Improved Formulation for Geometric Properties of Arbitrary Polyhedra

Abstract

T is necessary to compute various geometric properties in a e nite volume discretization scheme used in computational e uid dynamics. These properties often include face areas and normal vectors, face centers, cell volumes and cell centroids, etc. In a computational grid composed of relatively simple cell types, such as tetrahedra, hexahedra, and prisms, it is relatively easy to compute these properties. For a computational grid with arbitrary polyhedral cells, such as cut cells generated through cell cutting in an adaptive Cartesian grid method, 1 it is not trivial to calculate the geometric properties,especiallythecellcentroids.InarecentpaperbyBruner, 2 aformulawasgiventocomputethecellcentroidofan N-facedpolyhedron. In Bruner’ s formula, the cell centroid is computed through a surface integral over the N faces bounding the cell. The surface integral, however, requires the evaluation of a Gaussian quadrature. The computation of the quadrature needs three quadrature points for a triangle and four points for a quadrilateral. The computation of the Gaussian quadrature for an arbitrary polygonal face is not straightforward. In this Note, a new formulation is derived for the cell centroid of an N-faced polyhedron for which each face is an arbitrary planar polygon. The new centroid formulation does not need the evaluation of a Gaussian quadrature. Only surface areas, normals, and face centers are required.