Diffraction from arbitrarily shaped bodies of revolution: analytical regularization

Abstract

A mathematically rigorous and numerically efficient approach, based on analytical regularization, for solving the scalar wave diffraction problem with a Dirichlet boundary condition imposed on an arbitrarily shaped body of revolution is described. Seeking the solution in an integral-equation formulation, the singular features of its kernel are determined, and the initial equation transformed so that its kernel can be decomposed into a singular canonical part and a regular remainder. An analytical transformation technique is used to reduce the problem equivalently to an infinite system of linear algebraic equations of the second kind. Such system can be effectively solved with any prescribed accuracy by standard numerical methods. The matrix elements of this algebraic system are expressible in the terms of the Fourier coefficients of the remainder. Due to the smoothness of the remainder a robust and efficient technique is obtained to calculate the matrix elements. Numerical investigations of structures, such as the prolate spheroid and bodies obtained by rotation of "Pascal's Limaçon" and of the "Cassini Oval", exhibit the high accuracy and wide possibilities of the approach.

Diffraction From a Grating on a Chiral Medium: Application of Analytical Regularization Method

“…The freshest explanation of the ARM methodology by Yu. A. Tuchkin is included in [16,17]. As we mentioned, ARM is rather a philosophy than concrete techniques.…”

“…These faults are more pronounced in the resonant domain, which is of great interest to technical applications. The idea of Analytical Regularization Method (ARM) can be utilized for mathematically equivalent transformation of a first-kind operator equation to a second-kind Fredholm equation, which allows efficient numerical solution with preassigned accuracy [3,[15][16][17]. The brief explanation of the principal ideas of ARM is presented in Appendix.…”

Coupling and scattering from axisymmetric bodies, open and closed: Regularisation methods

“…for Laplace's equation) on the unit sphere. Then the analysis, described more fully in [3], shows that, if…”

“…Although it would seem that there are many ways of converting (10) to such a format, the form (12) is optimal for numerical calculations because of the decay rate of the coefficients. A careful discussion of this point, that is crucial for the effective application of analytical regularization methods, is given in [3]. It turns out that this choice of rescaling, determined for the closed surface S , is also the most effective for the open surface 0 S .…”

A New Analytically Regularizing Method for the Analysis of the Scattering by a Hollow Finite-Length Pec Circular Cylinder

“…It is interesting to observe that the regularizing procedure is in general neither trivial nor unique, and that the computational cost of numerical algorithm is strictly related to the selected regularizing scheme. As a matter of fact, different solutions have been proposed in the literature, ranging from the explicit inversion of the most singular part of the integral operator [14,[22][23][24]29] to the adoption of a Nystrom-type discretization scheme [17,25,30].…”