Solution Of Boundary Integral Equations Research Articles

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.

On Numerical Solutions of Boundary Integral Equations

Elastic boundary value problems can be formulated in the form of boundary integral equations, whose numerical solutions are obtained by boundary element methods. This paper deals with error estimation of boundary element solutions. Especially, approximate solutions obtained by constant or linear elements are investigated for cases of two-dimensional elastic bodies with smooth boundaries, which may contain cracks. In the analysis, the boundary element solutions of an infinite plane with a Griffith crack or a circular hole are derived in analytic forms. Examinations of them lead to estimation of residuals and errors.

The Boundary Integral Equation Method for Plates Resting on a Two‐Parameter Foundation

AbstractIn the paper the boundary integral equation method for solving the problem of a plate resting on a two‐parameter foundation is presented. Three boundary value problems are considered, and three cases are to be distinguished as regards a characteristic parameter of the foundation. The boundary integral equation solution for a clamped circular plate under a concentrated load is compared to the analytical one.

Boundary integral equation solution of moving boundary phase change problems

AbstractBoundary integral equation methods are presented for the solution of some two‐dimensional phase change problems. Convection may enter through boundary conditions, but cannot be considered within phase boundaries. A general formulation based on space‐time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation. The latter is pursued and applied in detail. An elementary, noniterative system is constructed, featuring linear interpolation over elements on a polygonal boundary. Nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution. The system performs well against basic analytical solutions, using these values in the interphase jump condition, with the simplest formulation of the surface normal at boundary vertices. Because the discretized surface changes automatically to fit the scale of the problem, the method appears to offer many of the advantages of moving mesh finite element methods. However, it only requires the manipulation of a surface mesh and solution for surface variables. In some applications, coarse meshes and very large time steps may be used, relative to those which would be required by fixed grid domain methods. Computations are also compared to original lab data, describing two‐dimensional soil freezing with a time‐dependent boundary condition. Agreement between simulated and measured histories is good.

Boundary integral equation solution of viscous flows with free surfaces

Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green’s Theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.

Boundary integral equation solutions for solitary wave generation, propagation and run-up

The boundary integral equation method (BIEM) is developed as a tool for studying two-dimensional, nonlinear water wave problems, including the phenomena of wave generation, propagation and run-up. The wave motions are described by a potential flow theory. Nonlinear free-surface boundary conditions are incorporated in the numerical formulation. Examples are given for either a solitary wave or two successive solitary waves. Special treatment is developed to trace the run-up and run-down along a shoreline. The accuracy of the present scheme is verified by comparing numerical results with experimental data of maximum run-up.

Some boundary integral equation solutions for three-dimensional stress concentration problems

The boundary integral equation (BIE) method for three-dimensional linear, elastic stress analysis is applied to some stress concentration problems associated with transverse circular holes in either hollow or solid circular cylinders subject to axial tension or torsion, also offset-oblique holes in cylinders subject to internal pressure. Satisfactory agreement is obtained with some previously published experimental results, although computed maximum stress concentration factors are generally higher than those obtained experimentally. The BIE method is shown to be a very useful tool for solving three-dimensional problems of engineering stress analysis.

Inverse scattering for an exterior Dirichlet problem

In this paper we consider scattering due to a metallic cylinder which is in the field of a wire carrying a periodic current. The aim of the paper is to obtain information such as the location and shape of the cylinder with a knowledge of a far-field measurement in between the wire and the cylinder. The same analysis is applicable in acoustics in the situation that the cylinder is a soft-wall body and the wire is a line source. The associated direct problem in this situation is an exterior Dirichlet problem for the Helmholtz equation in two dimensions. We present an improved low-frequency estimate for the solution of this problem using integral equation methods, and our calculations on inverse scattering are accurate to this estimate. The far-field measurements are related to the solutions of boundary integral equations in the low-frequency situation. These solutions can be expressed in terms of mapping function which maps the exterior of the unknown curve onto the exterior of a unit disk. The coefficients of the Laurent expansion of the conformal transformations can be related to the far-field coefficients. The first far-field coefficient leads to the calculation of the distance between the source and the cylinder. The other coefficients are determined by placing the source in a different location and using the corresponding new far-field measurements.

Boundary integral equation solution of the single sided linear induction pump problem

This paper utilizes the Boundary Integral Equation Method to analyse the single sided linear induction pump. The advantage offered by the method is that transverse edge effects are not neglected. The equations necessary for the electromagnetic field and force analysis of the pump are developed and the numerical solution of the equations is described. Particular attention is given to means of calculating the electromagnetic fields within the molten metal secondary. Typical field and force distributions are presented.

Boundary integral equation solutions to moving interface between two fluids in porous media

The boundary integral equation method is formulated for and applied to problems concerning a moving interface between two fluids in porous media. The mixing between two fluids is assumed to be insignificant; a sharp interface exists between them. Numerical and experimental results are presented for the tilting of a vertical interface in a Hele‐Shaw cell. During the initial stage of the interfacial movement, nonlinear effects are clearly demonstrated. The BIEM results are also compared to experimental results for a transient salt water intrusion problem.

An efficient algorithm for the numerical evaluation of boundary integral equations

The usual approach to the solution of boundary integral equations is to represent the unknown vector by a piecework constant, linear, or quadratic function over the given mesh subdivision. These representations have the advantages of consistency, ability to integrate the equations for the given functional approximations, and in general, improved accuracy as the degree of approximation is increased. While adequate for many problems, special requirements arise for certain nonlinear problems, e.g. plasticity, where the integral equations must be solved for each load increment. In the present paper a special numerical algorithm is outlined in which the unknown vector is represented as a combination of a Fourier series and piecewise linear function. The piecewise linear function is used only in high gradient regions of the unknown vector thus permitting an excellent representation with relatively few Fourier terms. The algorithm is compared with a linear representation alone for two problems which show the effects of multiple connectivity, sharp corners and discontinuous loading. For comparable accuracy both problems show a significant improvement in computer time required.

Boundary Integral Equation Solution of Non-linear Plane Potential Problems

This paper presents the boundary integral equation (BIE) formulation, and numerical solution procedure for two-dimensional problems governed by Laplac's equation and subject to non-linear boundary conditions. The introduction of non-linear terms constitutes a fundamental extension of the BIE method, as previous applications have been restricted entirely to linear problems. Furthermore, non-linearities necessitate the use of iterative solution techniques which present the conceptual disadvantage that a solution is not guaranteed. However, such difficulties were not encountered with the Newton—Raphson method employed in this study. The various features of the BIE technique are illustrated by the application to a physical problem which is of significance in heat exchanger design.

Boundary integral equation solution to axisymmetric potential flows: 2. Recharge and well problems in porous media

The boundary integral equation method (BIEM) is employed to solve both steady and transient axisymmetric flow problems in porous media. The problems analyzed here are governed by Laplace's equation; however, the unsteady and nonlinear behavior result from the presence of a free surface. Both finite and infinite domains are easily handled with the BIEM. Results are presented for a variety of well and recharge problems. Comparisons of BIEM results to a linearized theory show excellent agreement for recharge problems where the linearized theory is valid. In addition, results were obtained for cases where the linearized theory cannot be used. The BIEM solutions for steady state well problems are in excellent agreement with solutions obtained by a finite element method, as well as the analytic solution using the Dupuit assumption. Finally, the BIEM yields solutions to a variety of transient well problems. It is concluded that the BIEM is both an accurate and efficient method for solving well and recharge problems.

Boundary integral equation solution to axisymmetric potential flows: 1. Basic formulation

In this paper the two‐dimensional boundary integral equation method (BIEM) is extended to problems of axisymmetric flow governed by Laplace's equation. An ‘axisymmetric’ Green's function is defined which leads to a calculation procedure that parallels the two‐dimensional theory. The result is a numerical method which is efficient and accurate for a variety of axisymmetric potential flow problems.

Iterative Lösung gewisser Randwertprobleme und Integralgleichungen

Iterative Lösung gewisser Randwertprobleme und Integralgleichungen

Numerical stress concentrations for stepped shafts in torsion with circular and shaped fillets

A computer programme has been developed to solve for torsion of solids of revolution by numerical solution of an exact boundary integral equation which is valid for any shape and any degree of connectivity. The programme gives solutions directly at the boundary and requires only simple boundary data defining the contour to be input. The basic integral equation has been differentiated to give internal stresses directly, when required. The computed results have been checked experimentally by photoelastic techniques and are compared with analytical solutions for the circular cylinder and the cone. Comprehensive charts are given for stress concentrations in stepped shafts with circular fillets and fillets constructed of two blending radii. Fillets giving stress concentrations close to unity have also been investigated.