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Meshless Method using a RBF Interpolation Blending Scheme for inviscid Compressible Flows

Michael F. Harris
University of Central Florida, Mechanical and Aerospace Engineering Dept., Orlando, FL

Alain J. Kassab
Department of Mechanical and Aerospace Engineering, University of Central Florida, 4000 Central Florida Blvd, Orlando, Florida, USA

Eduardo Divo
Department of Mechanical Engineering, Embry-Riddle Aeronautical University, 600 S Clyde Morris Blvd, Daytona Beach, Florida, USA

DOI: 10.1615/
pages 771-777


The study of Radial basis functions (RBF) in the use of solving partial differential equations has grown in interest in recent years. These functions are capable of interpolating scattered data with impressive accuracy even when discontinuities in the data exist. The use of infinitely smooth RBF, such as the multiquadrics, inverse multiquadrics, and Gaussian, require the shape parameter in these functions to be chosen properly, otherwise poor accuracy or instabilities will occur. The shape parameter can vary greatly depending on the field, number of nodes, regions of steep gradients, shocks, or discontinuities. Generally, in RBF interpolation methods, the shape parameter is chosen to be arbitrarily high which renders the RBF flatter and produces a high condition number for the interpolation matrix. This approach tends to fail for regions where steep gradients, shocks, or discontinuities are present. In such cases, the shape parameter must be chosen to be low rendering the RBF steep and producing a low condition interpolation matrix. This work presents a blending scheme that combines low and high shape parameters. The blending scheme is able to sense steep gradients or shocks and adjust the shape parameter accordingly to maintain local accuracy. A formulation of the blending scheme using the Localized RBF Collocation Meshless Method (LRC-MM) and its application to the solution of the inviscid compressible Euler equations is presented in this paper.

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