SOLVING NONLINEAR DIFFERENTIAL EQUATIONS OF MECHANICS WITH THE BOUNDARY ELEMENT METHOD AND RADIAL BASIS FUNCTIONS

The Boundary Element Method is a very effective method for solving linear differential equations. To use it also in the consideration of non-linear problems some different procedures were developed, among them the dual reciprocity method and the particular integral method. Both procedures use interpolation conditions for the approximation with radial basis functions. In this paper a method is presented which avoids problems connected with interpolation by means of quasi-interpolation. It is possible to solve differential equations of the kind Δ^mu=p(u) approximately; the application to two non-linear problems of plate theory yield good results. Hints to a theoretical examination of the method including sufficient conditions for feasibility and convergence are given. © 1997 by John Wiley & Sons, Ltd.     

A new DRBEM model for wave refraction and diffraction

A numerical model using the dual reciprocity boundary element method (DRBEM) is developed to study the combined refraction and diffraction of water waves propagating around islands or solid offshore structures over a seabed with a variable depth. Based on the well-known mild-slope equation, the model has been validated by comparison with both analytical solutions and standard numerical solutions available in the literature. The results show that a considerable improvement in terms of numerical efficiency has been achieved with the adoption of the DRBEM and the model has a great potential to be used in engineering practice to solve wave refraction and diffraction problems.     

Dual reciprocity BEM: local versus global approximation functions for diffusion, convection and other problems

The use of the global approximation functions (elements of Pascal's triangle, sine expansions and others) in the dual reciprocity boundary element method is compared to the better known local radial basis functions for convection, diffusion and other problems in which the volume integrals considered contain first and second derivatives of the problem variables, time derivatives and sums and products of functions, including nonlinear terms. It will be shown that whilst it is possible to obtain accurate solutions to the problems considered using the global functions, a successful solution to a given problem depends very much on the function chosen, as well as other factors.     

用边界元反分析构造平面残余应力场

基于固有应变概念,采用边界元方法,提出一种反方法构造连续的满足域内自平衡条件的平面残余应力场。考虑到反分析的稳定性,固有应变场用一系列光滑基函数(如多项式和三角函数)近似;为了识别由剪切固有应变引起的残余应力,求出对应于固有应变的位移特解与面力特解,将域内积分用双重互易边界元法转换为边界积分,保持了边界元法的优势;同时导出了灵敏度矩阵的显式表达,以提高反分析的效率。最后给出了两个算例验证方法的可行性。     

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.     

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.     

Modal analysis of plates using the dual reciprocity boundary element method

This paper presents a new method for determining the natural frequencies and mode shapes for the free vibration of thin elastic plates using the boundary element and dual reciprocity methods. The solution to the plate's equation of motion is assumed to be of separable form. The problem is further simplified by using the fundamental solution of an infinite plate in the reciprocity theorem. Except for the inertia term, all domain integrals are transformed into boundary integrals using the reciprocity theorem. However, the inertia domain integral is evaluated in terms of the boundary nodes by using the dual reciprocity method. In this method, a set of interior points is selected and the deflection at these points is assumed to be a series of approximating functions. The reciprocity theorem is applied to reduce the domain integrals to a boundary integral. To evaluate the boundary integrals, the displacements and rotations are assumed to vary linearly along the boundary. The boundary integrals are discretized and evaluated numerically. The resulting matrix equations are significantly smaller than the finite element formulation for an equivalent problem. Mode shapes for the free vibration of circular and rectangular plates are obtained and compared with analytical and finite element results.     

Solving general field equations in infinite domains with dual reciprocity boundary element method

The dual reciprocity boundary element method has been successfully employed to solve general field equations posed in a closed domain, i.e. interior problems. Up to now, however, little effort has been made to extend it to exterior problems (i.e. general field equations posed in an infinite region), which are commonly encountered in engineering practice. In this paper, the interpolation functions associated with exterior problems, which were proposed by Loeffler and Mansur (in Boundary Elements X, Vol. 2, Springer, 1988), are first examined. We have found that the choice of the arbitrary constant, the inclusion of which is necessary in those interpolation functions, has clear effects on the accuracy of the numerical results. A mapping transformation, through which any exterior problem can be solved by solving an equivalent interior problem, is then proposed. Although there are certain regularity conditions attached to such a mapping, they can be easily satisfied if the unknown function satisfies certain regularity conditions at infinity in the original exterior problem. A successful application of this mapping transformation to a transient heat transfer problem demonstrates the good performance of this approach.     

Pressure correction DRBEM solution for heat transfer and fluid flow in incompressible viscous fluids

The Dual Reciprocity Boundary Element Method (DRBEM) is used to solve incompressible laminar viscous fluid flows and heat transfer. The DRBEM is extended to develop a pressure correction scheme to solve the incompressible Navier-Stokes equations. The velocity field is then used as input to the DRBEM solution of the energy transport equation, thereby retaining the boundary only discretization feature of the BEM for the solution of this problem. Numerical results for the proposed DRBEM solution for laminar flow and heat transfer in a channel are obtained for several Reynolds numbers and compare well with previously published data.     

Transient non‐linear heat conduction–radiation problems—a boundary element formulation

A novel boundary‐only formulation for transient temperature fields in bodies of non‐linear material properties and arbitrary non‐linear boundary conditions has been developed. The option for self‐irradiating boundaries has been included in the formulation. Heat conduction equation has been partially linearized by Kirchhoff's transformation. The result has been discretized by the dual reciprocity boundary element method. The integral equation of heat radiation has been discretized by the standard boundary element method. The coupling of the resulting two sets of equations has been accomplished by static condensation of the radiative heat fluxes arising in both sets. The final set of ordinary differential equations has been solved using the Runge–Kutta solver with automatic time step adjustment. The algorithm proved to be robust and stable. Numerical examples are included. Copyright © 1999 John Wiley & Sons, Ltd.     

The use of dual reciprocity boundary elements for simulation of groundwater flow and pollutant transport

Abstract

The dual reciprocity boundary element method (DRBEM) has been established as an effective numerical tool in the modeling of various engineering problems. Here the application of the DRBEM in groundwater flow and pollutant transport is described. Several cases analyzed prove that DRBEM is an effective numerical tool in simulating these problems.     

The boundary element solution of the Laplace and biharmonic equations subjected to noisy boundary data

This study investigates the numerical solution of the Laplace and biharmonic equations subjected to noisy boundary data. Since both equations are linear, they are numerically discretized using the Boundary Element Method (BEM), which does not use any solution domain discretization, to reduce the problem to solving a system of linear algebraic equations for the unspecified boundary values. It is shown that when noisy, lower-order derivatives are prescribed on the boundary, then a direct approach, e.g. Gaussian elimination, for solving the resulting discretized system of linear equations produces an unstable, i.e. unbounded and highly oscillatory, numerical solution for the unspecified higher-order boundary derivatives data. In order to overcome this difficulty, and produce a stable solution of the resulting system of linear equations, the singular value decomposition approach (SVD), truncated at an optimal level given by the L-curve method, is employed. © 1998 John Wiley & Sons, Ltd.