Deterministic and Stochastic Modeling in Computational Electromagnetics Integral and Differential Equation Approaches

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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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