Hybrid Approximation Functions in the Dual Reciprocity Boundary Element Method

Partridge, P. W.; Sensale, B.

doi:10.1002/(sici)1099-0887(199702)13:2<83::aid-cnm34>3.0.co;2-n

Cited by 30 publications

(12 citation statements)

References 9 publications

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“…For the numerical solution of the Navier±Stokes equations and their integral form given by (19) we have divided the entire domain into smaller subregions or subdomains which are formed by four linear boundary elements [9]. The four corners of each subdomain are points of discontinuity that require special attention as the tractions on both sides may not be the same.…”

Section: Numerical Solutionmentioning

confidence: 99%

“…One of the most popular is the dual reciprocity method (DRM) [9] that has been particularly attractive in recent years because of the advances in multi-dimensional interpolation with radial basis functions (RBF) [6], global functions such as those presented by Cheng et al [2] and augmented thin plate splines or hybrid functions applied by Partridge and Sensale [10] to the solution of some elasticity problems.…”

Section: Introductionmentioning

confidence: 99%

“…Additionally, when the nonlinear terms include derivatives of the velocities another interpolation must be used on the velocity ®eld. Thus, the resulting interpolation functions are differentiated to obtain the convective terms [9]. This commonly used interpolation does not satisfy the continuity equation, and this is more notorious in coarse meshes.…”

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confidence: 99%

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Conservative interpolation for the boundary integral solution of the Navier-Stokes equations

Florez

Power

Chejne

2000

Computational Mechanics

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The Dual Reciprocity Method (DRM) is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals. In this approach the non-linear terms are usually approximated by mathematical interpolation applied to the convective terms of the form of the Navier±Stokes equations. In this paper we introduce a conservative interpolation scheme that satis®es the continuity equation and performs better than pure mathematical interpolation. The new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the non-linear terms in the Navier±Stokes equations for moderate Reynolds number. IntroductionThe application of the boundary element method (BEM) to non-linear partial differential equations such as the Navier± Stokes set, leads to domain integrals that can be evaluated by using the cell integration approach (Cell-BEM) [1]. Although this approach is effective and general it makes the BEM lose its boundary only nature, resulting in a numerical scheme that is several orders of magnitude more time consuming than other domain techniques as the ®nite differences (FDM) and ®nite element methods (FEM).A handful of methods have been developed to take the domain integrals to the boundary. One of the most popular is the dual reciprocity method (DRM) [9] that has been particularly attractive in recent years because of the advances in multi-dimensional interpolation with radial basis functions (RBF) [6], global functions such as those presented by Cheng et al. [2] and augmented thin plate splines or hybrid functions applied by Partridge and Sensale [10] to the solution of some elasticity problems.The DRM approximates the nonlinear and nonhomogeneous terms of a partial differential equation as a series of vector valued interpolation functions, and ®nds a particular solution for the problem that can be used together with the Green's identities to convert the domain integrals into boundary integrals. Additionally, when the nonlinear terms include derivatives of the velocities another interpolation must be used on the velocity ®eld. Thus, the resulting interpolation functions are differentiated to obtain the convective terms [9]. This commonly used interpolation does not satisfy the continuity equation, and this is more notorious in coarse meshes. The only solution to overcome this de®ciency is to increase the mesh density especially in the zones where high velocity gradients are expected. On the other hand, a mass conservative interpolation method would allow to obtain better results even in coarse meshes and it would also improve the convergence properties of the numerical solution method.The BEM solution of large problems can be dramatically improved by using domain decomposition techniques. The original domain is divided into sub-regions, and in each one the full integral representation formula is applied. At the common interface between the adjacent sub-regions, the corresponding matching conditions are enforced to ensure the conti...

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Section: Numerical Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Conservative interpolation for the boundary integral solution of the Navier-Stokes equations

Florez

Power

Chejne

2000

Computational Mechanics

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“…One of the most popular is the dual reciprocity method (DRM) [2] that has been particularly attractive in recent years because of the advances in multi-dimensional interpolation with radial basis functions (RBF) [3], global functions such as those presented by Cheng et al [4] and augmented thin-plate splines or hybrid functions applied by Partridge and Sensale [5] to the solution of some elasticity problems. The DRM approximates the non-linear and non-homogeneous terms of a partial di erential equation as a series of vector-valued interpolating functions, and ÿnds a particular solution for the problem that can be used together with the Green's identities to convert the domain integrals into boundary integrals.…”

Section: Introductionmentioning

confidence: 99%

Multi-domain dual reciprocity BEM approach for the Navier-Stokes system of equations

Florez

Power

Chejne³

2000

Commun. Numer. Meth. Engng.

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This work presents a subdomain decomposition method as an alternative to improve the performance of the dual reciprocity boundary element method (DRBEM) in the BEM numerical solution of the Navier-Stokes equations. In the traditional DRBEM, the domain integrals that arise from the non-linear terms in the NavierStokes equations are approximated by a series of particular solutions and a set of collocation nodes distributed over the integration domain. In the present approach a subdomain technique is used in which the integration domain is divided into small quadrilateral elements whose four edges are linear discontinous boundary elements. The domain integrals in each subdomain are transformed into boundary integrals by dual reciprocity with augmented thin-plate splines i.e. r 2 log r, plus three additional linear terms from a Pascal triangle expansion. It will be shown that this multi-domain technique is e cient and promising for the solution of high Reynolds number problems.

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“…Moreover, the accuracy of the DRM is dependant on the choice of the used radial basis function [22,23]. As an example to the use of the DRM in the BEM is the work of Partridge and Sensale [24] who used the DRM to treat body forces in elasticity problems.…”

Section: Introductionmentioning

confidence: 99%

Derivation of particular solutions for linear loading in the thick plate theory

Rashed

2000

Computational Mechanics

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In this paper, the particular solutions in the theory of thick plates are derived. These new particular solutions are the analytical solutions of the governing differential equations of equilibrium due to linear domain loading. Uniform domain loading can be considered as a special case of the present formulation. The expressions for the displacement and the traction kernels are derived and given in explicit form. Also the necessary kernels for the internal stress resultants are also derived and given. The derived particular solutions are veri®ed analytically. A new set of boundary integral equation formulation for thick plates using the new particular solutions are outlined. IntroductionMany practical applications in engineering structures, such as side walls of water tanks and retaining walls (see for example, [1]), can be modeled using plates under linear domain loading. In the ®eld of structural design, the most popular method to analyze such structures is the ®nite element method [2] which requires the discretization of the problem domain leading to high computational requirements. In the past two decades, the boundary element method (BEM) has emerged as a powerful tool to solve engineering problems [3,4,5]. The main advantage of the BEM is the boundary only discretization of the considered problem. In the presence of body force terms, such as domain loading, a domain discretization is required to deal with such terms, which losses the main advantage of the BEM. Many researchers have studied methods of transformation of such domain integrals to the boundary. The ®rst technique is the direct application of Green's identities. Cruse [6], with the help of representing the fundamental solution in terms of the Galerkin tensor, used this technique to deal with body forces in elasticity problems. Stippes and Rizzo [7] extended the formulation in Ref. [6] to thermal stress problems. Hartmann and Zotemantel [8] used similar technique to transform domain integrals due to uniform loading in thin plates to the boundary. Vander Weee Èn [9] and El-Zafrany [10, 11] used Green's second identity to treat uniform loading in thick plates. Costa and Brebbia [12] applied the same technique for thin plates on elastic foundations and Rashed et al. [13, 14] for thick plates resting on elastic foundations. Antes and Steinfeld [15] used the Green ®rst identity to transform domain integrals due to body forces to the boundary for static and dynamic analysis of elasticity problems. Recently Rashed [16] extended the formulation in Ref. [9] to treat thick plates under linear loading.A logical extension to the above mentioned technique is the multiple reciprocity method (MRM) [17]. The MRM employs Green's second identity in a recursive way to transform domain integrals to a series of boundary integrals. In case of uniform body forces, only one application of Green's second identity is enough to transform such domain integrals to the boundary. Neves and Brebbia [18] used the MRM to treat body forces for elasticity problems. Recently, Rashed ...

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Hybrid Approximation Functions in the Dual Reciprocity Boundary Element Method

Cited by 30 publications

References 9 publications

Conservative interpolation for the boundary integral solution of the Navier-Stokes equations

Conservative interpolation for the boundary integral solution of the Navier-Stokes equations

Multi-domain dual reciprocity BEM approach for the Navier-Stokes system of equations

Derivation of particular solutions for linear loading in the thick plate theory

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