Nam Mai-Duy, Cam Minh Tri Tien, Dmitry Strunin and Warna Karunasena
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…
Abstract
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.
Details
Keywords
Nam Mai‐Duy and Thanh Tran‐Cong
This paper is concerned with the application of radial basis function networks (RBFNs) as interpolation functions for all boundary values in the boundary element method (BEM) for…
Abstract
This paper is concerned with the application of radial basis function networks (RBFNs) as interpolation functions for all boundary values in the boundary element method (BEM) for the numerical solution of heat transfer problems. The quality of the estimate of boundary integrals is greatly affected by the type of functions used to interpolate the temperature, its normal derivative and the geometry along the boundary from the nodal values. In this paper, instead of conventional Lagrange polynomials, interpolation functions representing these variables are based on the “universal approximator” RBFNs, resulting in much better estimates. The proposed method is verified on problems with different variations of temperature on the boundary from linear level to higher orders. Numerical results obtained show that the BEM with indirect RBFN (IRBFN) interpolation performs much better than the one with linear or quadratic elements in terms of accuracy and convergence rate. For example, for the solution of Laplace's equation in 2D, the BEM can achieve the norm of error of the boundary solution of O(10−5) by using IRBFN interpolation while quadratic BEM can achieve a norm only of O(10−2) with the same boundary points employed. The IRBFN‐BEM also appears to have achieved a higher efficiency. Furthermore, the convergence rates are of O(h1.38) and O(h4.78) for the quadratic BEM and the IRBFN‐based BEM, respectively, where h is the nodal spacing.
Details
Keywords
To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.
Abstract
Purpose
To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.
Design/methodology/approach
The proposed method is based on a Cartesian grid and a 1D integrated‐radial‐basis‐function scheme. The employment of integration to construct the RBF approximations representing the field variables facilitates a fast convergence rate, while the use of a 1D interpolation scheme leads to considerable economy in forming the system matrix and improvement in the condition number of RBF matrices over a 2D interpolation scheme.
Findings
The proposed method is verified by considering several test problems governed by second‐ and fourth‐order PDEs; very accurate solutions are achieved using relatively coarse grids.
Research limitations/implications
Only 1D and 2D formulations are presented, but we believe that extension to 3D problems can be carried out straightforwardly. Further, development is needed for the case of non‐rectangular domains.
Originality/value
The contribution of this paper is a new effective collocation formulation based on RBFs for solving PDEs.