14-17 May 2018
New Orleans
US/Central timezone

RBF-FD approximations based on polyharmonic splines basis with supplementary polynomials applied in a pore-scale problem

16 May 2018, 17:07
New Orleans

New Orleans

Poster + 3 Minute Pitch MS 2.08: Recent Advances in Multiscale Methods and Uncertainty Quantification Parallel 8-H


Luis Guilherme Cunha Santos (Unicamp)


The Radial Basis Function generated Finite Differential (RBF-FD) is a meshless method that has attracted attention in the last decades by its flexibility in the numerical approximation of PDEs, simplicity of computational implementation and ease in the approach of complex geometries. It has already been successfully applied to various engineering problems such as heat transfer, electrostatics, vibration, seismic modeling, and in particular, fluid dynamics problems [1, 2, 3, 4, 5]. In this way, o RBF-FD is a good candidate in a range of increasing applications of numerical simulations including seismic exploration of oil and gas, among others [6]. In this work, we present applications of RBF-FD with polyharmonic splines basis (PHS) with supplementary polynomials, RBF(PHS)-FD for short, in two benchmarks using the vorticity and stream-function formulation: i) problem of natural convection of air in a square cavity for some values of Rayleigh number; ii) problem of driven flow in a square cavity for several Reynolds number values. In both problems, we discretize the domain in uniform point clouds and non-uniform point cloud. Finally, after the validation of the benchmark results, the RBF (PHS) -FD is applied and discussed for a problem of flow of a fluid in the pore scale with a complex geometry. We will also add some comments about the use of multiscale meshless method for porous media transport problems.


[1]  V. Bayona and N. Flyer and B. Fornberg and G. Barnett, On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs. Journal of Computational Physics, v. 332, pp. 257-273, 2017.

[2]  N. Flyer, G. Barnett and L. Wicker, Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations. Journal of Computational Physics, v. 316, pp. 39-62, 2016.

[3]  N. Flyer, B. Fornberg, V. Bayona and G. Barnett, On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy. Journal of Computational Physics, v. 321, pp. 21-38, 2016.

[4] C. Shu, H. Ding and S. Yeo. Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 192, 941-954, 2003.

[5] B. Martin and B. Fornberg. Seismic modeling with radial basis function generated finite differences (RBF-FD) - a simplifica treatment of interfaces, Journal of Computational Physics, v. 335, 828-845, 2017.

[6]  B. Fornberg and N. Flyer, Primer on Radial Basis Functions with Applications to the Geosciences, Society for Industrial and Applied Mathematics, Philadelphia, 2015.

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Primary authors

Luis Guilherme Cunha Santos (Unicamp) Eduardo Abreu (University of Campinas, Sao Paulo, Brazil)

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