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Deterministic computational models are those for which all inputs are precisely known, whereas stochastic modeling reflects uncertainty or randomness in one or more of the data inputs. Many problems in computational engineering therefore require both deterministic and stochastic modeling to be used in parallel, allowing for different degrees of confidence and incorporating datasets of different kinds. In particular, non-intrusive stochastic methods can be easily combined with widely-used deterministic approaches, enabling this more robust form of data analysis to be applied to a range of computational challenges.

Deterministic and Stochastic Modeling in Computational Electromagnetics provides a rare treatment of parallel deterministic-stochastic computational modeling and its beneficial applications. Unlike other works of its kind, which generally treat deterministic and stochastic modeling in isolation from one another, it aims to demonstrate the usefulness of a combined approach and present particular use-cases in which such an approach is clearly required. It offers a non-intrusive stochastic approach which can be incorporated with minimal effort into virtually all existing computational models.

Readers will also find:

A range of specific examples demonstrating the efficiency of deterministic-stochastic modeling
Computational examples of successful applications including ground penetrating radars (GPR), radiation from 5G systems, transcranial magnetic and electric stimulation (TMS and TES), and more
Introduction to fundamental principles in field theory to ground the discussion of computational modeling

Deterministic and Stochastic Modeling in Computational Electromagnetics is a valuable reference for researchers, including graduate and undergraduate students, in computational electromagnetics, as well as to multidisciplinary researchers, engineers, physicists, and mathematicians.

Dragan Poljak, Ph.D, is Professor in the Department of Electronics and Computing Technology, University of Split, Croatia. He is a Senior Member of the IEEE and author of three books and more than one hunderd and fifty articles on subjects related to computational electromagnetics.

Anna Šušnjara, Ph.D, is a Postdoctoral Researcher in the Department of Electronics and Computing Technology, University of Split, Croatia. She is a member of the IEEE and has authored or co-authored more than 40 journal and conference papers on subjects related to computational electromagnetics.

1. Least Action Principle in electromagnetics 2

1.1. Hamilton principle 2

1.2. Newton equation of motion from Lagrangian 5

1.3. Noether’s theorem and conservation laws 7

1.4. Equation of continuity from Lagrangian 10

1.5. Lorentz force from Gauge Invariance 14

2. Fundamental Equations of Engineering Electromagnetics 17

2.1. Derivation of two canonical Maxwell equation 17

2.2. Derivation of two dynamical Maxwell equation 18

2.3. Integral form of Maxwell equations, continuity equations and Lorentz force 21

2.4. Phasor form of Maxwell equations 22

2.4. Continuity (interface) conditions 24

2.5. Poynting theorem 25

3. Variational methods in electromagnetics 40

3.1. Analytical methods 40

3.2. Capacity of insulated charged sphere 40

3.3. Spherical Grounding resistance 42

3.4. Variational basis for numerical methods 43

4. Outline of numerical methods 47

4.1. Variational basis for numerical methods 50

4.2. The Finite Element Method (FEM) 51

4.2.1 Basic concepts of FEM – One dimensional FEM 52

4.3.2 Linear and quadratic elements 74

4.3.2 Quadratic elements 75

4.3.4 Numerical solution of integral equations over unknown sources 76

5. Wire Configurations - Frequency Domain Analysis 79

5.1. Single wire in a presence of a lossy half-space 79

5.1.1 Horizontal dipole above a homogeneous lossy half-space 79

5.1.2 Horizontal dipole buried in a homogeneous lossy half-space 84

5.2 Horizontal dipole above a multi-layered lossy half-space 88

5.2.1 Integral equation formulation 88

5.2.2 Radiated field 93

5.2.3 Numerical results 95

5.3 Wire Array above a multilayer 114

5.3.1. Formulation 116

5.3.2 Numerical procedures 118

5.3.3 Computational examples 120

5.4. Wires of arbitrary shape radiating over a layered medium 137

5.4.1. Curved single wire in free space 139

5.4.2. Curved single wire in a presence of a lossy half-space 140

5.4.3. Multiple curved wires 142

5.4.5. Electromagnetic field coupling to arbitrarily shaped aboveground wires 151

5.4.5. Buried wires of arbitrary shape 161

5.5. Complex Power of Arbitrarily Shaped Thin Wire Radiating above a Lossy Half-space 168

5.5.1. Theoretical background 169

5.5.2. Numerical results 172

6. Wire Configurations - Time Domain Analysis 185

6.1 Single Wire above a Lossy Ground 186

6.1.1. Case of perfectly conducting ground (PEC) gound and dielectric half-space 190

6.1.2 Modified reflection coefficient for the case of an imperfect ground 191

6.2 Numerical solution of Hallen equation via Galerkin-Bubnov Indirect Boundary Element Method (GB-IBEM) 199

6.2.1 Computational examples 202

6.3 Application to Ground penetrating Radar (GPR) 205

6. 3.1 Transient Field due to Dipole Radiation Reflected from the Air-Earth Interface 207

6. 3.2 Transient Field Transmitted into a Lossy Ground due to Dipole Radiation 214

6.4 Simplified Calculation of Specific Absorption (SA) in Human Tissue 221

6.4.1 Calculation of specific absorption (SA) 222

6.4.2 Numerical results 223

6.5 Time Domain Energy Measures 229

6.6 Time Domain Analysis of Multiple Straight Wires above a Half-space by means of Various Time Domain Measures 234

6.6.1 Theoretical background 235

6.6.2 Numerical results 237

7. Bioelectromagnetics – Exposure of Humans in GHz Frequency Range 280

7.1 Assessment of Sab in a planar single layer tissue 280

7.1.1 Analysis of Dipole Antenna in Front of Planar Interface 282

7.1.2. Calculation of Absorbed Power Density 285

7.1.3 Computational Examples 285

7.2. Assessment of Transmitted Power Density in a Single Layer Tissue 289

7.2.1 Formulation 290

7.2.2 Results for current distribution 294

8. Multiphysics Phenomena 330

8.1. Electromagnetic-Thermal modeling of the Human Exposure to HF Radiation 330

8.1.1. Electromagnetic Dosimetry 330

8.1.2. Thermal Dosimetry 332

8.1.3. Computational examples 336

8.2. Magnetohydrodynamics (MHD) Models for Plasma Confinement 337

8.2.1. Grad Shafranov Equation 338

8.2.2. Transport Phenomena Modeling 349

8.3. Schrodinger Equation 358

8.3.1 Derivation of Schrördinger equation 359

8.3.2 Analytical solution of Schrördinger equation 360

8.3.3 FDM solution of Schrördinger equation 361

8.3.4 FEM solution of Schrördinger equation 362

8.3.5 Neural netwok approach to the solution of Schrördinger equation 364

9. Methods for stochastic analysis 372

9.1. Uncertainty quantification framework 373

9.1.1. Uncertainty quantification (UQ) of model input parameters 373

9.1.2. Uncertainty propagation (UP) 374

9.1.3. Monte Carlo method 375

9.2. Stochastic collocation method 376

9.2.1. Computation of stochastic moments 377

9.2.2. Interpolation approaches 378

9.2.3. Collocation points selection 379

9.2.4. Multidimensional stochastic problems 379

9.3. Sensitivity analysis 383

9.3.1. “One-at-a-time” (OAT) approach 384

9.3.2. ANalysis Of VAriance (ANOVA) based method 384

10. Stochastic-deterministic electromagnetic dosimetry 389

10.1. Internal stochastic dosimetry for a simple body model exposed to low frequency field 390

10.2. Internal stochastic dosimetry for a simple body model exposed to electromagnetic pulse 393

10.3. Internal stochastic dosimetry for a realistic three-compartment human head exposed to high frequency plane wave 396

10.4. Incident field stochastic dosimetry for base station antenna radiation 401

11. Stochastic-deterministic thermal dosimetry 411

11.1. Stochastic sensitivity analysis of bioheat transfer equation 412

11.2. Stochastic thermal dosimetry for homogeneous human brain 414

11.3. Stochastic thermal dosimetry for three-compartment human head 421

11.4. Stochastic thermal dosimetry below 6 GHz for 5G mobile communication systems 424

12. Stochastic-deterministic modelling in biomedical applications of electromagnetic fields430

12.1. Transcranial Magnetic Stimulation 430

12.2. Transcranial Electric Stimulation 435

12.2.1. Cylinder representation of human head 436

12.2.2. A 3-compartment human head model 438

12.2.3. A 9-compartment human head model 441

12.3. Neuron’s action potential dynamics 447

12.4. Radiation efficiency of implantable antennas 453

13. Stochastic-deterministic modelling of wire configurations in frequency and time domain 1

13.1. Ground penetrating radar 1

13.1.1. The transient current induced along the GPR antenna 2

13.1.2. The transient field transmitted into a lossy soil 5

13.2. Grounding systems 10

13.2.1. Test case #1: soil and lighting pulse parameters are random variables 12

13.2.2. Test case #2: soil and electrode parameters are random variables 13

13.2.3. Test case #3: soil, electrode and lighting pulse parameters are random variables 14

13.3. Air-traffic control systems 17

13.3.1. Runway covered with snow 19

13.3.2. Runway covered with vegetation 21

14. A note on stochastic modelling of plasma physics phenomena 488

14.1. Tokamak current diffusion equation 488

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