Extended Abstract: Advances in computer technology, scientific visualization, and discretization methods have made possible the numerical simulation of complex physical phenomena. More often than not, this involves the numerical solution of systems of partial differential equations utilizing finite difference, finite element or finite volume methods over realistic three-dimensional geometries. Inherent to any of these solution methods is, in addition to the discrete representation of the differential and/or integral operators in the equations, the discrete representation of the solution domain. This discrete domain over which the discrete equations are solved is called a grid [or a mesh]. One problem with realistic geometries is that they tend to be complex and as a result, the construction and design of grid structures over such geometries often becomes a very difficult task. In fact, the generation of a valid grid is usually the most labor intensive part of any computational field simulation. Volume grid cells are usually constructed using a trivariate polynomial map defined on a reference domain. The simplest and most popular trivariate is the trilinear. The map and its Jacobian are represented in Bezier form and a pyramid algorithm is utilized to simultaneously compute points and geometric characteristics associated with the map. In addition, sufficient conditions are given for the positivity of the Jacobian determinant and an iterative algorithm for solving the inversion problem is derived. The convergence and the accuracy of numerical solutions to partial differential equations strongly depend on the quality of the grids on which these solutions are computed. First and second order geometric characteristics for hexahedral volume grids are formulated and applied to evaluate the quality of three-dimensional grid structures. Examples measuring the Jacobian and the orthogonality of geologic grids are given.

Bio-Sketch: Vasilis G. Zafiris is a tenured Associate Professor of Computer & Mathematical Sciences at the University of Houston-Downtown. He received his B.S. in Aerospace Engineering and M.S. in Applied Mathematics both from Mississippi State University, and his Ph.D. in Applied Mathematics from the University of Houston-Central. His primary research interests focus on Scientific and Geometric Computing. He has published in journals and proceedings and has presented his work in national and international conferences. Before joining the university in 1999, he worked at Landmark Graphics in the Reservoir Simulation Group, at Los Alamos National Lab in the Earth and Environmental Science Group, and at Lawrence Livermore National Lab in the Scientific Computing Group. More recently, he worked as a consultant for Landmark Graphics and for GX Technologies.