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

The application of deterministic-stochastic models in some areas of computational electromagnetics is presented. Namely, in certain problems there is an uncertainty in the input data set as some system properties are partly or entirely unknown. Thus, a simple stochastic collocation (SC) method is used to determine the relevant statistics about the given responses. The SC approach also provides the assessment of the related confidence intervals in the set of calculated numerical results. The expansion of statistical output in terms of mean and variance over a polynomial basis, via SC method, is shown to be robust and efficient approach providing a satisfactory convergence rate. The presented stochastic framework provides means for sensitivity analysis enabling a better insight into the relationship between the input parameters and the output of interest. This chapter provides certain computational examples from the previous work by the authors illustrating successful application of SC technique in the areas of: human exposure to electromagnetic fields, transcranial magnetic stimulation (TMS), transient analysis of buried wires and design of instrumental landing system (ILS).

On the Various Applications of Stochastic Collocation in Computational Electromagnetics

This paper deals with thermal analysis of realistic models of the human eye and brain using the finite element method. The research presented in this paper is the sequel to the electromagnetic dosimetry model presented in the previous work by the authors. The paper presents the numerical results for the specific absorption rate (SAR) and the related temperature increase in various models of the human eye and the brain/head exposed to high-frequency (HF) electromagnetic (EM) radiation. Based on the numerical results for the induced electric field, the distribution of SAR in the human brain and human eye is determined, subsequently used as input to the thermal model. The thermal dosimetry model of both the brain and eye are based on the form of Pennes’ bioheat transfer equation, numerically solved using the finite element method (FEM). The comparison between the extracted models and the compound models of both the eye and brain, placed inside the realistic head model is presented. In case of the human eye, generally, comparable results were obtained for both SAR and temperature increase, while the compound eye model is found to be more suitable when the polarization of incident wave is considered. Moreover, the extracted eye model underestimated the temperature rise, attributed to better heat exchange than the compound model. The results for the compound eye indicate that in some situations, the eye lens could be omitted from simulation, facilitating the model preparation. The numerical results for all three brain models showed similar distributions of SAR and temperature rise. Also, the obtained results show that the peak SAR does not exceed the basic restriction limit for localized SAR, for occupational exposure. The thermal dosimetry assessment of the human brain exposed in four considered scenarios indicates the temperature should not exceed 0.1°C. Finally, the use of a geometrically simplified model may also be found useful in the initial dosimetry assessment prior to dealing with models with more anatomical features.

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On the Use of Compound and Extracted Models in Thermal Dosimetry Assessment

The paper deals with the uncertainty quantification of the transient axial current induced along the human body exposed to electromagnetic pulse radiation. The body is modeled as a straight wire antenna whose length and radius exhibit random nature. The uncertainty is propagated to the output transient current by means of the stochastic collocation method. The stochastic approach is entirely nonintrusive and serves as a wrapper around the deterministic code. The numerical deterministic model is based on the time domain Hallen integral equation solved by means of the Galerkin-Bubnov indirect boundary element method (GB-IBEM). The stochastic moments, i.e., the mean and the variance of the transient current, are calculated. Confidence margins are obtained for the whole duration of the transient response as well as for the maximal current value. The presented approach enables the estimation of the probability for the induced current to exceed the basic restrictions prescribed by regulatory bodies. The sensitivity analysis of the input parameters indicates to which extent the variation of the input parameter set influences the output values which is particularly interesting for the design of the human equivalent antenna.

/pdf/uncertainty-quantification-for-the-transient-response-of-3d45im8ivl.pdf

Uncertainty Quantification for the Transient Response of Human Equivalent Antenna Using the Stochastic Collocation Approach

In this paper, human body exposure to high frequency (HF) electromagnetic field is analysed. Small loop antenna as a transmitter and human body at defined distance are analytically and numerically modelled. The analysis is carried out for simplified cylinder and realistic human body model settled in free space and exposed to HF radiation in terms of induced current density and induced electric field inside the body at frequency of 6.78 MHz. Analytical approach involves the use of the approximate sinusoidal current distribution and analytical evaluation of field integrals. The results obtained analytically are validated by numerical simulations in commercial software FEKO, based on Method of Moments (MoM). It is shown that there is no significant difference in results while using different approaches (analytical and numerical) which is excellent for theoretical calculations limited on simplified models so both approaches can be used for human body exposure estimation in defined scenarios.

A Simplified Analytical Model for Human Exposure to Electromagnetic Radiation of HF Wireless Power Transmitter

This chapter deals with realistic models of the body exposed to static and low frequency (LF) fields. The human head models exposed to static electric field are handled via finite element method (FEM) and boundary element method (BEM). Whole body exposure to LF fields is analyzed by using the quasi-static formulation and BEM. Realistic body models with organs included are presented. Finally, pregnant woman/foetus exposed to power line electric field is analyzed.

Dragan Poljak

Papers

On the Various Applications of Stochastic Collocation in Computational Electromagnetics

On the Use of Compound and Extracted Models in Thermal Dosimetry Assessment

Uncertainty Quantification for the Transient Response of Human Equivalent Antenna Using the Stochastic Collocation Approach

A Simplified Analytical Model for Human Exposure to Electromagnetic Radiation of HF Wireless Power Transmitter

Realistic Models for Static and Low Frequency (LF) Dosimetry