Numerical Methods for Engineering: An introduction using MATLAB® and computational electromagnetics examples (Electromagnetic Waves)

✍ Scribed by Karl F. Warnick

• Publisher: Scitech Publishing
• Year: 2020
• Tongue: English
• Leaves: 377
• Edition: 2
• Category: Library

✦ Synopsis

The revised and updated second edition of this textbook teaches students to create modeling codes used to analyze, design, and optimize structures and systems used in wireless communications, microwave circuits, and other applications of electromagnetic fields and waves. Worked code examples are provided for key algorithms using the MATLAB technical computing language.

The book begins by introducing the field of numerical analysis and providing an overview of the fundamentals of electromagnetic field theory. Further chapters cover basic numerical tasks, finite difference methods, numerical integration, integral equations and the method of moments, solving linear systems, the finite element method, optimization methods, and inverse problems.

Developing and using numerical methods helps students to learn the theory of wave propagation in a concrete, visual, and hands-on way. This book fills the missing space of current textbooks that either lack depth on key topics or treat the topic at a level that is too advanced for undergraduates or first-year graduate students.

Presenting the topic with clear explanations, relevant examples, and problem sets that move from simple algorithms to complex codes with real-world capabilities, this book helps its readers develop the skills required for taking a mathematical prescription for a numerical method and translating it into a working, validated software code, providing a valuable resource for understanding the finite difference method, the method of moments, the finite element method, and other tools used in the RF and wireless industry.

✦ Table of Contents

Cover
Contents
About theAuthor
Preface
1 Introduction
1.1 Scientific Computation, Numerical Analysis, and Engineering
1.2 Computational Electromagnetics
1.2.1 Applications of CEM Tools
1.2.2 Types of CEM Methods
1.2.3 Mesh and Grid Generation
1.3 Accuracy and Efficiency
1.4 Programing Languages
1.5 Writing and Debugging Numerical Codes
References
2 Fundamentals of Electromagnetic Field Theory
2.1 Electromagnetic Field and Source Quantities
2.2 Maxwell's Equations
2.2.1 Constitutive Relations
2.2.2 Impressed and Induced Currents
2.2.3 Magnetic Currents
2.3 Coordinate Systems
2.3.1 Rectangular Coordinates
2.3.2 Cylindrical Coordinates
2.3.3 Spherical Coordinates
2.4 Gradient, Curl, and Divergence
2.5 Laplacian
2.6 Wave Propagation
2.7 Electromagnetic Boundary Conditions
2.8 Time-and Frequency-Domain Representations
2.9 PlaneWaves
2.9.1 Wave Vector
2.9.2 Position Vector
2.9.3 PlaneWave in Vector Notation
2.9.4 Characteristic Impedance
2.10 Propagating, Standing, and EvanescentWaves
2.11 Bessel Functions
2.11.1 Hankel Functions
2.11.2 CylindricalWaves
2.12 Power and Energy
2.13 InitialValue Problems and BoundaryValue Problems
2.13.1 Modes
2.13.2 1-D, 2-D, and 3-D Boundary Value Problems
2.13.2.1 1-D Problems
2.13.2.2 2-D Problems and the Transverse Electric (TE) and Transverse Magnetic (TM) Polarizations
2.13.2.3 3-D Problems
2.13.3 Radiation and Scattering Problems
2.13.4 Inverse Problems
2.13.5 Green's Functions and Radiation Integrals
2.13.6 Formulating and Solving Boundary Value Problems
2.13.7 The Equivalence Principle
2.14 Other Topics
Problems
References
3 Basic Numerical Tasks
3.1 Introduction to MATLAB Programing
3.1.1 Vectors and Arrays
3.1.1.1 Colon Operator
3.1.1.2 Vector and Array Operations
3.1.2 Working with Plots
3.1.2.1 3-D Plots
3.1.3 Scripts
3.1.3.1 Comments
3.1.3.2 ParameterValues and Plot Commands
3.1.4 Functions
3.1.5 Arguments and Structure Arrays
3.1.6 Speeding Up MATLAB Codes
3.1.7 Other MATLAB Commands
3.2 Numerical Differentiation
3.2.1 Code Example: Central Difference Rule
3.3 Numerical Integration
3.3.1 Code Example: Midpoint Integration Rule
3.4 Interpolation
3.5 Curve Fitting
3.5.1 Code Example: Polynomial Fitting
3.6 Newton's Method
Problems
References
4 Finite Difference Methods
4.1 Basic Components of Finite Difference Solvers
4.1.1 Grid
4.1.2 Stencil
4.1.3 Boundary Conditions
4.1.4 Sources
4.1.5 Solution Method
4.2 Wave Equation: 1-D FDTD Method
4.2.1 1-D Grid
4.2.2 Update Equation for the 1-DWave Equation
4.2.3 Initial Condition
4.2.4 Boundary Conditions for the 1-D FDTD Method
4.2.4.1 First-Order Mur Absorbing Boundary Condition
4.2.5 Hard and Soft Sources
4.2.6 Source Turn-On Functions
4.2.7 Code Example: 1-D FDTD Algorithm
4.2.7.1 Numerical ResultsWith the 1-D FDTD Code
4.2.8 Stability
4.2.9 Accuracy
4.2.9.1 r =1 Case
4.2.9.2 r <1 Case
4.3 Laplace's Equation: 2-D Finite Difference Method
4.3.1 Example: 2-D FD Method on a 4-by-4 Grid
4.3.2 Waveguide Modes
4.3.3 Spectrum of the Laplacian
4.3.4 Numerical Implementation
4.3.5 2-D FD for Transmission Lines with Dielectric Materials
4.4 2-D Finite Difference Time-Domain (FDTD) Method
4.4.1 2-D EM Problems
4.4.2 Yee Cell and 2-D FDTD Method for TM Polarized Fields
4.4.3 Dielectric and Conductive Materials
4.4.4 Anisotropic Materials
4.4.5 Stability Criterion
4.4.6 Boundary Conditions for the 2-D FDTD Method
4.4.6.1 First-Order Mur Boundary Condition
4.4.6.2 Perfectly Matched Layer
4.4.6.3 UPML for theTMz Polarization
4.4.7 Preprocessing
4.5 FDTD Modeling for Scattering Problems
4.5.1 Incident Field
4.5.2 Scattered Field
4.5.3 Scattered Field Formulation
4.5.3.1 PEC Scatterer
4.5.3.2 Dielectric Scatterer
4.5.4 Scattering Amplitudes, ScatteringWidth, and Radar Cross Section
4.5.5 Bistatic and Monostatic Scattering
4.6 Postprocessing the FDTD Solution
4.6.1 Frequency-Domain (Phasor) Fields
4.6.1.1 Two Equation, Two Unknown Method
4.6.1.2 Fourier Transform Method
4.6.2 Near Field to Far Field Transformation
4.6.2.1 2-D Green's Function
4.6.2.2 2-D Radiation Integral
4.6.2.3 Far-Field Radiation Integral
4.6.2.4 Implementation of the Near-to-Far Transformation
4.6.3 Other Types of Postprocessing
4.6.3.1 Antenna Parameters
4.6.3.2 Impedances
4.7 CodeVerification
4.7.1 Scattering by a Circular Cylinder
4.8 3-D FDTD Method
4.8.1 PEC Cavity
4.8.2 Stability Criterion
4.9 Summary
Problems
References
5 Numerical Integration
5.1 Types of Integration Rules
5.2 Newton–Cotes Quadrature Rules
5.2.1 Error Analysis of the Midpoint Rule
5.2.1.1 Code Example: Midpoint Rule Error as a Function of Integrand Frequency
5.2.2 Higher Order Newton–Cotes Rules
5.2.3 Extended Newton–Cotes Rules
5.2.3.1 Extended Midpoint Rule
5.2.3.2 Extended Trapezoidal Rule
5.2.3.3 Extended Simpson's Rule
5.2.3.4 Refining Extended Rules
5.2.4 Romberg Integration
5.3 Gaussian Quadrature
5.3.1 Orthogonal Polynomials and Gaussian Quadrature
5.3.2 Gauss–Legendre Quadrature (w(x)
5.4 Nonclassical Gaussian Quadrature Rules
5.4.1 Lanczos Algorithm
5.4.2 Weights and Nodes
5.5 Implementation
5.6 Other Methods for Singular Integrands
5.6.1 Singularity Subtraction
5.6.2 Transformation Rules
5.6.2.1 Code Example: Integration Using a Transformation Rule
5.7 Multidimensional Integrals
5.7.1 Monte Carlo Integration
5.8 MATLAB's Built-in Numerical Integration Functions
Problems
References
6 Integral Equations and the Method of Moments
6.1 Integral Operators
6.1.1 First and Second Kind Integral Equations
6.1.2 Solving Integral Equations Numerically
6.1.3 Smooth Kernels and Operator Conditioning
6.1.4 Singular Kernels and Conditioning for Non-Fredholm Operators
6.2 Integral Equations in Electromagnetics
6.2.1 Electric Field Integral Equation, 2-D Transverse Magnetic Polarization (TM-EFIE)
6.2.2 Electric Field Integral Equation, 2-D Transverse Electric Polarization (TE-EFIE)
6.2.3 Magnetic Field Integral Equation
6.2.4 Combined Field Integral Equation
6.2.5 Thin-Wire Integral Equations
6.2.5.1 Pocklington's Integral Equation
6.2.5.2 Hallén's Integral Equation
6.3 Method ofWeighted Residuals
6.3.1 Basis Functions
6.3.2 MoM Implementation
6.3.3 Mesh Types
6.4 Method of Moments for the TM-EFIE
6.4.1 1-D Mesh Generation for Simple Geometries
6.4.2 Path Integrals
6.4.3 Testing Integration
6.4.4 Source Integration
6.4.5 Incident Field
6.4.6 Physical Interpretation of the Method of Moments
6.4.7 Mesh Element Density
6.4.8 MoM Code Overview
6.4.9 1-D Mesh Generation for Complex Geometries
6.4.9.1 Polyarc Representation
6.4.10 Postprocessing
6.4.11 Scattering Amplitudes
6.4.12 MoM Implementation
6.4.12.1 Numerical Results
6.5 Accuracy and Efficiency of the Method of Moments
6.5.1 Computational Cost
6.5.2 Error Analysis
6.5.2.1 Norms
6.5.2.2 Error Measures
6.5.3 Sources of Error
6.5.3.1 Physical Problem
6.5.3.2 Mesh Representation
6.5.3.3 MoM Implementation
6.5.3.4 Current Solution Error
6.5.3.5 Suboptimal MoM Implementations
6.5.3.6 Scattering Amplitude Error
6.6 Dielectric Structures
6.6.1 2-D Volume Method of Moments
6.6.1.1 Numerical Example
6.7 2.5-Dimensional Methods
6.8 3-D Electric Field Integral Equation
6.8.1 Rooftop Functions
6.8.2 Rao–Wilton–Glisson (RWG) Basis
6.8.3 Method of Moments
6.8.3.1 Duffy's Transform
6.8.3.2 Incident Field
6.8.3.3 Postprocessing
Problems
References
7 Solving Linear Systems
7.1 Linear Spaces and Linear Operators
7.1.1 Linear Spaces
7.1.2 Norms on Linear Spaces
7.1.3 Linear Operators
7.1.4 Operator Norms
7.1.5 Range, Null Space, and Rank
7.1.6 Operator Inverse and Adjoint
7.1.7 Classes of Operators
7.1.8 Eigenvalues and Eigenvectors
7.1.9 Field of Values
7.1.10 Matrix Decompositions
7.1.10.1 Unitary Diagonalization
7.1.10.2 Other Matrix Decompositions
7.1.10.3 Decomposition Using Projection Operators
7.1.11 Other Matrix Formulas
7.2 Direct and Iterative Solution Methods
7.3 LU Decomposition
7.4 Iterative Methods
7.4.1 Stationary Iterations
7.4.1.1 Diagonal Dominance
7.4.2 Implementation of Iterative Algorithms
7.4.2.1 Sparse Matrices
7.5 Krylov Subspace Iterations
7.5.1 Conjugate Gradient Method
7.5.2 Residual Error
7.5.3 Condition Number
7.5.4 CGNE Algorithm
7.5.5 Other Krylov Subspace Methods
7.5.6 Convergence of Krylov Subspace Iterations
7.5.6.1 Examples
7.5.6.2 Other Factors Affecting Convergence Rates
7.5.7 Preconditioners
7.6 Multiscale Problems
7.6.1 Fast Algorithms
7.6.2 Reduced-Order Representations
Problems
References
8 Finite Element Method
8.1 Variational Principles in Mathematical Physics
8.1.1 Operators and Functionals
8.1.2 Variational Principles
8.1.3 Variational Calculus
8.1.4 Euler–Lagrange Equation
8.1.5 Variational Principles for PDEs
8.1.6 Variational Principles for Self-Adjoint, Positive Definite Operators
8.1.7 Functionals in Mathematical Physics
8.1.8 Rayleigh–Ritz Method
8.2 Overview of the Finite Element Method
8.2.1 Mesh Types and Mesh Generation
8.2.2 Basis Functions
8.2.3 Variational Principle and Rayleigh–Ritz Procedure
8.2.4 Linear System Solution
8.3 Laplace's Equation: 1-D FEM
8.3.1 Functional Form of the Generalized Laplace Equation
8.3.2 Mesh Representation
8.3.3 Rayleigh–Ritz Procedure
8.3.4 Element Stiffness Matrix
8.3.5 Basis Functions and Shape Functions
8.3.6 Evaluating the Element Stiffness Matrix
8.3.7 Assembly of the Global Stiffness Matrix
8.3.8 Example: Five-Element Mesh
8.3.9 Comparison of FEM and FD
8.3.10 Sparse Matrix and Dense Matrix Methods
8.4 Helmholtz Equation: 2-D FEM
8.4.1 Boundary Conditions for FEM
8.4.1.1 Boundary Terms in the Functional
8.4.2 Rayleigh–Ritz Method for the Helmholtz Functional
8.4.3 Eigenvalue Problems (Unknown k)
8.4.4 Scattering Problems (Known k)
8.4.5 FEM Formulation of the 2-D Scattering Problem
8.4.6 Triangular Mesh
8.4.7 Basis Functions and Shape Functions
8.4.8 Evaluating Element Matrices
8.4.9 Matrix Assembly
8.5 Finite Element Method–Boundary Element Method
8.5.1 Boundary Element Method
8.5.1.1 Extinction Theorem
8.5.1.2 FEM–BEM Linear System
8.5.2 Implementation
8.6 Numerical Results
Problems
References
9 Optimization Methods
9.1 Introduction
9.1.1 Optimization Problems
9.1.2 Local and Global Optimization
9.2 Classes of Optimization Problems
9.2.1 Convex Optimization
9.2.2 Types of Optimization Algorithms
9.2.3 Common Optimization Algorithms
9.2.4 Gradient and Gradient-Free Methods
9.3 One-Dimensional Optimization
9.3.1 Golden Section Search
9.3.2 Tolerance Parameter
9.3.3 Inverse Quadratic Interpolation
9.3.4 Brent's Method
9.4 Nelder–Mead Simplex Method
9.4.1 Initial Simplex
9.4.2 Simplex Transformations
9.4.3 Termination
9.4.4 Implementation Details
9.4.5 Numerical Example
9.5 Practical Considerations for Optimization
Problems
References
10 Inverse Problems
10.1 Introduction
10.2 Types of Inverse Problems
10.2.1 Inverse Scattering
10.2.2 Imaging
10.2.3 Inverse Source Problems
10.2.4 Design Synthesis
10.2.5 Applications
10.3 Ill-Posed Problems
10.3.1 Regularization
10.3.2 Resolution
10.3.3 Types of Inverse Scattering Methods
10.3.3.1 Optimization-Based Inverse Scattering Methods
10.3.3.2 Approximation-Based Methods
10.3.3.3 Other Inverse Scattering Methods
10.4 BornApproximation Methods
10.4.1 Iterated Born Approximation
10.4.2 Diffraction Tomography
10.4.3 Holographic Backpropagation Tomography
10.4.4 Simulating Forward Scattering Data
10.4.5 Implementing the Holographic Backpropagation Tomography Method
10.4.6 Numerical Examples of Holographic Backpropagation Tomography
10.5 Regularized Sampling
10.5.1 Discretization of the Regularized Sampling Equation
10.5.2 Implementing the Regularized Sampling Method
10.5.3 Numerical Examples of the Regularized Sampling Method
Problems
References
Index
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