Multiple reciprocity hybrid boundary node method for potential problems

Meshless methods have some obvious advantages such as they do not require meshes in the domain and on the boundary, only some nodes are needed in the computation. Furthermore, for the boundary-type meshless methods, the nodes are even not needed in the domain and only distributed on the boundary. Practice shows that boundary-type meshless methods are effective for homogeneous problems. But for inhomogeneous problems, the application of these boundary-type meshless methods has some difficulties and need to be studied further.The hybrid boundary node method (HBNM) is a boundary-only meshless method, which is based on the moving least squares (MLS) approximation and the hybrid displacement variational principle. No cell is required either for the interpolation of solution variables or for numerical integration. It has a drawback of ‘boundary layer effect’, so a new regular hybrid boundary node method (RHBNM) has been proposed to avoid this pitfall, in which the source points of the fundamental solutions are located outside the domain. These two methods, however, can only be used for solving homogeneous problems. Combining the dual reciprocity method (DRM) and the HBNM, the dual reciprocity hybrid boundary node method (DRHBNM) has been proposed for the inhomogeneous terms. The DRHBNM requires a substantial number of internal points to interpolate the particular solution by the radial basis function, where approximation based only on boundary nodes may not guarantee sufficient accuracy.Now a further improvement to the RHBNM, i.e., a combination of the RHBNM and the multiple reciprocity method (MRM), is presented and called the multiple reciprocity hybrid boundary node method (MRHBNM). The solution comprises two parts, i.e., the complementary and particular solutions. The complementary solution is solved by the RHBNM. The particular solution is solved by the MRM, i.e., a sum of high-order homogeneous solutions, which can be approximated by the same-order fundamental solutions. Compared with the DRHBNM, the MRHBNM does not require internal points to obtain the particular solution for inhomogeneous problems. Therefore, the present method is a real boundary-only meshless method, and can be used to deal with inhomogeneous problems conveniently. The validity and efficiency of the present method are demonstrated by a series of numerical examples of inhomogeneous potential problems.     

弹性地基板弯曲的杂交边界点法

基于双参数Pasternak弹性地基模型,将杂交边界点法与双互易法结合,用于弹性地基板弯曲问题的分析。将地基反力与横向载荷一起作为非齐次项,利用径向基函数插值得到特解,而齐次方程的通解则使用杂交边界点法求解。该方法无论插值还是积分都不需要网格,域内点仅用来插值非齐次项,因而仍是一种边界类型的无网格方法。数值算例表明：该文方法在分析弹性地基板弯曲问题时,具有计算精度高和收敛速度快等优点。     

A meshless hybrid boundary-node method for Helmholtz problems

By coupling the moving least squares (MLS) approximation with a modified functional, the hybrid boundary node-method (hybrid BNM) is a boundary-only, truly meshless method. Like boundary element method (BEM), an initial restriction of the present method is that non-homogeneous terms accounting for effects such as distributed loads are included in the formulation by means of domain integrals, and thus make the technique lose the attraction of its ‘boundary-only’ character.This paper presents a new boundary-type meshless method dual reciprocity-hybrid boundary node method (DR-HBNM), which is combined the hybrid BNM with the dual reciprocity method (DRM) for solving Helmholtz problems. In this method, the solution of Helmholtz problem is divided into two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by means of hybrid BNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of hybrid BNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The proposed method in the paper retains the characteristics of the meshless method and BEM, which only requires discrete nodes constructed on the boundary of a domain, several nodes in the domain are needed just for the RBF interpolation. The parameters that influence the performance of this method are studied through numerical examples and known analytical fields. Numerical results for the solution of Helmholtz equation show that high convergence rates and high accuracy are achievable.     

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.     

A hybrid radial boundary node method based on radial basis point interpolation

The hybrid boundary node method (HBNM) retains the meshless attribute of the moving least squares (MLS) approximation and the reduced dimensionality advantages of the boundary element method. However, the HBNM inherits the deficiency of the MLS approximation, in which shape functions lack the delta function property. Thus in the HBNM, boundary conditions are implemented after they are transformed into their approximations on the boundary nodes with the MLS scheme.This paper combines the hybrid displacement variational formulation and the radial basis point interpolation to develop a direct boundary-type meshless method, the hybrid radial boundary node method (HRBNM) for two-dimensional potential problems. The HRBNM is truly meshless, i.e. absolutely no elements are required either for interpolation or for integration. The radial basis point interpolation is used to construct shape functions with delta function property. So unlike the HBNM, the HRBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables, and boundary conditions can be applied directly and easily, which leads to greater computational precision. Some selected numerical tests illustrate the efficiency of the method proposed.     

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     

Development of hybrid boundary node method in two-dimensional elasticity

As a truly meshless method, the Hybrid Boundary Node Method (HBNM) does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. It has been applied to solve the potential problems. This paper presents a further development of the HBNM to the 2D elastic problems.In this paper, the hybrid displacement variational formulations have been coupled with the Moving Least Squares (MLS) approximation. The rigid body movement method is employed to solve the hyper-singular integrations. The ‘boundary layer effect’, which is the main drawback of the original HBNM, has been circumvented by an adaptive integration scheme.In the present method, the source points of the fundamental solution are arranged directly on the boundary. Thus, the uncertain scale factor taken in the Regular Hybrid Boundary Node Method (RHBNM) can be avoided. The parameters that influence the performance of this method are studied through several numerical examples and the known analytical solutions. The treatment of singularity and further integration has been given by a series of effective approaches. The computation results obtained by the present method are shown that good convergence and high accuracy with a small node number are achievable.     

DRM—MFS for two-dimensional finite elasticity

This paper presents a meshless method for the solution of problems in finite elasticity. The method is based on coupling the method of fundamental solutions (MFS) with dual reciprocity method (DRM). The solution is obtained by adding the homogeneous solution generated by MFS to the particular solution obtained by the radial basis function employed in DRM. The procedure has the advantage of eliminating domain integration. The proposed method is tested through two numerical examples that confirm its efficiency and accuracy.     

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.     

Development of a meshless hybrid boundary node method for Stokes flows

The meshless hybrid boundary node method (HBNM) is a promising method for solving boundary value problems, and is further developed and numerically implemented for incompressible 2D and 3D Stokes flows in this paper. In this approach, a new modified variational formulation using a hybrid functional is presented. The formulation is expressed in terms of domain and boundary variables. The moving least-squares (MLS) method is employed to approximate the boundary variables whereas the domain variables are interpolated by the fundamental solutions of Stokes equation, i.e. Stokeslets. The present method only requires scatter nodes on the surface, and is a truly boundary type meshless method as it does not require the ‘boundary element mesh’, either for the purpose of interpolation of the variables or the integration of ‘energy’. Moreover, since the primitive variables, i.e., velocity vector and pressure, are employed in this approach, the problem of finding the velocity is separated from that of finding pressure. Numerical examples are given to illustrate the implementation and performance of the present method. It is shown that the high convergence rates and accuracy can be achieved with a small number of nodes.     

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 meshless solution to two-dimensional convection–diffusion problems

An integral equation domain decomposition method has been implemented in a meshless fashion. The method exploits the advantage of placing the source point always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green's identities and the remaining equations are the derivatives of the first equation in respect to space coordinates. Radial basis function interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). Dual reciprocity method (DRM) has been applied to convert the domain integrals into boundary integrals, though the approach is general and can be applied without the DRM. The accuracy and robustness of the method has been tested on a convection–diffusion problem. The results obtained using the current approach have been compared with previously reported results obtained using the finite element method (FEM), and the DRM multi-domain approach (DRM-MD) showing similar level of accuracy.     

The evaluation of domain integrals in complex multiply-connected three-dimensional geometries for boundary element methods

The treatment of domain integrals has been a topic of interest almost since the inception of the boundary element method (BEM). Proponents of meshless methods such as the dual reciprocity method (DRM) and the multiple reciprocity method (MRM) have typically pointed out that these meshless methods obviate the need for an interior discretization. Hence, the DRM and MRM maintain one of the biggest advantages of the BEM, namely, the boundary-only discretization. On the other hand, other researchers maintain that classical domain integration with an interior discretization is more robust. However, the discretization of the domain in complex multiply-connected geometries remains problematic. In this research, three methods for evaluating the domain integrals associated with the boundary element analysis of the three-dimensional Poisson and nonhomogeneous Helmholtz equations in complex multiply-connected geometries are compared. The methods include the DRM, classical cell-based domain integration, and a novel auxiliary domain method. The auxiliary domain method allows the evaluation of the domain integral by constructing an approximately C ¹ extension of the domain integrand into the complement of the multiply-connected domain. This approach combines the robustness and accuracy of direct domain integral evaluation while, at the same time, allowing for a relatively simple interior discretization. Comparisons are made between these three methods of domain integral evaluation in terms of speed and accuracy. This work was partially supported by the United States Department of Energy (DOE) grants DE-FG03-97ER14778 and DE-FG03-97ER25332. This financial support does not constitute an endorsement by the DOE of the views expressed in this paper.     

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.     

The DRM sub-domain decomposition approach for two-dimensional thermal convection flow problems

This work presents a domain decomposition boundary integral equation method for the solution of the coupling of the momentum and energy equations governing the motion of a viscous fluid due to natural convection. The domain integrals in the proposed integral representation formula of both equations are transformed into surface integrals at the contour of each sub-region via the dual reciprocity method (DRM). Finally, some examples showing the accuracy, the efficiency and the flexibility of the proposed method are presented.     

A semi-analytic boundary element method for parabolic problems

A new semi-analytic solution method is proposed for solving linear parabolic problems using the boundary element method. This method constructs a solution as an eigenfunction expansion using separation of variables. The eigenfunctions are determined using the dual reciprocity boundary element method. This separation of variables-dual reciprocity method (SOV-DRM) allows a solution to be determined without requiring either time-stepping or domain discretisation. The accuracy and computational efficiency of the SOV-DRM is found to improve as time increases. These properties make the SOV-DRM an attractive technique for solving parabolic problems.     

A hybrid boundary node method

A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi‐sphere in the 3‐D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2‐D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.     

Trefftz‐based dual reciprocity method for hyperbolic boundary value problems

A new dual reciprocity‐type approach to approximating the solution of non‐homogeneous hyperbolic boundary value problems is presented in this paper. Typical variants of the dual reciprocity method obtain approximate particular solutions of boundary value problems in two steps. In the first step, the source function is approximated, typically using radial basis, trigonometric or polynomial functions. In the second step, the particular solution is obtained by analytically solving the non‐homogeneous equation having the approximation of the source function as the non‐homogeneous term. However, the particular solution trial functions obtained in this way typically have complicated expressions and, in the case of hyperbolic problems, points of singularity. Conversely, the method presented here uses the same trial functions for both source function and particular solution approximations. These functions have simple expressions and need not be singular, unless a singular particular solution is physically justified. The approximation is shown to be highly convergent and robust to mesh distortion. Any boundary method can be used to approximate the complementary solution of the boundary value problem, once its particular solution is known. The option here is to use hybrid‐Trefftz finite elements for this purpose. This option secures a domain integral‐free formulation and endorses the use of super‐sized finite elements as the (hierarchical) Trefftz bases contain relevant physical information on the modeled problem. Copyright © 2015 John Wiley & Sons, Ltd.     

An improved hybrid boundary node method for solving steady fluid flow problems

This paper describes the application of an improved hybrid boundary node method (hybrid BNM) for solving steady fluid flow problems. The hybrid BNM is a boundary type meshless method, which combined the moving least squares (MLS) approximation and the modified variational principle. It only requires nodes constructed on the boundary of the domain, and does not require any ‘mesh’ neither for the interpolation of variables nor for the integration. As the variables inside the domain are interpolated by the fundamental solutions, the accuracy of the hybrid BNM is rather high. However, shape functions for the classical MLS approximation lack the delta function property. Thus in this method, the boundary condition cannot be enforced easily and directly, and its computational cost is high for the inevitable transformation strategy of boundary condition. In the method we proposed, a regularized weight function is adopted, which leads to the MLS shape functions fulfilling the interpolation condition exactly, which enables a direct application of essential boundary conditions without additional numerical effort. The improved hybrid BNM has successfully implemented in solving steady fluid flow problems. The numerical examples show the excellent characteristics of this method, and the computation results obtained by this method are in a well agreement with the analytical solutions, which indicate that the method we introduced in this paper can be implemented to other problems.     

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.