A General 2 D Meshless Interpolating Boundary Node Method Based on the Parameter Space

The presented study proposed an improved interpolating boundary node method (IIBNM) for 2D potential problems. The improved interpolating moving least-square (IIMLS) method was applied to construct the shape functions, of which the delta function properties and boundary conditions were directly implemented. In addition, any weight function used in the moving least-square (MLS) method was also applicable in the IIMLS method. Boundary cells were required in the computation of the boundary integrals, and additional discretization error was not avoided if traditional cells were used to approximate the geometry. The present study applied the parametric cells created in the parameter space to preserve the exact geometry, and the geometry was maintained due to the number of cells. Only the number of nodes on the boundary was required as additional information for boundary node construction. Most importantly, the IIMLS method can be applied in the parameter space to construct shape functions without the requirement of additional computations for the curve length.

Meshless methods only require the scattered points in the domain or on the boundary, and the unknown fields are approximated by some approximation methods, of which the moving least-square (MLS) approximation [25] is the most commonly used.However, the MLS approximation does not have the delta function property and thus generates certain difficulties in many meshless methods, such as in the BNM and hybrid BNM, wherein the boundary conditions cannot be directly applied as in the BEM and requires the application of new sets of virtual nodal values to overcome this problem.Lancaster and Salkauskas further developed an interpolating moving least-square (IMLS) method, wherein specific singular functions were used as weight functions [25], thereby resulting in the computation of the inverse of the singular matrix and a reduction in the computational efficiency.Wang et al. [26,27] recently proposed an improved interpolating moving least-square (IIMLS) method.It was applied in the improved interpolating element-free Galerkin method (IIEFG) [26] and the improved interpolating boundary element-free method (IIBEFM) [27].Li [28] proposed the application of the IIMLS method into the boundary elementfree method (BEFM) for 3D potential problems and discussed the details of the refurbished IIMLS method.The IIMLS method has overcome drawbacks within the MLS and IMLS.Firstly, the shape functions obtained by the IIMLS method 2 Mathematical Problems in Engineering exhibit the delta function property, thereby resulting in easier boundary conditions implementation.Secondly, the weight function does not have any restrictions and any weight functions used in the MLS can be directly used in the IIMLS method.Therefore, the present study applied the IIMLS method as the approximation method to construct the shape functions of the proposed meshless method.
Most meshless methods, such as the EFG and BNM, cannot avoid the integral cells used in the approximation of the geometry.These cells are only applied for integration purposes and should be distinguished from the employed elements.However, models with complicated geometries often exhibit additional discretization errors because the integral cells are not always able to reproduce the exact geometry, which is a similar problem observed in the FEM or BEM with elements.Therefore, meshless methods with integral cells cannot overcome the second above-mentioned FEM-or BEM-related problem.
Hughes et al. [29] initially proposed an isogeometric analysis (IGA) that has recently received widespread research attention.The same parametric functions used in CAD to model the geometry are also used to approximate unknown fields in numerical analyses, of which most take the form of nonuniform rational B-splines (NURBS).In the IGA, the geometry of the model can be exactly preserved to avoid discretization errors.The idea has also been applied in other numerical methods, such as in the boundary element method [2,[30][31][32][33][34] and the boundary face method [35].The present study focused on an improved interpolating boundary node method (IIBNM) based on the parameter space for 2D potential problems, also called the improved interpolating boundary element-free method (IIBEFM).Unlike the wellknown isogeometric analysis in FEM or BEM, the present study only applied the parametric function to reproduce the exact geometry and approximated the unknown fields by the IIMLS method.Thus, each curve only requires the parametric function, thereby characterizing the presented method as a general meshless method that is independent of the type of parametric function.Compared with IIBNM (IIBEFM), the proposed method used the parametric function to reproduce the exact geometry, and the integrations were performed in the parameter space.Thus, the discretization error was avoided.In addition, the IIMLS was also performed in parameter space and the curve lengths between boundary points were not needed.
This paper is organized as follows.The improved interpolating moving least-square method is first introduced in the second section, followed by the parameter spacederived cells in the third section.The fourth section presents the formulations of the IIBNM for 2D potential problems.Finally, some numerical examples are presented to show the accuracy of the proposed method.

The Improved Interpolating Moving
Least-Square Method Suppose x = { 1 , . . .,   } ∈ Ω is a point, Ω is a bounded domain in space R  ( = 1, 2, 3), and y = { 1 , . . .,   } is a point in the local approximation domain (x) of x, which can either be an evaluation point x or be a nodal point y  . is the number of nodal points y  in the domain of (x).For a given function (x), it can be approximated by IIMLS [26,27] as where In ( 7) and (8), (x − y  ) are weight functions.{  (x)}  =1 is a given set of basis functions with  1 (x) ≡ 1.
where  is defined from 1 to .

Improved Interpolating Boundary Node Method for 2D Potential Problems
The well-known regularized boundary integral equation for 2D potential problems can be written as follows: where Γ is the boundary of the bound domain Ω, x and y are points on the boundary, and  and  are the potential and normal flux on the boundary, respectively. * (x, y) and  * (x, y) are fundamental solutions of the Laplace equation, which can be written in the 2D case as follows: where (x, y) = ‖x − y‖ is the distance between x and y and n is the unit outward normal at the boundary point.The field of  and  at the boundary can be separately interpolated by the IIMLS method within the parameter space of each curve.Namely, the influence domain of an evaluation point is truncated at the corner of the boundary.Supposing that a boundary has  distributed nodes x  (  ), then the field of  and  on the boundary can be interpolated by the IIMLS method in the parameter space as follows: where Φ  () is the contribution from node   to the evaluated point  obtained in the 1D parameter space.Φ  () = 0 if   is out of the influence domain of .  and   are nodal values of  and  at   .In the real computation, the parameter coordinate  is defined on each smooth curve separately, and ( 18) are performed only on the local nodes in the influence domain of .In parameter space, Φ can be written as Substituting ( 18) into ( 16) presents where  and  are the parameter coordinates of points x and y, respectively.By discretizing the boundary into cells in the parameter space, (20) can be rewritten as follows: where Γ  is the kth cell and  is defined as For each node x  (  ) on the boundary, an equation like (21) can be presented as follows: Since Φ  (  ) exhibits the interpolating property, ( 22) can be rewritten as follows: Finally, (21) can be written in matrix form as follows: where In ( 25) and ( 26),   is the set of cells that contributes to Φ  () and   is the set of cells that contain node x  (  ).Equations ( 24) to ( 28) are very similar to the equations in the BEM.Actually, the methods used in the BEM for the treatment of singular integrals can also be applied in the IIBNM.The present study employed the following basis functions {  ()}  =1 : In addition, the Gaussian weight function was applied as the weight function in the IIMLS method.

Numerical Examples
Numerical examples were examined to justify the accuracy of the proposed method.For the purpose of error estimation, a formula is defined as follows: where  ()  and  ()  refer to the exact and numerical solutions, respectively, and || max is the maximum value of  over  nodes.In this paper,  are the potential  or the derivatives of .
Example 1 (Dirichlet problems on a circular ring).This example presents concentric a circular ring centered at  1 = 0 and  2 = 0 with radii of  1 = 1.0 and  2 = 2.0, which is shown in Figure 1(a).Dirichlet boundary conditions were imposed on the boundaries and the following solutions were employed as the analytical solutions.
(i) Linear solution is (iv) Cubic solution is Five sets of nodes were employed to discretize the model: (a) 40 nodes (Figure 1(b)), (b) 80 nodes, (c) 160 nodes, (d) 320 nodes, and (e) 640 nodes.The relative errors of , / 1 , and / 2 with respect to the number of nodes are plotted in Figures 2, 3, and 4, respectively.The proposed method exhibited good convergence rates for all the four different analytical solutions and presented very high accuracy.
Example 2 (Dirichlet problems on an elliptical ring).The present example examined an elliptical ring with Dirichlet boundary conditions, as presented in Figure 5(a).The inner ellipse had a major radius and minor radius of 2 and 1, whereas the outer ellipse had a major radius and minor radius of 8 and 4.
The analytical solution for this example was derived as follows: Two discretization methods were applied to test the proposed method.In discretization method 1 (Figure 5(b)), the numbers of nodes distributed on the inner and outer curves were the same and seven sets of nodes were used to discretize the model presented in Table 1.Conversely, discretization method 2 (Figure 5(c)) exhibited nearly equal curve distances between the two nearest nodes, of which the number of nodes for each discretization is presented in Table 1.
The relative errors of , / 1 , and / 2 with respect to the number of nodes for the two different discretization methods are plotted in Figures 6 and 7, wherein the error exhibited a decrease following an increase in the number of nodes.Both discretization methods exhibited a high convergence rate.
The contour of potential  computed by total 1940 nodes was shown in Figure 8, and the results computed by traditional BEM with 1938 linear elements were also plotted for the purpose of comparison.It could be observed that the results of the proposed methods have good agreement with those obtained by BEM.
Example 3 (potential flow problem).The present example considered the potential flow problem around a cylinder of radius  = 1 in an infinite domain.Here, due to the symmetry of the problem, only part of the upper left quadrant of the field was modeled as presented in Figure 9, where  = 4 and  = 2 in the figure.The exact solution for this problem is given as follows [36]: where  represents the stream function.
The boundary conditions of this problem are as follows: Curve (1): The relative errors of ,  1 = / 1 , and  2 = / 2 were computed from 115 nodes in the domain, as plotted in Figure 10.According to Figure 10, the proposed method exhibited a high accuracy to obtain a good rate of convergence.
The contours of stream function  computed by 405 boundary nodes are shown in Figure 11.BEM with 400 linear elements is also applied to solve this problem.Good agreement can be observed in these two methods.Example 4 (mixed problem on a complex model).The present example offers a complex model with mixed boundary conditions, as presented in Figure 12.Neumann boundary conditions were imposed on line  2 = ±8 and Dirichlet boundary conditions were imposed on the rest of the boundaries.The model employed the following analytical solution: A total of 737 nodes were distributed on the boundaries for the IIBNM.The BEM with a total of 714 linear elements was also applied to solve the problem.The numerical results of , / 1 , and / 2 on line  2 = −1.0 between points { 1 = 1.0,  2 = −1.0}and { 1 = 8.5,  2 = −1.0}are plotted in Figures 13, 14, and 15, respectively.The contour of  was shown in Figure 16 and the results were compared with BEM.The figures exhibited very high accuracy of IIBNM.Likewise, the IIBNM results were well in agreement with those obtained by the BEM and the analytical solution.

Conclusions
The present study proposed a meshless improved interpolating boundary node method based on the parameter space.The improved interpolating moving least-square method was employed as the approximation method to construct the shape function of the meshless method.The shape functions exhibited delta function properties as compared to those obtained by the moving least-square method.In addition, any weight function used in the MLS was also applicable in the IIMLS method.Cells were created in the parameter space, which are called parametric cells, to avoid the generation of integral cell discretization errors.The cells can produce exact geometry no matter the number of distributed nodes on the boundary.In addition, the IIMLS method was also performed in the parameter space, thereby eliminating the evaluation for the curve length, which simplifies the procedures.The proposed method can be extended to other problems and 3D problems.Coupling the presented method with the fast multipole method [37][38][39] for large-scale computations is under research.It is also possible to solve problems with body force by coupling the presented method with methods for domain integrals [40][41][42] in the BEM.

Figure 9 :
Figure 9: Model of flow around a cylinder.

Figure 10 :
Figure 10: Relative error of for potential flow problem.

Figure 11 :
Figure 11: Contour of  for potential flow problem: (a) proposed method and (b) BEM.

Table 1 :
Information of the discretization.