Finite elements in CFD: What lies ahead

Abstract

SUMMARYThe current state of the art of Finite Element Methods in Computational Fluid Dynamics is reviewed. The aim of this review is to point out what appear currently as the main shortcomings of Finite Element Methods, so as to concentrate the efforts to remove them. The analysis of flows using Finite Elements will only be successful if all steps involved in it are optimized. Therefore, I deem it necessary to describe explicit and implicit flow solvers, unstructured multigrid methods. adaptive refinement schemes, grid generation and graphics in 3-D, the effective use of supercomputer hardware and memory; as well as the combination of structured and unstructured grids. INTRODUCTTONComputational fluid dynamics (CFD) has always been at the forefront of the development of numerical techniques for the simulation of physical phenomena. Several reasons have contributed to this leadership among the many disciplines that can be numbered under the heading of 'Computational Mechanics'. The first is the inherently non-linear behaviour of fluids (advection, turbulence) which must bc accounted for. The second is thc mixed hyperbolic/elliptic character of the partial differential equations describing fluid motion. This mixed hyperbolie/elliptic character implies an increased algorithmic complexity. The third reason is the need of engineers designing new aeroplanes to obtain much more accurate performaiice estimates (and therefore more accurate results for the simulations) than their colleagues designing bridges. The fourth reason is the size of typical problems and follows from the first three. A problem in structural mechanics is considered large if it exceeds 5 x lo3 nodes, whereas 'large' in C F D means more than 5 x los gridpoints.For 25 years C F D grew around Finite Difference Methods, as these were simple to understand and code, easy to vectorize, and the structured grids typically associated with them described appropriately the simple geometric complexity of the fields that were solved. However, as computers became bigger and faster, attempts were made to simulate more and more complex flow domains, and it soon became clear that structured grids where not flexible enough to describe these domains. It was at this point in time that unstructured grids, and Finite Element Techniques-a natural way of discretizing operators on them-entered the scene of CFD. Since then, unstructured grids have become a driving force within CFD, and many new developments could * Invited paper presented at the First World Congress on Compulalional Mechanics, Austin, Texas, September 22-26, 1986 0029-598 1/87/09174 1-1 6$08.00 only become a reality by using them, such as domain splitting, adaptive refinement by enrichment within the same grid, directional refincmcnt and thc solution of the flowfield around a complete aircraft with engines.However impressive the entry of Finite Elements into the field of CFD may have been, many more developments are still needed in order to transform what are now still rudimentary methods into efficient pr...

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