Iterative and self-adaptive finite-elements in electromagnetic modeling / Magdalena Salazar-Palma ... [and others].

Format:

Book

Contributor:

Salazar-Palma, Magdalena.

Series:

Artech House antenna library

Language:

English

Subjects (All):

Electromagnetism--Mathematics.

Electromagnetism.

Numerical analysis.

Finite element method.

Iterative methods (Mathematics).

Physical Description:

xxxvii, 770 pages : illustrations ; 24 cm.

Place of Publication:

Boston : Artech House, [1998]

Summary:

Ensure the accuracy of your results when applying the Finite Element Method (FEM) to electromagnetic and antenna problems with this self-contained reference. It provides you with a solid understanding of the method, describes its key elements and numerical techniques, and identifies various approaches to using the FEM in solving real-world microwave field problems.

Authored by four leading industry professionals, this information-packed, practical guide presents you with powerful new techniques for automatically generating self-adaptive meshes -- saving you hours of computation time you'd otherwise spend producing adequate meshes by hand. You also learn a groundbreaking new method that allows you to solve scattering and radiation problems from open regions without the need for hybrid procedures or absorbing boundary conditions.

Contents:

1.1 Numerical Methods in Electromagnetics 1

1.1.1 Classification of Methods by Technique 4

1.1.1.1 Analytical Methods 4

1.1.1.2 Numerical Methods 5

1.1.2 Classification of Numerical Methods by Type of Formulation 6

1.1.2.1. Methods Based on Partial Differential Equation Formulations 6

1.1.2.2. Methods Based on Integral Formulations 9

1.1.2.3. Comparison between Numerical Methods 11

1.2 The Finite Element Method in Electromagnetics 12

1.2.1 Variational Calculus and Variational Methods of Approximation 15

1.2.2 The Origin of the Finite Element Method and Its Development 19

1.2.3 The Finite Element Method in the Field of Electromagnetic Engineering 23

Chapter 2 The Finite Element Method 25

2.2 General Presentation of the Finite Element Method as Applied to Linear Boundary Value Problems 26

2.3 Definition of the Continuous Problem 31

2.3.1 Definition of the Domain of the Problem 32

2.3.2 Classical or Strong Formulation of the Problem 36

2.3.3 Weak Formulation of the Problem 42

2.3.3.1. The Weighted Residual Method 42

2.3.3.2. Variational Principles 45

2.4 Discretization of an Integral Form 50

2.4.1 Approximation of a Function 51

2.4.2 Discretization of a Weak Formulation or Variational Formulation of the Weighted Residual Type 52

2.4.3 Discretization of a Variational Principle: Ritz Method 59

2.4.4 Convergence and Other Properties of the Variational Methods of Approximation 62

2.5 Discretization of the Continuous Problem by Means of the Finite Element Method 63

2.5.1 Approximation of a Function by means of the Finite Element Method 64

2.5.1.1 Discretization of the Domain 66

2.5.1.2 Description of the Finite Elements 74

2.5.2 Discretization by the Finite Element Method 115

2.5.2.1 Calculation of the Local Integral Forms: Numerical Integration 119

2.5.2.2 Computation of the Global Integral Form: Assembly Process 123

2.5.2.3 Enforcement of the Essential Boundary Conditions: Global System of Equations 127

2.5.2.4 Matrix Storage and Solution of the Global System of Equations 128

2.5.2.5 Postprocessing of the Solution 132

2.5.3 Convergence of the Finite Element Method 134

2.6 Flow Diagram of a Finite Element Analysis 140

2.7 Public Domain and Commerical Software Packages for the Analysis of Electromagnetic Problems Utilizing the Finite Element Method 140

Chapter 3 Application of the Finite Element Method to the Analysis of Waveguiding Problems 145

3.2 Quasi-Static Analysis of Transmission Lines 149

3.2.1 Description of the Structures to be Analyzed 150

3.2.2 Circuital and Electromagnetic Characterization of TEM and Quasi-TEM Multiconductor Transmission Lines 153

3.2.2.1 Two-Conductor Transmission Line 153

3.2.2.2 Multiconductor Transmission Line in an Inhomogeneous Anisotropic Medium with Dielectric and Magnetic Losses and Imperfect Conductors 164

3.2.3 Application of the Finite Element Method to the Quasi-Static Analysis of Transmission Lines 168

3.2.3.1 Application of the Finite Element Method to the Direct Formulation 169

3.2.3.2 Application of the Finite Element Method to the Dual Standard Formulation 210

3.2.3.3 Application of the Finite Element Method to the Mixed Formulation 212

3.3 Full-Wave Analysis of Waveguiding Structures Utilizing the Finite Element Method 216

3.3.1 Description of the Geometry and Configuration of the Structures To Be Analyzed 217

3.3.2 Survey of Various Formulations of the Waveguiding Problem Utilizing the Finite Element Method 217

3.3.3 Full-Wave Analysis of Waveguiding Structures Using the Finite Element Method 221

3.3.3.1 Formulations Using Longitudinal Field Components: Lagrange Elements 222

3.3.3.2 Formulations Using Transverse and Longitudinal Field Components: Lagrange/Curl-Conforming Elements 227

Chapter 4 Self-Adaptive Mesh Algorithm 247

4.2 Self-Adaptive Techniques, Error Estimates, and Refinement Procedures 248

4.3 Application of a Self-Adaptive Mesh Algorithm to the Quasi-Static Analysis of Transmission Lines 253

4.3.1 Local and Global Error Estimates 254

4.3.2 Refinement Strategy 258

4.3.3 Element Subdivision Algorithms 259

4.3.4 Self-Adaptive Algorithm 263

4.3.5 Validation of the Self-Adaptive Algorithm 264

4.4 Extension of the Self-Adaptive Algorithm to the Full-Wave Analysis of Waveguiding Structures 283

5.2 Quasi-Static Analysis of Transmission Lines 291

5.2.1 Finite-Thickness Coupled Microstrip Lines 291

5.2.2 Finite-Thickness Coupled Striplines 293

5.2.3 Zero-Thickness Coupled Microstrip Lines 296

5.2.4 Three Coupled Microstrip Lines 298

5.2.5 Microstrip Line with Undercutting 299

5.2.6 Symmetric Coplanar Waveguide with Broadside-Coupled Lines 300

5.2.7 V-Grooved Microstrip Line 304

5.2.8 Suspended Stripline with Supporting Grooves 305

5.2.9 Microstrip Line Near a Dielectric Edge 309

5.2.10 Electro-Optical Coupler 314

5.2.11 Dissipative Structures 319

5.3 Full-Wave Analysis of Guiding Structures 320

5.3.1 Shielded Microstrip Line: Case A 320

5.3.2 Shielded Microstrip Line: Case B 323

5.3.3 Shielded Microstrip Line: Effect of Walls 323

5.3.4 Shielded Microstrip Line: Losses 325

5.3.5 Bilateral Circular Finline 325

5.3.6 Double Semicircular Ridge Guide 327

5.3.7 Coplanar Line with Anisotropic Substrate 329

5.3.8 Microstrip Line with Anisotropic Substrate 332

5.3.9 Finline with Anisotropic Substrate 332

5.3.10 Suspended Coplanar Waveguide 334

5.3.11 Coupled Microstrip Lines 335

Chapter 6 Application of Finite Element Method for the Solution of Open-Region Problems 337

6.2 Statement of the Problem 337

6.2.2 The Finite Element Method and Open-Region Problems 337

6.2.3 Nonlocal Boundary Conditions 344

6.2.4 Comments on Solution of Linear Equations 351

6.2.5 Applications 353

6.3 Two-Dimensional Electrostatic Problems 353

6.3.2 Formulation 354

6.3.3 Numerical Results 363

6.3.3.1 Circular Cylinder 363

6.3.3.2 Square Cylinder 365

6.3.3.3 Semicircular Cylinder 366

6.3.3.4 Bow-Tie Cylinder 368

6.4 TM Scattering 370

6.4.2 Formulation 371

6.4.3 Numerical Results 379

6.4.3.1 Elliptic Cylinder 379

6.4.3.2 Square Cylinder 382

6.4.3.3 Semicircular Cylinder 385

6.5 TE Scattering 388

6.5.2 Formulation 388

6.5.3 Numerical Results 395

6.5.3.1 Square Cylinder 395

6.5.3.2 Circular Cylinder 395

6.5.3.3 Semi-Circular Cylinder 398

Chapter 7 Finite Element Analysis of Three-Dimensional Electromagnetic Problems 401

7.2 Spurious Modes and Curl-Conforming Elements 401

7.2.1 Origin of Spurious Modes 402

7.2.1.1 Some Mathematical Concepts Related to the Spurious Modes 403

7.2.1.2 Some Early Ideas Regarding Spurious Modes 409

7.2.2 Solution to the Problem of Spurious Modes 411

7.2.2.1 At the Formulation Stage 412

7.2.2.2 At the Discretization Stage 415

7.3 Analysis of Three-Dimensional Cavity Resonances Using the Finite Element Method 422

7.3.1 Finite Element Method Formulation 422

7.3.1.1 Variational Formulation 425

7.3.1.2 Discretization by Curl-Conforming Elements 437

7.3.2 Dimension of the Vector Space Spanned by the Spurious Modes 442

7.3.3 Numerical Results 447

7.4 Analysis of Discontinuities in Waveguides Using the Finite Element Method 460

7.4.1 Finite Element Formulation 461

7.4.1.1 Variational Formulation 461

7.4.1.2 Computation of the Scattering Parameters 465

7.4.2 Numerical Results: Application to Rectangular Waveguides 466

7.5 Analysis of Scattering and Radiation from Three-Dimensional Open-Regions Using the Finite Element Method 480

7.5.1 The Method 480

7.5.1.2 Description of the Method 481

7.5.1.3 Finite Element Formulation and Features of the Iterative Method 483

7.5.2 Numerical Results 488

7.5.2.1 Radiation 488

7.5.2.2 Scattering 489

Appendix A A Mathematical Overview 501

A.1 Some Concepts of Functional Analysis 501

A.1.1 Dimension of a Space.

Finite-Dimensional Spaces 501

A.1.2 Functional Forms. Linear and Bilinear Operators 506

A.1.3 Hilbert and Sobolev Spaces 509

A.1.4 H(div,[Omega]) Spaces 517

A.1.5 H(curl,[Omega]) Spaces 518

A.1.6 H[superscript 1]([Omega])[times]H(curl,[Omega]) Space 519

A.2 Weak Integral Formulations by the Weighted Residual Method: Integration by Parts. Essential and Natural Boundary Conditions 520

A.3 An Overview of Variational Calculus 523

A.3.1 Variational Principles: Properties 523

A.3.2 Generalized, Complementary, and Mixed Variational Principles by Means of Lagrange Multipliers 527

Appendix B Definitions of Convergence 529

B.1 Types of Convergence 529

B.2 Some General Conclusions Regarding Convergence 531

Appendix C Topics Related to Finite Elements 533

C.1 Mapping Between Parent Finite Elements and Real Finite Elements: Properties 533

C.2 Lagrange Ordinary Elements 545

C.2.1 Rectangular Parent Elements 545

C.2.2 Simplex Parent Elements 546

C.3 Generation of One-Dimensional Infinite Elements for Asymptotic Approximation of the Unknown of Type (1/r) 547

C.4 Some Topics Related to Div-Conforming and Curl-Conforming Elements 551

C.4.1 Div-Conforming Triangular Parent Elements 551

C.4.1.1 First Order Elements 551

C.4.1.2 Second Order Elements 553

C.4.2 Curl-Conforming Simplex Parent Elements 555

C.4.2.1 Triangular Elements 555

C.4.2.2 Tetrahedral Elements 564

C.4.3 On the Assembly of Div-Conforming and Curl-Conforming Elements 567

C.5 Some General Conclusions Regarding the Use of Lagrange Elements, Div-Conforming and Curl-Conforming Elements 572

C.5.1 Two-Dimensional Deterministic Problems. Quasi-Static Analysis of Transmission Lines. Lagrange Elements Versus Div-Conforming Elements 572

C.5.2 Two-Dimensional Eigenvalue Problems. Full Wave Analysis of Waveguiding Structures. Lagrange Elements Versus Lagrange/Curl-Conforming Elements 574

C.5.3 Three-Dimensional Problems 576

Appendix D Maxwell's Equations in a Source-Free Region Specialized to Waveguiding Structures 577

D.2 Electromagnetic Characterization of Media in Electromagnetic Structures 577

D.3 Steady-State Maxwell's Equations in a Source-Free Waveguiding Structure 582

D.3.1 Steady-State Maxwell's Equations in a Source-Free Structure 582

D.3.2 Specialization to Waveguiding Structures 587

Appendix E Weak Formulations for the Quasi-Static Analysis of Waveguiding Structures and Their Finite Element Discretization 597

E.2 Direct Formulation. Lagrange Elements 597

E.2.1 Weak Formulation 597

E.2.2 Discretization by Means of Lagrange Finite Elements 601

E.3 Mixed Formulation. Div-Conforming Elements 604

E.3.1 Weak Formulation 604

E.3.2 Discretization by Means of Div-Conforming Finite Elements 607

Appendix F Weak Formulations for the Full-Wave Analysis of Waveguiding Structures and Their Finite Element Discretization 613

F.2 Formulation Utilizing the Longitudinal Components of the Electric and the Magnetic Fields 613

F.2.1 Inhomogeneous and Anisotropic Structures 613

F.2.1.1 Weak Formulation 613

F.2.1.2 Discretization by Means of Lagrange Finite Elements 622

F.2.2 Homogeneous and Isotropic Structures 626

F.2.2.1 Weak Formulation 626

F.2.2.2 Discretization by Means of Lagrange Finite Elements 630

F.3 Formulation Utilizing the Longitudinal and Transverse Components of the Electric or Magnetic Field 632

F.3.1 Standard Formulation 632

F.3.1.1 Weak Formulation 632

F.3.1.2 Discretization by Means of Lagrange/Curl-Conforming Elements 639

F.3.2 Nonstandard Formulation 646

F.3.2.1 A Variant of the Previous Weak Formulation 646

F.3.2.2 Discretization by Means of Lagrange/Curl-Conforming Elements 648

Appendix G Computation of Error Estimates and Indicators 651

G.2 Computation of Local Error Estimates for the Quasi-Static Analysis of Transmission Lines Using the Direct Formulation and Lagrange Elements 653

G.2.1 First-Order Straight Lagrange Triangular Element 658

G.2.2 Second-Order Lagrange Triangular Element 660

G.2.2.1 Straight or Subparametric Element 660

G.2.2.2 Curved or Isoparametric Element 663

G.2.3 Infinite Elements. Computation of the Residue at an Interface with an Ordinary Element 667

G.2.3.1 First-Order Infinite Element 670

G.2.3.2 Serendipity and Complete Second-Order Infinite Elements 671

G.3 Computation of Local Error Indicators for the Full-Wave Analysis of Waveguiding Structures 671

G.3.1 Formulation by Means of the Longitudinal Components of the Electric or the Magnetic Field and a Lagrange Element-Based Finite Element Method 671

G.3.1.1 First-Order Lagrange Straight Triangular Element 673

G.3.1.2 Second-Order Lagrange Straight Triangular Element 674

G.3.2 Formulation by Means of the Transverse and Longitudinal Components of the Electric or Magnetic Field and a Lagrange/Curl-Conforming Element-Based Finite Element Method 677.

Notes:

Includes bibliographical references (pages 697-742) and index.

ISBN:

089006895X

OCLC:

38738846