DOI: 10.1615/ICHMT.2015.IntSympAdvComputHeatTransf
ISBN Print: 978-1-56700-429-8
ISSN: 2578-5486
A COMPARISON BETWEEN GLOBAL AND LOCALIZED RBF MESHLESS METHODS FOR PROBLEMS INVOLVING CONVECTIVE HEAT TRANSFER
要約
Meshless methods are a relative newcomer to the field of computational methods; the
term "meshless method" refers to the class of numerical techniques that rely on either global or
localized interpolation on non-ordered spatial point distributions. They have the following
advantages: (1) domain and boundary discretization is bypassed; (2) domain integration is not
required; (3) custom points (e.g. randomly generated or imported from a file) can be used as the
domain; (4) exponential convergence for smooth boundary shapes and boundary data can be
realized; (5) multi-dimensional problems are naturally handled; (6) implementation is comparatively
easy. This paper extends the method developed by Sarler and Vertnik [2006] to solve problems
coupled with convective heat transfer.
Few studies have been carried out to compare global and localized RBF meshless methods side by
side. See, for example, Islam et al. [2012], who demonstrated the advantages of the localized
approach for the case of the diffusion-reaction equation in three-dimensions. Here we make a
comparison between global and localized radial basis function (RBF) methods after establishing the
accuracy of each one based on the solution to benchmark fluid flow problems, including the lid
driven cavity, natural convection, and flow over a backward step. Global RBF-based methods have
some well-known drawbacks, including poor conditioning of the ensuing algebraic set of equations.
While these drawbacks can be addressed, to some extent, by domain decomposition and appropriate
pre-conditioning, our results favor the localized approach.
The attractive feature of the localized RBF method is that it allows field variable derivatives of any
order to be estimated by simple inner products of vectors that can be pre-built and stored. Since the
multiquadric functions can be evaluated at a setup stage when these vectors are being built, the
computational burden of having to evaluate fractional powers and complicated functions at every
step of an iteration or time-marching scheme can be avoided. In addition, the memory demands of
the localized approach are minimal, as no global collocation matrix needs to be allocated; only small
vectors are stored for every one of the data centres. We conclude that localized methods offer
tremendous advantages over global RBF-based meshless methods in terms of data preparation,
parallelizability, and the possibility for a truly autonomous approach at the problem setup stage.