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ABSTRACT
Construction and optimization methods of spherical hexagonal-pentagonal geodesic grids are investigated. The objective is to compare grid structures on common ground.
The distinction between two types of hexagonal-pentagonal grids is made. Three conventional grid optimization methods are summarized. In addition, three new optimization methods are proposed. Six desirable conditions for an ideal grid are described, and the grid optimization methods are organized in view of such conditions.
Interval uniformity, area uniformity, isotropy, and bisection of cell faces are systematically investigated for optimized grids. There are compensations of preferable grid features in each optimization method, and an optimal method cannot be decided based only on the research of grid features. It is suggested that grid optimization methods should be selected based on research of numerical schemes.
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1. Introduction
Geodesic dome, which was first developed by Buck- minster Fuller, is a generic name of polyhedra, which are composed of plane triangles inscribed inside a sphere. It is desirable that component triangles are as equilateral as possible to construct geodesic domes with sufficient strength. Sadourny et al. (1968) and William- son (1968) independently recognized the possibility of using the structure, which approximates a sphere, as a spherical grid system for meteorological applications. They proposed integration schemes of the barotropic vorticity equation on such triangular grid systems. The grid systems they used are composed of geodesics-in other words, arcs of great circles. We call such grid sys- tems, which are composed of geodesics and approximate a sphere, the "spherical geodesic grid" in this paper.
A regular polyhedron is ideal for the construction of spherical geodesic grids since distortion of constructed triangular grid systems possibly impairs accuracy of nu- merical schemes (e.g., Tomita et al. 2001). Only the tetrahedron, octahedron, and icosahedron are the regu- lar polyhedra constituted of regular triangles. It is con- sidered that the icosahedron is the best for the starting point because it has more faces than the others.
Various grid construction methods based on the icosahedron have been proposed (e.g., Sadourny et al. 1968; Williamson 1968; Cullen 1974; Baumgardner and Frederickson 1985; Heikes and Randall 1995a). Con- structed triangular grid systems were directly used for shallow-water models by several researchers (e.g., Stu- hne and Peltier 1996; Giraldo 2000)....