Open Access

ARTICLE

A Regularized Method of Fundamental Solutions for 3D and Axisymmetric Potential Problems

Csaba Gáspár¹

1 Széchenyi István University, Györ, Hungary. E-mail: gasparcs@sze.hu

Computer Modeling in Engineering & Sciences 2014, 101(6), 365-386. https://doi.org/10.3970/cmes.2014.101.365

Abstract

The Method of Fundamental Solutions (MFS) is investigated for 3D potential problem in the case when the source points are located along the boundary of the domain of the original problem and coincide with the collocation points. This generates singularities at the boundary collocation points, which are eliminated in different ways. The (weak) singularities due to the singularity of the fundamental solution at the origin are eliminated by using approximate but continuous fundamental solution instead of the original one (regularization). The (stronger) singularities due to the singularity of the normal derivatives of the fundamental solution are eliminated by solving special auxiliary subproblems (desingularization). The desingularization idea is similar to a previously published technique and is completely independent of the applied regularization technique. The presented method produces well-conditioned or moderately ill-conditioned matrices in the resulting linear system of algebraic equations, while the accuracy remains acceptable. No boundary mesh structure is needed. The method is generalized to 3D axisymmetric potential problems in a natural way, despite in this case the fundamental solution does not remain a radial function. The use of extremely ill-conditioned matrices is still avoided.

---