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Fluid Dynamics Seminar

Monday, March 31, 2008, 4:00 PM
Cullimore Lecture Hall, Room 611
New Jersey Institute of Technology

High order vortex methods and field interpolation problems

Louis Rossi

Department of Mathematics, University of Delaware

Abstract

Recent computing hardware trends toward multi-core processors favor high-performance algorithms that are high-order, parallelizable and adaptive. Vortex methods are well-suited to make the most of these improvements for fluid flow simulations. Vortex methods are numerical schemes for approximating solutions to the Navier-Stokes equations using a linear combination of moving basis functions to approximate the vorticity field of a fluid. Typically, the basis function velocity is determined through a Biot-Savart integral applied at the basis function centroid. Since vortex methods are naturally adaptive, they are advantageous in flows dominated by localized regions of vorticity such as jets, wakes and boundary layers. While they have been successful in numerous engineering application, the complexity of understanding grid-free methods make their analysis a uniquely mathematical endeavor. One recent outcome of rigorous analysis is a new naturally adaptive high order method with basis functions that deform as they move according to flow properties. This new class of methods is unusual because the basis functions do not move with the physical flow velocity at the basis function centroid as is usually specified in vortex methods. One of the leading edge research problems associated with high accuracy methods is ``field interpolation'' which is the process of projecting extremely deformed elements onto a configuration of regular elements to prevent catastrophic growth of interpolation errors. Recent progress in this area brings together ideas from radial basis function interpolation, pre-conditioners, image processing, and partial differential equations.