UNIVERSIDADE ESTADUAL DE CAMPINAS
Faculdade de Engenharia Elétrica e de Computação
Francisco Teixeira Orlandini
A study on the construction of
𝐻(𝑐𝑢𝑟𝑙, Ω)-conforming
elements and their application on waveguide problems
Estudo sobre a construção de elementos
𝐻(𝑐𝑢𝑟𝑙, Ω)-conformes e suas aplicações em problemas de
guia de onda
Campinas
2018
Faculdade de Engenharia Elétrica e de Computação
Francisco Teixeira Orlandini
A study on the construction of
𝐻(𝑐𝑢𝑟𝑙, Ω)-conforming
elements and their application on waveguide problems
Estudo sobre a construção de elementos
𝐻(𝑐𝑢𝑟𝑙, Ω)-conformes e suas aplicações em problemas de
guia de onda
Dissertation presented to the Faculty of Elec-trical Engineering of the State University of Campinas in partial fullfilment of the require-ments for the degree of Master in Electrical Engineering.
Dissertação apresentada à Faculdade de En-genharia Elétrica e de Computação da Uni-versidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do tí-tulo de Mestre em Engenharia Elétrica, na Área de Telecomunicações e Telemática.
Supervisor: Prof. Dr. Hugo Enrique Hernández Figueroa Co-orientador Prof. Dr. Philippe Remy Bernard Devloo Este exemplar corresponde à versão
final da dissertação defendida pelo aluno Francisco Teixeira Orlandini, e orientada pelo Prof. Dr. Hugo Enrique Hernández Figueroa
Campinas
2018
Agência(s) de fomento e nº(s) de processo(s): CAPES, 1643848; FUNCAMP, 5149 Ficha catalográfica
Universidade Estadual de Campinas Biblioteca da Área de Engenharia e Arquitetura
Rose Meire da Silva - CRB 8/5974
Or5s
Orlandini, Francisco Teixeira,
1991-OrlA study on the construction of H(curl,Ω)-conforming elements and their application on waveguide problems / Francisco Teixeira Orlandini. – Campinas, SP : [s.n.], 2018.
OrlOrientador: Hugo Enrique Hernández Figueroa.
OrlCoorientador: Philippe Remy Bernard Devloo.
OrlDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação.
Orl1. Fotônica. 2. Guias de ondas óticos. 3. Método dos elementos finitos. 4. Método dos elementos finitos - Programa de computador. I. Hernandez-Figueroa, Hugo Enrique, 1959-. II. Devloo, Philippe Remy Bernard, 1958-. III. Universidade Estadual de Campinas. Faculdade de Engenharia Elétrica e de Computação. IV. Título.
Informações para Biblioteca Digital
Título em outro idioma: Estudo sobre a construção de elementos H(curl,Ω)-conformes e suas aplicações em problemas de guias de onda
Palavras-chave em inglês: Photonic
Optical waveguides Finite element method
Finite element method - Computer program
Área de concentração: Telecomunicações e Telemática Titulação: Mestre em Engenharia Elétrica
Banca examinadora:
Hugo Enrique Hernández Figueroa [Orientador] Renato Cardoso Mesquita
Lucas Heitzmann Gabrielli Data de defesa: 21-08-2018
Programa de Pós-Graduação: Engenharia Elétrica
Candidato: Francisco Teixeira Orlandini Data da defesa: 21 de Agosto de 2018
T´ıtulo da Tese: ”A study on the construction of H(curl,Ω)-conforming ele-ments and their application on waveguide problems” (”Estudos sobre a contruc¸ ˜ao de elementos H(curl,Ω)-conformes e suas aplicac¸ ˜oes em problemas de guia de onda”)
Prof. Dr. Hugo Enrique Hern ´andez Figueroa (Presidente, FEEC/UNICAMP) Prof. Dr. Renato Cardoso Mesquita( UFMG)
Prof. Dr. Lucas Heitzmann Gabrielli (FEEC/UNICAMP)
A ata da defesa, com as respectivas assinaturas dos membros da Comiss ˜ao Julgadora, encontra-se no processo de vida acad ˆemica do aluno.
There is a reason for the cliché regarding this kind of text: after all that happened during the period in which I’ve developed this work, emotions can be overwhelming. So,
without further ado...
First of all, I would like to express my gratitude to both of my advisors, Prof. Hugo Figueroa and Prof. Philippe Devloo. During my MSc, I was extremely privileged to count on excellent researchers to help me learn about all the different areas involved in this work. With Prof. Figueroa I was able to learn more on the intricacies of the physics behind the photonic devices. On several discussions with Prof. Luciano Prado I was able to understand the fundamental importance of numerical methods on the design of these devices. Prof. Devloo helped me nearly daily to see the different aspects of the Finite Element Method, and Prof. Sonia Gomes endured several discussions with me, and I have to thank her for the mathematical aspects of this work: although there is still a lot for me to learn. My friends Thiago Quinelato and Omar Durán taught me a lot, mainly during coffee breaks, and now you can rest assured that you will be able to enjoy your coffee in a more contemplative fashion. I want to thank all of you, for your time and patience (I know patience was necessary).
Working at LabMeC and at SimWorX were great experiences for me, and these environments have a great deal of participation in my wish to keep on researching in this area. Thank you for all the challenges, it was (and it still is) a great pleasure.
The support I have received from my family cannot be put down in words, neither could I express how thankful I am. I promise one day I will be able to properly explain what I study and what all these formulae are about. Until there, well, I will make up for it.
Speaking of support, I should also mention my gratitude for the financial support received by CAPES and by ANP through FUNCAMP. It was crucial for me to be able to fully concentrate on the tasks needed to make this project happen.
Outside office hours, the company of my friends were essential, and also it was full of learning and thoughtful moments: it has been incredible to be around brilliant people that were able to take any kind of mixtures of emotions, from moments of pure despair to episodes of euphoria, and then turn into something that, at least, provided lessons to be learned (and most of the time made us laugh quite a lot). Life is great when you are around.
I would also like to thank you that are reading this right now, because this means a lot to me. I really hope you like it.
—
Abstract
The advances in the fabrication of photonic waveguides in the past decades have led the scientific community to seek for numerical methods that could assist in the process of designing such devices. The design of photonic waveguides often require relative errors of 10⊗14 on the propagation constant in order to calculate the dispersion parameters. In this context, a hierarchical strategy for constructing 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements is introduced, for application in a Finite Element Method (FEM) scheme for modal analysis of electromagnetic waveguides. The hierarchical 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements are used for the transversal component of the electric field, coupled with scalar 𝐻1(Ω) elements for its longitudinal component. The Nédélec elements of the first kind were chosen for this work, and the ease of integration with p-adaptivity schemes motivated the hierarchical construction of the FE basis. The scheme is assessed by means of the analysis of well-known waveguides. The performance of higher order elements, uncommon in the Computational Electromagnetics community, is evaluated, and the importance of an accurate representation of curved geometries when using higher order elements is stressed. As a real-world scenario, the modal analysis of a Photonic Crystal Fiber illustrates the accuracy and the generalized eigenvalue problem size when dealing with a design process requiring high precision on the dispersion parameters.
Os avanços na fabricação de guias de ondas fotônicos nas últimas décadas levou a comuni-dade científica a buscar métodos numéricos para facilitar este processo. No caso do design de guias de onda fotônicos, erros relativos na constante de propagação na ordem de 10⊗14 são necessários para que se possa calcular os parâmetros de dispersão. É neste contexto que uma estratégia hierárquica para a construção de elementos 𝐻(𝑐𝑢𝑟𝑙; Ω)-conformes de alta ordem é apresentada, para aplicação no Método dos Elementos Finitos (MEF) para análise modal de guias de onda eletromagnéticos. Os elementos 𝐻(𝑐𝑢𝑟𝑙; Ω)-conformes hierárquicos são utilizados na aproximação da componente transversal do campo elétrico, enquanto elementos 𝐻1(Ω)-conformes são utilizados para a componente longitudinal. Neste trabalho, foi escolhida a primeira família dos elementos de Nédélec, e a construção hierárquica foi motivada pela facilidade de integração com esquemas de adaptabilidade-p. A avaliação do esquema proposto se deu através de guias de onda bastante estudados na literatura. O desempenho dos elementos de alta ordem, raros dentro da comunidade de Eletromagne-tismo Computacional(EC), é avaliado; e a importância de uma representação precisa de geometrias curvas ao se utilizar elementos de alta ordem é realçada. Finalmente, em um cenário de interesse da comunidade de EC, a análise modal de uma fibra de cristal fotônico é utilizada para ilustrar a precisão e o tamanho do problema matricial de autovalor gerado pelo esquema em um cenário no qual uma precisão alta nos parâmetros de dispersão é necessária.
Palavras-chaves: guias de onda fotônicos; método dos elementos finitos; elementos de
List of Figures
Figure 2.1 – An interface between two dielectric media with different constitutive
parameters. . . 21
Figure 3.1 – A partition 𝒯h over a domain Ω. . . 32
Figure 4.1 – The reference element. . . 38
Figure 4.2 – The basis functions for 𝑘 = 1. . . 43
Figure 4.3 – The basis functions associated with edge ˆ𝑒2 for 𝑘 = 1, 2, 3 and their tangential traces (𝑠 is the parametric coordinate of edge ˆ𝑒2). . . 45
Figure 4.4 – The basis functions associated with internal dofs. . . 46
Figure 4.5 – A 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming basis function on two adjacent elements. . . . 47
Figure 4.6 – Elements 𝐾1 and 𝐾2 and their edges. . . 47
Figure 5.1 – Convergence of 𝑢h for the model problem . . . 51
Figure 5.2 – Convergence of ∇× 𝑢h for the model problem. . . 52
Figure 5.3 – The geometry of a metallic rectangular waveguide. . . 55
Figure 5.4 – Convergence of the 𝑛eff for the TE10 mode in the WR-90 waveguide filled with air at 𝑓 = 25 𝐺𝐻𝑧. . . 57
Figure 5.5 – Convergence of the relative error of the dominant eigenvalue for a rectangular waveguide with 𝑎 = 1Û𝑚 and 𝑏 = 𝑎/2 for Ú = 1299.41𝑛𝑚 . 59 Figure 5.6 – Plot of electromagnetic fields in the WR-90 waveguide . . . 60
Figure 5.7 – Geometric mesh used in the approximation of fields in Figure 5.6. . . . 60
Figure 5.8 – The cross-section view of a step-index optical fiber. . . 61
Figure 5.9 – The refractive-index profile of a step-index optical fiber. . . 62
Figure 5.10–The 𝑏 Ü-diagram for the step-index fiber. . . 63
Figure 5.11–One Finite element mesh used to discretize the step-index fiber. . . 64
Figure 5.12–Details of two triangulations of the step-index fiber with linear-mapped elements . . . 65
Figure 5.13–Convergence of the first eigenvalue for the step-fiber using linear mapped elements . . . 67
Figure 5.14–Convergence of the first eigenvalue for the step-fiber using quadratic mapped elements . . . 68
Figure 5.15–Convergence of the first eigenvalue for the step-fiber using the curved mapped elements from NeoPZ. . . 69
Figure 5.16–Comparison of convergence rates for the real part of the effective index of a step-index optical fiber. The polynomial order 𝑘 = 4 was used in all three approximations. The geometry was described with elements obtained by linear mapping (blue curve), quadratic mapping (green curve) and exact mapping (red curve). . . 71
Figure 5.18–The dispersion curves for the first three LP modes on the step-fiber. . . 73 Figure 5.19–The geometry of the microstructured fiber . . . 74 Figure 5.20–The Finite Element mesh of the microstructured fiber . . . 75 Figure 5.21–Convergence of the effective index 𝑛eff of the sixth non-degenerated
mode in the microstructured fiber obtained using the curved mapped elements from NeoPZ. . . 76 Figure 5.22–Electric field plot of the sixth(up) and third(down) non-degenerated
electromagnetic modes of the microstructured fiber . . . 79 Figure 5.23–Example of hp-refined mesh for the microstructured fiber. The refinement
was performed manually based on a inspection of the solution. . . 80 Figure A.1 – A general computational domain surrounded by a PML. . . 90
List of Tables
Table 5.1 – Number of degrees of freedom for the model problem. . . 51
Table 5.2 – Convergence rates of 𝑢h for the model problem. . . 52
Table 5.3 – Convergence rates of ∇× 𝑢h for the model problem. . . 53
Table 5.4 – Number of degrees of freedom for the WR-90 approximation. . . 58
Table 5.5 – Convergence rates of the approximated 𝑛eff for the WR-90 waveguide. . 58
Table 5.6 – Number of degrees of freedom for the step-index fiber convergence as-sessment. . . 66
Table 5.7 – Convergence rates of the approximated 𝑛eff for the linear-mapped step-index optical fiber. . . 67
Table 5.8 – Convergence rates of the approximated 𝑛eff for the quadratic-mapped step-index optical fiber. . . 68
Table 5.9 – Convergence rates of the approximated 𝑛eff for the exactly mapped step-index optical fiber. . . 69
Table 5.10–The comparison of the absolute errors of the approximated 𝑛eff for all meshings of the step-index optical fiber. . . 70
Table 5.11–Number of degrees of freedom for the convergence assessment of the effec-tive index 𝑛eff of the sixth non-degenerated mode in the microstructured fiber at Ú0 = 1.45Û𝑚. . . 76
Table 5.12–Convergence rates of 𝑛eff for the exactly mapped microstructured fiber. 77 Table 5.13–Comparison of number of dofs needed to obtain a low error on the approximated 𝑛eff for the exactly mapped microstructured fiber between meshes with and without hp-adaptivity. . . 78
Electromagnetics
𝐸 Electric field (time harmonic)
𝐸t Transversal component of 𝐸 in a
waveguide
𝐸z Transversal component of 𝐸 in a
waveguide
Ûr Relative magnetic permeability
𝜖r Relative electric permittivity
Ú Wavelength 𝑓 Linear frequency
𝑘 Wavenumber
Ñ Propagation constant
Finite Element Method
Ω Two-dimensional euclidean domain
𝑑Ω Boundary of Ω 𝐿2(Ω) {︂ã: Ω ⊃ C ♣ ∫︁ Ωãã * 𝑑Ω < ∞ }︂ [𝐿2(Ω)]2 {︂ã: Ω ⊃ C2 ♣ ∫︁ Ωã≤ ã * 𝑑Ω < ∞ }︂ 𝐻1(Ω) ¶ ã ∈ 𝐿2(Ω) ♣ ∇ ã ∈ [𝐿2(Ω)]2♢ 𝐻1 0(Ω) ¶ ã ∈ 𝐻1(Ω) ♣ ã ♣dΩ= 0 ♢ 𝐻(𝑐𝑢𝑟𝑙; Ω) ¶ ã ∈ [𝐿2(Ω)]2 ♣ ∇× ã ∈ 𝐿2(Ω) ♢ 𝐻0(𝑐𝑢𝑟𝑙; Ω) ¶ ã ∈ 𝐻(𝑐𝑢𝑟𝑙; Ω) ♣ ∇× ã ♣dΩ= 0 ♢ ˆ 𝐾 Reference element 𝐾 General element
Fcurl Covariant Piola mapping
𝜙 Vector-valued shape function in Ω
ˆ
𝜙 Scalar-valued basis function in ˆ𝐾
𝜙 Vector-valued shape function in Ω
ˆ
𝜙 Vector-valued basis function in ˆ𝐾
Pk Space of polynomials of total degree
𝑘
˜Pk Space of homogeneous polynomials of total degree 𝑘
𝐹K Geometric mapping from ˆ𝐾 to 𝐾.
Contents
Contents . . . 13
1 Introduction . . . 15
1.1 Objectives . . . 16
1.2 Organization the contents . . . 16
1.2.1 Body . . . 17
1.2.2 Appendix . . . 18
2 A mathematical characterization of the waveguide problem . . . 19
2.1 The partial differential equation . . . 19
2.2 The boundary and interface conditions . . . 20
2.3 The variational form . . . 21
3 Relevant function spaces, formulations for the waveguide problem and Fi-nite Element aspects . . . 24
3.1 The choice of an approximation space . . . 24
3.2 Relevant Hilbert spaces for the waveguide problem . . . 25
3.3 Some formulations for the waveguide problem . . . 26
3.3.1 An 𝐻1(Ω) formulation for homogeneous waveguides . . . 26
3.3.2 An 𝐻1(Ω) × 𝐻1(Ω) formulation for inhomogeneous waveguides . . . 27
3.3.3 An [𝐻1(Ω)]2 formulation for inhomogeneous waveguides . . . 28
3.4 An 𝐻(𝑐𝑢𝑟𝑙; Ω) × 𝐻1(Ω) formulation for the waveguide problem . . . 29
3.5 The Finite Element Method: an overview . . . 32
3.6 The discretized form of the 𝐻(𝑐𝑢𝑟𝑙; Ω) × 𝐻1(Ω) formulation for the waveg-uide problem . . . 34
4 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming spaces and the construction of their hierarchical shape functions . . . 37
4.1 The reference element and the covariant Piola mapping . . . 37
4.2 Construction of the 𝐻(𝑐𝑢𝑟𝑙; ˆ𝐾) basis functions . . . 39
4.2.1 The reference element . . . 40
4.2.2 The polynomial space ℛk . . . 40
4.2.3 The degrees of freedom . . . 41
4.3 The hierarchical 𝐻(𝑐𝑢𝑟𝑙; Ω) elements . . . 42
5 Verification of the FE scheme and applications on computational electro-magnetics . . . 48
5.1 A model problem in two dimensions . . . 48
5.1.1 Expected convergence of 𝐻(𝑐𝑢𝑟𝑙; Ω) elements . . . 49
5.1.2 Numerical results . . . 50
waveguide problem . . . 54
5.2.2 Metallic rectangular waveguide . . . 55
5.2.2.1 Numerical results . . . 57
5.2.3 Step-index optical fiber . . . 60
5.2.3.1 Numerical results . . . 63
5.2.4 Six air-hole microstructured fiber . . . 74
5.2.4.1 Numerical results . . . 74
Conclusion . . . 81
Bibliography . . . 83
Appendix
88
APPENDIX A Perfectly Matching Layers . . . 89A.1 PML representation . . . 89
A.2 Definition of PML parameters . . . 90
APPENDIX B Implementation aspects . . . 92
B.1 Topics concerning the NeoPZ framework . . . 92
B.1.1 Implementation of 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements . . . 92
B.1.2 Implementation of the 𝐻(𝑐𝑢𝑟𝑙; Ω) × 𝐻1(Ω) formulation . . . 93
B.2 Other topics regarding implementation choices . . . 93
B.2.1 Dimensionless Helmholtz Equation . . . 93
B.2.2 Imposition of homogeneous Dirichlet boundary conditions . . . 94
B.2.3 Solver and data structure . . . 95
15
1 Introduction
The advances on the fabrication of photonic waveguides in the past twenty years have led the scientific community to seek for numerical methods that could assist on the process of design of such devices.
One type of analysis present in the design process is the modal analysis, in which the electromagnetic fields compatible with the structure of the device are analyzed, regardless of how one would excite these fields in the structure.
In the modal analysis of photonic waveguides, often the approximations of the propagation constant Ñ of these electromagnetic modes are desirable to present a challenging low relative error. In Mores et al. (2010), the analysis of a parametric amplifier demands the calculation of the so-called high-order dispersion parameters, which are essentially the derivatives of order m of the propagation constant. In this specific kind of application, a typical value of m is 𝑚 = 6, with a relative error of 10⊗2. Such requirement implies the need of a high precision on the propagation constant, with errors lower than 10⊗10. Similar requirements also appear on the analysis of the corner effect of high-contrast dielectric materials (CHIANG, Y.; CHIOU; CHANG, H., 2002), due to the abrupt variation of the electromagnetic fields near the dielectric corners.
Many were the approaches for this issue. Yen-Chung Chiang, Chiou, and Hung-Chun Chang (2002) develop an hp-adaptive algorithm for the Finite Differences Method (FDM). Recently, algorithms based on the pseudospectral method have been drawing the attention of the Computational Electromagnetics (CEM) researchers as an alternative for high precision approximations, using high-order polynomials with global support over the computational domain (CHIANG, Y.; CHIOU; CHANG, H., 2002; CHIANG, P.-J.; CHANG, H.-C., 2011).
The Finite Element Method (FEM) has also been applied for this class of problems (MORES et al., 2010; KOSHIBA; TSUJI, 2000; WIEDERHECKER, 2008). One class of elements that are often used in CEM applications of FEM are the 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements (NÉDÉLEC, 1980), often called edge elements among the CEM community. Works as Lee, Sun, and Cendes (1991) and Peterson (1994) have shown that there are FEM formulations using these elements that allow the confinement of spurious modes to the solutions associated with Ñ = 0. Vardapetyan and L. Demkowicz (2003) even present a formulation that, in practical terms, does not present spurious modes, for they are confined in Ñ = ∞.
The high-order Finite Elements, well known in Applied Mathematics (RAPETTI; BOSSAVIT, 2009), Computational Geosciences (WHEELER, M. F.; WHEELER, J. A.;
PESZYNSKA, 2000; BANGERTH et al., 2006) and in Computational Mechanics (DE-VLOO, P. B., 1991) are not commonly employed in most of the works using the FEM technique in CEM, thus turning to mesh refinement in order to achieve acceptable errors and thus generating a large algebraic system due to the low convergence rates of low-order elements. In Garcia-Castillo, Pardo, and L.F. Demkowicz (2008) the use of higher-order elements in the CE community is discussed in terms of p-adaptivity — in which the order of approximation can be varied within an element, thus allowing a higher approximation order for elements presenting higher errors in the mesh —, stating that the development of higher order 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements was still an open issue when the paper was published.
The present work proposes a hierarchical implementation of high order 𝐻(𝑐𝑢𝑟𝑙; Ω) elements for the modal analysis of dielectric waveguides. In what concerns computational aspects, the FE scheme is implemented in the FEM framework NeoPZ (DEVLOO, P., 1997). The algorithms in NeoPZ allied with the hierarchic characteristic of the proposed elements allows simplifies the application of hp-adaptive algorithms in the scheme (DEVLOO, P. R. B.; BRAVO; RYLO, 2009; CALLE; DEVLOO, P. R.; GOMES, 2015).
1.1
Objectives
The present work has the following objectives:
1 To develop and implement shape functions for 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming spaces that can be used in the modal analysis of inhomogeneous and transverse-anisotropic waveguides. The FEM formulation applied uses the 𝐻(𝑐𝑢𝑟𝑙; Ω) and 𝐻1(Ω) elements to approximate the transverse and the longitudinal components of the electric field, respectively, in the cross-section of a waveguide. Open waveguides are dealt with the use of uniaxial perfectly matched layers (UPMLs) (SACKS et al., 1995);
2 The developed shape functions are to be implemented in an hierarchical fashion in order to provide an easy integration in p-adaptive algorithms;
3 To analyze the efficiency of the developed strategy and the application of high order elements in waveguide problems. In order to evaluate its performance and robustness, waveguides of increasing complexity are studied, comparing the obtained results with results obtained by the CEM community for a fair assessment of the scheme.
1.2
Organization the contents
1.2. ORGANIZATION THE CONTENTS 17
1.2.1
Body
In Chapter 2, the waveguide problem is posed as a boundary-value problem. Interface and boundary conditions are discussed, and a variational form of the problem is derived.
Chapter 3 dwells on the variational form of the waveguide problem and
intro-duces the concept of approximation spaces, reviewing the relevant Hilbert spaces for the present work. After a comparison of different formulations for the waveguide problem, the
𝐻(𝑐𝑢𝑟𝑙; Ω) × 𝐻1(Ω) formulation that is investigated in this work is derived and discussed. Then, the Finite Element Method is presented as a manner of generating an algebraic problem from the variational form through the spatial discretization of the domain, and the discretized form of the 𝐻(𝑐𝑢𝑟𝑙; Ω) × 𝐻1(Ω) formulation is derived.
Having the discretized form of the variational formulation and the 𝐻1(Ω) elements available from Philippe Remy Bernard Devloo, Bravo, and Rylo (2009), the missing piece for the analysis of the waveguide problem is the construction of the finite-dimensional
𝐻(𝑐𝑢𝑟𝑙; Ω) approximation space. In Chapter 4, the 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming spaces are
discussed: the construction of a Finite Element family of 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements via a reference element and the covariant Piola transform is thoroughly detailed following Nédélec (1980) and Schneebeli (2003). In sequence, the methodology for constructing the hierarchical 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming spaces is finally presented.
Chapter 5 is dedicated to the analysis, validation and applications of the proposed
Finite Element scheme. First, a model problem is presented for the verification of the implementation of 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements, showing the expected convergence rates.
The metallic rectangular waveguide is used as means of a first analysis of the performance of the developed elements in the 𝐻(𝑐𝑢𝑟𝑙; Ω)×𝐻1(Ω) formulation. Subsequently, a step-index optical fiber is analyzed for testing the scheme with curved elements and validating the implemented Perfectly Matched Layers (PMLs). In this tests, the importance of correctly describing the waveguide geometry when using higher order elements is brought to discussion.
In order to evaluate the performance of the scheme in real-world challenging applications, a Photonic Crystal Fiber (PCF) waveguide is analyzed and the obtained results are compared with the results presented in Po-Jui Chiang and Hung-Chun Chang (2011). Finally, an preliminary example making use of the hp-adaptivity capabilities of the scheme is exhibited, showing the reduction on the number of equations needed to obtain a satisfactory result, even though no proper refinement strategy has been developed in this work.
1.2.2
Appendix
Appendix A presents an overview of the PMLs, a strategy that allows the
truncation of the computational domain when analyzing problems of radiation/scattering, or, as it is the case in the present work, the analysis of open waveguides. The mathematical principles behind the PMLs are discussed, as well as the methodology for the choice of the PMLs parameters in the experiments of this work.
Finally, Appendix B exposes aspects regarding the code implementation. The task of implementing the 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements in the NeoPZ framework is described, providing a brief look on how new families on Finite elements can be incorporated in NeoPZ. Topics concerning the code precision and performance are discussed, such as the dimensionless Helmholtz Equation, the imposition of homogeneous Dirichlet boundary conditions and the choice of solvers and matrix storage format. Lastly, the developed user interface of the software is shown.
19
2 A mathematical characterization of the
waveguide problem
The mathematical modeling of a physical phenomenon is a step of fundamental importance in its study, for it is the description of the scenario one desires to understand. In this chapter, the boundary-value problem that represents the waveguide is introduced, and it is shown how it applies to a diversity of waveguide configurations, regardless of anisotropy of the dielectrics, losses and so forth; and the relevant boundary and interface conditions for the present work are described. Finally, a variational form of the problem is presented, given its adequacy for the Finite Element Method, as it will be shown in Section 3.5.
2.1
The partial differential equation
The inhomogeneous Helmholtz wave equation for the electric field 𝐸 : R3 ⊃ C3 can be derived from the time-harmonic Maxwell’s equations, along with the constitutive relations between the electromagnetic fluxes and fields.
∇×[︁Û⊗1(∇× 𝐸)]︁⊗ æ2𝜖𝐸 = 0 . (2.1)
In Equation (2.1), Û and 𝜖 are tensors representing, respectively, the magnetic permeability and the electric permittivity of the space, and æ is the angular frequency of the electromagnetic wave. Unless stated otherwise, all development from now on do not assume conditions such as isotropy or homogeneity to hold.
Let 𝑘0 denote the wave propagation constant in vacuum, and Û0 = 𝐼Û0 and 𝜖0 = 𝐼𝜖0 represent the constant constitutive properties in such medium. These quantities are related to the angular frequency by the equation:
𝑘02 = æ2Û0𝜖0. (2.2)
Applying Equation (2.2) in 2.1 results in
∇×[︁Û⊗r1(∇× 𝐸)]︁⊗ 𝑘02𝜖r𝐸 = 0 , (2.3)
which is the equation from which the waveguide analysis will be developed. The fact that one can write Ó = ÓrÓ0, (Ó = Û, 𝜖), i.e., write the permittivity(permeability) of a general medium as a relative permittivity(permeability) is used in this equation and it will be adopted throughout the present work
Assuming that the waveguide has a cross-section Ω contained in the 𝑥𝑦-plane and its longitudinal axis is along the 𝑧 direction, a field 𝐸 propagating in the waveguide can be written, with the spatial dependencies explicitly stated, as
𝐸(𝑥, 𝑦, 𝑧) = 𝐸(𝑥, 𝑦) 𝑒(⊗jβz). (2.4)
The electric fields of Equation (2.4) are the quantities of interest in waveguide problems, along with the propagation constant in the 𝑧-direction Ñ, for waveguides are structures that aim to guide an electromagnetic wave, as opposed to devices such as antennas, which irradiate waves. Given that the propagation constant 𝑘0 is a function of the frequency of the field and Û and 𝜖 are constitutive parameters of the media in Ω, Equations 2.3 and 2.4 can be developed to express an eigenvalue problem with 𝐸 as its eigenvector and Ñ as its eigenvalue (given a Ñ value, of course, 𝑘0 can be an eigenvalue). The study of the eigenvalues and eigenvectors of a waveguide, in the context of electromagnetics, is called modal analysis. For the description of the waveguide problem to be complete, the boundary conditions for the waveguide must be analyzed.
2.2
The boundary and interface conditions
If the waveguide cross-section Ω ⊆ R2 presents a dielectric discontinuity as repre-sented in Figure 2.1, the electric field must satisfy the condition
(𝐸1⊗ 𝐸2) × 𝑛 = 0 , in Ω, (2.5)
meaning that the tangential component of the electric field is continuous between different dielectric media (BALANIS, 1989). Although the interface condition is obtainable directly from Faraday’s law of induction, it is explicitly stated here not only for clarity purposes, but for its relationship with the finite-dimensional 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming subspace that will be introduced in Section 3.2, and will be used in the FEM formulation of the waveguide problem.
The boundary conditions considered in this chapter will be restricted to the perfect
electric conductor(PEC) and the perfect magnetic conductor(PMC) conditions. Let Γ denote the boundary of Ω, and let the subscripts of Equations 2.6 and 2.7 refer to the boundary condition of a section of Γ such that Γ = ΓP EC∪ ΓP M C and ΓP EC ∩ ΓP M C = ∅.
𝐸×𝑛 = 0 on ΓP EC, (2.6)
(∇× 𝐸) ×𝑛 = 0 on ΓP M C. (2.7)
Equation (2.6) states that the tangential component of the electric field vanishes on a PEC. Equation (2.7), allied with Maxwell’s equations, shows that the tangential
2.3. THE VARIATIONAL FORM 21
Figure 2.1 – An interface between two dielectric media with different constitutive parame-ters.
component of the magnetic field vanishes on a PMC. The modal analysis on a waveguide will not deal with sources of any kind, so the stated boundary conditions are sufficient for most cases, provided that the waveguide shielding is assumed to be a perfect conductor. However, one must consider the case of open waveguides, i.e., structures that are not revolved by any shielding, as is the case of microstrip lines, optical fibers and others. In computational electromagnetics, such cases are generally dealt with the application of perfectly matching layers(PMLs)(BERENGER, 1994). This subject is discussed in Appendix A, but for now it suffices to say that the PML can be interpreted as a fictitious absorbing material that surrounds a truncated computational domain and presents little or no reflection of the incoming electromagnetic waves.
Now that the waveguide problem is characterized as a boundary-value problem, the variational form of the problem must be presented before applying the FEM method.
2.3
The variational form
A solution for Equation (2.3) must be second-order differentiable, due to the term ∇× ∇× 𝐸 appearing in the equation. While methods such as the Finite Difference Method would work on the PDE directly, this is not the case for a Finite Element Method formulation, that is based on a variational form of the problem. Therefore, one ought to use the so-called weak form of the problem, in the sense that some restrictions are
relaxed, for the method to be applied. This is usually done by the Ritz method or the Galerkin method (MONK, 2003b). In this work, the latter one is chosen. In this section, the variational formulation is presented and discussed.
The formulation is obtained multiplying Equation (2.3) by the complex conjugate of a sufficiently smooth vector function 𝜙 (the criteria on smoothness will be discussed in the next section) and integrating over Ω.
∫︁ Ω [︁ ∇×(︁Û⊗r1∇× 𝐸)︁≤ 𝜙*⊗ 𝑘20𝜖r𝐸≤ 𝜙 *]︁ 𝑑Ω = 0 . (2.8)
Where the superscript * denotes the complex conjugate. Making use of the vector identity ∇≤ (𝐴 × 𝐵) = 𝐵 ≤ ∇× 𝐴 ⊗ 𝐴 ≤ ∇× 𝐵, and decomposing the boundary Γ in Γ = ΓP EC+ ΓP M C, one can write:
∫︁ Ω [︁(︁ Û⊗r1∇× 𝐸)︁≤ ∇× 𝜙*⊗ 𝑘02𝜖r𝐸≤ 𝜙 *]︁ 𝑑Ω + ∫︁ ΓP EC [︁(︁ Û⊗r1∇× 𝐸)︁× 𝜙*]︁≤ 𝑛 𝑑𝑠 + ∫︁ ΓP M C [︁(︁ Û⊗r1∇× 𝐸)︁× 𝜙*]︁≤ 𝑛 𝑑𝑠 = 0 , (2.9)
where Gauss’s theorem was used to take the last two integrals to the boundaries. The invariance over circular shifts of the scalar product of the boundaries integrals allows Equation (2.9) to be rewritten as ∫︁ Ω [︁(︁ Û⊗r1∇× 𝐸 )︁ ≤ ∇× 𝜙*⊗ 𝑘02𝜖r𝐸≤ 𝜙 *]︁ 𝑑Ω + ∫︁ ΓP EC (𝑛 × 𝜙* ) ≤(︁Û⊗r1∇× 𝐸 )︁ 𝑑𝑠+ ∫︁ ΓP M C (𝑛 × 𝜙* ) ≤(︁Û⊗r1∇× 𝐸)︁ 𝑑𝑠 = 0 . (2.10)
Choosing 𝜙 such that it satisfies 𝑛 × 𝜙* = 0 on Γ
P EC: ∫︁ Ω [︁(︁ Û⊗r1∇× 𝐸)︁≤ ∇× 𝜙*⊗ 𝑘02𝜖r𝐸≤ 𝜙 *]︁ 𝑑Ω + ∫︁ ΓP M C (𝑛 × 𝜙* ) ≤(︁Û⊗r1∇× 𝐸)︁ 𝑑𝑠 = 0 . (2.11) It is clear that the solution of (2.11) is allowed to be less smooth than the analytic solution of (2.3). Equation (2.3) must be satisfied at every point of Ω, while the solution of (2.11) is sought in an integral sense. Also, in order to satisfy (2.3), a solution must be at least twice continuously differentiable, another factor relaxed in 2.11. Monk (2003b) shows that Equation (2.11), for a Lipschitz-bounded domain Ω, has a unique solution for an appropriate choice of function space for 𝜙1. Finally, applying Equation (2.4) in Equation (2.11) allows to state the variational form of the problem as:
Find non-trivial (Ñ, 𝐸(𝑥, 𝑦)) ∈ (C × 𝒳 ) such that:
∫︁ Ω {︁[︁ Û⊗r1∇× (︁ 𝐸𝑒(⊗jβz))︁]︁≤ ∇× 𝜙*⊗ 𝑘02𝜖r𝐸𝑒(⊗jβz)≤ 𝜙 *}︁ 𝑑Ω + ∫︁ ΓP M C (𝑛 × 𝜙* ) ≤[︁Û⊗r1∇× (︁ 𝐸𝑒(⊗jβz))︁]︁ 𝑑𝑠 = 0 , ∀𝜙 ∈ 𝒳 , (2.12) 1
2.3. THE VARIATIONAL FORM 23
.
Equation (2.12) shall be used on the next chapter to develop a Finite Element formulation for the waveguide problem. The function space 𝒳 has not been defined yet: all that has been said about it up to this point is that its functions satisfy 𝜙 ×𝑛 = 0, ∀𝜙 ∈ 𝒳 and some smoothness criteria. Its choice is of fundamental importance for the development of this work. Its characteristics and a systematic way of constructing it will be discussed in the next two chapters.
3 Relevant function spaces, formulations for
the waveguide problem and Finite Element
aspects
The Finite Element Method(FEM) will be presented in this chapter. A brief overview of the method based on Ciarlet (1978) will be given, demonstrating how it allows a solution for the variational form of a boundary-value problem to be approximated based on a spatial discretization of the domain and functions with local support. These functions reside in finite-dimensional subspaces of Hilbert spaces, so the relevant function spaces for the current work will be introduced. Different FEM formulations for the waveguide will be presented and discussed. The chosen formulation for this work will then be derived from the variational problem.
3.1
The choice of an approximation space
The Galerkin method provides a variational form of a boundary-value problem. In the context of this work, Equation (2.12) is a variational form for the waveguide problem. However, there are no means to solve it without gaining more insight on the function space 𝒳 in which a solution is being sought.
Equation (2.12) was chosen to finish Chapter 2 as it introduces the concept of a variational form for the waveguide problem when combined with Equation (2.4). From that variational form, different formulations for the waveguide problem will be developed in this chapter. These different mathematical representations will arise from the choice of the state variable being approximated and from simplifications and assumptions on the physical phenomena(if one is seeking TE modes only, the variational form can be further simplified based on the knowledge that 𝐸z = 0, and it could also be formulated in terms of
𝐻z as opposed to the other components of 𝐸) and from the choice of the approximation
space 𝒳 .
The choice of the function space 𝒳 is heavily dependent on the problem in question. When approximating an electric field, one may seek a function space containing functions with a well-defined curl, for a better representation of the magnetic field. If the boundary-value problem represents a fluid flow, on the other hand, it might be more interesting to ensure that the functions in 𝒳 present a well-defined divergent. It is worth noting that interface conditions such as (2.5), although obtainable from the original form of the problem, are not present in the variational form of the problem. Therefore, they are either
3.2. RELEVANT HILBERT SPACES FOR THE WAVEGUIDE PROBLEM 25
guaranteed by certain characteristics of 𝒳 , or the variational form must be modified in order to contemplate these needs.
Before the discussion of different formulations for the waveguide problem, some function spaces that will be used in the formulations will be presented. These spaces belong to a class called Hilbert spaces, and are widely used in FEM formulations for a wide range of problems.
3.2
Relevant Hilbert spaces for the waveguide problem
On this section, some relevant function spaces will be introduced. First and foremost, the spaces are:
𝐿2(Ω) = {︂ ã : Ω ⊃ C ⧹︃ ⧹︃ ⧹︃ ⧹︃ ∫︁ Ωãã * 𝑑Ω < ∞ }︂ , 𝐻1(Ω) ={︁ã ∈ 𝐿2(Ω)⧹︃⧹︃ ⧹︃∇ ã ∈ [𝐿2(Ω)]2 }︁ , 𝐻(𝑐𝑢𝑟𝑙; Ω) ={︁ã∈ [𝐿2(Ω)]2⧹︃⧹︃ ⧹︃∇× ã ∈ 𝐿2(Ω) }︁ . (3.1) (3.2) (3.3) ‖ã‖2L2 = ∫︁ Ωãã * 𝑑Ω, ‖ã‖2 H1 = ‖ã‖2L2 + ‖∇ ã‖2L2, ‖ã‖2Hcurl = ‖ã‖ 2 L2 + ‖∇× ã‖2L2. (3.4) (3.5) (3.6) Equation (3.1) introduces the Lebesgue space 𝐿2(Ω) over the body of complex numbers. This function space can be thought of as the space of square-integrable functions, and its norm is presented in Equation (3.4), where ã* represents the complex-conjugate of
ã. The 𝐿2(Ω) space has a vector counterpart whose definition is very similar to (3.1). The 𝐻1(Ω) space is defined in (3.2). An analysis of (3.5) shows that a polynomial function ã ∈ 𝐻1(Ω) must be continuous for its gradient to be square-integrable in Ω. With the 𝐻1(Ω) space, the concept of the trace of a function is introduced. For a domain Ω with a sufficiently smooth boundary Γ, the trace of a function, in the 𝐻1(Ω) sense, is defined by the continuous mapping (CIARLET, 1978):
tr : ã ∈ 𝐻1(Ω) ⊃ tr ã = ã ♣
Γ∈ 𝐿2(Γ)1. (3.7)
Finally, the 𝐻(𝑐𝑢𝑟𝑙; Ω) function space is presented in Equation (3.3). A function
𝜙 ∈ 𝐻(𝑐𝑢𝑟𝑙; Ω) presents a square-integrable curl. The 𝐻(𝑐𝑢𝑟𝑙; Ω) also has the notion of
1
The trace of an H1
(Ω) function resides in the H12(Γ) space. However, since H12(Γ) ⊆ L2(Γ), this
trace. The trace of a function 𝜙, in the 𝐻(𝑐𝑢𝑟𝑙; Ω) sense, is defined by:
tr : 𝜙 ∈ 𝐻(𝑐𝑢𝑟𝑙; Ω) ⊃ tr 𝜙 = 𝑛 × 𝜙 ♣Γ∈ 𝐿2(Γ)2, (3.8) where 𝑛 is the normal unit vector of Γ. Therefore, this trace is defined in terms of the tangential component of 𝜙 over the boundary.
While the 𝐻1(Ω) and the 𝐻(𝑐𝑢𝑟𝑙; Ω) had their traces defined, this was not done for the 𝐿2(Ω) space. The reason is that a general 𝜙 ∈ 𝐿2(Ω) space is not sufficiently smooth in order to be able to analyze the restriction of a function over the boundary. In fact, 𝜙 may not even be defined in all points of Ω, let alone Γ.
For 𝐻1(Ω) and 𝐻(𝑐𝑢𝑟𝑙; Ω), useful subspaces will also be defined:
𝐻01(Ω) ={︁ã∈ 𝐻1(Ω)⧹︃⧹︃ ⧹︃tr ã = 0 }︁ , 𝐻0(𝑐𝑢𝑟𝑙; Ω) = ¶ ã ∈ 𝐻0(𝑐𝑢𝑟𝑙; Ω) ♣ tr ã = 0 ♢ . (3.9) (3.10) The spaces defined in Equations 3.9 and 3.10 are useful for embedding homogeneous boundary conditions because of their vanishing trace, and for this property they are often employed in variational formulations.
3.3
Some formulations for the waveguide problem
In this section, different formulations for the waveguide problem will be presented. All the formulations are obtained from Equation (2.12), the variational form of the problem. As aforementioned, the differences among the formulations may arise by eventual simplifications or modifications performed on (2.12), or simply by the choice of the approximation space. The following list does not aim to be a complete guide of possible formulations for the waveguide, but to provide an introductory discussion on the choice of an approximation space, and to exhibit the reasons for the chosen formulation for this work. For the sake of notation, the domain Ω is assumed to contain only isotropic media, but apart from the two first formulations, that do not support anisotropic media3, the procedures for obtaining the anisotropic formulation are basically the same to the isotropic case, although lengthier.
3.3.1
An
𝐻
1(Ω) formulation for homogeneous waveguides
In Peter Peet Silvester (1969) there is one of the first applications of FEM in the context of computational electromagnetics, and it is precisely on the analysis of waveguides, 2
Similarly to the scenario in (3.7), this trace actually resides in H−12(Γ).
3
Actually, the formulation presented in Section 3.3.2 can treat anisotropic media, as long as their constitutive parameters can be written in a diagonalized form.
3.3. SOME FORMULATIONS FOR THE WAVEGUIDE PROBLEM 27
albeit restricting the discussion to isotropic and homogeneous waveguides. Such formulation can be obtained by simplifying (2.12): since homogeneous waveguides will present only
TE(or TM ) solutions, the variational form can be significantly simplified if expressed in terms of the longitudinal component of the magnetic (or electric) field, noting that Equation (2.12) can be easily modified to be expressed in terms of the magnetic field. The resulting FEM formulation, for TM modes, and a PEC boundary condition, reads as following:
Find non-trivial (𝑘t, 𝐸z) ∈ (C × 𝐻01(Ω)) such that:
∫︁ Ω∇t𝐸z≤ ∇t𝜙 * 𝑑Ω = ‖𝑘t‖2 ∫︁ Ω𝐸z𝜙 * 𝑑Ω, ∀𝜙 ∈ 𝐻1 0(Ω), (3.11)
where the subscript t denotes the transverse components, and 𝑘t is given by the dispersion
relation 𝑘t = 𝑘20Ûr𝜖r⊗ Ñ2. Given that none of the terms in Equation (3.11) depend on
the frequency, the propagation constant Ñ is obtained by the dispersion relation. The boundary condition is embedded on the choice of 𝐻1
0(Ω) ⊆ 𝐻1(Ω), and this is often the case with essential boundary conditions.
The formulation for TE modes reads as:
Find non-trivial (𝑘t, 𝐻z) ∈ (C × 𝐻1(Ω)) such that:
∫︁ Ω∇t𝐻z≤ ∇t𝜙 * 𝑑Ω ⊗ ∫︁ Γ𝜙 * ∇t𝐻z♣P EC≤ 𝑛 𝑑Γ = ‖𝑘t‖2 ∫︁ Ω𝐻z𝜙 * 𝑑Ω, ∀𝜙 ∈ 𝐻1(Ω). (3.12)
On Equation (3.12), ∇ 𝐻z♣P EC stands for the value that ∇ 𝐻z is expected to satisfy
at a PEC, i.e., ∇ 𝐻z♣P EC = 0. Therefore, this integral vanishes, and the only difference
between the TE and the TM is the space in which the variable is being sought, 𝐻1 0(Ω) and 𝐻1(Ω). Since the enforcement of boundary conditions in the following formulations will follow the same principle, the formulations will be presented for the PEC boundary condition in order to avoid repetition.
The formulations in Equations 3.11 and 3.12 are of simple implementation, and are based on 𝐻1(Ω), an approximation space whose application in FEM is well known since the earlier years of the method. However, the homogeneity of the guide imposes a severe limitation of these formulations for practical applications.
3.3.2
An
𝐻
1(Ω) × 𝐻
1(Ω) formulation for inhomogeneous waveguides
In Ahmed and Daly (1969) and Csendes and P. Silvester (1970), a new formulation is introduced in order to overcome this limitation. This formulation is known as 𝐸z-𝐻z
formulation, for the variables being approximated are the longitudinal components of both the magnetic and electric fields. The decomposition of the variational form (2.12) in terms of 𝐸z and 𝐻z is rather complicated: a simpler way to obtain this formulation is
manipulation. After the same procedure is applied, in a similar fashion, to the magnetic field wave equation, one obtains:
Find non-trivial (𝑘t, 𝐸z, 𝐻z) ∈ (C × 𝐻01(Ω) × 𝐻1(Ω)) such that:
∫︁ Ω 1 𝑘2 t ⋃︀ ⨄︀𝜖∇ 𝐸z≤ ∇ 𝜙*e+ Û ∇ 𝐻z≤ ∇ 𝜙*h +Ñ æ (︃ 𝜕𝐸z 𝜕𝑥 𝜙* h 𝜕𝑦 ⊗ 𝜕𝐸z 𝜕𝑦 𝜙* h 𝜕𝑥 + 𝜕𝐻z 𝜕𝑥 𝜙* e 𝜕𝑦 ⊗ 𝜕𝐻z 𝜕𝑦 𝜙* e 𝜕𝑥 ⎜ ⊗ 𝑘t2𝜖𝐸z𝜙 * e ⊗ 𝑘 2 tÛ𝐻z𝜙 * h ⋂︀ ⋀︀𝑑Ω, ∀𝜙e ∈ 𝐻01(Ω) , 𝜙h ∈ 𝐻1(Ω). (3.13)
The boundary conditions in the 𝐸z-𝐻z formulation are the same as the boundary
conditions on (3.11) and (3.12). Equation (3.13) can be manipulated in order to generate an eigenvalue problem having 𝑘0 as its eigenvalue for a given Ó = Ñ/𝑘0. This formula-tion, although more complicated, is able to deal with more realistic scenarios such as inhomogeneous media. One major disadvantage of this formulation, however, is that its
discretization with the Finite Element Method lead to the presence of spurious modes. Spurious modes are solutions for the waveguide problem that do not represent a physical solution, i.e., a solution that is allowed to exist in the numerical method that do not satisfy the original boundary-value problem. This issue is not exclusive of the 𝐸z-𝐻z
formulation, and even though it is a problem known for decades (CSENDES; SILVESTER, P., 1970), it is not one of a simple solution: distinguishing the spurious modes from the physical ones can be an exhausting task, let alone eliminate them.
For (3.13), it is known that the spurious modes are associated with the interface condition (2.5): the longitudinal components of the fields are continuous, as discussed in Section 3.2, but the continuity of the transverse components is imposed weakly, as the only variables present in the formulation are 𝐸z and 𝐻z. The enforcement of (2.5) by means of
introduction of Lagrange multipliers in Equation (3.13) is an option for eliminating the spurious modes in this formulation (JIN, J.-M., 2014).
3.3.3
An
[𝐻
1(Ω)]
2formulation for inhomogeneous waveguides
The 𝐸z-𝐻z formulation presents some drawbacks, such as the inability to deal with
general anisotropic media and the issue of spurious solutions. Approximating (2.12) with a field 𝐸 ∈ [𝐻1(Ω)]3, as it may seem intuitive at this point, has some disadvantages: in the same way that the spurious modes in the 𝐸z-𝐻z formulation were related to interface
conditions, a FEM formulation using functions in [𝐻1(Ω)]3 would suffer from difficulties on imposing interface and boundary conditions, for different conditions are required for the tangential and for the normal components of an electromagnetic field.
3.4. AN H(CU RL; Ω)× H1
(Ω) FORMULATION FOR THE WAVEGUIDE PROBLEM 29
One solution is to impose the divergence conditions from the Maxwell’s equations using a penalty term on the variational form4. An alternative is to use the divergent conditions to eliminate the longitudinal component of the fields, and write the variational form by means of the transverse components (GELDER, 1970). The formulation in terms of the magnetic field reads as:
Find non-trivial (Ñ, 𝐻t) ∈ (C × [𝐻1(Ω)] 2) such that: ∫︁ Ω [︁(︁ 𝜖⊗r1∇t× 𝐻t )︁ ≤ ∇t× 𝜙 * ⊗(︁𝑘02Ûr⊗ Ñ2𝜖 ⊗1 r )︁ 𝐻t≤ 𝜙 * + Û⊗1 r ∇t≤ (Ûr𝐻t) ∇t≤ (︁ 𝜖⊗1 r 𝜙 *)︁]︁ 𝑑Ω ⊗∑︁ i ∫︁ Γi Û⊗1 r 𝜖 ⊗1 r ∇≤ (Ûr𝐻t) 𝜙 * ≤ 𝑛, 𝑑Γi = 0 ,∀𝜙 ∈ [︁𝐻1(Ω)]︁2, (3.14)
where 𝑖 denotes the 𝑖-th subdomain of Ω where Ûr and 𝜖r are continuous.
The formulation of Equation (3.14) eliminates the spurious modes observed in the 𝐸z-𝐻z formulation and it is extensible for anisotropic media. However, it still suffers
from difficulties regarding interface and boundary conditions: the PEC boundary for the magnetic field represents a natural boundary condition, which is approximately satisfied. Unless explicitly enforced, this formulation will also suffer from spurious modes, although for different reasons (JIN, J.-M., 2014). This formulation also suffers with sharp metallic edges, requiring careful measures to be taken.
These limitations and considerations on scalar and vector formulations for the waveguide represent serious drawbacks on applying the Finite Element Method for the waveguide problem. For this reason, among others, one popular approach is to use
𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements, and a formulation for the waveguide problem using
this approximation space will be detailed in the next sections, for it is the formulation used throughout this work.
3.4
An
𝐻(𝑐𝑢𝑟𝑙; Ω)
×𝐻
1(Ω) formulation for the waveguide problem
In this section, a different formulation will be presented, following a brief discussion of the 𝐻(𝑐𝑢𝑟𝑙; Ω) space in the context of electromagnetics. The formulation will be derived and the imposition of interface and boundary conditions will be detailed.Nédélec (1980) discusses the use of 𝐻(𝑐𝑢𝑟𝑙; Ω) functions in the context of electro-magnetics, but its application on the waveguide problem would only appear by the end of 4
There is another option: the divergent condition would be automatically satisfied if the approximation space had functions differentiable up to the second order (JIN, J.-M., 2014). For the complications associated with this requirement on the approximation space, this approach is not the most preferred one.
that decade. Equation (3.3) seems promising in the context of electromagnetic fields, for the well-defined curl of the functions in this space, and Equation (3.8) suggests that the difficulty on imposing boundary and interface conditions that appeared on the previous formulations could be greatly alleviated. This is a major advantage over the previous formulations. The resulting formulation, presented in Lee, Sun, and Cendes (1991), will be derived on this section. First, the variational form of the waveguide problem will be repeated for their importance on this section.
Find non-trivial (Ñ, 𝐸(𝑥, 𝑦)) ∈ (C × 𝒳 ) such that:
∫︁ Ω {︁[︁ Û⊗r1∇×(︁𝐸𝑒(⊗jβz))︁]︁≤ ∇× 𝜙* ⊗ 𝑘2 0𝜖r𝐸𝑒(⊗jβz)≤ 𝜙 *}︁ 𝑑Ω + ∫︁ ΓP M C (𝑛 × 𝜙* ) ≤[︁Û⊗r1∇×(︁𝐸𝑒(⊗jβz))︁]︁ 𝑑𝑠 = 0 , ∀𝜙 ∈ 𝒳 , (2.12) for Ω ∈ R2.
For this formulation, the first step is to decompose the electric field in its transversal and longitudinal components, similarly as to what was done in the formulation presented in Section 3.3.3. However, this time, this procedure is not done in order to eliminate the
𝐸z variable. This formulation is able to deal with materials with transverse-anisotropy,
i.e., a restriction will be imposed over Ûr and 𝜖𝑟 such that they can be represented in
matricial form as:
Ûr= ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ Ûxx Ûxy 0 Ûyx Ûyy 0 0 0 Ûzz ⋂︀ ⎥ ⎥ ⎥ ⋀︀ = ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 1 0 0 1 0 0 ⋂︀ ⎥ ⎥ ⎥ ⋀︀Ûxy ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 1 0 0 1 0 0 ⋂︀ ⎥ ⎥ ⎥ ⋀︀ T + ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 0 0 1 ⋂︀ ⎥ ⎥ ⎥ ⋀︀Ûzz ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 0 0 1 ⋂︀ ⎥ ⎥ ⎥ ⋀︀ T, (3.15) 𝜖r = ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 𝜖xx 𝜖xy 0 𝜖yx 𝜖yy 0 0 0 𝜖zz ⋂︀ ⎥ ⎥ ⎥ ⋀︀ = ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 1 0 0 1 0 0 ⋂︀ ⎥ ⎥ ⎥ ⋀︀ 𝜖xy ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 1 0 0 1 0 0 ⋂︀ ⎥ ⎥ ⎥ ⋀︀ T + ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 0 0 1 ⋂︀ ⎥ ⎥ ⎥ ⋀︀𝜖zz ⋃︀ ⋁︀ ⋁︀ ⋁︀ ⨄︀ 0 0 1 ⋂︀ ⎥ ⎥ ⎥ ⋀︀ T. (3.16)
Even though this formulation is not able to deal with general anisotropic materials, in practical terms the restriction of Equations (3.15) and (3.16) do not represent a significant limitation of the method. Writing Equation (2.12) in terms of (3.15) and (3.16):
3.4. AN H(CU RL; Ω)× H1
(Ω) FORMULATION FOR THE WAVEGUIDE PROBLEM 31
Find non-trivial (𝑘2 0, 𝐸t, 𝐸z) ∈ (C × 𝐻0(𝑐𝑢𝑟𝑙; Ω) × 𝐻01(Ω)) such that: ∫︁ Ω [︁ Û⊗zz1(∇t× 𝐸t) ≤ (∇t× 𝜙) * + Û⊗1 xy (∇t𝐸z+ 𝑗Ñ𝐸t) ≤ (∇t𝜙+ 𝑗Ñ𝜙) *]︁ dΩ = ∫︁ Ω⊗𝑘 2 0(𝜖xy𝐸t≤ 𝜙 * + 𝜖zz𝐸z𝜙 * ) dΩ , ∀𝜙 ∈ 𝐻0(𝑐𝑢𝑟𝑙; Ω) , 𝜙 ∈ 𝐻01(Ω). (3.17)
Equation (3.17) shows the variational form with the decomposed electric field and can be seen as an eigenvalue problem on both 𝑘2
0 and Ñ. One may seek to impose a value of
𝑘0 and let Ñ be the eigenvalue of (3.17), which is often more desirable in practical settings than setting a fixed value of Ñ and seek 𝑘2
0. However, (3.17) is a quadratic eigenvalue problem on Ñ, which is not desirable for numerical purposes. In order to alleviate this issue, Lee, Sun, and Cendes (1991) introduces the following transformations:
𝑒t= Ñ𝐸t, (3.18)
𝑒z = ⊗𝑗𝐸z. (3.19)
Applying (3.18) into (3.17), we have: Find non-trivial (Ñ2, 𝑒 t, 𝑒z) ∈ (C × 𝐻0(𝑐𝑢𝑟𝑙; Ω) × 𝐻01(Ω)) such that: ∫︁ Ω {︂ Û⊗1 zz (∇t× 𝑒t) ≤ (∇t× 𝜙) * ⊗ 𝑘20𝜖xy𝑒t≤ 𝜙 * + Ñ2[︁Û⊗1 xyS(∇t𝑒z+ 𝑒t) ≤ (∇t𝜙+ 𝜙) * ⊗ 𝑘02𝜖zz𝑒z𝜙 *]︁}︂dΩ = 0, ∀𝜙 ∈ 𝐻0(𝑐𝑢𝑟𝑙; Ω) , 𝜙 ∈ 𝐻01(Ω), (3.20) where Û⊗1 xyS is defined as: Û⊗xy1S = Û ⊗1 xy ⋃︀ ⨄︀0 1 1 0 ⋂︀ ⋀︀. (3.21)
As anticipated, Equation (3.20) is an eigenvalue problem on Ñ2 with (𝑒
𝑡, 𝑒z) as
its eigenfunctions. However, no framework for obtaining its solution was presented yet: The function spaces 𝐻0(𝑐𝑢𝑟𝑙; Ω) and 𝐻01(Ω) are infinite-dimensional spaces, so even if one finds a candidate solution (Ñ2, 𝑒
t, 𝑒z), there are no means for testing this solution for every
function in 𝐻0(𝑐𝑢𝑟𝑙; Ω) and 𝐻01(Ω), as Equation (3.20) requires. Therefore, this variational form can be approximated seeking for a solution not in the function spaces 𝐻0(𝑐𝑢𝑟𝑙; Ω) and 𝐻1
0(Ω), but in finite-dimensional spaces 𝑈h ⊆ 𝐻0(𝑐𝑢𝑟𝑙; Ω) and 𝑉h ⊆ 𝐻01(Ω). Still, a
methodology for constructing the spaces 𝑈h and 𝑉h has not been presented yet, and that is
precisely the objective of the Finite Element Method, which is the subject of the following section.
Figure 3.1 – A partition 𝒯h over a domain Ω. Due to the curved nature of Ω, the resulting
domain is denoted by Ωh.
3.5
The Finite Element Method: an overview
This Section will discuss the generation of the finite-dimensional spaces 𝑈h and 𝑉h.
For purposes of generality, the discussion will develop over a general function space 𝒳 , except for commentaries regarding the specific spaces 𝐻0(𝑐𝑢𝑟𝑙; Ω) and 𝐻01(Ω).
Generating a basis for a finite-dimensional subspace 𝒳h ⊆ 𝒳 in an arbitrary domain
Ω is a challenging task which is addressed by the Finite Element Method (FEM). FEM can be regarded in the context of this work as a methodology for generating an basis for the subspace 𝒳h(Ω) that allows the variational form of a boundary-value problem to be
approximated in a discretized form as an algebraic problem. This procedure is done by a spatial discretization of Ω, i.e., Ω is divided into a finite number of closed subsets. A rigorous analysis of this discretization is performed by Ciarlet (1978), but for the objectives of this work it suffices to say that, for a given triangulation 𝒯h = ¶𝐾♢, the elements 𝐾
should not overlap and their union must be the closure of Ω. In the context of FEM, this partition is often called the geometrical meshing of Ω.
This procedure is illustrated in Figure 3.1: on this example, Ω is a domain with curvilinear boundaries. Due to this reason, the partition 𝒯his performed on an approximated
Ωh. Curved elements have been present in literature for decades (CIARLET, 1978), but
inexact geometrical representation is still present in many works, and for Maxwell’s problems is still an open field of discussion (MONK, 2003b). It should be noted that generally in FEM schemes without curved elements the geometry representation error tend to smaller values as the elements get smaller. For the purposes of this discussion, it will be assumed that Ω = ∪K∈𝒯h𝐾, except when explicitly noted.
The basis for the subspace 𝒳h(Ω) ⊆ 𝒳 (Ω) is then systematically constructed by
means of functions defined over the elements 𝐾. For this purpose, the concept of extension of a function must be introduced:
3.5. THE FINITE ELEMENT METHOD: AN OVERVIEW 33
For a function ã : 𝐾 ⊃ C, its extension over Ω, in this work, is defined as: ℰ(ã) : ∏︁ ⨄︁ ⋃︁ ã, ∀𝑥 ∈ ˚𝐾 0, ∀𝑥 ∈ Ω∖ ˚𝐾 , (3.22)
where ˚𝐾 denotes the interior of a given set 𝐾 ⊆ Ω.
The basis for 𝒳h(Ω) is then constructed by the sum of extensions of functions with
compact support on the elements 𝐾 ∈ 𝒯h, that is:
𝜙= ∑︁
K∈𝒯h
ℰ(𝜙k), 𝜙k∈ 𝒳h(𝐾) (3.23)
One important remark is that the extension as defined in (3.22) does not guarantee that a function 𝜙k ∈ 𝒳h(𝐾) belongs to 𝒳h(Ω), therefore, the sum presented in
Equa-tion (3.23) is also not guaranteed to belong to 𝒳h(Ω). A function ã ∈ 𝐻1(Ω), for instance,
is expected to satisfy some criteria of smoothness, since its gradient must be in 𝐿2(Ω), according to (3.2). Similarly, a function 𝜙 ∈ 𝐻(𝑐𝑢𝑟𝑙; Ω) has some requirements for its curl must be in 𝐿2(Ω).
In order to generate a function that belongs to a certain function space not only
locally (within the element) but globally, the Finite Element Method makes use of the
degrees of freedom(often abbreviated as dofs). The dofs are a set of linear functionals over a function space that will be used to construct the basis functions for the approximation
space. The dofs are a fundamental part of the FE methods. Ciarlet (1978), for instance, defines a Finite Element as: a geometry, a polynomial space and a set of dofs.
The specific details on how to construct the dofs are dependent on the function space they act upon, but for the purposes of this work it can be said that they will enforce some criteria of continuity, that they are linearly independent, and that they uniquely define a function in a given finite-dimensional space.
In order to illustrate the purpose of dofs, let us have two functions, ã1 ∈ 𝐻1(𝐾1) and ã2 ∈ 𝐻1(𝐾2), and let 𝐾1 and 𝐾2 be adjacent elements. An appropriate choice of dofs will guarantee that tr ã1 = tr ã2, by the definition of the trace operator in Equation (3.7). This will guarantee that functions with non-vanishing trace on the boundary of a given element are completed with a function defined in another element, as in Equation (3.23). This assures that the gradient of the global function is square-integrable. For a more detailed discussion on dofs on 𝐻1(Ω) functions, the reader is refereed to Ciarlet (1978).
Similarly, for constructing 𝐻(𝑐𝑢𝑟𝑙; Ω)-conforming elements, the main idea is related to the trace of the functions in the 𝐻(𝑐𝑢𝑟𝑙; Ω) sense, as defined in Equation (3.8). These functions also present a form of continuity between adjacent elements: the tangential component of a function 𝜙 ∈ 𝐻(𝑐𝑢𝑟𝑙; Ω) must be continuous over the interface between adjacent elements (NÉDÉLEC, 1980).
