Some comments on the use of radial basis functions in the dual reciprocity method

Golberg, Michael A.; Chen, C. S.; Bowman, H.; Power, H.

doi:10.1007/s004660050339

Cited by 64 publications

(37 citation statements)

References 18 publications

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“…An error and convergence analysis of the DRM applied to Poisson's equation have been recently reported by Golberg et al, 24 where they observed that a full understanding of the convergence of the DRM requires one to consider both the RBFs interpolation and BEM errors.…”

Section: Basic Concepts Of the Drm-md Approachmentioning

confidence: 98%

The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems

Popov¹,

Power²

1999

Int. J. Numer. Meth. Engng.

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SUMMARYThis work presents a multi-domain decomposition integral equation method for the numerical solution of domain dominant problems, for which it is known that the standard Boundary Element Method (BEM) is in disadvantage in comparison with classical domain schemes, such as Finite Di erence (FDM) and Finite Element (FEM) methods. As in the recently developed Green Element Method (GEM), in the present approach the original domain is divided into several subdomains. In each of them the corresponding Green's integral representational formula is applied, and on the interfaces of the adjacent subregions the full matching conditions are imposed.In contrast with the GEM, where in each subregion the domain integrals are computed by the use of cell integration, here those integrals are transformed into surface integrals at the contour of each subregion via the Dual Reciprocity Method (DRM), using some of the most e cient radial basis functions known in the literature on mathematical interpolation. In the numerical examples presented in the paper, the contour elements are deÿned in terms of isoparametric linear elements, for which the analytical integrations of the kernels of the integral representation formula are known. As in the FEM and GEM the obtained global matrix system possesses a banded structure. However in contrast with these two methods (GEM and non-Hermitian FEM), here one is able to solve the system for the complete internal nodal variables, i.e. the ÿeld variables and their derivatives, without any additional interpolation.Finally, some examples showing the accuracy, the e ciency, and the exibility of the method for the solution of the linear and non-linear convection-di usion equation are presented.

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Section: Basic Concepts Of the Drm-md Approachmentioning

confidence: 98%

The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems

Popov¹,

Power²

1999

Int. J. Numer. Meth. Engng.

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“…However, if rank de"ciency occurs for both matrices A and B, at least one diagonal entry in R will become zero, and the minimum generalized singular value will not approach zero theoretically. To examine equations (20) and (21) with equation (14), we can say that the generalized singular-value decomposition method can be used to extract the common part between two matrices as we expect in equation (14). Further, the common part, matrix P in equation (14), now is (R2W) in equations (20) and (21).…”

Section: Theorem 1 For the Helmholtz Equation Given Two Systems Havmentioning

confidence: 99%

Applications of the Generalized Singular-Value Decomposition Method on the Eigenproblem Using the Incomplete Boundary Element Formulation

Kuo

Yeih

2000

Journal of Sound and Vibration

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In this paper, four incomplete boundary element formulations, including the real-part singular boundary element, the real-part hypersingular boundary element, the imaginary-part boundary element and the plane-wave element methods, are used to solve the free vibration problem. Among these incomplete boundary element formulations, the real-part singular and the hypersingular boundary elements are of the singular type and the other two are of the regular type. When the incomplete formulation is used, the spurious eigensolution may be encountered. An auxiliary system, whose boundary conditions are linearly independent of those of the original system, is required in the proposed method in order to eliminate the spurious eigensolution. A mathematical proof is given to show that the spurious eigensolution will appear in both the original and auxiliary systems. As a result, one can eliminate spurious eigensolution by means of the generalized singular-value decomposition method. In addition to the spurious eigensolution problem, the regular boundary element formulation further su!ers ill-conditioned behaviors in the problem solver. It is explained analytically using a circular domain why ill-conditioned behaviors exist. Two main ill-conditioned problems are the numerical instability of solution and the nonexistence of solution. To solve the numerical instability of solution problem existing in these proposed regular formulations, the Tikhonov's regularization technique is used to improve the condition number of the leading coe$cient matrix; further, the generalized singular-value decomposition method is used to eliminate spurious eigensolutions. Numerical examples are given to verify the performance of these proposed methods, and the results match the analytical values well.

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“…Golberg et al [114] have given information on the use of RBFs in dual respiratory method (DRM), particularly in thin plates splines. They have pointed out that the omission of the linear terms could have biased the numerical results.…”

Section: Rbfns In Approximation and Interpolationmentioning

confidence: 99%

Where are Universals?

Peacock¹

2016

Metaphysica

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Radial basis function networks (RBFNs) have gained widespread appeal amongst researchers and have shown good performance in a variety of application domains. They have potential for hybridization and demonstrate some interesting emergent behaviors. This paper aims to o er a compendious and sensible survey on RBF networks. The advantages they o er, such as fast training and global approximation capability with local responses, are attracting many researchers to use them in diversi ed elds. The overall algorithmic development of RBF networks by giving special focus on their learning methods, novel kernels, and ne tuning of kernel parameters have been discussed. In addition, we have considered the recent research work on optimization of multi-criterions in RBF networks and a range of indicative application areas along with some open source RBFN tools.

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Some comments on the use of radial basis functions in the dual reciprocity method

Cited by 64 publications

References 18 publications

The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems

The DRM-MD integral equation method: an efficient approach for the numerical solution of domain dominant problems

Applications of the Generalized Singular-Value Decomposition Method on the Eigenproblem Using the Incomplete Boundary Element Formulation

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