The equal spacing of N points on a sphere with application to partition-of-unity wave diffraction problems

Abstract

This paper addresses applications involving the selection of a set of points on a sphere, in which the uniformity of spacing can be of importance in enhancing the computational performance and/or the accuracy of some simulation. For the authors, the motivation for this work arises from the need to specify wave directions in a partition-of-unity approach for numerical analysis of wave diffraction problems. A new spacing method is presented, based on a physical analogy in which an arbitrary number of charged particles are held in static equilibrium on a spherical surface. The new method, referred to in this paper as the Coulomb force method, offers an improvement over simpler methods, e.g., latitude/longitude and discretised cube methods, in terms of both the uniformity of spacing and the arbitrary nature of the number of points N that can be considered. A simple extension to the algorithm allows points to be biased towards a direction of choice. Numerical results of a wave scattering problem solved with a partition-of-unity boundary element method demonstrate the benefits of the algorithm.

Introduction

There are numerous applications in science and engineering in which a set of N points is to be spaced as uniformly as possible on a spherical surface. The motivation for the authors of this paper is the definition of a set of plane wave directions to form a basis for a partition-of-unity [1] finite or boundary element analysis of wave diffraction. The choice of such a set of directions, being a set of unit vectors on the unit sphere, presents a directly equivalent problem. Confining this discussion to the various three-dimensional wave diffraction algorithms that exist, an efficient spacing algorithm would be of benefit in the plane-wave methods of Perrey-Debain et al. [2], in the discontinuous enrichment method of Massimi et al. [3], in the variational theory of complex rays of Kovalevsky et al. [4], in the ultra weak variational formulation of Luostari et al. [5], as well as other Trefftz methods. It is not the intention of the authors to present in this paper a detailed review of such methods; the interested reader is referred to Bettess [6].

Researchers from a diverse set of fields have studied the problem of finding a uniform set of points on sphere. In Monte Carlo approaches, the desire is to produce a set of points that is statistically uniform; that is, a suitable χ² test shows no significant deviation from the uniform distribution. Possibly the simplest method to achieve a statistically uniform distribution of points on the unit-sphere was first devised for the unit-circle by von Neumann [7] and extended by Cook [8] for spheres of three dimensions and higher. A sample $\mathbf{x}$ is taken from the uniform distribution on ${[-1,1]}^n$, where n is the number of dimensions being considered. The sample is rejected if its Euclidean norm, $\parallel \mathbf{x}\parallel$, is greater than 1 and accepted if $\parallel \mathbf{x}\parallel \le 1$. Sampling continues until the desired number of points is obtained. The points are then normalised so that they are placed on the surface of the sphere. This method is adequate for circles and three-dimensional spheres. However, as n increases, the size of the space $\parallel \mathbf{x}\parallel >1$ becomes much larger than the space $\parallel \mathbf{x}\parallel \le 1$; this means the ratio of rejected to accepted points increases rapidly and most of the computational burden is on generating points that will be discarded.

A similar method, presented by Muller [9], uses sample points taken from the normal distribution. This is possible as the multivariate normal distribution is radially symmetric. Given a suitable normal distribution, this method has a lower ratio of rejected to accepted points compared to taking points from the uniform distribution. A family of methods, using the beta distribution, were developed for higher dimensional spheres [10], [11], [12], [13], [14]. The relationship between these efficient methods was presented by Harman and Vladimir [15].

In mathematics, the ‘uniform spacing’ of points ordinarily refers to points that fit the statistical, uniform distribution. Conversely, in the physical sciences, ‘uniform spacing’ of points refers to making the distance or angle between adjacent points equal by maximising or minimising some criterion. One such example of this is the Thomson Problem: determining the minimum energy configuration of N electrons on the surface of a sphere. This is often associated with the Tammes problem in which N points are arranged on the surface of a sphere so that the minimum distance between them is maximised. Erber and Hockney [16] presented equilibrium configurations of charges on a sphere for $2\le N\le 65$. Glasser and Every [17] extended these calculations to $N\le 101$. Morris et al. [18] developed a genetic algorithm that searches for the steepest-decent in energy; with this algorithm, configurations were extended to $N\le 200$. Saff and Kuijlaars [19] considered configurations of $N\to \infty$, stating that the general pattern of optimal configuration was the same for all values of N.

In the study of meteorology, spherical grids can be used to model the atmosphere. Kurihara [20] stated that a homogeneous density of grid points on a globe is desirable, presenting a new grid system that was almost homogeneous. Sahr et al. [21] later reviewed methods of so-called geodesic discrete global grid systems in which the globe, modelled as an oblate spheroid, is divided into cells; some of these approaches examined ways of making these cells of equal area. It can be desirable to find uniformly spaced points on other surfaces: in operational research, Rubinstein [22] and Smith [23] considered generating random vectors uniformly on the surface of complex, multidimensional surfaces.

This paper concentrates on presenting a new method of producing equally spaced points on the unit-sphere in three-dimensions. This method is valid for arbitrary N and can be modified to fix one or more points on the sphere; a modification to cluster the points towards one point is also shown. This new method is used in the partition-of-unity enrichment of a boundary element method simulation of an acoustic wave scattered by a sphere; these simulations benefit from the ability to specify the position of one point and choose an arbitrary N.