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Advances in viscous vortex methods—meshless spatial adaption based on radial basis function interpolation
Author(s) -
Barba L. A.,
Leonard A.,
Allen C. B.
Publication year - 2005
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.811
Subject(s) - interpolation (computer graphics) , radial basis function , mathematics , meshfree methods , discretization , vortex , algorithm , mathematical optimization , computer science , finite element method , mathematical analysis , physics , artificial neural network , thermodynamics , animation , computer graphics (images) , machine learning
Abstract Vortex methods have a history as old as finite differences. They have since faced difficulties stemming from the numerical complexity of the Biot–Savart law, the inconvenience of adding viscous effects in a Lagrangian formulation, and the loss of accuracy due to Lagrangian distortion of the computational elements. The first two issues have been successfully addressed, respectively, by the application of the fast multipole method, and by a variety of viscous schemes which will be briefly reviewed in this article. The standard method to deal with the third problem is the use of remeshing schemes consisting of tensor product interpolation with high‐order kernels. In this work, a numerical study of the errors due to remeshing has been performed, as well as of the errors implied in the discretization itself using vortex blobs. In addition, an alternative method of controlling Lagrangian distortion is proposed, based on ideas of radial basis function (RBF) interpolation (briefly reviewed here). This alternative is formulated grid‐free, and is shown to be more accurate than standard remeshing. In addition to high‐accuracy, RBF interpolation allows core size control, either for correcting the core spreading viscous scheme or for providing a variable resolution in the physical domain. This formulation will allow in theory the application of error estimates to produce a truly adaptive spatial refinement technique. Proof‐of‐concept is provided by calculations of the relaxation of a perturbed monopole to a tripole attractor. Copyright © 2005 John Wiley & Sons, Ltd.