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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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 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