Transforming the domain integrals to the boundary using approximate particular solutions: a boundary element approach for nonlinear problems (original) (raw)
Related papers
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.
Numerical solution of problems with non-linear boundary conditions
Mathematics and Computers in Simulation, 2003
In this paper, we are concerned with an elliptic problem in a bounded two-dimensional domain equipped with a non-linear Newton boundary condition. This problem appears, e.g. in the modelling of electrolysis procedures. We assume that the non-linearity has a polynomial behaviour. The problem is discretized by the finite element (FE) method with conforming piecewise linear or polynomial approximations. This problem has been investigated in [Num. Math. 78 (1998) 403; Num. Funct. Anal. Optimiz. 20 (1999) 835] in the case of a polygonal domain, where the convergence and error estimates are established. In [Feistauer et al., On the Finite Element Analysis of Problems with Non-linear Newton Boundary Conditions in Non-polygonal Domains, in press] the convergence of the FE approximations to the exact solution is proved in the case of a nonpolygonal domain with curved boundary. The analysis of the error estimates leads to interesting results. The non-linearity in boundary condition causes the decreas of the approximation error. Further decreas is caused by the application of the numerical integration in the computation of boundary integrals containing the non-linear terms. In [Feistauer et al., Numerical analysis of problems with non-linear Newton boundary conditions, in: Proceedings of the Third Conference of ENUMATH'99, p. 486], numerical experiments prove that this decreas is not the result of a poor analysis, but it really appears. In our paper, we give a brief of the results. The main attention is paid to the development of the error estimates for higher-order FE method. The error estimates are compared with experiments.
A boundary element approach for non-homogeneous potential problems
Computational Mechanics, 1991
This paper reports an implementation of a Boundary Element Method dealing with two-dimensional inhomogeneous potential problems. This method avoids the tedious calculation of the domain integral contributions to the boundary integral equations. This is achieved by applying approximate particular solutions which are obtained by expressing the source distribution in terms of a linear combination of radial basis functions. Numerical examples show that the method is efficient and can produce accurate results.
Communications in Numerical Methods in Engineering, 2007
In this paper, a sufficient condition, for transforming domain integrals into boundary integral is described. The transformation is accomplished by Green's and Gauss' theorems. It is shown that a wide range of domain integrals including some integrals in boundary element method (BEM) satisfy this sufficient condition and can be simply transformed to boundary. Although emphasis is made on potential and elastostatic problems, this method can also be used for many other applications. Using the present method, a wide range of 2D and 3D domain integrals over simply or multiply connected regions can be transformed exactly to the boundary. The resultant boundary integrals are numerically evaluated using an adaptive version of the Simpson integration method. Several examples are provided to show the efficiency and accuracy of the present method.
Some notes on the general boundary element method for highly nonlinear problems
Communications in Nonlinear Science and Numerical Simulation, 2005
We give a short review of the so-called general boundary element method for strongly nonlinear problems in heat and viscous flow and a brief discussion about opportunity and challenge of the boundary element method as a numerical tool, compared with other numerical techniques.
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.
Direct solution of ill-posed boundary value problems by radial basis function collocation method
International Journal for Numerical Methods in Engineering, 2005
Numerical solution of ill-posed boundary value problems normally requires iterative procedures. In a typical solution, the ill-posed problem is first converted to a well-posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill-posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method.
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.