Computing non-Newtonian fluid flow with radial basis function networks

Mai‐Duy, N.; Tanner, Roger I.

doi:10.1002/fld.977

Cited by 19 publications

(5 citation statements)

References 48 publications

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“…Regular FIGURE 8 Nonrectangular domain: a schematic diagram for boundary stencils. The stencil, which is associated with node 5, consists of regular nodes: (1,4,5,6,7,8,9) and irregular nodes: (10,11,12). The PDE is imposed at side nodes: (11,4,6,8) [Color figure can be viewed at wileyonlinelibrary.com] nodes are simply the intersection points of the stencil grid lines, while irregular nodes are generated from the intersection of the boundary and the stencil grid lines.…”

Section: Pdesmentioning

confidence: 99%

“…In Kansa's method, the PDE is discretised by means of point collocation and the field variable is approximated by a set of multiquadric functions with a differentiation process being employed to obtain basis functions for derivative terms in the PDE (DRBFs). Since then, the RBF solution to differential problems has received a great deal of attention (see, e.g., [4][5][6][7][8][9][10][11][12][13][14]). For a given node distribution, the solution accuracy can still be improved by changing the value of the RBF width.…”

Section: Introductionmentioning

confidence: 99%

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A symmetric integrated radial basis function method for solving differential equations

Mai‐Duy

Dalal

et al. 2018

Numerical Methods Partial

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In this article, integrated radial basis functions (IRBFs) are used for Hermite interpolation in the solution of differential equations, resulting in a new meshless symmetric RBF method. Both global and local approximation‐based schemes are derived. For the latter, the focus is on the construction of compact approximation stencils, where a sparse system matrix and a high‐order accuracy can be achieved together. Cartesian‐grid‐based stencils are possible for problems defined on nonrectangular domains. Furthermore, the effects of the RBF width on the solution accuracy for a given grid size are fully explored with a reasonable computational cost. The proposed schemes are numerically verified in some elliptic boundary‐value problems governed by the Poisson and convection‐diffusion equations. High levels of the solution accuracy are obtained using relatively coarse discretisations.

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Section: Pdesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A symmetric integrated radial basis function method for solving differential equations

Mai‐Duy

Dalal

et al. 2018

Numerical Methods Partial

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“…The RBF approximations can be constructed through differentiation (DRBF) or integration (IRBF). The governing equations of fluid dynamics have been successfully solved by the DRBF- and IRBF-based methods (Mai-Duy and Tanner, 2005, 2007; Dehghan and Shokri, 2008; Kosec and Šarler, 2008, 2013; Mohebbi et al , 2014; Ngo-Cong et al , 2017; Ebrahimijahan and Dehghan, 2021; Ebrahimijahan et al , 2020, 2022a, 2022b; Abbaszadeh et al , 2022; Mesgarani et al , 2022).…”

Section: Introductionmentioning

confidence: 99%

An effective high-order five-point stencil, based on integrated-RBF approximations, for the first biharmonic equation and its applications in fluid dynamics

Mai‐Duy

Tien

Strunin

et al. 2023

HFF

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Purpose The purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics. Design/methodology/approach The first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation. Findings These proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids. Originality/value The main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

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“…The RBF methods have been developed into three different types of discretisations: global (e.g. [4,5,6,7,8,9]), local (e.g. [10,11,12,13,14]) and compact local (e.g.…”

Section: Introductionmentioning

confidence: 99%