The application of integral equation methods to the numerical solution of some exterior boundary-value problems
08 Jun 1971-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (PROCEEDINGS OF THE ROYAL SOCIETY A MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES)-Vol. 323, Iss: 1553, pp 201-210
TL;DR: The application of integral equation methods to exterior boundary-value problems for Laplace's equation and for the Helmholtz (or reduced wave) equation is discussed in this article, where it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first.
Abstract: The application of integral equation methods to exterior boundary-value problems for Laplace’s equation and for the Helmholtz (or reduced wave) equation is discussed. In the latter case the straightforward formulation in terms of a single integral equation may give rise to difficulties of non-uniqueness; it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first. Finally, an outline is given of methods for transforming the integral operators with strongly singular kernels which occur in the second equation.
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TL;DR: In this paper, a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain and estimates of the error due to the finite boundary are obtained for several cases.
Abstract: Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace's equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used.
603 citations
Cites methods from "The application of integral equatio..."
...Such formulations for the Helmholtz equation can be found in Chertock [9], Kleinman and Roach [16], and Burton and Miller [ 8 ]....
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Book•
15 Jul 2007TL;DR: Important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research.
Abstract: Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research. Also, learning the fundamentals of linear elastic wave theory does not require a quantum leap for electromagnetic practitioners. Integral equation methods have been around for several decades, and their introduction to electromagnetics has been due to the seminal works of Richmond and Harrington in the 1960s. There was a surge in the interest in this topic in the 1980s (notably the work of Wilton and his coworkers) due to increased computing power. The interest in this area was on the wane when it was demonstrated that differential equation methods, with their sparse matrices, can solve many problems more efficiently than integral equation methods. Recently, due to the advent of fast algorithms, there has been a revival in integral equation methods in electromagnetics. Much of our work in recent years has been in fast algorithms for integral equations, which prompted our interest in integral equation methods. While previously, only tens of thousands of unknowns could be solved by integral equation methods, now, tens of millions of unknowns can be solved with fast algorithms. This has prompted new enthusiasm in integral equation methods.
473 citations
462 citations
01 Aug 2009
TL;DR: In this paper, the authors propose a fast multipole BEM for potential problems, including Stokes flow problems, Elastostatic problems, and Acoustic wave problems, which is a more general approach than conventional BEM.
Abstract: 1. Introduction 2. Conventional BEM for potential problems 3. Fast multipole BEM for potential problems 4. Elastostatic problems 5. Stokes flow problems 6. Acoustic wave problems.
394 citations
TL;DR: State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed and Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite- element methods are described.
Abstract: State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite-element methods are described. Stabilized, multiscale, and other wave-based discretization methods developed to reduce this error are reviewed. A review of finite-element methods for acoustic inverse problems and shape optimization is also given.
368 citations
Cites background from "The application of integral equatio..."
...A natural way of modeling the acoustic region exterior to a scattering/radiating object is to introduce a boundary element discretization of the surface S based on an integral representation of the exact solution in the exterior [35, 45, 176]....
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References
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Book•
01 Jan 1929
2,246 citations
"The application of integral equatio..." refers background in this paper
...I t is known tha t in either case, provided g is continuous, the solution exists and is unique (Kellogg 1929, pp. 311- 314)....
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TL;DR: In this article, a combined Helmholtz Integral Equation Formulation (CHIEF) was proposed to obtain an approximate solution of the exterior steadystate acoustic radiation problem for an arbitrary surface whose normal velocity is specified.
Abstract: Three different integral formulations have been used as a basis for obtaining approximate solutions of the exterior steady‐state acoustic radiation problem for an arbitrary surface whose normal velocity is specified: (1) the simple‐source formulation, adapted from potential theory; (2) the surface Helmholtz integral formulation, based on the integral expression for pressure in the field in terms of surface pressure and normal velocity; and (3) the interior Helmholtz integral formulation, in which the surface pressure is determined by making a certain integral vanish for all points interior to the radiating surface. For certain characteristic wavenumbers, it is shown that no solution of the simple‐source formulation exists in general and that there is no unique solution of the surface Helmholtz integral formulation. The interior Helmholtz integral formulation is subject to similar difficulties and has undesirable computational characteristics. A Combined Helmholtz Integral Equation Formulation (CHIEF) that overcomes the deficiencies of the first two methods and the undesirable computational characteristics of the third, is described. The significant improvement over the previous three methods, which is accomplished through the use of CHIEF, is illustrated by numerical examples involving spheres, finite cylinders, cubes, and a steerable array mounted in two different boxlike structures.
986 citations
"The application of integral equatio..." refers methods in this paper
...A method of a somewhat different character has been described by Schenck (1968) who uses, in addition to Green’s boundary formula, the corresponding interior formula applied at a selected set of interior points....
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TL;DR: In this paper, a short study of Fredholm integral equations related to potential theory and elasticity is presented, with a view to preparing the ground for their exploitation in the numerical solution of difficult boundary-value problems.
Abstract: This paper makes a short study of Fredholm integral equations related to potential theory and elasticity, with a view to preparing the ground for their exploitation in the numerical solution of difficult boundary-value problems. Attention is drawn to the advantages of Fredholm ’s first equation and of Green’s boundary formula. The latter plays a fundamental and hitherto unrecognized role in the integral equation formula of biharm onic problems.
312 citations
"The application of integral equatio..." refers background in this paper
...In such cases, however, the difficulty is readily overcome by replacing ] by A[cr] + const, x J (See Jaswon (1963) for a discussion of this phenomenon; a companion paper by Symm (1963) describes some applications of integral equation methods to two-dimensional problems.)...
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270 citations