Transforming the domain integrals to the boundary using approximate particular solutions: a boundary element approach for nonlinear problems (original) (raw)
On the general boundary element method
Engineering Analysis with Boundary Elements, 1998
The basic ideas of the general boundary element method (BEM) proposed by Liao (The quite general BEM for strongly nonlinear problems, in: C. A. Brebbia, S. Kim, T. -483) are further greatly generalized by introducing two nonzero parameters to construct homotopies. This general BEM is wLlid for strongly nonlinear problems, including even those whose governing equations and boundary conditions do not contain any linear terms. Therefore, it can greatly enlarge the application areas of the boundary element method as a numerical methodology. A two-dimensional nonlinear differential equation is used to verify the validity of the further generalized boundary element method. Moreover, this exarrLple illustrates that, by means of the proposed general boundary element method, iteration is not absolutely necessary for nonlinear problems. This shakes the absolutely governing place of iterative methodology of the boundary element method for nonlinear problems, and might be beneficial for us to understand the essence of solving nonlinear problems.
A comparison of domain integral evaluation techniques for boundary element methods
International Journal for Numerical Methods in Engineering, 2001
In many cases, boundary integral equations contain a domain integral. This can be evaluated by discretization of the domain into domain elements. Historically, this was seen as going against the spirit of boundary element methods, and several methods were developed to avoid this discretization, notably dual and multiple reciprocity methods and particular solution methods. These involved the representation of the interior function with a set of basis functions, generally of the radial type. In this study, meshless methods (dual reciprocity and particular solution) are compared to the direct domain integration methods. The domain integrals are evaluated using traditional methods and also with multipole acceleration. It is found that the direct integration always results in better accuracy, as well as smaller computation times. In addition, the multipole method further improves on the computation times, in particular where multiple evaluations of the integral are required, as when iterative solvers are used. The additional error produced by the multipole acceleration is negligible.
Integrals of radial basis functions for boundary element method
Communications in Numerical Methods in Engineering, 2000
The technique for transferring the domain integrals into equivalent boundary integrals, published in this journal by Wen, Aliabadi and Rooke, requires analytical expressions for ÿve integrals of radial basis functions. Exact solutions for these integrals are obtained by using Mathematica. The solutions are equivalent to the published one, but more convenient for computation. One of the published solutions is wrong.
A recursive application of the integral equation in the boundary element method
Engineering Analysis with Boundary Elements, 2011
This paper presents a recursive application of the governing integral equation aimed at improving the accuracy of numerical results of the boundary element method (BEM). Usually, only the results at internal domain points when using BEM are found using this approach, since the nodal boundary values have already been calculated. Here, it is shown that the same idea can be used to obtain better accuracy for the boundary results as well. Instead of locating the new source points inside the domain, they are positioned on the boundary, with different coordinates to the nodal points. The procedure is certainly general, but will be presented using as an example the two dimensional Laplace equation, for the sake of simplicity to point out the main concepts and numerical aspects of the method proposed, especially due to the determination of directional derivatives of the primal variable, which is part in hypersingular BEM theory.
Journal of Integral Equations and Applications, 1992
Recently, Galerkin and collocation methods have been analyzed in connection with the nonlinear boundary integral equation which arises in solving the potential problem with a nonlinear boundary condition. Considering this model equation, we propose here a discretized scheme such that the nonlinearity is replaced by its L 2 -orthogonal projection. We are able to show that this approximate collocation scheme preserves the theoretical L 2 -convergence. For piecewise linear continuous splines, our numerical experiments confirm the theoretical quadratic L 2 -convergence.
A convergence analysis of the performance of the DRM-MD boundary integral approach
International Journal for Numerical Methods in Engineering, 2007
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.
International Journal for Numerical Methods in Engineering, 1993
In the boundary integral equation method (BIEM), use of Lagrangian shape functions together with conforming boundary elements requires continuity of functions at the interelement boundary. When the flux or the traction is discontinuous due to the presence of corners or discontinuous boundary conditions, conforming elements can be a source of error. In this paper, we detail a multiple-node method in which this error can either be eliminated or substantially reduced. The paper is limited to the Laplace problem and the problem of elastostatics in two dimensions. However, the method can be easily extended to problems of other types and to higher dimensions.
The Boundary Element Method for Potential Problems
Fundamental Concepts and Models for the Direct Problem
This chapter presents the boundary element method applied to potential problems. The integral equation is obtained for the Laplace equation and discretized into boundary elements. Constant, linear, and quadratic boundary elements are considered. The method is applied to some numerical examples and results are compared to analytical solutions. A convergence study is carried out in order to access the behaviour of the method with mesh refinement.
Recent Advances and Emerging Applications of the Boundary Element Method
Applied Mechanics Reviews, 2012
Workshop on the Emerging Applications and Future Directions of the Boundary Element Method," University of Akron, Ohio, September 1-3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green's functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.