Mathematical Problems in Engineering

Volume 2016, Article ID 3564632, 13 pages

http://dx.doi.org/10.1155/2016/3564632

## Numerical Investigation on Convergence Rate of Singular Boundary Method

State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 210098, China

Received 19 November 2015; Accepted 6 April 2016

Academic Editor: George Tsiatas

Copyright © 2016 Junpu Li et al. 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

The singular boundary method (SBM) is a recent boundary-type collocation scheme with the merits of being free of mesh and integration, mathematically simple, and easy-to-program. Its essential technique is to introduce the concept of the source intensity factors to eliminate the singularities of fundamental solutions upon the coincidence of source and collocation points in a strong-form formulation. In recent years, several numerical and semianalytical techniques have been proposed to determine source intensity factors. With the help of these latest techniques, this short communication makes an extensive investigation on numerical efficiency and convergence rates of the SBM to an extensive variety of benchmark problems in comparison with the BEM. We find that in most cases the SBM and BEM have similar convergence rates, while the SBM has slightly better accuracy than the direct BEM. And the condition number of SBM is lower than BEM. Without mesh and numerical integration, the SBM is computationally more efficient than the BEM.

#### 1. Introduction

The boundary element method (BEM) [1–4] and the method of fundamental solutions (MFS) [5–8] are two important numerical methods for science and engineering applications. However, it is a sophisticated mathematical and time-consuming issue for numerical integration over the singularities in the BEM. As for the MFS, the location of fictitious boundary is vital for its numerical accuracy and reliability and remains an open issue to optimally determine especially for complex-shaped or multiconnected domain problems.

To remedy these drawbacks in the BEM and MFS, several numerical schemes have been proposed, such as the boundary collocation method (BCM) [9], the modified collocation Trefftz method (MCTM) [10–12], the regularized meshless method (RMM) [13–16], the modified method of fundamental solutions (MMFS) [17], the boundary distributed source (BDS) method [18–20], and the nonsingular method of fundamental solutions [21].

The singular boundary method (SBM) [22] was proposed by the authors in 2009 and is a strong-form boundary discretization technique. In order to regularize the singularities of fundamental solutions upon the coincidence of source and collocation points, the concept of the source intensity factors (SIFs) was first introduced, which is also called the origin intensity factors (OIFs) in some literatures [22]. Under the extensive studies, four techniques have been proposed to determine the source intensity factors of both the fundamental solution and its derivative, namely, inverse interpolation technique (IIT), semianalytical technique with subtracting and adding-back desingularization (SAT1), semianalytical technique with integral mean value approach (SAT2), and semianalytical technique with the empirical formula (SAT3). Numerical investigation shows that the SBM can provide accurate solutions in potential [22], Helmholtz [23], acoustic and elastic waves [24, 25], and water wave problems [26] with arbitrarily complex-shaped computational geometries.

This short communication will make a comparison on numerical efficiency and convergence rate under the extensive benchmark testing with the best approach to determining the source intensity factors. A brief outline of the paper is as follows: Section 2 will describe the techniques to determine SIFs which will be used in Section 3 for solving 2D and 3D Laplace, Helmholtz, and modified Helmholtz equations. Section 3 will make a comparison on numerical efficiency and convergence rate under the extensive benchmark testing in comparison with the direct BEM. At last, Section 4 will make a conclusion.

#### 2. The Latest Techniques to Determine Source Intensity Factors

This section will describe the techniques to determine source intensity factors for solving 2D and 3D Laplace, Helmholtz, and modified Helmholtz equations.

##### 2.1. SBM for Laplace Equations

This section describes the SBM for Laplace equations. Consider the Laplace equations:where denotes the Laplacian operator, the solutions are the potentials in domain , is the known function, and is the unit outward normal on physical boundary. and represent the essential boundary (Dirichlet) and the natural boundary (Neumann) conditions.

By adopting the fundamental solution of Laplace equation, the solution is approximated by a linear combination of fundamental solutions with respect to different source points as below:where is the number of source points and is the th unknown coefficient. The fundamental solution will have singularities when . To solve this problem, we introduce the concept of source intensity factors in SBM. We place all computing nodes on the same physical boundary. So the source points and the collocation points are the same set of boundary nodes. When , we use source intensity factors replacing the singular terms in formulation (2). Thus the SBM formulation can be expressed aswhere and are defined as the source intensity factors corresponding to the fundamental solutions and the unit outward normal of fundamental solutions, namely, the diagonal elements of the SBM interpolation matrix.

Therefore, to solve all kinds of physical and mechanical problems with formulations (3), the key issue is to determine the source intensity factors.

In recent years, four techniques have been proposed to determine the abovementioned source intensity factors. And the merits and demerits of these techniques have already been extensively investigated in some literatures [22–26]. In this study, we only list the best formulations of these techniques to investigate the numerical efficiency and convergence rates of the SBM in comparison with the BEM under several typical benchmark problems.

In this study, we use formulas (4) and (5) to determine the source intensity factors for 2D and 3D Laplace problems with Neumann boundary condition.

The formulation can be expressed aswhere is one inner point inside the domain , is the sample solution in 2D problems, and is the sample solution in 3D problems. is the fundamental solution of Laplace equation, , and . is the corresponding influence area of source point . For 2D problem, is half length of the curve between source points and as shown in Figure 1(a). And, for 3D problem, it is the corresponding infinitesimal area as shown in Figure 1(b).