H. Power

Dual reciprocity method using compactly supported radial basis functions

Compactly supported radial basis functions have been used to interpolate the forcing term in the DRM. The resulting DRM matrix is sparse and our approach is very accurate and efficient. Our proposed method is especially attractive for large scale problems. Copyright

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

This 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 Difference (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 Greens integral representational formula is applied, and on the interfaces of the adjacent subregions the full matching conditions are imposed. n nIn 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 efficient radial basis functions known in the literature on mathematical interpolation. In the numerical examples presented in the paper, the contour elements are defined 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 field variables and their derivatives, without any additional interpolation. n nFinally, some examples showing the accuracy, the efficiency, and the flexibility of the method for the solution of the linear and non-linear convection–diffusion equation are presented. Copyright

An O(N) Taylor series multipole boundary element method for three-dimensional elasticity problems

Abstract It has been previously reported in the open literature that multipole boundary element strategy based on Taylor expansions can result in computer codes which require O(N log N) operations for problems with N degrees of freedom. In this work we present a multipole BEM strategy developed for three-dimensional elasticity problems which is also based on Taylor expansions but requires only O(N) operations and O(N) memory. Results are presented for size of problems in which N=O(104).

On the convergence of the dual reciprocity boundary element method

Abstract This paper presents a study of the convergence properties of the dual reciprocity method (DRM). DRM is one of the most popular techniques used to transform volume integrals that arise, for example, from the inhomogeneous term of Poissons equation, into equivalent boundary integrals in the boundary element method (BEM). The transformation is carried out by expanding the inhomogeneous term into approximating functions whose particular solutions can be easily obtained. In the present paper, interpolation functions are derived from the approximating functions, and their properties are studied theoretically and numerically. The results obtained confirm that the interpolation converges to the real function.

DRM-MD approach for the numerical solution of gas flow in porous media, with application to landfill

Abstract In the present work two different Dual Reciprocity Method (DRM) formulations were developed, for convection–diffusion flow of a mixture of gases in a multi-layer porous media, with an application to landfills. The first method treated the whole problem domain as a single one, while the second formulation, the Dual Reciprocity Method Multi-Domain decomposition (DRM-MD), divides the initial domain into a large number of subdomains. The main advantage of the single domain formulation, in relation to domain methods, is that only surface elements are necessary, so the input data is reduced. A drawback of this approach is that it results in a fully populated matrix system, limiting it to small or medium size problems. On the other hand, the DRM-MD formulation extends the range of applications of the technique to large problems, since the final matrix of the system is sparse (band diagonal) and the matrix coefficients of geometrically similar subregions are calculated only once.

A novel boundary integral formulation for three-dimensional analysis of thin acoustic barriers over an impedance plane

This article presents a three-dimensional formulation for the analysis of acoustic barriers over an impedance plane as infinitely thin structures. The barriers are therefore modeled as simple surfaces rather than volumetric structures. Using this approach, the problems caused by near-singular integrations and near-degenerate systems of equations are averted, and mesh generation is made easier. A dual-boundary-element method is used in the analysis, involving the simultaneous solution of standard and hypersingular boundary integral equations. An optimization procedure is used to speed up the assembling of the system of equations, increasing the applicability of the method to a wider range of frequencies.

Three-dimensional scattering of seismic waves from topographical structures

Abstract A direct boundary element method for calculating the three-dimensional scattering of seismic waves from irregular topographies and buried valleys due to incident P-, S- and Rayleigh waves is presented. It has been formulated with isoparametric quadratic boundary elements and the comparison with other results show that the method is accurate and efficient. The study of the behaviour of two types of mountains for different incidences is also shown. For some incidences, factors of vertical amplification can reach up to 20 times the incident motion and factors of horizontal amplification could be as high as four times the free-field motion. The largest amplifications have been found in mountains with vertical walls while mountains with smooth slopes exhibit little amplification with factors smaller than four. Results in the time domain show how the duration of motion could be incremented compared with the free-field motion and illustrate the great amplification of the incident wavelet at some sites of the mountains.

The DRM subdomain decomposition approach to solve the two-dimensional Navier–Stokes system of equations

Abstract This work presents a domain decomposition boundary integral equation method for the solution of the two-dimensional Navier–Stokes system of equations. It is known that, within this topic, the standard boundary element method is at a disadvantage when compared with classical domain schemes, such as finite difference and finite elements methods. In the present approach the original domain is divided into several subdomains. In each of them the integral representation formula of a non-homogeneous Stokes flow field is applied, and on the interface of the adjacent sub-regions the full matching conditions of the problem are imposed. The domain integrals resulting from the non-homogeneous terms of the formulation are transformed into surface integrals at the contour of each sub-region via the dual reciprocity method. Finally, some examples showing the accuracy, efficiency and flexibility of the proposed method are presented.

Numerical solution of convection–diffusion problems at high Péclet number using boundary elements

This paper describes a boundary element scheme for solving steady-state convection–diffusion problems at high Peclet numbers. A special treatment of the singular integrals is included which uses discontinuous elements and a regularization procedure. Transformations are performed to avoid directly evaluating Bessel functions for Cauchy principal value and hypersingular integrals. Test examples are solved with values of Peclet number up to 107 to assess the numerical scheme.

Comparison between continuous and discontinuous boundary elements in the multidomain dual reciprocity method for the solution of the two-dimensional Navier–Stokes equations

Abstract Multidomain decomposition techniques are 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 Navier–Stokes 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 either isoparametric linear discontinuous or linear continuous 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. In the present work we compare the numerical results obtained by using both kind of boundary elements, continuous and discontinuous, in each subdomain.