The equal spacing of N points on a sphere with application to partition-of-unity wave diffraction problems
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 is taken from the uniform distribution on , where n is the number of dimensions being considered. The sample is rejected if its Euclidean norm, , is greater than 1 and accepted if . 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 becomes much larger than the space ; 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 . Glasser and Every [17] extended these calculations to . Morris et al. [18] developed a genetic algorithm that searches for the steepest-decent in energy; with this algorithm, configurations were extended to . Saff and Kuijlaars [19] considered configurations of , 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.
Section snippets
Partition-of-unity boundary element method
The following section gives a brief derivation of the partition-of-unity boundary element method (PU-BEM). For a more thorough introduction to the topic of boundary elements, the authors recommend [24], [25].
Let be an infinite acoustic domain containing a smooth scatterer of boundary . Assuming time dependence, the wave equation is reduced to the Helmholtz equation:where is the Laplacian operator, is the wave potential at q, and k is the
Discretised cube boundary method
While in two dimensions, the uniform spacing of directions around the unit circle is a trivial problem, the move to three dimensions presents a greater difficulty, since it is not generally possible, and certainly not intuitive, to define a uniform division of the solid angle. There are also some trivial cases relating to the vertices and/or faces of the platonic solids. But in order to take full advantage of the plane wave basis methods in wave modelling, considerably larger numbers of
Example solutions
Fig. 6a shows a solution for the case M=8 as determined by the discretised cube boundary method; lines have been added to help show how these are the vertices of a cube. Fig. 6b shows the same case but determined by the Coulomb force method; the lines added show that this appears like two faces of a cube that are rotated 45° from each other.
This is an interesting case as both methods produce an equally spaced distribution of points with an equal minimum distance between points – approximately
Numerical results
The following numerical results are from simulations of a plane wave impinging a unit-radius, perfectly scattering sphere. Assuming the incident wave is propagating in the direction , the scattered acoustic potential at any point can be found with the analytical solution [27]:where jn is the spherical Bessel function of the first kind, hn is the spherical Hankel function of the first kind, Pn is the Legendre function of the
Conclusions
This paper has presented a new method for producing evenly spaced distributions of arbitrary numbers of points on a spherical surface. Although this has widespread application in science and engineering, the motivation is the efficient solution of partition-of-unity finite and boundary element problems in 3D wave scattering.
The method is a simple one, based on the use of an explicit time stepping scheme to converge to a static equilibrium state for a set of charged particles on a spherical
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