Lattice Boltzmann Method with Improved Radial Basis Function Method
Date
2019-11
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Abstract
The Lattice Botlzmann method is an alternative numerical method of resolving problems relating to flow and heat transfer (i.e. Navier Stokes equations). This method consists of two steps, collision as well as streaming. In turn, streaming is non-local and functions on square or cubic grids. In general, the streaming process is undertaken by shifting the distribution function to the exact adjacent node (perfect shift). It is notable that this process does not entail interpolation, which makes lends a sense of precision to the advection process. On the other hand, the prefect shift in the streaming step impacts the applicability of LBM for problems with complex geometry. In literature, several works have been proposed to resolve this issue. Conventional methods (finite volume, finite difference, and finite element methods) are employed to solve the streaming step in the lattice Boltzmann method. Unlike the perfect shift, these methods tend to suffer from a dissipation error because they utilize a lower order approximation of the derivative. This dissipation error significantly impacts the accuracy of lattice Boltzmann method, thus making it comparable or even inferior to the accuracy of the conventional method of Navier Stokes solvers. This might render lattice Boltzmann method unfeasible because it has the same accuracy as that of the conventional methods and need a greater number of variables (distribution functions) to solve for. In order to resolve the problem, the meshless method has been used in literature to enhance the accuracy of the streaming. In this regard, Shu et al. proposed a meshless lattice Boltzmann method based on the least-square formulation of the Taylor series. Although the accuracy of this approach is good, it significantly depends on the mesh structure. To overcome this problem, Musavi and Ashrafizaadeh proposed the radial basis function based on the weak formulation of the advection equation. Against this backdrop, the present work proposes the usage of radial basis function interpolation in order to improve the accuracy. In general, the accuracy of radial basis function based methods depends on the shape parameter. Two algorithms have been proposed in the present work to solve this issue. The second algorithm is superior since its accuracy is independent of any parameter, including the shape parameter. The second approach is used to solve the streaming step in the lattice Boltzmann method. The combined approach is validated using simple problems. According to the findings, the proposed approach is reliable and consistent, whereas its high accuracy can be compared to the prefect shift.
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Keywords
Radial Basis Function, Lattice Boltzmann Method, Partition of Unity, Hermite, Meshless, RBF, LBM, PUM
Citation
Bawazeer, S. (2019). Lattice Boltzmann Method with Improved Radial Basis Function Method (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.