A global meshless collocation particular solution method (integrated Radial Basis Function) for two-dimensional Stokes flow problems

Bustamante, C. A.; Power, H.; Sua, You-Hui; Florez, W. F.

doi:10.1016/j.apm.2012.09.069

Cited by 29 publications

(13 citation statements)

References 18 publications

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“…f .x n , y n / In this study, all the Equations in (5)-(9) for MHD convection Stokes flow are considered as Poisson's type and solved iteratively approximating the right hand sides using RBF approximation. That is, L D r 2 is the Laplace operator for all of the Equations (5)- (9). B D I for the Equations (5)- (7), B D @ @n for the pressure Equation (8), and in the Equation (9), B D I and B D @ @n for the vertical and parallel walls of the cavity, respectively.…”

Section: Numerical Methods (Rbf Approximation)mentioning

confidence: 99%

MHD Stokes flow and heat transfer in a lid‐driven square cavity under horizontal magnetic field

Gürbüz

Tezer‐Sezgin

2017

Math Methods in App Sciences

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This study considers the steady flow of a viscous, incompressible and electrically conducting fluid in a lid‐driven square cavity under the effect of a uniform horizontally applied magnetic field. The governing equations are obtained from the Navier‐Stokes equations including buoyancy and Lorentz force terms and the energy equation including Joule heating and viscous dissipation terms. These equations are solved iteratively in terms of velocity components, stream function, vorticity, temperature, and pressure by using radial basis function approximation. Particular solution, which is approximated by radial basis functions to satisfy both differential equation and boundary conditions, becomes the solution of the differential equation itself. Vorticity boundary conditions are obtained from stream function equation using finite difference scheme. Normal derivative of pressure is taken as zero on the boundary. The numerical results are obtained for several values of Hartmann number and Grashof number for the Stokes approximation (Re << 1). The results show that when the viscous dissipation is present, the flow and isolines concentrate through the cold wall forming boundary layers as Grashof number increases. An increase in the magnetic field intensity retards the effect of buoyancy force in the square cavity, whereas the movement of the upper lid causes buoyancy force to be dominant. The solution is obtained in a considerably low computational expense through the use of radial basis function approximations for the MHD equations. Copyright © 2017 John Wiley & Sons, Ltd.

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Section: Numerical Methods (Rbf Approximation)mentioning

confidence: 99%

MHD Stokes flow and heat transfer in a lid‐driven square cavity under horizontal magnetic field

Gürbüz

Tezer‐Sezgin

2017

Math Methods in App Sciences

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“…Radial basis functions were originally used to generate meshless global interpolants , evolving in a method to solve PDEs . Studies concerning its mathematical implications and its applicability have emerged in the last couple of decades, where different aspects of the method are discussed, for example, interpolation matrix's conditionality, optimal shape parameter, node distribution, and boundary conditions.…”

Section: Radial Basis Functionsmentioning

confidence: 99%

“…Radial basis functions were originally used to generate meshless global interpolants [38,39], evolving in a method to solve PDEs [40,41]. Studies concerning its mathematical implications [40,[42][43][44][45][46] and its applicability [47][48][49][50][51][52][53][54][55] have emerged in the last couple of decades, where different aspects of the method are discussed, for example, interpolation matrix's conditionality, optimal shape parameter, node distribution, and boundary conditions. The method starts by assuming that any given function f .x 0 / can be expressed as a linear combination of known RBF [15] as follows:…”

Section: Radial Basis Functionsmentioning

confidence: 99%

Educating local radial basis functions using the highest gradient of interest in three dimensional geometries

Palacio‐Betancur

Villada‐Gil

Pablo

et al. 2016

Numerical Meth Engineering

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Summary We present a novel methodology to effectively localize radial basis function approximation methods in three dimensions. The local scheme requires shape parameter‐dependent functions that can be used to approximate gradients of scattered data and to solve partial differential operators. The optimum shape parameter is obtained from the highest gradient of interest, where a known analytical function, when boundary conditions are not present, or a shape parameter‐free global approximation are used to educate the localized scheme. The later option is applicable to problems where the operator needs to be solved multiple times, like in time evolution or stochastic integration. Past shape parameter's optimizations, for two‐dimensional domains, based on the condition number of the interpolant matrix, were unable to provide satisfactory approximations. The applicability of our method is illustrated in the context of an analytical expression interpolation and during a Ginzburg–Landau relaxation of a free energy functional. In general, the optimum shape parameter depends on geometry, node distribution, and density, whereas the approximation errors decrease as the node density and the local stencil size increase. The effective localization of radial basis functions motivates its use in moving boundary problems and accelerates solutions through sparse matrix solvers. Copyright © 2016 John Wiley & Sons, Ltd.

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“…In both cases, no external body force (source) was considered. In this spatial homogeneous framework we refer the papers by Bustamante and co-authors [13] where a meshfree method for Stokes flow was presented and tested. In [14] a dual reciprocity boundary element method combined with radial basis functions (RBF) was developed and numerically tested for Stokes flows (in a velocity-vorticity formulation).…”

Section: Introductionmentioning

confidence: 99%

Meshfree methods for nonhomogeneous Brinkman flows

Martins

Rebelo

2014

Computers & Mathematics with Applications

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a b s t r a c tWe present a meshfree method for the numerical solution of non homogeneous Brinkman systems with Dirichlet and traction boundary conditions. The aim of the paper is to propose a representation for the fluid flow as a linear superposition of unsteady Stokeslets and elastodynamic P and S waves. A theoretically study for the proposed choice of basis functions is given and several numerical examples are presented in order to illustrate and discuss the feasibility of the method.

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A global meshless collocation particular solution method (integrated Radial Basis Function) for two-dimensional Stokes flow problems

Cited by 29 publications

References 18 publications

MHD Stokes flow and heat transfer in a lid‐driven square cavity under horizontal magnetic field

MHD Stokes flow and heat transfer in a lid‐driven square cavity under horizontal magnetic field

Educating local radial basis functions using the highest gradient of interest in three dimensional geometries

Meshfree methods for nonhomogeneous Brinkman flows

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