F. J. Rizzo

Hypersingular Boundary Integral Equations: Some Applications in Acoustic and Elastic Wave Scattering

Interpretation in terms of Hadamard finite-part integrals, even for integrals in three dimensions, is given, and this concept is compared with the Cauchy Principal Value, which, by itself, is insufficient to render meaning to the hypersingular integrals. Motivation for this work is given in the context of scattering of time-harmonic waves by cracks. A numerical example is given for the problem of acoustic scattering by a rigid screen in three spatial dimensions

A method of solution for certain problems of transient heat conduction

This paper develops a numerical treatment of classical boundary value problems for ar- bitrarily shaped plane heat conducting solids obeying Fouriers law. An exact integral formula defined on the boundary of an arbitrary body is obtained from a fundamental singular solu- tion to the governing differential equation. This integral formula is shown to be a means of numerically determining boundary data, complementary to given data, such that the Laplace transformed temperature field may subsequently be generated by a Greens type integral identity. The final step, numerical transform inversion, completes the solution for a given problem. All operations are ideally suited for modern digital computation. Three illustra- tive problems are considered. Steady-state problems, for which the Laplace transform is un- necessary, form a relatively simple special case. A FORMULATION of the various transient boundary value problems associated with isotropic solids obeying Fouriers law of heat conduction is developed. An exact in- tegral formula is derived relating boundary heat flux and boundary temperature, in the Laplace transform space, that corresponds to the same admissible transformed temperature field throughout the body. Part of the boundary data in the formula is known from the description of a well posed bound- ary value problem. As is shown, the remaining part of the boundary data is obtainable numerically from the formula it- self regarded as a singular integral equation. Once both trans- formed temperature and heat flux are known everywhere on the boundary, the transformed temperature throughout the body is obtainable by means of a Greens type integral identity. This identity yields the field directly in terms of the mentioned boundary data. The final step, transform in- version, although done approximately also, is accomplished by a technique particularly well suited to the class of problems under investigation. The main feature of the solution procedure suggested is its generality. It is applicable to solids occupying domains of rather arbitrary shape and connectivity. Boundary data may be prescription of temperature, or heat flux, or parts of each corresponding to a mixed type problem. Also, a linear combination of temperature and flux may be given corre- sponding to the so-called convection boundary condition. The same boundary formula described previously is applicable in every case. Approximations in the transform space are made only on the boundary, in contrast to finite difference procedures, and the approximations made are conceptually simple, natural to make, and give rise, as is shown, to very ac- curate data for a relatively crude boundary approximation pattern. Problems posed for composite bodies, i.e., two or more heat conducting solids bonded together, are particularly amenable to the present treatment. One computer program is employed which utilizes only data describing the domain geometry, boundary temperature or flux, material properties, and a sequence of values of the transform parameter neces- sary for the inversion scheme. Output is the transformed temperature at any desired field point. A second program in-

On boundary integral equations for crack problems

A ubiquitous linear boundary-value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle. One typical example is the scattering of scalar waves by a curved crack or rigid strip (Neumann boundary condition) in two dimensions. This problem can be reduced to an integrodifferential equation, which is often regularized. We adopt a more direct approach, and prove that the problem can be reduced to a hypersingular boundary integral equation. (Similar reductions will obtain in more complicated situations.) Computational schemes for solving this equation are described, with special emphasis on smoothness requirements. Extensions to three-dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed.

A Method for Stress Determination in Plane Anisotropic Elastic Bodies

Using a fundamental solution to the appropriate field equations of linear anisotropic elasticity, a real variable integral formula of the Somigliana type is derived. The formula relates an elastic displace ment field to boundary traction and displacement vectors; all refer to an arbitrary equilibrated stress state present in an orthotropic body of arbitrary shape and connectivity. A fundamental relation between boundary traction and displacement is then derived which is a mechanism for determining (numerically in practice) that part of such boundary data not initially given from a knowledge of that part which is given. Once all boundary quantities are known, the field solution is given by the integral formula and its first derivatives. Finally, the body may be composite; i.e., it may contain inclusions or inhomogeneities which may be isotropic or rigid as special cases. Numerical techniques are indicated and several problems are solved for illustration.

HYPERSINGULAR INTEGRALS : HOW SMOOTH MUST THE DENSITY BE ?

Hypersingular integrals are guaranteed to exist at a point x only if the density function f in the integrand satises certain conditions in a neighbourhood of x. It is well known that a su- cient condition is that f has a Holder-continuous rst derivative. This is a stringent condition, especially when it is incorporated into boundary-element methods for solving hypersingular in- tegral equations. This paper is concerned with nding weaker conditions for the existence of one-dimensional Hadamard nite-part integrals: it is shown that it is sucien t for the even part of f (with respect to x) to have a Holder-continuous rst derivative { the odd part is al- lowed to be discontinuous. A similar condition is obtained for Cauchy principal-value integrals. These simple results have non-trivial consequences. They are applied to the calculation of the tangential derivative of a single-layer potential and to the normal derivative of a double-layer potential. Particular attention is paid to discontinuous densities and to discontinuous boundary conditions. Also, despite the weaker sucien t conditions, it is rearmed that, for hypersingular integral equations, collocation at a point x at the junction between two standard conforming boundary elements is not permissible, theoretically. Various modications to the denition of nite-part integral are explored.

On the numerical solution of two-dimensional potential problems by an improved boundary integral equation method

Abstract The use of piecewise quadratic polynomial approximations in the boundary integral equation method for the solution of boundary value problems involving Laplaces equation and certain Poisson equations is described. To illustrate various features of this technique the results of several numerical experiments are presented.

On time-harmonic elastic-wave analysis by the boundary element method for moderate to high frequencies

Abstract Certain inherent difficulties in obtaining accurate boundary element solutions to problems of elastodynamics at moderate to high frequencies are examined. Some procedures for overcoming or compensating for these difficulties are described. In addition, results demonstrating the effectiveness of these procedures are presented. Finally, the use of these procedures in solving transient problems via Fourier-transform inversion is illustrated.

Hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions

Abstract A weakly singular form of the hypersingular boundary integral equation (BIE) (traction equation) for 3-D elastic wave problems is developed in this paper. All integrals involved are at most weakly singular and except for a stronger smoothness requirement on boundary elements, regular quadrature and collocation procedures used for conventional BIEs are sufficient for the discretization of the original hypersingular BIE. This weakly singular form of the hypersingular BIE is applied to the composite BIE formulation which uses a linear combination of the conventional BIE and the hypersingular BIE to remove the fictitious eigenfrequencies existing in the conventional BIE formulation for elastic wave problems. Numerical examples employing different types of boundary elements clearly demonstrate the effectiveness and efficiency of the developed formulation.

Scattering of elastic waves from thin shapes in three dimensions using the composite boundary integral equation formulation

In this paper, the composite boundary integral equation (BIE) formulation is applied to scattering of elastic waves from thin shapes with small but finite thickness (open cracks or thin voids, thin inclusions, thin-layer interfaces, etc.), which are modeled with two surfaces. This composite BIE formulation, which is an extension of the Burton and Miller’s formulation for acoustic waves, uses a linear combination of the conventional BIE and the hypersingular BIE. For thin shapes, the conventional BIE, as well as the hypersingular BIE, will degenerate (or nearly degenerate) if they are applied individually on the two surfaces. The composite BIE formulation, however, will not degenerate for such problems, as demonstrated in this paper. Nearly singular and hypersingular integrals, which arise in problems involving thin shapes modeled with two surfaces, are transformed into sums of weakly singular integrals and nonsingular line integrals. Thus, no finer mesh is needed to compute these nearly singular integrals...

Optic nerve axoplasm and papilledema

A detailed review of optic nerve axoplasm is presented. A number of hypotheses have been postulated for the pathogenesis of papilledema associated with increased intracranial pressure. These hypotheses, mechanical and nonmechanical, are critically evaluated in relation to five essential features of papilledema. Theories, as well as clinical and experimental studies, of axonal transport are reviewed, and a new hypothesis is proposed: Papilledema is primarily a mechanical, nonvascular phenomenon in which an excess amount of extracellular fluid is present in the prelaminar region of the optic disc and the accumulation of that fluid results from the leakage of axoplasm from optic nerve fibers which are compressed posterior to the lamina cribrosa of the optic disc. The authors believe that this is the only existing hypothesis consistent with all the known facts about papilledema. Discussions by Drs. J. Terry Ernest, Thomas R. Hedges, and S. S. Hayreh follow the review.