The dual reciprocity boundary element method for solving Cauchy problems associated to the Poisson equation

This paper presents an application of the dual reciprocity method (DRM) to a class of inverse problems governed by the Poisson equation. Here the term inverse refers to the fact that the boundary conditions are not fully specified, i.e. they are not known for the entire boundary of the solution domain. In order to investigate the ability of the DRM to reconstruct the unknown boundary conditions using overspecified conditions on the accessible part of the boundary we consider some test problems involving circular, annular and square domains. Due to the ill-posed nature of the problem, i.e. the instabilities in the solution of these problems, the DRM is combined with the Tikhonov regularization method.     

An analysis of the linear advection–diffusion equation using mesh-free and mesh-dependent methods

The numerical solution of advection–diffusion equations has been a long standing problem and many numerical methods that attempt to find stable and accurate solutions have to resort to artificial methods to stabilize the solution. In this paper, we present a meshless method based on thin plate radial basis functions (RBF). The efficiency of the method in terms of computational processing time, accuracy and stability is discussed. The results are compared with the findings from the dual reciprocity/boundary element and finite difference methods as well as the analytical solution. Our analysis shows that the RBFs method, with its simple implementation, generates excellent results and speeds up the computational processing time, independent of the shape of the domain and irrespective of the dimension of the problem.     

Domain decomposition boundary element method with overlapping sub-domains

A numerical approach based on the domain decomposition boundary element method (BEM) with overlapping sub-domains has been developed. The approach simplifies the assembly of the equations arising from the BEM sub-domain methods, reduces the size of the system matrix, produces a closed system of equations when continuous elements are used, and reduces any problems arising from near-singular or singular integrals which otherwise may appear in the integral equations. The overlapping numerical approach is tested on three different problems, i.e., the Poisson equation, and a one-dimensional and two-dimensional convection–diffusion problems. The approach is implemented in combination with the dual reciprocity method (DRM) with two different radial basis functions (RBFs), though the approach is general and can be applied with other BEM formulations. The results are compared with the previous results obtained using the dual reciprocity method–multi domain (DRM–MD) approach, showing comparable accuracy and convergence.     

First kind Bessel function (J-Bessel) as radial basis function for plane dynamic analysis using dual reciprocity boundary element method

This paper presents a new radial basis function (RBF) for the boundary element method in the analysis of plane transient elastodynamic problems. The dual reciprocity method (DRM) is reconsidered by using the first kind Bessel (J-Bessel) function as a new generation of RBFs to approximate the inertia term. Employing the initial value theorem of Laplace transform, the particular solution kernels of the proposed RBFs corresponding to displacement and traction, with no singular terms, has been explicitly derived. Furthermore, the limiting values of the particular solution kernels have been evaluated. To illustrate the validity and accuracy of the present RBFs, three numerical examples are examined and compared to the results of analytical and other RBFs reported in the literature. In comparison with other RBFs, J-Bessel RBFs represent more accurate results, using a smaller degree of freedom, and hence they are more efficient.     

The solution of elastostatic and dynamic problems using the boundary element method based on spherical Hankel element framework

In this paper, new spherical Hankel shape functions are used to reformulate boundary element method for 2‐dimensional elastostatic and elastodynamic problems. To this end, the dual reciprocity boundary element method is reconsidered by using new spherical Hankel shape functions to approximate the state variables (displacements and tractions) of Navier's differential equation. Using enrichment of a class of radial basis functions (RBFs), called spherical Hankel RBFs hereafter, the interpolation functions of a Hankel boundary element framework has been derived. For this purpose, polynomial terms are added to the functional expansion that only uses spherical Hankel RBF in the approximation. In addition to polynomial function fields, the participation of spherical Bessel function fields has also increased robustness and efficiency in the interpolation. It is very interesting that there is no Runge phenomenon in equispaced Hankel macroelements, unlike equispaced classic Lagrange ones. Several numerical examples are provided to demonstrate the effectiveness, robustness and accuracy of the proposed Hankel shape functions and in comparison with the classic Lagrange ones, they show much more accurate and stable results.     

A convergence analysis of the performance of the DRM‐MD boundary integral approach

In this article, we study the performance of the dual reciprocity multi‐domains approach (DRM‐MD) when the shape functions of the boundary elements, for both the approximation of the geometry and the surface variables of the governing equations, are quadratic functions. A series of tests are carried out to study the consistency of the proposed boundary integral technique. For this purpose a limiting process of the subdivision of the domain is performed, reporting a comparison of the computed solutions for every refining scheme. Furthermore, the DRM‐MD is solved in its dual reciprocity approximation using two different radial basis interpolation functions, the conical function r plus a constant, i.e. (1+r), and the augmented thin plate spline. Special attention is paid to the contrast between numerical results yielded by the DRM‐MD approach using linear and quadratic boundary elements towards a full understanding of its convergence behaviour. Copyright © 2006 John Wiley & Sons, Ltd.     

Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation

In this paper the diffusion equation is solved in two-dimensional geometry by the dual reciprocity boundary element method (DRBEM). It is structured by fully implicit discretization over time and by weighting with the fundamental solution of the Laplace equation. The resulting domain integral of the diffusive term is transformed into two boundary integrals by using Green's second identity, and the domain integral of the transience term is converted into a finite series of boundary integrals by using dual reciprocity interpolation based on scaled augmented thin plate spline global approximation functions. Straight line geometry and constant field shape functions for boundary discretization are employed. The described procedure results in systems of equations with fully populated unsymmetric matrices. In the case of solving large problems, the solution of these systems by direct methods may be very time consuming. The present study investigates the possibility of using iterative methods for solving these systems of equations. It was demonstrated that Krylov-type methods like CGS and GMRES with simple Jacobi preconditioning appeared to be efficient and robust with respect to the problem size and time step magnitude. This paper can be considered as a logical starting point for research of iterative solutions to DRBEM systems of equations. © 1998 John Wiley & Sons, Ltd.     

Differential quadrature Trefftz method for Poisson-type problems on irregular domains

In this paper differential quadrature Trefftz method (DQTM), a new meshless method based on coupling the dual reciprocity method (DRM) with the differential quadrature method (DQM) and the Trefftz method, is used to analyze Poisson-type interior and exterior problems. In this method, the DRM is used to construct equivalent equations to the original differential equation. Then the DQM is employed to approximate the particular solutions, while Trefftz method leads to a boundary-only formulation for homogeneous solution. As a result, an inherently meshless, integration-free, boundary-only DQ Trefftz collocation technique is developed for solving Poisson-type problems. Due to the flexibility in choosing points on boundaries, the new method also works well on irregular domains. Numerical results show that the present method works efficiently with quite few points on both uniform and irregular domains.     

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

We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris, Polyzos and Beskos (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results confirmed by a numerical experiment.     

Plane stress and plate bending coupling in BEM analysis of shallow shells

In this paper the shear deformable shallow shells are analysed by boundary element method. New boundary integral equations are derived utilizing the Betti's reciprocity principle and coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Two techniques, direct integral method (DIM) and dual reciprocity method (DRM), are developed to transform domain integrals to boundary integrals. The force term is approximted by a set of radial basis functions. Several examples are presented to demonstrate the accuracy of the two methods. The accuracy of results obtained by using boundary element method are compared with exact solutions and the finite element method. Copyright © 2000 John Wiley & Sons, Ltd.     

Transformation of domain integrals to boundary integrals in BEM analysis of shear deformable plate bending problems

In this paper two techniques, dual reciprocity method (DRM) and direct integral method (DIM), are developed to transform domain integrals to boundary integrals for shear deformable plate bending formulation. The force term is approximated by a set of radial basis functions. To transform domain integrals to boundary integrals using the dual reciprocity method, particular solutions are employed for three radial basis functions. Direct integral method is also introduced in this paper to evaluate domain integrals. Three examples are presented to demonstrate the accuracy of the two methods. The numerical results obtained by using different particular solutions are compared with exact solutions. Received 27 January 1999     

A dual reciprocity hybrid radial boundary node method based on radial point interpolation method

A novel truly meshless method called dual reciprocity hybrid radial boundary node method (DHRBNM) is developed in present, which combines dual reciprocity method (DRM), hybrid boundary node method (HBNM) and radial point interpolation method (RPIM). Compared to the dual reciprocity hybrid boundary node method (DHBNM), RPIM is exploited to replace the moving least square in DHRBNM, unlike HBNM, the shape function obtained by present method has the delta function property, so the boundary conditions can be applied directly and easily, and computational expense is greatly reduced. In order to get the interpolation property of different basis function in DRM, different approximate functions are applied in DRM for comparison, and the accuracy and efficiency of them are discussed. Besides, RPIM is also exploited in DRM, which can greatly improve the accuracy of present method. Moreover, the accuracy of DRM is greatly influenced by the nodes number and their location, hence, some examples are investigated to show that the internal node number is equal to boundary node number and they are arranged parallel to the high gradient direction of the problem are the best choice. Finally, DHBNM is applied for comparison and some selected numerical examples are given to illustrate that the present method is efficient and less computational expense than that of DHBNM.     

An efficient implementation of the radial basis integral equation method

An efficient and accurate implementation of the meshless radial basis integral equation method (RBIEM) is proposed. The proposed implementation does not involve discretization of the subdomains’ boundaries. By avoiding the boundary discretization, it was hypothesised that a significant source of error in the numerical scheme is avoided. The proposed numerical scheme was tested on two problems governed by the Poisson and Helmholtz equations. The test problems were selected such that the spatial gradients of the solutions were high to examine the robustness of the numerical scheme. The dual reciprocity method (DRM) and the cell integration technique were used to treat the domain integrals arising from the source terms in the partial differential equations. The results showed that the proposed implementation is more accurate and more robust than the previously suggested implementation of the RBIEM. Though the CPU time usage of the proposed scheme is lower, the difference to the previously proposed scheme is not significant. The proposed scheme is easier to implement, since the task of keeping track of boundary elements and boundary nodes is not needed. The proposed implementation of the RBIEM is promising and opens up possibilities for efficient implementation in three-dimensional problems. This is currently under investigation.     

Multi-domain dual reciprocity for the solution of inelastic non-Newtonian flow problems at low Reynolds number

An efficient boundary element solution of the motion of inelastic non-Newtonian fluids at low Reynolds number is presented in this paper. For the numerical solution all the domain integrals of the boundary element formulation have been transformed into equivalent boundary integrals by means of the dual reciprocity method (DRM). To achieve an accurate approximation of the non-linear and non-Newtonian terms two major improvements have been made to the DRM, namely the use of augmented thin plate splines as interpolation functions, and the partition of the entire domain into smaller subregions or domain decomposition. In each subregion or domain element the DRM was applied together with some additional equations that ensure continuity on the interfaces between adjacent subdomains. After applying the boundary conditions the final systems of equations will be sparse and the approximation of the nonlinear terms will be more localised than in the traditional DRM. This new method known as multidomain dual reciprocity (MD-DRM) has been used to solve several non-Newtonian problems including the pressure driven flow of a power law fluid, the Couette flow and two simulations of industrial polymer mixers. Received 7 February 2001     

A boundary element method for analysis of thermoelastic deformations in materials with temperature dependent properties

In this paper, the boundary element method (BEM) for solving quasi‐static uncoupled thermoelasticity problems in materials with temperature dependent properties is presented. The domain integral term, in the integral representation of the governing equation, is transformed to an equivalent boundary integral by means of the dual reciprocity method (DRM). The required particular solutions are derived and outlined. The method ensures numerically efficient analysis of thermoelastic deformations in an arbitrary geometry and loading conditions. The validity and the high accuracy of the formulation is demonstrated considering a series of examples. In all numerical tests, calculation results are compared with analytical and/or finite element method (FEM) solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

A dual reciprocity BEM approach using new Fourier radial basis functions applied to 2D elastodynamic transient analysis

In this paper, a new boundary element analysis for two-dimensional (2D) transient elastodynamic problems is proposed. The dual reciprocity method (DRM) is reconsidered by employing new radial basis functions (RBFs) to approximate the domain inertia terms. These new RBFs, which are in the form of ζ+κ sin (ωr+α), are called Fourier RBFs hereafter. Using the method of variation of parameters, the particular solution kernels of Fourier RBFs corresponding to displacement and traction, whose a few terms are singular, has been explicitly derived. Therefore, a new simple smoothing trick has been employed to resolve the singularity problem. Moreover, the limiting values of the particular solution kernels have been evaluated. In order to find the unknown parameters of Fourier RBFs, an optimization problem seeking for the optimum value of the Houbolt scheme parameter β that minimizes the mean squared error (MSE) function of the problem is established. Since the MSE function of the proposed RBFs is a function of five unknown parameters (i.e., ζ, κ, ω, α, and β), the genetic algorithm (GA) has been used to solve the necessary optimization problem. In order to illustrate the validity, accuracy, and superiority of the present study, several numerical examples are examined and compared to the results of analytical and other RBFs reported in the literature. Compared to other RBFs, Fourier RBFs show more accurate and stable results. Moreover, these results are obtained using less degree of freedom without any additional internal points that are commonly used to improve the accuracy of the results.     

Some comments on the use of Radial Basis Functions in the Dual Reciprocity Method

We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris et al. (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results confirmed by a numerical experiment.     

Iterative schemes for the solution of system of equations arising from the DRM in multi domain approach,and a comparative analysis of the performance of two different radial basis functions used in the interpolation

The aim of this work is to carry out a systematic experimental study of the use of iterative techniques in the context of solving the linear system of equations arising from the solution of the diffusion–convection equation with variable velocity field through the use of the dual reciprocity method in multi domains (DRM-MD). We analyse the efficiency and accuracy of the computed solutions obtained from the DRM-MD integral equation numerical approach applying various iterative algorithms. For every iterative method tested, we consider a set of different preconditioners, depending on the features of the input matrix to be solved with the chosen method.To check the accuracy of the solutions obtained through the selected iterative methods they are contrasted against the solutions obtained applying some direct methods, as SVD, Golub's method and Cholesky decomposition. The numerical results are also compared with a benchmark analytical solution. Furthermore, we present a comparative analysis of the linear systems of algebraic equations obtained from DRM-MD considering two approximating functions. The conical function r plus a constant, i.e. (1+r) and the augmented thin plate spline, both of them radial basis functions (RBFs).     

Numerical comparisons of two meshless methods using radial basis functions

The recent advance in the development of various kinds of meshless methods for solving partial differential equations has drawn attention of many researchers in science and engineering. One of the domain-type meshless methods is obtained by simply applying the radial basis functions (RBFs) as a direct collocation, which has shown to be effective in solving complicated physical problems with irregular domains. More recently, a boundary-type meshless method that combines the method of fundamental solutions and the dual reciprocity method with the RBFs has been developed. In this paper, the performances of these two meshless methods are compared and evaluated. Numerical results indicate that these two methods provide a similar optimal accuracy in solving both 2D Poisson's and parabolic equations.     

Solving the elliptic Monge–Ampère equation by Kansa's method

The elliptic Monge–Ampère equation is a fully nonlinear partial differential equation, which originated in geometric surface theory and has been widely applied in dynamic meteorology, elasticity, geometric optics, image processing and others. The numerical solution of the elliptic Monge–Ampère equation has been a subject of increasing interest recently. In this paper, Kansa's method (with multiquadric basis functions) is introduced and used to solve numerically the Monge–Ampère equation. Kansa's method is a meshfree method which uses the combination of some radial basis functions (RBFs) to approximate the solution of the partial differential equation. We prove the classical consistency of the Kansa's method for the elliptic Monge–Ampère equation. Finally, we also present some numerical experiment to demonstrate the effectiveness of Kansa's method for the elliptic Monge–Ampère equation.