Autore:
Chiara Pelletti
Relatori:
Prof. Agostino Monorchio
Prof. Raj Mittra
Prof. Giuliano Manara
Numerically Efficient Techniques for
Electromagnetic Scattering Calculations in the
Frequency Domain
Anno 2011
UNIVERSITÀ DI PISA
Scuola di Dottorato in Ingegneria “Leonardo da Vinci”
Corso di Dottorato di Ricerca in
Ingegneria dell’Informazione
To my husband and to my family,
Acknowledgements
I would like to thank my husband and my family who have always supported me and believed in me. Without the encouragement and support of Giacomo I would never made it through my way towards completing my PhD.
I would like to express my gratitude to Prof. Agostino Monorchio for his advices, his good heart and because he gave me the chance to grow with experiences abroad. I would like to express my hearty thanks to Prof. Raj Mittra who stood by my side, believing in me, stimulating me. He made grow my mind in a way I would never think possible, thanks to his infinite experience; I will never forget the countless discussions we had about electromagnetics and his patience; I will always hold him up as a professional model. I would also like to thank my friends and colleagues at the Electromagnetic and Communication Laboratory (ECL), Kadappan Panayappan for his kindness and helpfulness, Arash Rashidi, Dr. Jonathan Bringuier and all the members of my group at Penn State. Without their technical support and valuable contributions I would never be able to complete this dissertation.
I would also like to thank my colleagues at the Microwave and Radiation Laboratory (MRL) in Pisa.
Abstract
This thesis work represents a collection of numerical techniques, applied in the frequency domain, aimed at overcoming some of the issues emerging from conventional numerical methods. In particular, the method developed in the second chapter circumvents the need to employ the Green’s function to formulate the problem and having to deal with the singularities of the same in the Z matrix computation. Furthermore, since the formulation deals with closed-form expressions for the fields generated by the basis functions, as opposed to vector and scalar potentials typically employed in the conventional MoM, the so-called ‘low frequency’ problem, which plagues the latter owing to the presence of the 1/ω factor in the scalar potential term, is no longer a concern in this novel technique.
The third chapter will introduce the Dipole Moment (DM) formulation and its application to the solution of scattering problems. It will be demonstrated that the dipole moment representation of a scatterer neither suffers from the singularity problem associated with the Green’s functions nor does it experience any difficulties at low frequencies.
Dealing with multiscale objects, which have features that are both small and large compared to the wavelength, is highly challenging and often forces us to compromise the accuracy (relaxing the numerical discretization process when attempting to capture the small-scale features) in order to cope with the limited available resources in terms of CPU memory and time. In the last part of the elaborate, a new scheme combining the Recursive Update Frequency Domain (RUFD) method with the Dipole Moment (DM), will be deployed to solve multiscale problems in a numerically efficient manner.
The formulation will be demonstrated to be efficient both in terms of run time and memory requirement compared to conventional commercial solvers.
Sommario
Questo lavoro di tesi si prefigge lo scopo di sviluppare algoritmi numerici nel dominio della frequenza al fine di superare alcuni dei limiti che caratterizzano le tecniche convenzionali di analisi elettromagnetica. In particolare, nel primo capitolo, è riportata una formulazione MoM la quale, impiegando espressioni analitiche per il campo elettromagnetico radiato da un dipolo elettricamente corto, riesce ad evitare l’utilizzo di equazioni a potenziali misti e di conseguenza i relativi problemi a bassa frequenza e il trattamento della singolarità proprio della funzione di Green.
Nel terzo capitolo verrà introdotto l’algoritmo di tipo Dipole Moment e la sua applicazione alla risoluzione di problemi di scattering elettromagnetico. In particolare verrà dimostrato che la suddetta tecnica permette di risolvere agevolmente problemi caratterizzati da perdite e spessori finiti senza incorrere in matrici numericamente mal condizionate.
La soluzione di problemi di tipo multiscala, i quali coinvolgono contemporaneamente oggetti sia piccoli che grandi rispetto alla lunghezza d’onda, risulta spesso fortemente onerosa dal punto di vista computazionale costringendo a sacrificare l’accuratezza del risultato finale a causa delle limitate risorse di calcolo a disposizione. Al fine di superare i problemi precedentemente citati, nell’ultima parte dell’elaborato verrà introdotta una tecnica ibrida che combina il metodo Dipole Moment con un nuovo algoritmo ricorsivo nel dominio della frequenza, il RUFD. Tale metodo si è infatti dimostrato essere vantaggioso sia in termini di tempo che di occupazione di memoria rispetto ai principali solutori elettromagnetici disponibili in commercio.
List of Acronyms
EM: Electromagnetic MoM: Method of Moments
FDTD: Finite Difference Time Domain FEM: Finite Element Method
FM: Finite Method
CPU: Central Processing Unit IE: Integral Equation Method GF: Greens Function
EFIE: Electric Field Integral Equation PEC: Perfect Electric Conductor CFIE: Combined Field Integral Equation FMG: Fast Matrix Generation
EDM: Equivalent Dipole Moment DMA: Dipole Moment Approach DM: Dipole Moment
CBFM: Characteristic Basis Function Method RUFD: Recursive Update in the Frequency Domain CEM: Computational Electromagnetics
DoF: Degrees of Freedom
MFIE: Magnetic Field Integral Equation RWG: Rao Wilton Glisson
RCS: Radar Cross Section TE: Transverse Electric TM: Transverse Magnetic
CBF: Characteristic Basis Function PWS: Plane Wave Spectrum
SVD: Singular Value Decomposition RHS: Right Hand Side
RAM: Random Access Memory
DM/RUFD: Dipole Moment Recursive Update Frequency Domain PML: Perfectly Matched Layers
FIT : Finite Integration Technique ABC: Absorbing Boundary Conditions
Contents
Contents
i
Introduction
iii
1. Moment Method solution of Maxwell’s equations
1
1.1 Maxwell’s equations
. . .
11.2 Description of a scattering problem: source-field relationships
. . .
31.3 The surface equivalence principle
. . .
61.4 TheMethod of Moments
.
.
.
. . . .
111.4.1 Numerical solution to the EFIE
.
. . . .
151.4.2 Numerical solutionto the CFIE
.
. . . .
212. Efficient evaluation of the matrix elements in the context of the MoM 29
2.1 Limitations in the conventional MoM formulation
.
.
.
.
. . . .
292.1.1 Low-frequency breakdown
. . .
302.1.2 Numerical treatment
of
Green’s functions singularities. . .
312.2 Fields radiated by a rectangular basis function
. .
.
. . .
342.2.1 Closed-form expressions for the fields radiated by a sinusoidal current
.
34distribution 2.2.2 Fields radiated by a rooftop basis function
.
.
. . .
382.2.3 Generalized field expressions
. . .
412.2.4 Numerical solution to the CFIE
. . .
. . . .
442.3 Numerical results
. . . . .
.
. . . . .
.
. . . .
.
. . . .
472.3.1 3λ square conducting plate
. . .
.
. . . .
.
. . .
472.3.2 λ conducting cube
. . .
.
. . .
.
. . . .
.
. . .
502.3.3 λ/2 conducting cylinder
.
. . .
.
. . . .
.
. . . .
512.3.4 90° corner reflector
. . .
.
. . .
.
. . . .
.
. . . .
532.3.5 λ/4 dielectric cube
. . .
.
. . .
.
. . . .
.
. . .
542.3.6 Low-frequency performance
. . .
.
. . . .
.
. . . .
553. A new Dipole-Moment-based approach for solving MoM-type problems 59
3.1 Scattering by an electrically small sphere and dipole moments.
. . .
59ii
Contents
3.1.2 Dipole moments and scattering from a small dielectric sphere
. . . .
663.2 The Dipole Moment Approach (DMA)
.
. . . .
. . . .
. . . . .
683.3 The Characteristic Basis Function Method (CBFM) applied to the Dipole Moment Approach (DMA)
.
. . . .
. . . .
. . . .
. . .
773.3.1. CBFM basic theory
. . . .
. . . .
. . . .
. . . .
773.3.2. The CBFM applied to the DMA
. . . .
. . . .
. . . .
833.4 Numerical results
. . .
.
. . .
.
. . . .
.
. . . .
863.4.1. λ/4 in length λ/100 in diameter conducting wire
. . . .
863.4.2. λ/4 in length λ/100 in diameter conducting bend wire
. . . .
873.4.3. λ/4 on the side, λ/100 in length square dielectric plate
. . . .
893.4.4. Dielectric sphere λ/20 in radius
. .
.
. . . .
903.4.5. Dielectric coated conducting plate
. .
. . . .
914. On the hybridization of RUFD algorithm with DM for solving multiscale
problems
93
4.1 The Recursive Update in the Frequency Domain (RUFD) method
. . .
934.1.1. Convergence of the RUFD scheme
. . . .
. . . .
. . . .
964.2 The Hybrid DM/RUFD technique
. . .
974.2.1. Iterative DM/RUFD hybrid scheme
. . .
994.2.2. Self-consistent DM/RUFD hybrid scheme
. . .
. . .
1014.3 Numerical results
. . .
1054.3.1. λ square dielectric-layered conducting plate
. . . .
1054.3.2. λ/2 square PEC sheet in the presence of a 3λ/20 conducting wire
. . . .
1074.3.3. λ square PEC sheet in the presence of a λ/2 conducting wire
. . . .
1084.3.4. λ square dielectric-coated PEC sheet in the presence of a λ/2 PEC wire 110
Conclusions
113
Appendix
115
Bibliography
119
Introduction
After the establishment of Maxwell’s theory in 1864 [1], [2], the first electromagnetic (EM) experiments involved simple shapes as spheres, cylinders, plates, etc. [3]-[8]. Due to the market demand for analyzing real-world problems, solutions to more complex geometries were needed. As a result, several approximate techniques for solving Maxwell’s equations were developed. Ray theory, diffraction theory and perturbation techniques [9]-[17] were introduced to provide an approximate result to Maxwell’s equations in the high-frequency regime. While circuit theory, interpreted as a reduction of Maxwell’s theory in the low frequency limit, has been successfully employed to analyze complex structures.
Following the advent of computer technology in the sixties, several full-wave approaches have been developed for the analysis of a wide variety of electromagnetic (EM) problems arising in scattering, antennas and microwave circuits design. To mention some of the most popular: the Method of Moments (MoM) [18]-[20], the Finite Difference Time Domain (FDTD) algorithm [21]-[22] and the Finite Element Method (FEM) [24], [25]. These methods basically transform Maxwell’s equations expressed in integral, differential or integro-differential form into algebraic equations and can operate in the time or frequency domains. Their computational efficiency mostly depends upon the number of unknowns they require to accurately describe a given problem and on how efficiently the set of equations to be solved is built.
The Method of Moments is often privileged in terms of efficiency with respect to Finite Methods (FM) because only surface current densities are considered as unknowns to solve for in a given problem. However, conventional MoM with sub-sectional basis functions can become inefficient for the analysis of structures large in terms of wavelengths. As a matter of fact, the size of the related impedance matrix grows rapidly as the problem becomes large, placing an heavy burden on the Central Processing Unit (CPU) in terms of memory and time.
Furthermore, the field representation in terms of the vector and scalar potentials aimed at solving Integral Equations (IE) involves issues related to the highly oscillating nature of the Green’s Functions (GFs). In particular, its singular and hyper-singular behavior requires the evaluation of integrals which involve extraction processes and complex numerical integration techniques [26]. One of the major contributions to the MoM matrix filling-time results to be the repeated numerical treatment of the GFs.
Another drawback in the conventional MoM formulation involves the numerical solution of Maxwell’s equations at low frequencies, plagued with numerous problems. By following the potential method for solving Maxwell’s equations, a combination of the contribution from the current J associated with the vector potential A and that from the charge ρ related to the scalar potential φ is involved in the solution of the Electric Field Integral Equation
(EFIE)
.
At very low frequencies, if the EFIE is solved through the Moment Method and some correspondent basis functions [27], the contribution from the vector potential to the impedance matrix results much smaller with respect to that of the scalar potential due to the lack of consideration of the frequency scaling of the irrotational component of the current. This makes the impedance matrix nearly singular and difficult to invert al low frequencies [28], [29].iv
Introduction
By making use of loop-star or loop-tree basis [30]-[34] with frequency normalization, the problem of nearly singular matrices at very low frequencies is only partially solved, since the matrix still results ill-conditioned. Therefore, there does not exist a method that can solve Maxwell’s equations rapidly all the way from zero frequency to microwave frequencies. This is a remarkable point, because there is a growing need to analyze EM phenomena in antennas and circuits whose components or parts can be a tiny to a sizable fraction of the wavelength.
Other limits in the conventional Method of Moments involve difficulties in the accurate treatment of thin-wired and thin-surfaced structures (shells), especially those with finite losses, for which a large number of unknowns is often required and lead to matrix ill-conditioning, as well as a lack of a universal formulation for the analysis of dielectrics and Perfect Electric Conductor (PEC) type of materials.
This thesis, placed in context of the Method of Moments, is a collection of numerical techniques aimed at solving or mitigating some of the above mentioned issues avoiding the use of the conventional formulation based on the use of the potentials.
As a starting point, the conventional MoM and its application to the solution of the EFIE and the Combined Field integral Equation (CFIE) integral equation is exposed in the first chapter.
Next, a numerical technique for an efficient calculation of the matrix elements in the context of the Method of Moments will be introduced. The method, based on closed form expressions for the fields radiated by rooftop basis functions, totally bypasses the singularity issues related to the treatment of the GFs in the calculation of the impedance matrix elements.
Furthermore, in contrast to some techniques recently proposed to accelerate the direct solution of the MoM system of equations, as the Fast Matrix Generation (FMG) [35] or the Equivalent Dipole Moment (EDM) [36] method, our expressions are valid at arbitrary distance between source and observation points. In particular, the conventional MoM formulation is not recalled for the treatment of the near terms in the process of calculating the matrix elements, therefore a unique formulation is employed. Also, the absence of the potentials makes the technique suitable for low frequency analysis avoiding the use of special basis functions.
The Dipole Moment Approach (DMA) and its application to the analysis of electromagnetic scattering is introduced in the third chapter. This novel technique, which totally bypasses the use of Green’s-function-based integral equation formulation, helps to eliminate two of the key sources of difficulties in the conventional MoM, namely the singularity and low-frequency problems. Specifically, it is shown that there are no singularities that we need to be concerned with in the Dipole Moment (DM) formulation; hence, this obviates the need for special techniques to integrate it. Furthermore, the DM formulation goes smoothly and uniformly to the static limit as ω→ 0; hence, no special basis functions (e.g., the loop-trees) are needed for low frequency analysis.
Introduction
v
Another salutary feature in the presented approach comprises the ability to handle thin and lossy structures, whether they be metallic, dielectric-type or even combinations thereof. A number of numerical examples will be presented to show that the DM formulation can handle these types of objects with ease and without running into ill-conditioning problems, even for very thin wire-like or surface structures, which conventionally lead to ill-conditioned MoM matrices. The Characteristic Basis Function Method (CBFM) [37] will be introduced and applied in this context to reduce the number of unknowns.
In the last chapter a new general-purpose Computational Electromagnetics (CEM) algorithm, called RUFD (Recursive Update in the Frequency Domain) [38], for solving electromagnetic radiation and scattering problems will be introduced. This technique, which shares many attributes with the FDTD method, generates the solution of Maxwell's equations in the frequency rather than in the time domain. Therefore it is well suited for dealing with dispersive media, as well as for deriving solutions to problems involving high-Q structures. It is also considerably more efficient, in comparison to the FDTD algorithm, which requires long run times when an accurate solution is desired at low frequencies. A substantial part of the final chapter will be dedicated to hybridizing the Recursive Update Frequency Domain method with the DM approach to solve multiscale problems in a numerically efficient manner. The direct solution of multiscale problems by means of conventional CEM methods is highly challenging, even with the availability of modern supercomputers, because of the large number of DoFs (degrees of freedom) introduced when attempting to accurately describe objects with fine features, which share the computational domain with other large ones. Therefore some techniques to combine the powerfulness of the DMA in describing objects that are small in comparison to the wavelength with the RUFD, which will be used to independently represent the large object, will be presented.
