On the use of quasi-equidistant source points over the sphere surface for the method of fundamental solutions

Journal of Computational and Applied Mathematics, 359 (2019), 55-68
https://doi.org/10.1016/j.cam.2019.03.019

The method of fundamental solutions is broadly used in science and engineering to
numerically solve the direct time-harmonic scattering problem. In 2D the choice of
source points is usually made by considering an inner pseudo-boundary over which
equidistant source points are placed. In 3D, however, this problem is much more
challenging, since, in general, n equidistant points over a closed surface do not exist.
In this paper we discuss a method to obtain a quasi-equidistant point distribution over
the unit sphere surface, giving rise to a Delaunay triangulation that might also be used
for other boundary element methods. We give theoretical estimates for the expected
distance between points and the expect area of each triangle. We illustrate the feasibility
of the proposed method in terms of the comparison with the expected values for
distance and area. We also provide numerical evidence that this point distribution leads
to a good conditioning of the linear system associated with the direct scattering problem,
being therefore an adequated choice of source points for the method of fundamental
solutions.