Iterative Boundary Element Method Research Articles

A numerical wave tank model for cavitating hydrofoils

In this paper, an iterative boundary element method (IBEM) for both 2-D and 3-D cavitating hydrofoils moving steadily inside a numerical wave tank (NWT) is presented and some extensive numerical results are given. The cavitating hydrofoil part, the free surface part and the wall parts of NWT are solved separately, with the effects of one on the others being accounted for in an iterative manner. The cavitating hydrofoil surface, the free surface, the bottom surface and the side walls are modelled by a low-order potential based panel method using constant strength dipole and source panels. Second-order correction on the free surface in 2-D are included into the calculations by the method of small perturbation expansion both for potential and for wave elevation. The source strengths on the free surface are expressed in terms of perturbation potential by applying first-order (linearized) and second-order free surface conditions. The IBEM is applied to a 2-D (NACA16006 and NACA0012) and a 3-D rectangular cavitating hydrofoil and the effect of number of iterations, the effect of the depth of the hydrofoil from finite bottom and the effect of the walls of NWT, on the results are discussed.

An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput. Maths. Math. Phys. 31 (1991) 45] for obtaining approximate solutions to the ill-posed Cauchy problem for the Helmholtz equation is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularising criterion is also proposed.

An iterative boundary element algorithm for a singular Cauchy problem in linear elasticity

In this paper a new iterative boundary element method (BEM) is developed to solve singular Cauchy problems in isotropic linear elasticity. In order to deal with the difficulties associated with the instability caused by the prescription of the boundary data and the low rate of convergence manifested by standard numerical techniques, the iterative algorithm proposed by Kozlov et al. [1] is combined with the singularity subtraction technique, see Portela et al. [2]. A regularising stopping criterion is developed and, moreover, for a given physical geometry and loading of a cracked elastic medium, the convergence and the stability of the algorithm proposed are numerically investigated for both exact and noisy data. The conclusion of this paper is that if a singular problem is encountered then a singularity based technique, such as that described in Sect. 2, should be employed. In addition, if this singular problem is such that inverse analysis in necessary, such as in crack problems, then a suitable regularisation method should be applied to the regular problem obtained by taking into account the singularity, and not to the singular problem with the singularity ignored.

Conjugate Gradient-Boundary Element Method for the Cauchy Problem in Elasticity

In this paper, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the boundary element method (BEM) for obtaining stable approximate solutions to the Cauchy problem in linear elasticity is analysed. An efficient stopping criterion for the CGM proposed by Nemirovskii in 1986 is employed and in addition the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data.

An iterative boundary element method for Cauchy inverse problems

We are interested in this paper in recovering an harmonic function from the knowledge of Cauchy data on some part of the boundary. A new inversion method is introduced. It reduces the Cauchy problem resolution to the determination of the resolution of a sequence of well-posed problems. The sequence of these solutions is proved to converge to the Cauchy problem solution. The algorithm is implemented in the framework of boundary elements. Displayed numerical results highlight its accuracy, as well as its robustness to noisy data.

Solution of stokes flow using an iterative DRBEM based on compactly-supported, positive-definite radial basis function

This paper describes an iterative dual reciprocity boundary element method (DRBEM) based on the compactly-supported, positive definite radial basis function for the solution of Stokes flow problems. The method involves the solution of Laplace equations for vorticity, and Poisson equations for velocity with the solenoidal components of vorticity as right-hand sides. These equations are solved using iterative BEM and DRBEM. For the DRBEM, the compactly supported, positive definite radial basis function is used to approximate the body force term to convert it to boundary integrals. A relaxation method is devised such that the resulting algebraic equations are solved without the need of assembling a matrix. Hence, large systems of equations such as the fine discretization needed for Stokes flow problems can be handled easily. The results of Stokes flow in a square cavity and a circular cavity are presented and compared with other numerical model results. The iterative DRBEM solution is found to be satisfactory.

An iterative algorithm for singular Cauchy problems for the steady state anisotropic heat conduction equation

In this paper, we propose an alternating iterative algorithm to solve a singular Cauchy problem for the anisotropic heat conduction equation. The numerical algorithm is based on the boundary element method (BEM), modified to take into account the form of the singularity, without substantially increasing the amount of computation involved. Two test examples, the first with a singularity caused by an abrupt change in the boundary conditions and the second with a singularity caused by a sharp re-entrant corner, are investigated. The numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed, the iterative BEM is efficient in dealing with the difficulties arising from both the instabilities produced by the boundary condition formulation and the slow rate of convergence of standard numerical methods around the singular point.

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.

An iterative boundary element method for solving the one-dimensional backward heat conduction problem

In this paper, the iterative algorithm proposed by V.A. Kozlov and V.G. Maz'ya [Leningrad Math. J. 5 (1990) 1207–1228] is numerically implemented using the boundary element method (BEM) in order to solve the backward heat conduction problem (BHCP). The convergence and the stability of the numerical method are investigated and a stopping criterion is proposed. The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data.

An iterative bem for the cauchy steady state heat conduction problem in an anisotropic medium with unknown thermal conductivity tensor

In this paper we propose an alternating iterative boundary element method (BEM) to simultaneously predict the unknown conductivity coefficients and the unknown boundary data for a Cauchy steady state heat conduction problem in an anisotropic medium. This complex ill-posed problem is obtained by combining a Cauchy inverse thermal problem with a parameter estimation problem. The numerical algorithm is based on an iterative (BEM) combined with a least squares technique. The numerical results obtained confirm that provided that an appropriate stopping regularization criterion is imposed, the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularization criterion to cease the iterative process is proposed and a variable relaxation factor is used to increase the rate of convergence of the algorithm.

An Iterative Boundary Element Method for the Solution of a Cauchy Steady State Heat Conduction Problem

An Iterative Boundary Element Method for the Solution of a Cauchy Steady State Heat Conduction Problem

Solution of Poisson's equation by iterative DRBEM using compactly supported, positive definite radial basis function

In the numerical solution of three-dimensional boundary value problems, the matrix size can be so large that it is beyond a computer's capacity to solve it. To overcome this difficulty, an iterative dual reciprocity boundary element method (DRBEM) is developed to solve Poisson's equation without the need of assembling a matrix. The DRBEM procedure requires that the right hand side of Poisson's equation be approximated by a radial basis function interpolation. In the iterative solution, it is found that only compactly supported, positive definite radial basis functions lead to converged results.

The boundary element solution of the Cauchy steady heat conduction problem in an anisotropic medium

In this paper the iterative algorithm proposed by Kozlov et al. for the Cauchy problem for the Laplace equation is extended in order to solve the Cauchy steady-state heat conduction problem in an anisotropic medium. The iterative algorithm is numerically implemented using the boundary element method (BEM). The convergence and the stability of the numerical method, as well as various types of accuracy, convergence and stopping criteria, are investigated. The numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed, then the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularization criterion to cease the iterative process is proposed and the rate of convergence of the algorithm is improved by using various relaxation procedures between iterations. A new concept of a variable relaxation factor is proposed. Analytical formulae for the coefficients of the matrices resulting from the direct application of the BEM in an anisotropic medium are also presented. Copyright © 2000 John Wiley & Sons, Ltd.

A fast iterative boundary element method for solving closed crack problems

This paper describes an iterative hybrid boundary element method for solving non-linear closed crack problems. The original crack problem is split into two separate sub-problems: (1) the exterior body (without crack) which is modeled using displacement discontinuity boundary elements, and (2) the crack in an infinite domain which is modeled using either a dislocation density or displacement discontinuity approach. Iteration is performed between the exterior body and the crack faces until all boundary conditions are satisfied. Several advantages of this method over previous methods are discussed and are demonstrated in example problems.

An alternating method for an inverse Cauchy problem

In this paper, an iterative boundary element method based on our relaxed algorithm introduced in [8] is used to solve numerically a class of inverse boundary problems. A dynamical choice of the relaxation parameter is presented and a stopping criterion based on our theoretical results is used. The numerical results show that the algorithm produces a reasonably approximate solution and improves the rate of convergence of Kozlov's scheme [10].

A fast convergent iterative boundary element method on PVM cluster

This paper reports a fast convergent boundary element method on a Parallel Virtual Machine (PVM) (Geist et al., PVM: Parallel Virtual Machine, A Users' Guide and Tutorial for Networked Parallel Computing. MIT Press, Cambridge, 1994) cluster using the SIMD computing model (Single Instructions Multiple Data). The method uses the strategy of subdividing the domain into a number of smaller subdomains in order to reduce the size of the system matrix and to achieve overall speedup. Unlike traditional subregioning methods, where equations from all subregions are assembled into a single linear algebraic system, the present scheme is iterative and each subdomain is handled by a separate PVM node in parallel. The iterative nature of the overall solution procedure arises due to the introduction of the artificial boundaries. However, the system equations for each subdomain is now smaller and solved by direct Gaussian elimination within each iteration. Furthermore, the boundary conditions at the artificial interfaces are estimated from the result of the previous iteration by a reapplication of the boundary integral equation for internal points. This method provides a consistent mechanism for the specification of boundary conditions on artificial interfaces, both initially and during the iterative process. The method is fast convergent in comparison with other methods in the literature. The achievements of this method are therefore: (a) simplicity and consistency of methodology and implementation; (b) more flexible choice of type of boundary conditions at the artificial interfaces; (c) fast convergence; and (d) the potential to solve large problems on very affordable PVM clusters. The present parallel method is suitable where (a) one has a distributed computing environment; (b) the problem is big enough to benefit from the speedup achieved by coarse-grained parallelisation; and (c) the subregioning is such that communication overhead is only a small percentage of total computation time.

An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation

In this paper a new numerical method which does not require a regularization parameter is developed for solving the backward heat conduction problem (BHCP). The inverse and ill-posed BHCP for the heat equation is approximated with a convergent sequence of elliptic Cauchy problems for which the stable algorithm of the Kozlov et al. [1] type is accommodated. Further, a boundary element method (BEM) is developed for obtaining a stable solution of the BHCP.

Power iterative multiple reciprocity boundary element method for solving three-dimensional Helmholtz eigenvalue problems

The multiple reciprocity boundary element method (MRBEM) has been employed to solve the three-dimensional Helmholtz equation, ▿ 2 gf + k 2 gf = 0. In the present technique, the Helmholtz equation is arranged as ▿ 2 gf + k 0 2 gf + gf/ γ = 0 where k 0 is an estimate of k and λ is equal to ( k 2 − k 0 2) −1. As the term gf/λ is treated as a source, the power iteration technique with Wielandt's spectral shift is used to find the value of λ. The boundary integral equation is formulated with the fundamental solution to ▿ 2 gf + k 0 2 gf + δ i = 0. The domain integral related to the above source is transformed into a series of boundary integrals, with the aid of the higher order fundamental solutions based on the spherical Bessel functions. The eigenvalue k 2 can also be described using only boundary integrals. Test calculations demonstrate that the present technique is efficient for finding k 2 and easier to handle than the conventional determinant search scheme.

An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation

In this paper the boundary element method (BEM) is used iteratively in order to implement numerically the alternating algorithm proposed by Kozlov et al. (Kozlov, V. A., Maz'ya, V. G. & Fomin, A. V. Comput. Maths. Phys., 1991, 31, 45–52) for solving the Cauchy problem for the Laplace equation. Various types of convergence and accuracy criteria, and boundary condition formulations are investigated which confirm that the iterative BEM produces a convergent and accurate numerical solution with respect to the number of boundary elements and number of iterations.

A alternating boundary element method for solving cauchy problems for the biharmonic equation

In this study the Cauchy problems that can be associated to the biharmonic equation are classified into 5 distinct ill-posed formulations. The biharmonic equation is numerically discretised using the boundary element method (BEM) and the degree of ill-posedness of the inverse formulations is characterised by the condition number of the senstitivity matrix. The biharmonic Cauchy problems are mathematically algorithmised by accomodating the iterative method developed by Kozlov et al. [1] which is convergent for Cauchy problems of elliptic equations in general. The numerical implementation of the mathematical algorithm is based on an iterative application of the BEM for a sequence of mixed well-posed direct biharmonic problems. It is shown that the iterative BEM produces a convergent and accurate numerical solution with respect to increasing the number of boundary elements and the number of iterations. Care was taken to ensure that all the unspecified boundary data is accurately approximated and this was accomplished by a further application of the singular value decomposition truncated at an optimal level given by the L-curve method. Furthermore, the iterative alternating algorithm has a regularizing character and the stability of the numerical solution was shown by imposing various amounts of noise in the input data.