Numerical Methods in Electromagnetics (eBook)

Special Volume

E.J.W. TER MATEN, W.H.A. SCHILDERS (Autoren)

eBook Download: EPUB

2005 | 1. Auflage
928 Seiten
Elsevier Science (Verlag)
978-0-08-045915-8 (ISBN)

Infos

This special volume provides a broad overview and insight in the way numerical methods are being used to solve the wide variety of problems in the electronics industry. Furthermore its aim is to give researchers from other fields of application the opportunity to benefit from the results wich have been obtained in the electronics industry.

* Complete survey of numerical methods used in the electronic industry
* Each chapter is selfcontained
* Presents state-of-the-art applications and methods
* Internationally recognised authors

Inhalt

front cover 
1 
copyright 
6 
front matter 
7 
General Preface 7
Preface 9
Contents of Volume XIII 13
Contents of the Handbook 14
Introduction to Electromagnetism 21
List of symbols 21
Preface 25
The microscopic Maxwell equations 27
Potentials and fields, the Lagrangian 31
The macroscopic Maxwell equations 38
Wave guides and transmission lines 62
From macroscopic field theory to electric circuits 67
Gauge conditions 78
The geometry of electrodynamics 86
Outlook 101
Acknowledgements 113
Integral theorems 113
Vector identities 118
References 119
body 
123 
Discretization of Electromagnetic Problems: The ``Generalized Finite Differences'' Approach 123
Chapter I: Preliminatires: Euclidean Space 
127 
Chapter II: Rewriting the Maxwell Equations 
145 
Chapter III: Discretizing 
165 
Chapter IV: Finite Elements 
185 
References 1 211
Finite-Difference Time-Domain Methods 217
Introduction 2 217
Maxwell's equations 223
The Yee algorithm 226
Nonuniform Yee grid 240
Alternative finite-difference grids 244
Numerical dispersion 249
Algorithms for improved numerical dispersion 268
Numerical stability 279
Introduction 279
Complex-frequency analysis 279
Stable range: 0< =xi<
Unstable range: 1 < xi<
Example of calculating a stability bound: 3D cubic-cell lattice 281
Courant factor normalization and extension to 2D and 1D grids 282
Examples of calculations involving numerical instability in a 1D grid 283
Example of calculation involving numerical instability in a 2D grid 285
Linear instability when the normalized Courant factor equals 1 287
Generalized stability problem 287
Boundary conditions 288
Variable and unstructured meshing 288
Lossy, dispersive, nonlinear, and gain materials 288
Alternating-direction-implicit time-stepping algorithm for operation beyond the Courant limit 289
Introduction 289
Formulation of the Zheng et al. algorithm 290
Unsimplified system of time-stepping equations 290
Subiteration 1 290
Subiteration 2 291
Simplified system of time-stepping equations 292
Subiteration 1 292
Subiteration 2 293
Proof of numerical stability 294
Numerical dispersion 295
Additional accuracy limitations and their implications 295
Perfectly matched layer absorbing boundary conditions 296
Introduction to absorbing boundary conditions 296
Introduction to impedance-matched absorbing layers 297
Berenger's perfectly matched layer 297
Two-dimensional TEz case 297
Field-splitting modification of Maxwell's equations 297
Plane-wave solution within the Berenger medium 299
Reflectionless matching condition 300
Structure of an FDTD grid employing Berenger's PML ABC 300
Two-dimensional TMz case 301
Three-dimensional case 301
Stretched-coordinate formulation of Berenger's PML 302
An anisotropic PML absorbing medium 304
Perfectly matched uniaxial medium 304
Relationship to Berenger's split-field PML 307
A generalized three-dimensional formulation 307
Inhomogeneous media 309
Theoretical performance of the PML 310
The continuous space 310
The discrete space 310
Grading of the PML loss parameters 310
Polynomial grading 311
Geometric grading 311
Discretization error 312
Efficient implementation of UPML in FDTD 312
Derivation of the finite-difference expressions 313
Computer implementation of the UPML 315
Numerical experiments with Berenger's split-field PML 317
Outgoing cylindrical wave in a two-dimensional open-region grid 317
Outgoing spherical wave in a three-dimensional open-region lattice 318
Dispersive wave propagation in metal waveguides 320
Dispersive and multimode wave propagation in dielectric waveguides 321
Numerical experiments with UPML 323
Current source radiating in an unbounded two-dimensional region 323
Highly elongated domains 326
Microstrip transmission line 328
UPML terminations for conductive and dispersive media 329
Summary and conclusions 330
References 331
Discretization of Semiconductor Device Problems (I) 335
Fluid models for transport in semiconductors 335
A nonlinear block iterative solution of the semiconductor device equations: the Gummel map 347
Mixed formulation of second order elliptic problems 353
Application to continuity equations 397
Other approaches 405
Discretization schemes for Energy-Transport and Energy-Balance models 422
Numerical results 439
References 3 
453 
Discretization of Semiconductor Device Problems (II) 461
Introduction 4 
461 
Introduction to the hydrodynamical models of silicon semiconductors 464
Numerical methods 482
Applications to 1D problems 498
Application of adaptive mesh refinement 518
References 4 
536 
Modelling and Discretization of Circuit Problems 541
Preface 5 
545 
DAE-Systems - the Modelling Aspect 547
DAE-index - the Structural Aspect 563
Numerical Integration Schemes 581
Numerical Treatment of Large Problems 603
Periodic Steady-State Problems 635
References 5 
667 
Simulation of EMC Behaviour 679
Introduction 6 
679 
Derivation of Kirchhoff equations 680
Interaction integrals 691
Numerical integration 692
Analytical integration 705
Regularisations 711
Taylor expansion 724
Analytical integration of the inner integrals for vector valued basis functions 733
Analytical integration of integrals over a triangle 745
Solution of Kirchhoff's equations 748
Linear algebra 754
Matrix condensation 759
Boundary singularities 762
Basis functions 764
Legendre polynomials 767
Inner products 769
References 6 
771 
Solution of Linear Systems 773
What to expect in this chapter? 773
Direct solution method 774
Iterative solution methods 798
Preconditioning 827
Example 7 834
References 7 
836 
Reduced-Order Modeling 843
Introduction to the problem of model reduction 843
Time-invariant linear dynamical systems 846
Krylov-subspace techniques 856
Schur interpolation 870
Hankel-norm model reduction 882
Second-order linear dynamical systems 900
Semi-second-order dynamical systems 907
Concluding remarks 8 
909 
References 8 910
index 
915 

Textauszug

Introduction to Electromagnetism

Wim Magnus

IMEC, Silicon Process and Device Technology Division (SPDT), Quantum Device Modeling Group (QDM), Kapeldreef 75, Flanders, B-3001 Leuven, Belgium

E-mail address: [e-mail]

Wim Schoenmaker

MAGWEL N.V., Kapeldreef 75, B-3001 Leuven, Belgium

E-mail address: [e-mail]

Publisher Summary

This chapter provides an overview on electromagnetism. Electromagnetism, formulated in terms of the Maxwell equations and quantum mechanics, formulated in terms of the Schrödinger equation, constitutes the physical laws by which the bulk of natural experiences are described. Apart from the gravitational forces, nuclear forces and weak decay processes, the description of the physical facts starts with these underlying microscopic theories. The ambition of physicists, chemists and engineers, to provide tools for performing calculations, does not only boost progress in technology but also has a strong impact on the formulation of the equations that represent the physics knowledge and hence provides a deeper understanding of the underlying physics laws. With the advent of powerful computer resources, it has become feasible to extract information from these basic laws with unprecedented accuracy. In particular, the complexity of realistic systems manifests itself in the non-trivial boundary conditions, such that without computers, reliable calculations are beyond reach.

List of symbols

1 Preface

Electromagnetism, formulated in terms of the Maxwell equations, and quantum mechanics, formulated in terms of the Schrödinger equation, constitute the physical laws by which the bulk of natural experiences are described. Apart from the gravitational forces, nuclear forces and weak decay processes, the description of the physical facts starts with these underlying microscopic theories. However, knowledge of these basic laws is only the beginning of the process to apply these laws in realistic circumstances and to determine their quantitative consequences. With the advent of powerful computer resources, it has become feasible to extract information from these basic laws with unprecedented accuracy. In particular, the complexity of realistic systems manifests itself in the non-trivial boundary conditions, such that without computers, reliable calculation are beyond reach.

The ambition of physicists, chemists and engineers, to provide tools for performing calculations, does not only boost progress in technology but also has a strong impact on the formulation of the equations that represent the physics knowledge and hence provides a deeper understanding of the underlying physics laws. As such, computational physics has become a cornerstone of theoretical physics and we may say that without a computational recipe, a physics law is void or at least incomplete. Contrary to what is sometimes claimed, that after having found the unifying theory for gravitation and quantum theory, there is nothing left to investigate, we believe that physics has just started to flourish and there are wide fields of research waiting for exploration.

This volume is dedicated to the study of electrodynamic problems. The Maxwell equations appear in the form

     (1.1)

where Δ describes the near-by field variable correlation of the field that is induced by a source or field disturbance. Near-by correlations can be mathematically expressed by differential operators that probe changes going from one location the a neighboring one. It should be emphasized that “near-by” refers to space and time.

One could “easily” solve these equations by construction the inverse of the differential operator. Such an inverse is usually known as a Green function.

There are two main reasons that prevent a straightforward solution of the Maxwell equations. First of all, realistic structure boundaries may be very irregular, and therefore the corresponding boundary conditions cannot be implemented analytically. Secondly, the sources themselves may depend on the values of the fields and will turn the problem in a highly non-linear one, as may be seen from Eq. (1.1) that should be read as

     (1.2)

The bulk of this volume is dedicated to find solutions to equations of this kind. In particular, Chapters II, III, IV and V are dealing with above type of equations. A considerable amount of work deals with obtaining the details of the right-hand side of Eq. (1.2), namely how the source terms, being charges and currents depend in detail on the values of the field variables.

Whereas, the microscopic equation describe the physical processes in great detail, i.e., at every space–time point field and source variables are declared, it may be profitable to collect a whole bunch of these variables into a single basket and to declare for each basket a few representative variables as the appropriate values for the fields and the sources. This kind of reduction of parameters is the underlying strategy of circuit modeling. Here, the Maxwell equations are replaced by Kirchhoff's network equations. This is the starting point for Chapter VI.

The “basket” containing a large collection of fundamental degrees of freedom of field and source variables, should not be filled at random. Physical intuition suggests that we put together in one basket degrees of freedom that are “alike”. Field and source variables at near-by points are candidates for being grabbed together, since physical continuity implies that a all elements in the basket will have similar values.1

The baskets are not only useful for simplifying the continuous equations. They are vital to the discretization schemes. Since any computer has only a finite memory storage, the continuous or infinite collection of degrees of freedom must be mapped onto a finite subset. This may be accomplished by appropriately positioning and sizing of all the baskets. This procedure is named “grid generation” and the construction of a good grid is often of great importance to obtain accurate solutions.

After having mapped the continuous problem onto a finite grid one may establish a set of algebraic equations connecting the grid variables (basket representatives) and explicitly reflecting the non-linearity of the original differential equations. The solution of large systems of non-linear algebraic equations is based on Newton's iterative method. To find the solution of the set of non-linear equations F(x)=0, an initial guess is made: x=xinit=x0. Next the guess is (hopefully) improved by looking at the equation:

     (1.3)

where the matrix A is

     (1.4)

In particular, by assuming that the correction brings us close to the solution, i.e., , where , we obtain that

     (1.5)

Next we repeat this procedure, until convergence is reached. A series of vectors, xinit=x0,x1,x2,…,xn−1,xn=xfinal, is generated, such that |F(xfinal)|<ε, where ε is some prescribed error criterion. In each iteration a large linear matrix problem of the type A|x〉=|b〉 needs to be solved.

2 The microscopic Maxwell equations

2.1 The microscopic Maxwell equations in integral and differential form

In general, any electromagnetic field can be described and characterized on a microscopic scale by two vector fields E(r,t) and B(r,t) specifying respectively the electric field and the magnetic induction in an arbitrary space point r at an arbitrary time t. All dynamical features of these vector fields are contained in the well-known Maxwell equations (Maxwell [1954a], Maxwell [1954b], Jackson [1975], Feynman, Leighton and Sands [1964a])

     (2.1)

     (2.2)

     (2.3)

     (2.4)

They describe the spatial and temporal behavior of the electromagnetic field vectors and relate them to the sources of electric charge and current that may be present in the region of interest. Within the framework of a microscopic description, the electric charge density ρ and the electric current density J are considered spatially localized distributions residing in vacuum. As such they represent both mobile charges giving rise to macroscopic currents in solid-state devices, chemical solutions, plasmas, etc., and bound charges that are confined to the region of an atomic nucleus. In turn, the Maxwell equations in the above presented form explicitly refer to the values taken by E and B in...

Produkt-Details

• Erscheint lt. Verlag: 4.4.2005
• Mitarbeit: Herausgeber (Serie): Philippe G. Ciarlet
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-045915-3 / 0080459153
• ISBN-13: 978-0-08-045915-8 / 9780080459158