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

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.

/pdf/hypersingular-integrals-how-smooth-must-the-density-be-2y0iy05vpg.pdf

Hypersingular integrals: how smooth must the density be?

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.

A Method for Stress Determination in Plane Anisotropic Elastic Bodies

SUMMARY A boundary integral equation formulation for thin bodies which uses CBIE (conventional BIE) only is well known to be degenerate. A mixed formulation for a thin rigid scatterer which combines CBIE and HBIE (hypersingular BIE) is motivated by examining the discretized form of the integral equations, and this formulation is shown to be nondegenerate for thin non-rigid inclusion problems. A near-singular integration procedure, useful for singular integrals as well, is presented. Finally, numerical examples for acoustic wave scattering from rigid and soft scatterers are presented.

Frank J. Rizzo

Papers

Hypersingular integrals: how smooth must the density be?

A Method for Stress Determination in Plane Anisotropic Elastic Bodies

Boundary integral equations for thin bodies

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

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