see Figure 5).

Figure 5: Intraelement field and frame field of a hexahedron HFS-FEM element for 3D elastic problem (the source points and centroid of 20-node element are omitted in the figure for clarity and clear view, which is similar to that of the 8-node element).

According to (50) and (49), the corresponding stress fields can be expressed as where The components are given byAs a consequence, the traction can be written in the form in which

To link the unknown and the nodal displacement , the frame is defined as where the symbol “” is used to specify that the field is defined on the element boundary only, is the matrix of shape functions, and is the nodal displacements of elements. Taking the surface 2-3-7-6 of a particular 8-node brick element (see Figure 5) as an example, matrix and vector can be expressed as where the shape functions are expressed as where () can be expressed by natural coordinate and is the natural coordinate of the -node of the element (Figure 6).

Figure 6: Typical linear interpolation for the frame fields of 3D brick elements.
4.4.2. Modified Functional for Hybrid Finite Element Method

In the absence of body forces, the hybrid functional used for deriving the present HFS-FEM is written as [22]By applying the Gaussian theorem, (108) can be simplified asDue to the satisfaction of the equilibrium equation with the constructed intraelement fields, we have the following expression for HFS finite element model:The functional (110) contains only boundary integrals of the element and will be used to derive HFS-FEM formulation for the three-dimensional elastic problem in the following section.

4.4.3. Element Stiffness Matrix

Substituting (95), (102), and (104) into the functional (110), we have where To enforce interelement continuity on the common element boundary, the unknown vector should be expressed in terms of nodal DOF . The stationary condition of the functional with respect to and yields again, respectively, (33) and (34).

4.4.4. Numerical Integral over Element

Considering a surface of the 3D hexahedron element, as shown in Figure 6, the vector normal to the surface can be obtained by where and are the tangential vectors in the -direction and -direction, respectively, calculated bywhere is the number of nodes of the surface and are the nodal coordinates. Thus the unit normal vector is given bywhereis the Jacobian of the transformation from Cartesian coordinates () to natural coordinates ().

For the matrix, we introduce the matrix function Then we can getand we rewrite it to the component form as Using the relationship and the Gaussian numerical integration, we can obtainwhere and are, respectively, the number of surface of the 3D element and the number of Gaussian integral points in each direction of the element surface. Similarly, we can calculate the matrix byIt should be mentioned that the calculation of vector in equation is the same as that in the conventional FEM, so it is convenient to incorporate the proposed HFS-FEM into the standard FEM program. Besides, the stress and traction estimations are directly computed from (99) and (100), respectively. The boundary displacements can be directly computed from (104) while the displacements at interior points of element can be determined from (95) plus the recovered rigid-body modes in each element, which is introduced in the following section.

4.4.5. Recovery of Rigid-Body Motion Terms

As in Section 2, the least square method is employed to recover the missing terms of rigid-body motions. The missing terms can be recovered by setting for the augmented internal fieldand using a least-square procedure to match and at the nodes of the element boundaryThe above equation finally yieldswhere

5. Thermoelasticity Problems

Thermoelasticity problems arise in many practical designs such as steam and gas turbines, jet engines, rocket motors, and nuclear reactors. Thermal stress induced in these structures is one of the major concerns in product design and analysis. The general thermoelasticity is governed by two time-dependent coupled differential equations: the heat conduction equation and the Navier equation with thermal body force [44]. In many engineering applications, the coupling term of the heat equation and the inertia term in Navier equation are generally negligible [44]. As a consequence, most of the analyses are employing the uncoupled thermoelasticity theory which is adopted in this topic [4552]. In this section, the HFS-FEM is presented to solve 2D and 3D thermoelastic problems with considering arbitrary body forces and temperature changes [53]. The method used herein is similar to that in Section 4.

5.1. Basic Equations for Thermoelasticity

Consider an isotropic material in a finite domain and let () denote the coordinates in Cartesian coordinate system. The equilibrium governing equations of the thermoelasticity with the body force are expressed as where is the stress tensor, is the body force vector, and . The generalized thermoelastic stress-strain relations and kinematical relation are given as where is the strain tensor, is the displacement vector, is the temperature change, is the shear modulus, is the Poisson’s ratio, is the Kronecker delta, and is the thermal constant with being the coefficient of linear thermal expansion. Substituting (128) into (127), the equilibrium equations may be rewritten as

For a well-posed boundary value problem, the following boundary conditions, either displacement or traction boundary condition, should be prescribed as where is the boundary of the solution domain , and are the prescribed boundary values, and is the boundary traction, in which denotes the boundary outward normal.

5.2. The Method of Particular Solution

For the governing equation (130) given in the previous section, the inhomogeneous term can be eliminated by employing the method of particular solution [16, 36, 44]. Using superposition principle, we decompose the displacement into two parts, the homogeneous solution and the particular solution as follows:in which the particular solution should satisfy the governing equation but does not necessarily satisfy any boundary condition. It should be pointed out that its solution is not unique and can be obtained by various numerical techniques. However, the homogeneous solution should satisfywith modified boundary conditionsFrom above equations, it can be seen that once the particular solution is known, the homogeneous solution in (135)–(137) can be solved by (135). In the following section, RBF approximation is described to illustrate the particular solution procedure, and the HFS-FEM is presented which can be used for solving (135)–(137).

5.3. Radial Basis Function Approximation

RBF is to be used to approximate the body force and the temperature field in order to obtain the particular solution. To implement this approximation, we may consider two different ways: one is to treat body force and the temperature field separately as in Tsai [54]. The other is to treat as a whole [53]. Here we demonstrated that the performance of the latter one is usually better than the former one.

5.3.1. Interpolating Temperature and Body Force Separately

The body force and temperature are assumed to be by the following two equations: where is the number of interpolation points, are the basis functions, and and are the coefficients to be determined by collocation. Subsequently, the approximate particular solution can be written as follows: where and are the approximated particular solution kernels. Once the RBF is selected, the problem of finding a particular solution will be reduced to solve the following equations:

To solve (140), the displacement is expressed in terms of the Galerkin-Papkovich vectors [43, 5557]Substituting (142) into (140), one can obtain the following biharmonic equation:If taking the Spline Type RBF , one can get the following solutions: wherefor two-dimensional problem and wherefor three-dimensional problem, where represents the Euclidean distance of the given point from a fixed point <