Dual-reciprocity BEM based on global interpolation functions

Cheng, Alexander H.‐D.; Lafe, Olu; Grilli, S. T.

doi:10.1016/0955-7997(94)90024-8

Cited by 49 publications

(17 citation statements)

References 3 publications

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“…One of the most popular is the dual reciprocity method (DRM) [7,8] that has been particularly attractive in recent years because of the advances in multidimensional interpolation with radial basis functions (RBF) [9], global functions such as those presented by Cheng et al [10] and augmented thin plate splines [11] or hybrid functions applied by Partridge and Sensale [12] to the solution of elasticity problems. The first comparison of application between the usual conical function, (1 ϩ R), and the augmented thin plate splines functions as well as report of improvement of accuracy in the DRM approach when subdomains are used was reported by Popov and Power [13].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of thermal convection problems using the multidomain boundary element method

Florez

Power

Chejne

2002

Numerical Methods Partial

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The multidomain dual reciprocity method (MD-DRM) has been effectively applied to the solution of two-dimensional thermal convection problems where the momentum and energy equations govern the motion of a viscous fluid. In the proposed boundary integral method the domain integrals are transformed into equivalent boundary integrals by the dual reciprocity approach applied in a subdomain basis. On each subregion or domain element the integral representation formulas for the velocity and temperature are applied and discretised using linear continuous boundary elements, and the equations from adjacent subregions are matched by additional continuity conditions. Some examples showing the accuracy, the efficiency and flexibility of the proposed method are presented.

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Section: Introductionmentioning

confidence: 99%

Numerical solution of thermal convection problems using the multidomain boundary element method

Florez

Power

Chejne

2002

Numerical Methods Partial

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“…The augmentation terms, being global, depend only on the co-ordinates (x i , y i ). 6 As an example, consider the case of the linear RBF r augmented with the ®rst-order terms from the Pascal triangle (Figure 1), 1, x and y, in such a way that a 1 1, a 2 x i and a 3 y i , hybrid function RAPT2:…”

Section: Implementation Of Hybrid Functionsmentioning

confidence: 99%

“…5 The radial basis functions were shown to be`local' in behaviour, interpolating only in the neighbourhood of a particular point. Cheng et al 6 used for the ®rst time the`global' functions, those which interpolate over the entire geometry, although these functions (Pascal triangle expansions, sine and cosine series and others) were mentioned by Nardini and Brebbia. 1 Karur and Ramachandran 7 studied the radial basis functions for Poisson problems, considering r, r 3 , r 2 log r and others.…”

Section: Introductionmentioning

confidence: 98%

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Hybrid Approximation Functions in the Dual Reciprocity Boundary Element Method

Partridge¹,

Sensale²

1997

Commun. Numer. Meth. Engng.

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SUMMARYThe dual reciprocity boundary element method traditionally uses the linear radial basis function r for interpolation. Recently, however, the use of the r function has been questioned both in relation to accuracy and in relation to the number and position of internal nodes required to obtain satisfactory solutions. Much research has been done in an attempt to ®x criteria for choosing which approximation function should be used. One of the alternatives recently suggested is the augmented thin plate spline function, which consists of a thin plate spline function, r 2 logr, augmented with the ®rst three terms of a Pascal triangle expansion. In this paper families of similar functions are obtained by augmenting radial basis functions with appropriate global expansions: these functions will be called hybrid approximate functions. It will be shown that using an appropriate hybrid function accurate results can be obtained for many body forces or pseudo body forces in elasticity without the need for internal nodes.

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“…In fact, that difficulty has occurred in previous work when using polynomial particular solutions in the DRM [5]. However, if the points are chosen on a rectangular grid in R 2 and a box grid in R 3 , then one can find Lagrange interpolants by using the forms of the one dimensional interpolants discussed previously.…”

Section: Multi-dimensional Interpolationmentioning

confidence: 99%

A mesh‐free approach to solving the axisymmetric Poisson's equation

Chen

Muleshkov

Golberg³

et al. 2004

Numerical Methods Partial

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In this article, we extend previous work of Karageorghis and Fairweather (Int J Numer Methods Engng 44 (199), 1653–1669) on the method of fundamental solutions (MFS) for solving Laplace's equation in axisymmetric geometry to the case of Poisson's equation. When the boundary condition and source term are axisymmetric, the problem reduces to solving the Poisson's equation in cylindrical coordinates in the two‐dimensional (r, z) region of the original three‐dimensional domain S. Hence, the original boundary value problem is reduced to a two‐dimensional one. To make use of the MFS, it is necessary to calculate a particular solution, which can be subtracted off, so that the MFS can be used to solve the resulting Laplace problem. This presents a novel problem, since the axisymmetric Poisson operator does not have constant coefficients, so previous methods based on radial basis functions cannot be used. To overcome this, the source term is approximated by a two‐dimensional polynomial in r and z as in Goldberg et al. (Numer Methods Partial Differential Eq 19 (2003), 112–133). One can then obtain polynomial particular solutions by the method of undetermined coefficients. The resulting Laplace equation is then solved by the axisymmetric MFS. Numerical results are given to show the accuracy and efficiency of the proposed numerical method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

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Dual-reciprocity BEM based on global interpolation functions

Cited by 49 publications

References 3 publications

Numerical solution of thermal convection problems using the multidomain boundary element method

Numerical solution of thermal convection problems using the multidomain boundary element method

Hybrid Approximation Functions in the Dual Reciprocity Boundary Element Method

A mesh‐free approach to solving the axisymmetric Poisson's equation

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