Error Estimations for Flow Characterization With Numerical and Analytical Solutions

Error Estimations for Flow Characterization With Numerical and Analytical Solutions

Irem Sanal
DOI: 10.4018/IJMTIE.2017070102
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Use of radial basis functions(RBFs) in the numerical solution of partial differential equations has gained popularity as it is meshless and can readily be extended to multi-dimensional problems. RBFs have been used in different context and emerged as a potential alternative for numerical solution of PDEs. In this article, a Flow Between Parallel Plates problem was solved using a Multiquadric Radial Basis Function Collocation Method (MQ-RBFCM), then, the results were compared with the analytic ones and the root mean square of the errors between the model and analytic results were calculated. Numerical results are presented for 5 different cases, where the number of inputs or definitions are increased to see whether changing the number of points makes the results better or not. Also, the absolute errors between the results were calculated to have a 3D model of the error rates and this has proven for which cases the MQ-RBFCM are better. As a result, RBF is shown to produce accurate results while requiring a much-reduced effort in problem preparation in comparison to traditional numerical methods.
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1. Introduction

Established numerical methods such as finite element methods FEM and finite volume methods FVM that are routinely used to resolve complex multiphysics interactions require significant effort in mesh generation. In fact, for most models of geometrically intricate components commonly encountered in engineering analysis, mesh generation is very time consuming, far from automated, and most often the most taxing part of the modeling process. The term meshless methods refer to a class of numerical techniques that rely on global interpolation on no ordered spatial point distributions that, as such, offer the hope of reducing the effort devoted to model preparation. These techniques have been under much development over the past few years (Belytscho, Lu, & Gu, 1994; Atluri & Shen, 2002; Atluri & Zhu, 1998; Liu, 2003; Melenk & Babuska, 1996; Kansa, 1990; Kansa, 1990; Kansa & Hon, 2000). Meshless methods use a nodal or point distribution that is not required to be uniform or regular in their spatial distribution due to the fact that most rely on global radial-basis functions RBF (Powell, 1992; Buhmann, 2003; Dyn, Levin, & Rippa, 1986). Moreover, care must be taken in the evaluation of derivatives in global RBF meshless methods. Although, very promising, these techniques can also be computationally intensive.

The basic idea of RBFCM is the construction of an unknown function, by the information that we get from the problem. An RBF depends on the number of nodes at which we will solve our problem and distances between these nodes. The use of RBF as a meshless procedure for numerical solution of PDEs is based on the collocation schemes. Due to the collocation technique, this method does not need to evaluate any integral (Cheng, Golberg, Kansa, & Zammito, 2003; Hoon, 2002; Kansa, 1990; Larsson & Fomberg, 2003; Hermite-Birkhoff, 1992). The RBFs methods performed well in many calculations including the numerical experiments that were reported by Franke (1982). For more information about the meshless method, we refer readers to Fasshuaer (2007) and the references therein.

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