Cutouts in Shallow Shells

1970 ◽  
Vol 37 (2) ◽  
pp. 374-383 ◽  
Author(s):  
J. Lyell Sanders
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Arbitrary Shape ◽  
Shallow Shell ◽  
Boundary Value ◽  
Integral Representations ◽  
Shallow Shells ◽  
General Solutions ◽  
Rayleigh Principle ◽  
Shell Equations

Integral representations of general solutions to the shallow shell equations are obtained by means of the Betti-Rayleigh principle. With the aid of these integral representations the cutout problem of shallow shells is reduced to a system of four coupled integral equations. The cutout can have an arbitrary shape. Cutouts with reinforced edges and other boundary-value problems for shallow shells can be reduced to integral equations by the method developed in the paper.

scholarly journals On the Solution of some Axisymmetric Boundary Value Problems by means of Integral Equations

1963 ◽  
Vol 13 (3) ◽  
pp. 235-246 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Value ◽  
Diffraction Theory ◽  
Integral Representations ◽  
Fredholm Integral Equations ◽  
Fredholm Equations ◽  
Abel Integral Equations ◽  
Spherical Caps ◽  
Method Show

This paper concludes a series of papers (1) on a group of axisymmetric boundary value problems in potential and diffraction theory by considering some potential problems for a circular annulus. The Dirichlet problem for an annulus has recently been considered by Gubenko and Mossakovskiǐ (2), who, by a somewhat complicated method, show it to be governed by either one of two Fredholm integral equations of the second kind. The purpose of the present paper is to show how the method developed in previous papers, by which certain integral representations of the potentials in problems for circular disks arid spherical caps are used to reduce such problems to the solutions of either single Abel integral equations or Abel and Fredholm equations, can be applied to both the Dirichlet and Neumann problems for the annulus to give reasonably straightforward derivations of the governing Fredholm equations.

scholarly journals Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

2020 ◽  
Vol 22 (3) ◽  
pp. 319-332
Author(s):  
Aleksandr N. Tynda ◽  
Konstantin A. Timoshenkov
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Double Layers ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

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