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.

The application of integral equation methods to the numerical solution of some exterior boundary-value problems

CHAPTER 4 – Integral Equation Solutions of Three-dimensional Scattering Problems

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.

https://ntrs.nasa.gov/api/citations/19800015552/downloads/19800015552.pdf

Boundary conditions for the numerical solution of elliptic equations in exterior regions

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.

https://www.olemiss.edu/sciencenet/benet/05eabeb.pdf

Heritage and early history of the boundary element method

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.

Integral Equation Methods for Electromagnetic and Elastic Waves

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.

Improved Integral Formulation for Acoustic Radiation Problems

The sonar equations provide a convenient way to calculate the detection range of actual and proposed sonar systems. They are simple, but approximate, relationships between parameters that describe the effects of source, target, and acoustic medium independently. The purpose of this paper is to derive an exact form of the active sonar equation by using the Helmholtz integral formulation to solve the boundary‐value problem with source and target both present in the medium. The resulting equation involves combinations of linear integral operators; however, it is suitable for solution by numerical techniques already developed for radiation from objects of arbitrary shape. Furthermore, it is shown that these integral operators reduce to multiplicative factors which represent general definitions of the source level, transmission loss, and target strength when the source‐to‐target distance is large. Thus this work establishes a basis for the sonar equations as the limiting form of an exact boundary‐value problem...

Helmholtz integral formulation of the sonar equations

The three‐dimensional acoustic scattering function of an object is very difficult and expensive to measure, even in controlled experiments. Usually one is limited to sampling this function at a limited number of monostatic angles and perhaps at a few bistatic angles in a single plane. However, the three‐dimensional function is controlled by relatively few degrees of freedom at low frequencies. By using a computational model to generate the far‐field propagator function for the object, and applying reciprocity conditions as a constraint, a least‐squares problem can be formulated to estimate the radiating part of the surface source strength. Singular value decomposition of the propagator function is used to identify the efficient radiating modes. Once the surface source density is determined by solving the least‐squares problem, the full three‐dimensional scattering function can be reconstructed. The theoretical basis for this technique will be presented, along with a discussion of results obtained to date. Practical issues dealing with the numerical implementation and computational efficiency will also be discussed.

A hybrid method for predicting the complete scattering function from limited data

: Much of the extensive literature on shell theory centers around the computation of on curves, usually thought of as the roots of a determinant D which relates the angular frequency w to the wave numbers for free waves traveling on the shell For a cylindrical shell, the free wave paths are actually helical curves on the surface of the shell, having both a circumferential and an axial wave number In this case, it is useful to indicate explicitly that the dispersion relation is a function of three variables, ie, D(w, m, n) = 0, where m is the number of half wavelengths of a wave traveling in the axial direction, and n is the number of full wavelengths of a wave traveling around the shell We shall refer to this function D(w, m, n) as the dispersion volume density, since it is a function of three variables Normally, one sees various two-dimensional displays of the dispersion relation, which are created by plotting any two of these variables against each other, while holding the third variable fixed Target strength, Martsam Active surveillance,

The Efficient Calculation and Display of Dispersion Curves for a Thin Cylindrical Shell Immersed in a Fluid

The classical Helmholtz integral is of current importance in numerical solutions for radiation from arbitrary vibrating surfaces over which the normal velocity is specified. Special conditions on frequency and normal velocity are derived here, under which the Helmholtz integral reduces to an explicit integral representation for the pressure field in the region exterior to an arbitrary vibrating surface. The frequency is required to be a characteristic frequency for the region enclosed by the vibrating surface subject to Neumann boundary conditions. The normal‐velocity configuration is required to satisfy either of two equivalent integral relations. It appears that for important geometries such as the finite cylinder and the rectangular parallelepiped these special situations can be identified a priori with minimal effort.

Harry A. Schenck

Papers

Improved Integral Formulation for Acoustic Radiation Problems

Helmholtz integral formulation of the sonar equations

A hybrid method for predicting the complete scattering function from limited data

The Efficient Calculation and Display of Dispersion Curves for a Thin Cylindrical Shell Immersed in a Fluid

Vanishing of the Surface‐Pressure Contribution to the Helmholtz Integral