A simple empirical formula of origin intensity factor in singular boundary method for two-dimensional Hausdorff derivative Laplace equations with Dirichlet boundary

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.

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Analysis of Two-Dimensional Thin Structures (From Micro- to Nano-Scales) Using the Singular Boundary Method

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2013.m454 ◽

2015 ◽

Vol 7 (5) ◽

pp. 597-609 ◽

Cited By ~ 1

Author(s):

Dejian Shen ◽

Yan Gu

Keyword(s):

Smart Materials ◽

Boundary Element Methods ◽

Singular Boundary ◽

Two Dimensional ◽

Thin Structures ◽

Micro Electro Mechanical Systems ◽

Structural Problems ◽

Boundary Method ◽

Potential Applications ◽

Singular Boundary Method

AbstractThis study investigates the applicability of the singular boundary method (SBM), a recent developed meshless boundary collocation method, for the analysis of two-dimensional (2D) thin structural problems. The troublesome nearly-singular kernels, which are crucial in the applications of SBM to thin shapes, are dealt with efficiently by using a non-linear transformation technique. Promising SBM results with only a small number of boundary nodes are obtained for thin structures with the thickness-to-length ratio is as small as 1E-9, which is sufficient for modeling most thin layered coating systems as used in smart materials and micro-electro-mechanical systems. The advantages, disadvantages and potential applications of the proposed method, as compared with the finite element (FEM) and boundary element methods (BEM), are also discussed.

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

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2013.m359 ◽

2015 ◽

Vol 7 (1) ◽

pp. 13-30 ◽

Cited By ~ 57

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.

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