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Journal ArticleDOI

Improved Integral Formulation for Acoustic Radiation Problems

01 Jul 1968-Journal of the Acoustical Society of America (Acoustical Society of America)-Vol. 44, Iss: 1, pp 41-58
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.
Citations
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Journal ArticleDOI
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.

1,127 citations


Cites methods from "Improved Integral Formulation for A..."

  • ...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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Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: The early history of the boundary element method up to the late 1970s can be traced to the early 1960s, when the electronic computers had become available as mentioned in this paper, and the full emergence of the numerical technique known as the boundary elements method occurred in the late1970s.
Abstract: This article explores the rich heritage of the boundary element method (BEM) by examining its mathematical foundation from the potential theory, boundary value problems, Green's functions, Green's identities, to Fredholm integral equations. The 18th to 20th century mathematicians, whose contributions were key to the theoretical development, are honored with short biographies. The origin of the numerical implementation of boundary integral equations can be traced to the 1960s, when the electronic computers had become available. The full emergence of the numerical technique known as the boundary element method occurred in the late 1970s. This article reviews the early history of the boundary element method up to the late 1970s.

555 citations

Book
15 Jul 2007
TL;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