On the supporting nodes in the localized method of fundamental solutions for 2D potential problems with Dirichlet boundary condition

This paper proposes a simple, accurate and effective empirical formula to determine the number of supporting nodes in a newly-developed method, the localized method of fundamental solutions (LMFS). The LMFS has the merits of meshless, high-accuracy and easy-to-simulation in large-scale problems, but the number of supporting nodes has a certain impact on the accuracy and stability of the scheme. By using the curve fitting technique, this study established a simple formula between the number of supporting nodes and the node spacing. Based on the developed formula, the reasonable number of supporting nodes can be determined according to the node spacing. Numerical experiments confirmed the validity of the proposed methodology. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters.


Introduction
The method of fundamental solutions (MFS) [1,2] is a simple, accurate and efficient boundary-type meshfree approach for the solution of partial differential equations, which uses the fundamental solution of a differential operator as a basis function. Chen et al. [3,4] demonstrated the equivalence between the MFS and the Trefftz collocation method [5] under certain conditions. condition. A reasonable number of supporting nodes can be directly given according to the size of node spacing. The proposed methodology is applicable to both regular and irregular distribution of nodes in arbitrary domain. This study consummates the LMFS and lays a foundation for simulating various engineering problems accurately and stably.
The rest of paper is organized as follows. In Section 2, we give the governing equation and boundary condition of 2D potential problems, and provide the numerical implementation of LMFS. Section 3 introduces the empirical formula of the node spacing and the number of supporting nodes in the LMFS. Section 4 investigates two numerical examples with irregular domain and nodes to illustrate the accuracy and reliability of the empirical formula. Section 5 summarizes some conclusions.

The LMFS for 2D potential problems
Considering the following 2D potential problem with the Dirichlet boundary condition: where 2  is the 2D Laplacian, ( , ) u x y the unknown variable,  the computational domain,    the boundary of domain, and ( , ) u x y the given function. According to the ideas of the LMFS, N ni nb  nodes i x ( 1, 2,..., ) iN  should be distributed inside the considered domain  and along its boundary  , where ni is the number of interior nodes, nb is the number of boundary nodes. Consider an arbitrary node (0) x (called the central node), we can find m supporting nodes () i x ( 1, 2,..., im  ) around the central node (0) x (see Figure 1 (a)). At the same time, the local subdomain s  can also be defined. Subsequently, the MFS formulation is implemented for the local subdomain. For this purpose, an artificial boundary s  is selected at a certain distance from the boundary of local subdomain, as shown in Figure 1 It should be pointed out that the artificial boundary is a circle centered at (0) x and with radius s R (see Figure 1 (b)), here s R is a parameter that should be manually fixed by the user.
In each local subdomain, we determine the unknown coefficients in which () i  represents the weighting function related with node () i x and is introduced as Then, a linear system can be formed The vector b in Eq (10) can be further expressed as (11) According to Eqs (9)-(11), the unknown coefficients 12 ( , , , ) T M      can now be calculated as To ensure the regularity of matrix A in Eq (12), 1 m  should be greater than M . For simplicity, we fixed 12 mM  in the computations. It should be pointed out that the matrix A with a small size given in Eq (10) is well-conditioned. In this study, the MATLAB routine " \ AB " is used to calculate 1  AB in order to avoid the troublesome matrix inversion. Substituting Eq (12) into Eq (4) as 0 i  , the numerical solution at central node (0) x is expressed as where subscript i of () j i c is used to distinguish the coefficients for different interior nodes. At boundary nodes with Dirichlet boundary condition, we have Using the given boundary data and combing Eqs (15) and (16), we have the following sparse system of linear algebraic equations ,  CU f (17) where NN  C represents the coefficient matrix, 12 ( , , , ) is a known vector composed by given boundary condition and zero vector. It should be pointed out that the system given in Eq (17) is well-conditioned, and standard solvers can be used to obtain its solution. In this study, the MATLAB routine " \  U C f " is used to solve this system of equation. After solving Eq (17), the approximated solutions at all nodes can be acquired.

Experiential formula of the supporting nodes in the LMFS
In this section, three different analytical solutions are provided to summarize the relationship between the number of supporting nodes (m) and the node spacing ( h  ). Without loss of generality, the node spacing is defined by Noted that the node spacing is equivalent to the total number of nodes, and is suitable for the arbitrary distribution of nodes including regular and irregular distributions. In the investigation, the equidistant nodes are used, thus the node spacing h  is the distance between two adjacent nodes. In all calculations, the artificial radius is fixed with 1.5 s R  , and the numerical results are calculated on a computer equipped with i5-5200 CPU@2.20GHz and 4GB memory. To estimate the accuracy of the present scheme, we adopt the root-mean-square error (RMSE) defined by  N is the total number of tested points, which refers to all nodes unless otherwise specified.
As shown in Figure 2, a rectangular with equidistant nodes is considered. To obtain a relatively universal formula, three different exact solutions are employed, as shown below: 22 ( , ) ,    By considering the above three cases for different analytical solutions, the following simple empirical formula can be carefully concluded: From Eq (23), we can easily determine the number of supporting nodes by the node spacing. In the next section, two complex examples will be provided to verify the present expression.

Numerical experiments
This section tests two numerical examples with complex boundaries. The following three different analytical solutions are used to to carefully validate the reliability of the empirical formula.

Example 1:
A gear-shaped domain is considered as shown in Figure 6. The LMFS uses 1334 nodes including 1134 internal nodes and 200 boundary nodes generated from the MATLAB codes, The node spacing is 0.0667m h  calculated from Eq (18). It can be known from the empirical formula (23) that the number of supporting nodes should be 6 m  , when the numerical error remains 1.0e 02 RMSE  .

Example 2:
In the second example, we consider a car cavity model. Figure 8 shows the problem geometry and the distribution of nodes. The total number of nodes is 11558 N  , including 11212 internal nodes and 346 boundary nodes. Nodes in this model are derived from the HyperMesh software. The node spacing is 0.02m h  calculated from Eq (18). It should be pointed out that According to the proposed empirical formula (23), the number of supporting nodes needs to meet 8 m  when the numerical error remains 1.0e 02 RMSE  .

Conclusions
In this paper, a simple, accurate and effective empirical formula is proposed to choose the number of supporting nodes. As a local collocation method, the number of supporting nodes has an important impact on the accuracy of the LMFS. This study given the direct relationship between the number of supporting nodes and the node spacing. By using the proposed formula, a reasonable number of supporting nodes can be determined according to the node spacing. Numerical results demonstrate the validity of the developed formula. The proposed empirical formula is beneficial to accuracy, simplicity and university for selecting the number of supporting nodes in the LMFS. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters.
It should be noted that the present study focuses on the 2D homogeneous linear potential problems with Dirichlet boundary condition. The proposed formula can not be directly applied to the 2D cases with Neumann boundary and 3D cases with Dirichlet or Neumann boundary condition. In addition, the LMFS depends on the fundamental solutions of governing equations. For the nonhomogeneous and/or nonlinear problems without the fundamental solutions, the appropriate auxiliary technologies should be introduced in the LMFS, and then the corresponding formula still need for further study. The subsequent works will be emphasized on these issues, according to the idea developed in this paper.