Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

https://www-dimat.unipv.it/sangalli/iga_for_maxwell.pdf

Isogeometric analysis in electromagnetics: B-splines approximation

The characteristic basis function method (CBFM) has been hybridized with the adaptive cross approximation (ACA) algorithm to construct a reduced matrix equation in a time-efficient manner and to solve electrically large antenna array problems in-core, with a solve time orders of magnitude less than those in the conventional methods. Various numerical examples are presented that demonstrate that the proposed method has a very good accuracy, computational efficiency and reduced memory storage requirement. Specifically, we analyze large 1-D and 2-D arrays of electrically interconnected tapered slot antennas (TSAs). The entire computational domain is subdivided into many smaller subdomains, each of which supports a set of characteristic basis functions (CBFs). We also present a novel scheme for generating the CBFs that do not conform to the edge condition at the truncated edge of each subdomain, and provide a minor overlap between the CBFs in adjacent subdomains. As a result, the CBFs preserve the continuity of the surface current across the subdomain interfaces, thereby circumventing the need to use separate ldquoconnectionrdquo basis functions.

Fast Analysis of Large Antenna Arrays Using the Characteristic Basis Function Method and the Adaptive Cross Approximation Algorithm

Quadrature by expansion

The Messenger Lectures at Cornell University were instituted in 1924 by Hiram J. Messenger ‘to provide a course or courses of lectures on the evolution of civilization for the special purpose of raising the moral standard of our political, business and social life’. In November 1964, Feynman gave seven lectures, extempore with the help of brief notes, on ‘The Character of Physical Law’. The transcripts were prepared and published by BBC in 1965.

The Character of Physical Law

This paper presents new singular curl- and divergence-conforming vector bases that incorporate the edge conditions. Singular bases complete to arbitrarily high order are described in a unified and consistent manner for curved triangular and quadrilateral elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester-Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. The curl (divergence) conforming singular bases guarantee tangential (normal) continuity along the edges of the elements allowing for the discontinuity of normal (tangential) components, adequate modeling of the curl (divergence), and removal of spurious modes (solutions). These singular high-order bases should provide more accurate and efficient numerical solutions of both surface integral and differential problems. Sample numerical results confirm the faster convergence of these bases on wedge problems.

/pdf/singular-higher-order-complete-vector-bases-for-finite-rzlca5c2v9.pdf

Singular higher order complete vector bases for finite methods

A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1/R singularities is presented. The numerical quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provided.

/pdf/machine-precision-evaluation-of-singular-and-nearly-singular-3z5bdbf029.pdf

Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions

A new Wiener-Hopf approach for the solution of impenetrable wedges at skew incidence is presented. Mathematical aspects are described in a unified and consistent theory for angular region problems. Solutions are obtained using analytical and numerical-analytical approaches. Several numerical tests from the scientific literature validate the new technique, and new solutions for anisotropic surface impedance wedges are solved at skew incidence. The solutions are presented considering the geometrical and uniform theory of diffraction coefficients, total fields, and possible surface wave contributions

/pdf/wiener-hopf-solution-for-impenetrable-wedges-at-skew-4dwh8illx4.pdf

Wiener-Hopf Solution for Impenetrable Wedges at Skew Incidence

[1] A general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the paper.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2007RS003673

Fredholm factorization of Wiener‐Hopf scalar and matrix kernels

The diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized Wiener-Hopf equations, and the solution is obtained using analytical and numerical-analytical approaches that reduce the Wiener-Hopf factorization to Fredholm integral equations of second kind. Mathematical aspects are described in a unified and consistent theory for angular region problems. The formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique. The solutions consider engineering applications in terms of GTD/UTD diffraction coefficients and total fields.

Guido Lombardi

Papers

Singular higher order complete vector bases for finite methods

Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions

Wiener-Hopf Solution for Impenetrable Wedges at Skew Incidence

Fredholm factorization of Wiener‐Hopf scalar and matrix kernels

The Wiener-Hopf Solution of the Isotropic Penetrable Wedge Problem: Diffraction and Total Field