MATLAB-based finite element programming in electromagnetic modeling

440 Pages·2019·66.56 MB·English

by  Kuzuoglu

MATLAB®-based Finite
Element Programming in
Electromagnetic Modeling
http://taylorandfrancis.com
MATLAB®-based Finite
Element Programming in
Electromagnetic Modeling
Özlem Özgün
Mustafa Kuzuo lu
ğ
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accu-
racy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not
constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the
MATLAB® software.
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Contents
Preface...............................................................................................................................................ix
Authors ..............................................................................................................................................xi
Chapter 1 Introduction ..............................................................................................................1
1.1 Mathematical and Numerical Modeling ............................................................1
1.2 Numerical Methods in Electromagnetics ..........................................................4
1.3 General Framework of Finite Element Method .................................................6
1.3.1 Finite Elements & Mesh Generation ....................................................7
1.3.2 Finite Element Formulation ...............................................................8
1.3.2.1 Construction of Element Matrix Equations ..........................9
1.3.2.2 Assembly of Global Matrix and Its Sparse Storage ............11
1.3.2.3 Imposition of Boundary Conditions & Mesh Truncation ...13
1.3.2.4 Solution of Global Matrix Equation & Interpretation
of Results ....................................................................................13
References ..............................................................................................................14
Chapter 2 MATLAB Primer .......................................................................................................17
2.1 Getting Started with MATLAB® ........................................................................17
2.1.1 Desktop Environment .........................................................................18
2.1.2 Help Facility ....................................................................................20
2.2 Basics ...............................................................................................................20
2.2.1 Creating Variables ..............................................................................20
2.2.2 Data Types and Numerical Precision .................................................21
2.2.3 Arithmetic Operators and Precedence Rules ......................................23
2.2.4 Built-In Mathematical Functions ...................................................24
2.2.5 Some Useful Commands ....................................................................24
2.3 Matrices and Arrays ........................................................................................25
2.3.1 Defining Matrices and Vectors ...........................................................25
2.3.2 Empty Matrix ..................................................................................26
2.3.3 Accessing Array Elements ...............................................................26
2.3.4 Colon Operator ...................................................................................27
2.3.5 Matrix Dimensions .............................................................................28
2.3.6 Matrix Transpose ......................................................................................28
2.3.7 Matrix Addition and Subtraction ............................................................30
2.3.8 Matrix Multiplication .........................................................................31
2.3.9 Matrix Division and Solution of Linear Matrix Equations ................32
2.3.10 Element-Wise Matrix Operations .......................................................35
2.3.11 Concatenating Matrices ......................................................................36
2.3.12 Elementary Matrices ..........................................................................37
2.4 Programming Scripts and Functions ...............................................................38
2.4.1 Script m-Files .....................................................................................38
2.4.2 Function m-Files.................................................................................39
2.4.3 Logical Statements .............................................................................40
2.4.4 Selection Statements ...........................................................................41
2.4.5 Speeding Up Performance .................................................................41
v
vi Contents
2.5 Input and Output Functions .............................................................................46
2.6 Advanced Mathematics ...............................................................................................48
2.6.1 Linear Algebra.......................................................................................48
2.6.2 Sparse Matrices ..................................................................................49
2.6.3 Iterative Matrix Solvers ......................................................................50
2.6.4 Integration and Differentiation ...........................................................50
2.6.5 Complex Numbers ..............................................................................50
2.6.6 Random Numbers and Statistics.....................................................51
2.6.7 Computational Geometry ...................................................................52
2.7 Plotting and Graphics ........................................................................................52
2.8 Getting More Information ...............................................................................54
References ..............................................................................................................60
Chapter 3 Mathematical Fundamentals of FEM ............................................................................61
3.1 Introduction ...............................................................................................................61
3.2 Mathematical Structures ............................................................................63
3.2.1 Algebraic Structures ...........................................................................65
3.2.2 Topological Structures........................................................................71
3.3 Approximation Theory ....................................................................................84
3.3.1 Approximate Solutions of Linear Operator Equations .......................84
3.3.2 Calculus of Variations and Stationary Points of Functionals .............96
3.4 Basic Concepts of Electromagnetic Theory ..................................................102
3.5 Weak Variational Forms of the BVPs in Electromagnetics ..........................109
3.5.1 Derivation of Weak Form of Scalar PDE .........................................109
3.5.2 Derivation of Weak Form of Vector PDE. .......................................110
References ................................................................................................................111
Chapter 4 One-Dimensional Finite Element Analysis ..............................................................113
4.1 Introduction .............................................................................................................114
4.2 Mesh Generation ................................................................................................116
4.3 Shape Functions .............................................................................................122
4.3.1 Linear Elements ..............................................................................122
4.3.2 Higher-Order Elements ....................................................................128
4.3.2.1 Quadratic Elements...........................................................128
4.3.2.2 Cubic Elements .................................................................131
4.4 Finite Element Formulation ...........................................................................135
4.4.1 Evaluation of Element Matrices ............................................................135
4.4.1.1 Linear Elements ................................................................139
4.4.1.2 Quadratic Elements...........................................................141
4.4.1.3 Cubic Elements .................................................................142
4.4.2 Global Matrix Assembly ......................................................................142
4.4.2.1 Linear Elements ................................................................142
4.4.2.2 Higher-Order Elements .....................................................148
4.4.3 Imposition of Boundary Conditions .................................................149
4.4.3.1 Dirichlet Boundary Conditions ...........................................149
4.4.3.2 Mixed and Neumann Boundary Conditions .....................152
4.4.4 Evaluation and Visualization of the Solution ...................................153
Contents vii
4.5 Electrostatic Analysis ....................................................................................154
4.5.1 Application 1: Calculation of Electrostatic Potential in a
Homogeneous Medium with Arbitrary Charge Density Function ...158
4.5.2 Application 2: Capacitor with Inhomogeneous Medium Having
Spatially-Varying Permittivity Profile ..............................................165
4.6 Time-Harmonic Electromagnetic Analysis ...................................................170
4.6.1 Scattering from a Lossy Dielectric Slab .............................................170
4.6.2 Scattering from a PEC-Backed Dielectric Slab................................180
4.6.3 Electromagnetic Wave Propagation Inside a Parallel-Plate
Waveguide.............................................................................................183
References......................................................................................................................193
Chapter 5 Two-Dimensional Finite Element Analysis ..............................................................195
5.1 Introduction ............................................................................................................196
5.2 Mesh Generation ................................................................................................198
5.3 Shape Functions .............................................................................................205
5.3.1 Linear Triangular Elements ................................................................209
5.3.2 Bilinear Quadrilateral Elements .............................................................216
5.3.3 Higher-Order Elements ........................................................................219
5.3.3.1 6-noded Quadratic Triangular Element ............................219
5.3.3.2 10-noded Cubic Triangular Element ................................221
5.3.3.3 8-noded Quadratic Quadrilateral Element
(Serendipity Element) ............................................................222
5.3.3.4 9-noded Quadratic Quadrilateral Element
(Lagrangian Element) .......................................................222
5.4 Finite Element Formulation ...........................................................................224
5.4.1 Evaluation of Element Matrices ............................................................224
5.4.1.1 Linear Triangular Elements ..............................................229
5.4.1.2 Bilinear Quadrilateral Elements .......................................234
5.4.2 Global Matrix Assembly .....................................................................242
5.4.3 Imposition of Boundary Conditions .................................................243
5.4.3.1 Implementation of Mixed Boundary Conditions
in Triangular Elements ..........................................................243
5.4.3.2 Implementation of Mixed Boundary Conditions
in Quadrilateral Elements .................................................246
5.5 Electrostatic Analysis ......................................................................................248
5.5.1 Parallel-Plate Capacitor Problem with Truncated
Computational Domain .......................................................................249
5.5.2 Parallel-Plate Capacitor Problem with Dielectric Inhomogeneity ....254
5.5.3 Microstrip Transmission Line ...............................................................256
5.6 Time-Harmonic Electromagnetic Analysis ...................................................260
5.6.1 Scattering from a PEC or Dielectric Object .....................................261
5.6.2 Mesh Truncation Methods................................................................270
5.6.2.1 Absorbing Boundary Conditions (ABCs) .........................275
5.6.2.2 Locally-Conformal PML Method (LC-PML) ...................277
5.6.3 Scattering from a PEC Object Coated by an Arbitrary Number
of Dielectric Layers .............................................................................282
5.6.4 Scattering from an Anisotropic Object ............................................282
5.6.5 Scattering from Periodic Structures .................................................283
5.6.6 Scattering from Rough Surfaces ......................................................286
viii Contents
5.6.7 Visualization of Modes in a Parallel-Plate Waveguide or 2-D Horn ...293
5.6.8 Scattering from an Object Inside a Parallel-Plate Waveguide .........296
5.6.9 Radiation from a Line-Source in a Parallel-Plate Waveguide .........300
5.6.10 Analysis of 3-D Waveguides with Uniform Cross-Section ..............305
References ................................................................................................................315
Chapter 6 Three-Dimensional Finite Element Analysis ...........................................................319
6.1 Introduction ............................................................................................................320
6.2 Mesh Generation ...............................................................................................323
6.3 Nodal Elements .................................................................................................328
6.3.1 Tetrahedral Element ............................................................................329
6.3.2 Hexahedral Element .............................................................................333
6.3.3 Triangular Prism Element...................................................................334
6.3.4 Higher-Order Elements ........................................................................335
6.3.4.1 Quadratic Tetrahedral Element .........................................335
6.3.4.2 Quadratic Hexahedral Element ..........................................336
6.3.4.3 Quadratic Triangular Prism Element ................................337
6.4 Edge (Vector) Elements .................................................................................338
6.4.1 Tetrahedral Edge Element ................................................................339
6.4.2 Hexahedral Edge Element ................................................................343
6.5 Finite Element Formulation with Nodal Elements ........................................344
6.5.1 Tetrahedral Element .........................................................................347
6.5.2 Other Elements .................................................................................350
6.6 Finite Element Formulation with Edge Elements..........................................355
6.6.1 Tetrahedral Edge Element ................................................................356
6.6.2 Hexahedral Edge Element ................................................................361
6.7 Electrostatic Analysis for Extraction of 3-D Interconnect
Capacitance Matrices.........................................................................................371
6.8 Time-Harmonic Electromagnetic Analysis ...................................................376
6.8.1 Scattering from a PEC or Dielectric Object .....................................376
6.8.2 Scattering from an Object Inside a Rectangular Waveguide .............387
6.8.3 Cavity Resonator ..............................................................................389
References.....................................................................................................................392
Chapter 7 Past and Present: A Selected Bibliography.................................................................393
7.1 Introduction ............................................................................................................393
Scientific Literature ......................................................................................................394
Appendix: List of Codes ..............................................................................................................417
Index ..............................................................................................................................................423
Preface
This is a programming-oriented and learner-centered textbook on the finite element method (FEM),
with special emphasis given to developing MATLAB programs for numerical modeling of electro-
magnetic boundary value problems. This book can be used by postgraduate or senior undergraduate
students in electrical and computer engineering, physics or other disciplines and aims to provide
the essential background, knowledge, and skills in FEM so that they are able to develop their own
finite element codes for specific applications. This book also serves as a comprehensive guide for
researchers and practitioners who have some introductory knowledge about numerical solution
techniques, and want to gain experience in FEM code development. Consequently, by means of
the approaches introduced in this book, readers can develop a deep understanding of finite element
programming in the context of canonical electromagnetic problems, supported by complete step-
by-step MATLAB programs with detailed descriptions, so that they can modify, adapt and apply
the provided programs and formulations to develop programs for other similar problems through
various exercises. Therefore, the book gradually strengthens readers’ confidence in their finite ele-
ment programming skills and strives to make them self-directed learners. Overall, this book aims
to make teaching and learning the finite element method simple, effective and productive.
This is a self-contained and self-sufficient textbook that can be used either as a sole resource
or as a supplement in finite element or computational electromagnetics courses. The book does
not require any prior knowledge of FEM. It starts with simple one-dimensional static and time-
harmonic problems and extends the developed theory to more complex two- or three-dimensional
problems. It provides sufficient theoretical background on the topic, and it thoroughly covers all
phases (pre-processing, main body and post-processing) in FEM. FEM formulations are obtained
for boundary value problems governed by a partial differential equation that is expressed in terms
of a generic unknown function, and then, these formulations are specialized to various electromag-
netic applications together with a post-processing phase. For example, the Laplace equation, which
is used to model various physical phenomena in many disciplines, is modeled and programmed by
FEM, and then, it is applied to electrostatic applications, such as capacitance computation. Since
the method is mostly described in a general context, readers from other disciplines can also use this
book and easily adapt the provided codes to their engineering problems. Readers, especially in elec-
tromagnetics, are guided to practice with the provided formulations and codes and to tailor them to
their own similar problems. Since programming is about a process, rather than simply the results of
running code, this book aims to carefully guide readers to get involved in the programming phases
of FEM and to develop an ability to utilize them in further applications. Due to its self-directed
nature, the book does not intend to cover every latest development, detail or application. After form-
ing a solid background on the fundamentals of FEM by means of canonical problems, readers are
guided to more advanced applications of FEM in electromagnetics through a survey chapter at the
end of the book.
The roadmap of the book can be summarized as follows:
Chapter 1: A brief introduction to the concepts of mathematical and numerical modeling, with
special emphasis given to numerical methods in electromagnetics, is given. Afterwards,
a detailed introduction to FEM, with an overview of the main steps of FEM applied to
a generic boundary value problem, is summarized. The main ingredients of FEM are
outlined.
Chapter 2: A concise tutorial for MATLAB programming is presented. Sufficient information
about MATLAB is provided so that the codes given in the book can easily be followed.
Chapter 3: Mathematical fundamentals of FEM are presented, so that a solid mathematical
background is established before going into the details of FEM. Approximate solution
ix