There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The electromagnetic fields within a detailed model of the human eye and its surrounding bony orbit are calculated for two different frequencies of plane-wave irradiation: 750 MHz and 1.5 GHz. The computation is performed with a finite-difference algorithm for the time-dependent Maxwell's equations, carried out to the sinusoidal steady state. The heating potential, derived from the square of the electric field, is used to calculate the temperatures induced within the eyeball of the model. This computation is performed with the implicit alternating-direction (IAD) algorithm for the heat conduction equation. Using an order-of-magnitude estimate of the heat-sinking capacity of the retinal blood supply, it is determined that a hot spot exceeding 40.4/spl deg/C occurs at the center of the model eyeball at an incident power level of 100 mW/cm/sup 2/ at 1.5 GHz.

Computation of the Electromagnetic Fields and Induced Temperatures Within a Model of the Microwave-Irradiated Human Eye

Transcranial magnetic stimulation in basic and clinical neuroscience: A comprehensive review of fundamental principles and novel insights.

We discuss the transmission line (TL) theory and its application to the problem of lightning electromagnetic field coupling to TLs. We start with the derivation of the general field-to-TL coupling equations for the case of a single-wire line above a perfectly conducting ground. The derived equations are solely based on the thin-wire approximation and they do take into account high-frequency radiation effects. Under the TL approximation, the general equations reduce to the Agrawal et al. field-to-TL coupling equations. After a short discussion on the underlying assumptions of the TL theory, three seemingly different but completely equivalent approaches that have been proposed to describe the coupling of electromagnetic fields to TLs are presented. The derived equations are then extended to deal with the presence of losses and multiple conductors and expressions for the line parameters, including the ground impedance and admittance, are presented. The time-domain representation of the field-to-TL coupling equations, which allows for a straightforward treatment of nonlinear phenomena as well as the variation in the line topology, is also described. Solution methods in the frequency domain and in the time domain are given and application examples with reference to lightning-induced voltages are presented and discussed. Specifically, the effects of ground losses and corona are illustrated and discussed. When the traveling voltage and current waves are originated from lumped excitation sources located at a specific location along a TL (direct lightning strike), both the corona phenomenon and ground losses result, in general, in an attenuation and dispersion of propagating surges along TLs. However, when distributed sources representing the action of the electromagnetic field from a nearby lightning illuminating the line are present, ground losses and corona phenomenon could result in important enhancement of the induced voltage magnitude.

A Review of Field-to-Transmission Line Coupling Models With Special Emphasis to Lightning-Induced Voltages on Overhead Lines

This chapter deals with fundamental concepts in electromagnetic theory and outlines some basics of numerical modeling. Thus, the chapter starts with Maxwell equations, continuity equation and Poynting theorem. Then, electromagnetic wave equations and potentials are derived, and finally, fundamentals of radiation are presented. The first part of the chapter ends with both analytical analysis of dipole antenna and Pocklington integro-differential equation approach. The second part of the chapter yields some introductory aspects of numerical methods. A brief description of finite element method (FEM), boundary element method (BEM) and numerical solution of integral equations over sources is given. Some simple computational examples are given, as well.

Theoretical Background: an Outline of Computational Electromagnetics (CEM)

Formulation involving the electric field integral equation (EFIE) and the related numerical solution via Method of Moments (MoM) variants requires tedious calculation of various double surface integrals arising from the use of vector triangular basis functions. The approach to evaluate such an integral strongly depends on the distance between source and observation points, respectively. In the case of a rather close interaction between the elements, due to a strong singularity, a purely numerical approach could fail resulting in the incorrect, spurious results. Hence, a combined analytical-numerical approach to extract the mentioned singularities is necessary. The present paper deals with an implementation of such an approach to the calculation of certain integrals types arising in the EFIE formulation for electromagnetic scattering from dielectric objects.

Some computational aspects of calculation of integrals arising within the framework of Method of Moments - application to bioelectromagnetics

This work is devoted to the impact of random environmental constraints regarding the transient impedance of horizontal grounding wire. Deterministic modeling relies on solving the Pocklington equation in the frequency domain (antenna theory). First, the method is based upon the scattered voltage analytically computed at the input terminal of the grounding electrode; transient impedance is then extracted from Inverse Fast Fourier Transform. Finally, a methodology is proposed to precisely assess the respective influence of different parameters jointly by means of stochastic collocation method and sensitivity analysis.

Statistical transient impedance of horizontal grounding systems: Application to sensitivity analysis with ANOVA approaches

The analysis of horizontal grounding electrode has been carried out using the antenna theory (AT) approach in the frequency and time domain , respectively. The formulation is based on the corresponding space-frequency and space-time Pocklington integro-differential equations. The integro-differential relationships are numerically handled via the Galerkin–Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM). Some illustrative computational examples related to frequency domain (FD) and time domain (TD) analysis are given in the paper.

Frequency Domain and Time Domain Response of the Horizontal Grounding Electrode Using the Antenna Theory Approach

The paper deals with the assessment of the influence of finite conductivity of the horizontal electrode to related transient response. Transient response of the grounding electrode buried in a lossy half-space is determined via analytical solution of the corresponding Pocklington equation in the frequency domain. Time domain response is obtained by means of Inverse Fast Fourier Transform (IFFT). The electrode is excited via an equivalent current source representing the lightning strike current. Presence of the earth-air interface is taken into account via a simplified reflection coefficient arising from the Modified Image Theory (MIT). The transient current at the center of the electrode is calculated for the case of perfectly conducting (PEC) electrode and for the electrodes made of copper and aluminum. Comparison of the results shows no discrepancy between these electrodes, justifying the use of a PEC electrode approximation.

Dragan Poljak

Papers

Theoretical Background: an Outline of Computational Electromagnetics (CEM)

Some computational aspects of calculation of integrals arising within the framework of Method of Moments - application to bioelectromagnetics

Statistical transient impedance of horizontal grounding systems: Application to sensitivity analysis with ANOVA approaches

Frequency Domain and Time Domain Response of the Horizontal Grounding Electrode Using the Antenna Theory Approach

Influence of the grounding electrode conductivity on the induced transient current