Iterative and self-adaptive finite-elements in electromagnetic modeling

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Publication Information:

Boston, Mass. : Artech House, 1998

ISBN:

9780890068953

Subject Term:

Electromagnetism -- Mathematics

Numerical analysis

Finite element method

Iterative methods (Mathematics)

Added Author:

Salazar-Palma, Magdalena

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000004124438	QC760 I84 1998	Open Access Book	Book	Searching... Unknown

This volume aims to assist readers in ensuring the accuracy of their results when applying the Finite Element Method (FEM) to electromagnetic and antenna problems. It outlines the method, describes its key elements and numerical techniques, and identifies various approaches to using the FEM in solving real-world microwave field problems.

Author Notes

Magdalena Salazar-Palma obtained a Ph.D. from the Universidad Politecnica de Madrid, Spain.

She is a professor titular in the Departmento de Senales, Sistemas y Radiocommunicaciones at the Universidad Politecnica de Madrid, Spain. She is a co-author of Iterative and Self-Adaptive Finite-Elements in Electromagnetic Modeling (Artech House, 1998). She is the chairperson of the Spain section of IEEE.

050

Preface	p. xv
Acknowledgements	p. xix
List of Figures	p. xxi
List of Tables	p. xxxiii
Chapter 1 Introduction	p. 1
1.1 Numerical Methods in Electromagnetics	p. 1
1.1.1 Classification of Methods by Technique	p. 4
1.1.1.1 Analytical Methods	p. 4
1.1.1.2 Numerical Methods	p. 5
1.1.2 Classification of Numerical Methods by Type of Formulation	p. 6
1.1.2.1. Methods Based on Partial Differential Equation Formulations	p. 6
1.1.2.2. Methods Based on Integral Formulations	p. 9
1.1.2.3. Comparison between Numerical Methods	p. 11
1.2 The Finite Element Method in Electromagnetics	p. 12
1.2.1 Variational Calculus and Variational Methods of Approximation	p. 15
1.2.2 The Origin of the Finite Element Method and Its Development	p. 19
1.2.3 The Finite Element Method in the Field of Electromagnetic Engineering	p. 23
Chapter 2 The Finite Element Method	p. 25
2.1 Introduction	p. 25
2.2 General Presentation of the Finite Element Method as Applied to Linear Boundary Value Problems	p. 26
2.3 Definition of the Continuous Problem	p. 31
2.3.1 Definition of the Domain of the Problem	p. 32
2.3.2 Classical or Strong Formulation of the Problem	p. 36
2.3.3 Weak Formulation of the Problem	p. 42
2.3.3.1. The Weighted Residual Method	p. 42
2.3.3.2. Variational Principles	p. 45
2.4 Discretization of an Integral Form	p. 50
2.4.1 Approximation of a Function	p. 51
2.4.2 Discretization of a Weak Formulation or Variational Formulation of the Weighted Residual Type	p. 52
2.4.3 Discretization of a Variational Principle: Ritz Method	p. 59
2.4.4 Convergence and Other Properties of the Variational Methods of Approximation	p. 62
2.5 Discretization of the Continuous Problem by Means of the Finite Element Method	p. 63
2.5.1 Approximation of a Function by means of the Finite Element Method	p. 64
2.5.1.1 Discretization of the Domain	p. 66
2.5.1.2 Description of the Finite Elements	p. 74
2.5.2 Discretization by the Finite Element Method	p. 115
2.5.2.1 Calculation of the Local Integral Forms: Numerical Integration	p. 119
2.5.2.2 Computation of the Global Integral Form: Assembly Process	p. 123
2.5.2.3 Enforcement of the Essential Boundary Conditions: Global System of Equations	p. 127
2.5.2.4 Matrix Storage and Solution of the Global System of Equations	p. 128
2.5.2.5 Postprocessing of the Solution	p. 132
2.5.3 Convergence of the Finite Element Method	p. 134
2.6 Flow Diagram of a Finite Element Analysis	p. 140
2.7 Public Domain and Commerical Software Packages for the Analysis of Electromagnetic Problems Utilizing the Finite Element Method	p. 140
Chapter 3 Application of the Finite Element Method to the Analysis of Waveguiding Problems	p. 145
3.1 Introduction	p. 145
3.2 Quasi-Static Analysis of Transmission Lines	p. 149
3.2.1 Description of the Structures to be Analyzed	p. 150
3.2.2 Circuital and Electromagnetic Characterization of TEM and Quasi-TEM Multiconductor Transmission Lines	p. 153
3.2.2.1 Two-Conductor Transmission Line	p. 153
3.2.2.2 Multiconductor Transmission Line in an Inhomogeneous Anisotropic Medium with Dielectric and Magnetic Losses and Imperfect Conductors	p. 164
3.2.3 Application of the Finite Element Method to the Quasi-Static Analysis of Transmission Lines	p. 168
3.2.3.1 Application of the Finite Element Method to the Direct Formulation	p. 169
3.2.3.2 Application of the Finite Element Method to the Dual Standard Formulation	p. 210
3.2.3.3 Application of the Finite Element Method to the Mixed Formulation	p. 212
3.2.4 Conclusions	p. 215
3.3 Full-Wave Analysis of Waveguiding Structures Utilizing the Finite Element Method	p. 216
3.3.1 Description of the Geometry and Configuration of the Structures To Be Analyzed	p. 217
3.3.2 Survey of Various Formulations of the Waveguiding Problem Utilizing the Finite Element Method	p. 217
3.3.3 Full-Wave Analysis of Waveguiding Structures Using the Finite Element Method	p. 221
3.3.3.1 Formulations Using Longitudinal Field Components: Lagrange Elements	p. 222
3.3.3.2 Formulations Using Transverse and Longitudinal Field Components: Lagrange/Curl-Conforming Elements	p. 227
3.3.4 Conclusions	p. 246
Chapter 4 Self-Adaptive Mesh Algorithm	p. 247
4.1 Introduction	p. 247
4.2 Self-Adaptive Techniques, Error Estimates, and Refinement Procedures	p. 248
4.3 Application of a Self-Adaptive Mesh Algorithm to the Quasi-Static Analysis of Transmission Lines	p. 253
4.3.1 Local and Global Error Estimates	p. 254
4.3.2 Refinement Strategy	p. 258
4.3.3 Element Subdivision Algorithms	p. 259
4.3.4 Self-Adaptive Algorithm	p. 263
4.3.5 Validation of the Self-Adaptive Algorithm	p. 264
4.4 Extension of the Self-Adaptive Algorithm to the Full-Wave Analysis of Waveguiding Structures	p. 283
4.5 Conclusions	p. 290
Chapter 5 Additional Examples	p. 291
5.1 Introduction	p. 291
5.2 Quasi-Static Analysis of Transmission Lines	p. 291
5.2.1 Finite-Thickness Coupled Microstrip Lines	p. 291
5.2.2 Finite-Thickness Coupled Striplines	p. 293
5.2.3 Zero-Thickness Coupled Microstrip Lines	p. 296
5.2.4 Three Coupled Microstrip Lines	p. 298
5.2.5 Microstrip Line with Undercutting	p. 299
5.2.6 Symmetric Coplanar Waveguide with Broadside-Coupled Lines	p. 300
5.2.7 V-Grooved Microstrip Line	p. 304
5.2.8 Suspended Stripline with Supporting Grooves	p. 305
5.2.9 Microstrip Line Near a Dielectric Edge	p. 309
5.2.10 Electro-Optical Coupler	p. 314
5.2.11 Dissipative Structures	p. 319
5.3 Full-Wave Analysis of Guiding Structures	p. 320
5.3.1 Shielded Microstrip Line: Case A	p. 320
5.3.2 Shielded Microstrip Line: Case B	p. 323
5.3.3 Shielded Microstrip Line: Effect of Walls	p. 323
5.3.4 Shielded Microstrip Line: Losses	p. 325
5.3.5 Bilateral Circular Finline	p. 325
5.3.6 Double Semicircular Ridge Guide	p. 327
5.3.7 Coplanar Line with Anisotropic Substrate	p. 329
5.3.8 Microstrip Line with Anisotropic Substrate	p. 332
5.3.9 Finline with Anisotropic Substrate	p. 332
5.3.10 Suspended Coplanar Waveguide	p. 334
5.3.11 Coupled Microstrip Lines	p. 335
5.4 Conclusions	p. 335
Chapter 6 Application of Finite Element Method for the Solution of Open-Region Problems	p. 337
6.1 Introduction	p. 337
6.2 Statement of the Problem	p. 337
6.2.1 Introduction	p. 337
6.2.2 The Finite Element Method and Open-Region Problems	p. 337
6.2.3 Nonlocal Boundary Conditions	p. 344
6.2.4 Comments on Solution of Linear Equations	p. 351
6.2.5 Applications	p. 353
6.3 Two-Dimensional Electrostatic Problems	p. 353
6.3.1 Introduction	p. 353
6.3.2 Formulation	p. 354
6.3.3 Numerical Results	p. 363
6.3.3.1 Circular Cylinder	p. 363
6.3.3.2 Square Cylinder	p. 365
6.3.3.3 Semicircular Cylinder	p. 366
6.3.3.4 Bow-Tie Cylinder	p. 368
6.3.4 Conclusion	p. 369
6.4 TM Scattering	p. 370
6.4.1 Introduction	p. 370
6.4.2 Formulation	p. 371
6.4.3 Numerical Results	p. 379
6.4.3.1 Elliptic Cylinder	p. 379
6.4.3.2 Square Cylinder	p. 382
6.4.3.3 Semicircular Cylinder	p. 385
6.4.4 Conclusion	p. 387
6.5 TE Scattering	p. 388
6.5.1 Introduction	p. 388
6.5.2 Formulation	p. 388
6.5.3 Numerical Results	p. 395
6.5.3.1 Square Cylinder	p. 395
6.5.3.2 Circular Cylinder	p. 395
6.5.3.3 Semi-Circular Cylinder	p. 398
6.5.4 Conclusion	p. 400
6.6 Summary	p. 400
Chapter 7 Finite Element Analysis of Three-Dimensional Electromagnetic Problems	p. 401
7.1 Introduction	p. 401
7.2 Spurious Modes and Curl-Conforming Elements	p. 401
7.2.1 Origin of Spurious Modes	p. 402
7.2.1.1 Some Mathematical Concepts Related to the Spurious Modes	p. 403
7.2.1.2 Some Early Ideas Regarding Spurious Modes	p. 409
7.2.2 Solution to the Problem of Spurious Modes	p. 411
7.2.2.1 At the Formulation Stage	p. 412
7.2.2.2 At the Discretization Stage	p. 415
7.3 Analysis of Three-Dimensional Cavity Resonances Using the Finite Element Method	p. 422
7.3.1 Finite Element Method Formulation	p. 422
7.3.1.1 Variational Formulation	p. 425
7.3.1.2 Discretization by Curl-Conforming Elements	p. 437
7.3.2 Dimension of the Vector Space Spanned by the Spurious Modes	p. 442
7.3.3 Numerical Results	p. 447
7.4 Analysis of Discontinuities in Waveguides Using the Finite Element Method	p. 460
7.4.1 Finite Element Formulation	p. 461
7.4.1.1 Variational Formulation	p. 461
7.4.1.2 Computation of the Scattering Parameters	p. 465
7.4.2 Numerical Results: Application to Rectangular Waveguides	p. 466
7.4.3 Conclusions	p. 477
7.5 Analysis of Scattering and Radiation from Three-Dimensional Open-Regions Using the Finite Element Method	p. 480
7.5.1 The Method	p. 480
7.5.1.1 Introduction	p. 480
7.5.1.2 Description of the Method	p. 481
7.5.1.3 Finite Element Formulation and Features of the Iterative Method	p. 483
7.5.2 Numerical Results	p. 488
7.5.2.1 Radiation	p. 488
7.5.2.2 Scattering	p. 489
7.5.3 Conclusions	p. 500
Appendix A A Mathematical Overview	p. 501
A.1 Some Concepts of Functional Analysis	p. 501
A.1.1 Dimension of a Space. Finite-Dimensional Spaces	p. 501
A.1.2 Functional Forms. Linear and Bilinear Operators	p. 506
A.1.3 Hilbert and Sobolev Spaces	p. 509
A.1.4 H(div,[Omega]) Spaces	p. 517
A.1.5 H(curl,[Omega]) Spaces	p. 518
A.1.6 H[superscript 1]([Omega])[times]H(curl,[Omega]) Space	p. 519
A.2 Weak Integral Formulations by the Weighted Residual Method: Integration by Parts. Essential and Natural Boundary Conditions	p. 520
A.3 An Overview of Variational Calculus	p. 523
A.3.1 Variational Principles: Properties	p. 523
A.3.2 Generalized, Complementary, and Mixed Variational Principles by Means of Lagrange Multipliers	p. 527
Appendix B Definitions of Convergence	p. 529
B.1 Types of Convergence	p. 529
B.2 Some General Conclusions Regarding Convergence	p. 531
Appendix C Topics Related to Finite Elements	p. 533
C.1 Mapping Between Parent Finite Elements and Real Finite Elements: Properties	p. 533
C.2 Lagrange Ordinary Elements	p. 545
C.2.1 Rectangular Parent Elements	p. 545
C.2.2 Simplex Parent Elements	p. 546
C.3 Generation of One-Dimensional Infinite Elements for Asymptotic Approximation of the Unknown of Type (1/r)	p. 547
C.4 Some Topics Related to Div-Conforming and Curl-Conforming Elements	p. 551
C.4.1 Div-Conforming Triangular Parent Elements	p. 551
C.4.1.1 First Order Elements	p. 551
C.4.1.2 Second Order Elements	p. 553
C.4.2 Curl-Conforming Simplex Parent Elements	p. 555
C.4.2.1 Triangular Elements	p. 555
C.4.2.2 Tetrahedral Elements	p. 564
C.4.3 On the Assembly of Div-Conforming and Curl-Conforming Elements	p. 567
C.5 Some General Conclusions Regarding the Use of Lagrange Elements, Div-Conforming and Curl-Conforming Elements	p. 572
C.5.1 Two-Dimensional Deterministic Problems. Quasi-Static Analysis of Transmission Lines. Lagrange Elements Versus Div-Conforming Elements	p. 572
C.5.2 Two-Dimensional Eigenvalue Problems. Full Wave Analysis of Waveguiding Structures. Lagrange Elements Versus Lagrange/Curl-Conforming Elements	p. 574
C.5.3 Three-Dimensional Problems	p. 576
Appendix D Maxwell's Equations in a Source-Free Region Specialized to Waveguiding Structures	p. 577
D.1 Introduction	p. 577
D.2 Electromagnetic Characterization of Media in Electromagnetic Structures	p. 577
D.3 Steady-State Maxwell's Equations in a Source-Free Waveguiding Structure	p. 582
D.3.1 Steady-State Maxwell's Equations in a Source-Free Structure	p. 582
D.3.2 Specialization to Waveguiding Structures	p. 587
Appendix E Weak Formulations for the Quasi-Static Analysis of Waveguiding Structures and Their Finite Element Discretization	p. 597
E.1 Introduction	p. 597
E.2 Direct Formulation. Lagrange Elements	p. 597
E.2.1 Weak Formulation	p. 597
E.2.2 Discretization by Means of Lagrange Finite Elements	p. 601
E.3 Mixed Formulation. Div-Conforming Elements	p. 604
E.3.1 Weak Formulation	p. 604
E.3.2 Discretization by Means of Div-Conforming Finite Elements	p. 607
Appendix F Weak Formulations for the Full-Wave Analysis of Waveguiding Structures and Their Finite Element Discretization	p. 613
F.1 Introduction	p. 613
F.2 Formulation Utilizing the Longitudinal Components of the Electric and the Magnetic Fields	p. 613
F.2.1 Inhomogeneous and Anisotropic Structures	p. 613
F.2.1.1 Weak Formulation	p. 613
F.2.1.2 Discretization by Means of Lagrange Finite Elements	p. 622
F.2.2 Homogeneous and Isotropic Structures	p. 626
F.2.2.1 Weak Formulation	p. 626
F.2.2.2 Discretization by Means of Lagrange Finite Elements	p. 630
F.3 Formulation Utilizing the Longitudinal and Transverse Components of the Electric or Magnetic Field	p. 632
F.3.1 Standard Formulation	p. 632
F.3.1.1 Weak Formulation	p. 632
F.3.1.2 Discretization by Means of Lagrange/Curl-Conforming Elements	p. 639
F.3.2 Nonstandard Formulation	p. 646
F.3.2.1 A Variant of the Previous Weak Formulation	p. 646
F.3.2.2 Discretization by Means of Lagrange/Curl-Conforming Elements	p. 648
Appendix G Computation of Error Estimates and Indicators	p. 651
G.1 Introduction	p. 651
G.2 Computation of Local Error Estimates for the Quasi-Static Analysis of Transmission Lines Using the Direct Formulation and Lagrange Elements	p. 653
G.2.1 First-Order Straight Lagrange Triangular Element	p. 658
G.2.2 Second-Order Lagrange Triangular Element	p. 660
G.2.2.1 Straight or Subparametric Element	p. 660
G.2.2.2 Curved or Isoparametric Element	p. 663
G.2.3 Infinite Elements. Computation of the Residue at an Interface with an Ordinary Element	p. 667
G.2.3.1 First-Order Infinite Element	p. 670
G.2.3.2 Serendipity and Complete Second-Order Infinite Elements	p. 671
G.3 Computation of Local Error Indicators for the Full-Wave Analysis of Waveguiding Structures	p. 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	p. 671
G.3.1.1 First-Order Lagrange Straight Triangular Element	p. 673
G.3.1.2 Second-Order Lagrange Straight Triangular Element	p. 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	p. 677
Bibliography	p. 697
Books	p. 697
Articles	p. 702
About the Authors	p. 743
Index	p. 745

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