Thursday,
September 3, 1998, 3:05 p.m. in PSA Room 104
Ali R. Baghai-Wadji
Vienna University of Technology, on leave with Motorola Inc., Scottsdale
Data Recycling in the Boundary Element Method
Abstract
Using Green's second integral theorem and appropriately
constructed Green's functions (GFs), a boundary value problem
formulated in terms of a system of partial differential
equations can be transformed into an equivalent system of
singular surface-integral equations. The discretization of
problems's boundary surface followed by the minimization
of weighted residuals leads to a finite dimensional matrix
equation Ax=b, or Ax=lambda x which has to be solved
numerically. In nearly all the applications a real-domain
representation of problems' GFs are used. The various
elements of A, i.e. (aij), are Fourier-type oscillating
integrals which are generally geometry, material, and frequency
dependent, and therefore, by varying any of these parameters, the
calculations must be repeated. Furthermore, the diagonal elements
(aii) are singular, and the near-diagonal elements, while being
theoretically regular, are "numerically" divergent or very
slowly convergent, and thus often the required accuracy must
be compromised. Along the wedges and at the corners of the
boundary the convergence problems are even more severe. These
drawbacks have prevented the "Boundary Element Method (BEM)"
from establishing itself in the same line as the other popular
techniques, such as the finite element, or the finite difference
time domain method.
In an effort to remove these shortcomings the speaker has
developed the Fast-MoM (Fast Method-of-Moments). Fast-MoM
is a technique for a systematic regularization of divergent
integrals and allows the generation of geometry, material and
frequency independent universal functions which can be used
repeatedly, once the class of problems is known (data recycling).
The Fast-MoM consists of the following steps which will be
discussed:
i) A constructive procedure for the diagonalization of PDEs,
along with a physical interpretation of the diagonalization
ii) Transformation into the spectral domain (SD)
iii) Calculation of the eigenvalues, eigenvectors and their
derivatives in the farfield in the SDiv) Construction of Green's functions (GFs): Two representations
of GFs will be discussed. The underlying theory allows to
convert oscillatory integrals into decaying ones and vice
versa. The discussion will be complemented by a third
representation of the GFs.
v) A unified multi-step procedure for the calculation of aij
(i not equal j) (mutual interaction elements) and aii
(self-action elements) will be presented. The procedure involves
a way for the regularization of divergent integrals and relies
on the distribution theory. Removable-, hyper-, and essential-
singularities will be treated, keeping in mind the numerical
implementation. Thereby, functions in Hardy space will be
transformed into the "band-limited" functions which allow,
reliably, achievement of virtually any desired accuracy using public
domain quadratures. It will be shown that both aij and aii can be
calculated from a set of universal functions (typically four).
Numerical results will include the solution of the Poisson equation
(electrostatic, magnetostatic), Helmholz equation (scalar waves)
and various generalizations including vector fields, and the wave
propagation in anisotropic and bi-anisotropic media. The results
will be compared with experimental data.
