Singularity problems from source functions as source nodes located near boundaries; numerical methods and removal techniques
| Autor: | Ming-Gong Lee, Zi-Cai Li, Hung-Tsai Huang, Li-Ping Zhang |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Source function
Physics Dirichlet problem Applied Mathematics Mathematical analysis General Engineering Boundary (topology) 02 engineering and technology 01 natural sciences Numerical integration 010101 applied mathematics Computational Mathematics 020303 mechanical engineering & transports Singularity 0203 mechanical engineering Secant method Trefftz method Method of fundamental solutions 0101 mathematics Analysis |
| Zdroj: | Engineering Analysis with Boundary Elements. 130:300-321 |
| ISSN: | 0955-7997 |
| DOI: | 10.1016/j.enganabound.2021.05.016 |
| Popis: | Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S . The source function q ln | P Q * ¯ | is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by q ln | P Q * ¯ | as the source node Q * is located near the boundary Γ ( = ∂ S ) . So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q * located near Γ . In this paper, an additional FS as, d 0 ln | P Q 0 ¯ | , is added to the original source nodes in the traditional MFS, and the point charge d 0 ( = q ) and the source node Q 0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q * is located inside S but near Γ , both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function q ln | P Q * ¯ | as the source node Q * is located near Γ . |
| Databáze: | OpenAIRE |
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