Fundamental Concepts and Approaches

This introductory chapter uses fluid mechanics as an example field problem for reference; the applicability of the concepts discussed is, however, not in any way limited to this area. Fluid mechanics is described by nonlinear equations, which cannot generally be solved analytically, but which have been solved using various approximate methods including expansion and perturbation methods, sundry particle and vortex tracing methods, collocation and integral methods, and finite difference, finite volume, and finite element methods. Generally the finite difference, finite volume, and finite element discretization methods have been the most successful, but to use them it is necessary to discretize the field using a grid (mesh). (The terms grid and mesh are used interchangeably throughout with identical meaning.) The mesh can be structured or unstructured, but it must be generated under some of the various constraints described below, which can often be difficult to satisfy completely. In fact, at present it can take orders of magnitude more person-hours to construct the grid than it does to construct and analyze the physical solution on the grid. This is especially true now that solution codes of wide applicability are becoming available. Computational fluid dynamics (CFD) is a prime example, and grid generation has been cited repeatedly as a major pacing item (cf. Thompson [1996]). The same is true for other areas of computational field simulation. The proceedings of the several international conferences on grid generation (Thompson [1982], Hauser and Taylor [1986], Sengupta, et al. [1988], Arcilla, et al. [1991], Eiseman, et al. [1994], Soni et al. [1996]) as well as those of the NASA conferences (Smith [1980], Smith [1992], Choo [1995]) provide numerous illustrations of application to CFD and some other fields. A recent comprehensive text is Carey [1997]. Joe F. Thompson