scholarly journals Numerical Investigation on Convergence Rate of Singular Boundary Method

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
Junpu Li ◽  
Wen Chen ◽  
Zhuojia Fu
Keyword(s):  
Short Communication ◽  
Convergence Rates ◽  
Fundamental Solutions ◽  
Benchmark Problems ◽  
Singular Boundary ◽  
Intensity Factors ◽  
Collocation Points ◽  
Boundary Method ◽  
Collocation Scheme ◽  
Singular Boundary Method

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.

An Improved Formulation of Singular Boundary Method

2012 ◽  
Vol 4 (5) ◽  
pp. 543-558 ◽  
Author(s):  
Wen Chen ◽  
Yan Gu
Keyword(s):  
Intensity Factor ◽  
Neumann Boundary ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Benchmark Problems ◽  
Singular Boundary ◽  
Boundary Method ◽  
New Formulation ◽  
Physical Boundary ◽  
Singular Boundary Method

AbstractThis study proposes a new formulation of singular boundary method (SBM) to solve the 2D potential problems, while retaining its original merits being free of integration and mesh, easy-to-program, accurate and mathematically simple without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). The key idea of the SBM is to introduce the concept of the origin intensity factor to isolate the singularity of fundamental solution so that the source points can be placed directly on the physical boundary. This paper presents a new approach to derive the analytical solution of the origin intensity factor based on the proposed subtracting and adding-back techniques. And the troublesome sample nodes in the ordinary SBM are avoided and the sample solution is also not necessary for the Neumann boundary condition. Three benchmark problems are tested to demonstrate the feasibility and accuracy of the new formulation through detailed comparisons with the boundary element method (BEM), MFS, regularized meshless method (RMM) and boundary distributed source (BDS) method.

Singular Boundary Method for Various Exterior Wave Applications

2015 ◽  
Vol 12 (02) ◽  
pp. 1550011 ◽  
Author(s):  
Zhuo-Jia Fu ◽  
Wen Chen ◽  
Ji Lin ◽  
Alexander H.-D. Cheng
Keyword(s):  
Fundamental Solution ◽  
Helmholtz Equation ◽  
Laplace Equation ◽  
International Workshop ◽  
Fundamental Solutions ◽  
Singular Boundary ◽  
Intensity Factors ◽  
Trefftz Method ◽  
Boundary Method ◽  
Singular Boundary Method

This paper presents an improved singular boundary method (ISBM) to various exterior wave applications. The SBM is mathematically simple, easy-to-program, meshless and introduces the concept of source intensity factors to eliminate the singularity of the fundamental solutions. In this study, we first derive the source intensity factors of the exterior Helmholtz equation by means of the source intensity factors of the exterior Laplace equation due to the same order of the singularities on their fundamental solutions. The source intensity factors of the exterior Laplace equation can be determined using the reference technique [Chen, W. and Gu, Y. [2011] "Recent advances on singular boundary method," Joint international workshop on Trefftz method VI and method of fundamental solution II, Taiwan]. Numerical illustrations demonstrate the efficiency and accuracy of the proposed scheme on four benchmark exterior wave examples.

scholarly journals Precorrected-FFT Accelerated Singular Boundary Method for High-Frequency Acoustic Radiation and Scattering

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 238
Author(s):  
Weiwei Li ◽  
Fajie Wang
Keyword(s):  
High Frequency ◽  
Acoustic Radiation ◽  
Low Frequency ◽  
Singular Boundary ◽  
Cpu Time ◽  
Interpolation Matrix ◽  
Collocation Points ◽  
Boundary Method ◽  
Frequency Regime ◽  
Singular Boundary Method

This paper presents a precorrected-FFT (pFFT) accelerated singular boundary method (SBM) for acoustic radiation and scattering in the high-frequency regime. The SBM is a boundary-type collocation method, which is truly free of mesh and integration and easy to program. However, due to the expensive CPU time and memory requirement in solving a fully-populated interpolation matrix equation, this method is usually limited to low-frequency acoustic problems. A new pFFT scheme is introduced to overcome this drawback. Since the models with lots of collocation points can be calculated by the new pFFT accelerated SBM (pFFT-SBM), high-frequency acoustic problems can be simulated. The results of numerical examples show that the new pFFT-SBM possesses an obvious advantage for high-frequency acoustic problems.

Solution of Two-Dimensional Stokes Flow Problems Using Improved Singular Boundary Method

2015 ◽  
Vol 7 (1) ◽  
pp. 13-30 ◽  
Author(s):  
Wenzhen Qu ◽  
Wen Chen
Keyword(s):  
Fundamental Solution ◽  
Stokes Flow ◽  
Singular Boundary ◽  
Two Dimensional ◽  
Intensity Factors ◽  
Flow Problems ◽  
Modified Method ◽  
Boundary Method ◽  
Singular Boundary Method

AbstractIn this paper, an improved singular boundary method (SBM), viewed as one kind of modified method of fundamental solution (MFS), is firstly applied for the numerical analysis of two-dimensional (2D) Stokes flow problems. The key issue of the SBM is the determination of the origin intensity factor used to remove the singularity of the fundamental solution and its derivatives. The new contribution of this study is that the origin intensity factors for the velocity, traction and pressure are derived, and based on that, the SBM formulations for 2D Stokes flow problems are presented. Several examples are provided to verify the correctness and robustness of the presented method. The numerical results clearly demonstrate the potentials of the present SBM for solving 2D Stokes flow problems.

Evaluating the Origin Intensity Factor in the Singular Boundary Method for Three-Dimensional Dirichlet Problems

2017 ◽  
Vol 9 (6) ◽  
pp. 1289-1311 ◽  
Author(s):  
Linlin Sun ◽  
Wen Chen ◽  
Alexander H.-D. Cheng
Keyword(s):  
Dirichlet Boundary ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Singular Boundary ◽  
Dirichlet Problems ◽  
Boundary Method ◽  
Interpolation Technique ◽  
Inverse Interpolation ◽  
New Formulation ◽  
Singular Boundary Method

AbstractIn this paper, a new formulation is proposed to evaluate the origin intensity factors (OIFs) in the singular boundary method (SBM) for solving 3D potential problems with Dirichlet boundary condition. The SBM is a strong-form boundary discretization collocation technique and is mathematically simple, easy-to-program, and free of mesh. The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions. Traditionally, the inverse interpolation technique (IIT) is adopted to calculate the OIFs on Dirichlet boundary, which is time consuming for large-scale simulation. In recent years, the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary, but the Dirichlet problem remains an open issue. This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary. Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique, the method of fundamental solutions, and the boundary element method.

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