Method Of Discrete Singularities Research Articles

Cauchy-Toeplitz matrices and some applications

Some properties of Cauchy-Toeplitz matrices of the form T n = [1/( i - j + 1 2 )] are investigated. It is shown that minimal singular values of T n converge to zero with increasing n. Some applications of Cauchy-Toeplitz matrices to the solution of singular integral equations by the method of discrete singularities are considered.

The method of discrete singularities in plane problems of the theory of elasticity with non-smooth boundaries

The numerical solution of a class of elasticity-theory problems that is broader compared with that considered earlier in /1–3/ is investigated, namely, plane problems with non-smooth boundaries by the method of discrete singularities (MDS). The MDS is the direction of numerical solution of boundary value problems that is substantially the Tikhonov regularization method /4/ based on boundary singular integral equations (SIE). It is best to combine the MDS with the method of finite elements in computations of geometrically complex objects when the solution for the low level superelements is obtained by using the MDS. The MDS includes the reduction of the problem to SIE, the parametric assignment of the contour, the investigation of the SIE properties, smoothing of the SIE kernels, extraction of the unique SIE solution, justification of the selection of two matched systems of points on the contour, passage from SIE to a system of linear algebraic equations and assurance of its determinancy and non-degeneracy, analysis of the convergence of the solution, and the application of quadrature formulas for Cauchy-type integrals. An elastic isotropic homogeneous medium is considered that occupies a simply-connected domain D with a piecewise-smooth closed contour Γ on a plane. The solution of plane elasticity theory problems in a known way /5, 6/ is reduced to determining two analytic functions, ϑ, ψ, say. The non-smoothness of the boundary influences the realization of the MDS, however, the essence of the method is conserved. The purpose of the paper being published is to study and give a foundation for what is new in the MDS for plane elasticity theory problems with non-smooth boundaries as compared with the smooth boundary case in question.

A numerical analysis of electromagnetic scattering of perfect conducting cylinders by means of discrete singularity method improved by optimization process

AbstractAs a method of solving the electromagnetic scattering by perfectly conducting cylinders, the discrete singularity method is discussed. For this method, a finite number of singular points is distributed within the scatterer and the wave function composed of a linear combination of solutions (each of which is derived from a Helmholtz equation with each single singularity among the many) is matched with the incident wave function on the boundary. the remaining important problem in this method is how to distribute these singularities in order to obtain the accurate solution with faster convergence and with fewer number of singular points for an arbitrary configuration of the scatterer. the purpose of this paper is to present a reasonable solution to this problem. the essential point of the method is that by making as unknown quantities, not only the unknown amplitude coefficients involved in the linear combination of the solutions, but also the coordinates of the singularities, the scattered wave is matched on the boundary in the sense of the least squares. This problem is linear in the amplitude coefficient but nonlinear in the singularity coordinate. Thus we solve this problem by using the algorithm by Marquardt et al. of a nonlinear optimization process. This paper considers the general description and indicates its effectiveness by showing a few examples of the scattering problem of rectangular cylinders.

The method of discrete singularities in plane problems of the theory of elasticity

Plane problems of the theory of elasticity are reduced to sets of singular integral equations for which a direct method of solution is developed, similar to the method of discrete vortices used in aerodynamics. Numerical solutions of a number of plane problems of the theory of elasticity are considered, stable numerical solutions are obtained, and their convergence is proved.

An approximate method for solving two-dimensional low-Reynolds-number flow past arbitrary cylindrical bodies

This paper presents an approximate method for solving Oseen's linearized equations for a two-dimensional steady flow of incompressible viscous fluid past arbitrary cylindrical bodies at low Reynolds numbers. The formulation is based on a discrete singularity method with a least squares criterion for satisfying the no-slip boundary condition. That is, sets of Oseenlets, sinks, sources and vortices are discretely distributed in the interior of the body, and then the least squares criterion attempts to minimize the integrated squares of velocities along the body contour, thus leading to a system of simultaneous algebraic equations. Complex-variable arithmetic, usually available on modern computers, makes the computation algorithm very simple. Furthermore, the method is applicable to cases that cannot be solved by classical analytical approaches. As examples of application, we computed the forces acting on a single circular cylinder, two circular cylinders of equal radius separated by a distance, an inclined elliptic cylinder and an inclined square cylinder all of which are immersed in uniform flow fields. The computed results agree very well with those of classical analytical methods.

Approximate computation of the cavitational flow around small span wings

The linearized problem of a completely cavitating wing of the small span λ is considered. The system of singular integral equations of a cavitating lifting surface is reduced to one-dimensional equations by using the Lawrence approximation. The method of discrete singularities is used for the numerical solution of this system. Dependences of the lift coefficient and the cavern length on the cavitation number are presented for rectangular wings for 0.25≤λ≤4.