Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Part 1 Thin plates: introduction the fundamentals of the small-deflection plate bending theory rectangular plates circular plates bending of plates of various shapes plate bending by approximate and numerical methods advanced topics buckling of plates vibration of plates. Part 2 Thin shells: introduction to the general linear shell theory geometry of the middle surface the general linear theory of thin shells the membrane theory of shells applications of the membrane theory to the analysis of shell structures moment theory of circular cylindrical shells the moment theory of shells of revolution approximate theories of shell analysis and their application advanced topics buckling of shells vibration of shells. Appendices: some reference data Fourier series expansion verification of relations of the theory of surfaces derivation of the strain-displacement relations verification of equilibrium equations.

Thin Plates And Shells: Theory Analysis And Applications

The aim of this paper is to describe the development of the method of fundamental solutions (MFS) and related methods over the last three decades. Several applications of MFS-type methods are presented. Techniques by which such methods are extended to certain classes of non-trivial problems and adapted for the solution of inhomogeneous problems are also outlined.

https://msvlab.hre.ntou.edu.tw/grades/meshless/MFS98.pdf

The method of fundamental solutions for elliptic boundary value problems

Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

A compact representation is given of the electric- and magnetic-type dyadic Green's functions for plane-stratified, multilayered, uniaxial media based on the transmission-line network analog along the aids normal to the stratification. Furthermore, mixed-potential integral equations are derived within the framework of this transmission-line formalism for arbitrarily shaped, conducting or penetrable objects embedded in the multilayered medium. The development emphasizes laterally unbounded environments, but an extension to the case of a medium enclosed by a rectangular shield is also included.

/pdf/multilayered-media-green-s-functions-in-integral-equation-icv3t901yt.pdf

Multilayered media Green's functions in integral equation formulations

The features of an advanced numerical solution capability for boundary value problems of linear, homogeneous, isotropic, steady-state thermoelasticity theory are outlined. The influence on the stress field of thermal gradient, or comparable mechanical body force, is shown to depend on surface integrals only. Hence discretization for numerical purposes is confined to body surfaces. Several problems are solved, and verification of numerical procedures is obtained by comparison with accepted results from the literature.

An advanced boundary integral equation method for three‐dimensional thermoelasticity

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

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

This paper develops a numerical treatment of classical boundary value problems for ar- bitrarily shaped plane heat conducting solids obeying Fourier's 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 Green's 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 Fourier's 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 Green's 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-

A method of solution for certain problems of transient heat conduction

Etude du rayonnement et de la diffusion d'ondes elastiques par des obstacles de forme arbitraire

A boundary integral equation method for radiation and scattering of elastic waves in three dimensions

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.

Frank J. Rizzo

Papers

An advanced boundary integral equation method for three‐dimensional thermoelasticity

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

A method of solution for certain problems of transient heat conduction

A boundary integral equation method for radiation and scattering of elastic waves in three dimensions

On Boundary Integral Equations for Crack Problems