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Systematic Determination of Eigenfields in Frequency Domain
Abstract
This paper addresses numerical procedures utilized to the accurate and robust calculation of thousands of eigenpairs for the Dirac billiard resonator. The main challenges posed by the present work are: first, the capability of the approaches to tackle the large-scale eigenvalue problem, second, the ability to accurately extract many, i.e. order of thousands, interior eigenfrequencies for the considered problem, and third, the efficient implementation of the underlying approaches. The eigenfield calculations are accomplished in two steps. Initially, the finite integration technique or the finite element method with higher order curvilinear elements is applied, and further, the (B-)Lanczos method with its variations is exploited for the eigenpair determination. The comparative assessment of the numerical results to the complementary measurements confirms the applicability of the approaches and points out the significant reductions of computational costs. Finally, all of the results indicate that the suggested techniques can be used for precise determination of many eigenfrequencies.
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This article presents experimental results on properties of waves propagating
in an unbounded and a bounded photonic crystal consisting of metallic cylinders
which are arranged in a triangular lattice. First, we present transmission
measurements of plane waves traversing a photonic crystal. The experiments are
performed in the vicinity of a Dirac point, i.e., an isolated conical
singularity of the photonic band structure. There, the transmission shows a
pseudodiffusive 1/L dependence, with $L$ being the thickness of the crystal, a
phenomenon also observed in graphene. Second, eigenmode intensity distributions
measured in a microwave analog of a relativistic Dirac billiard, a rectangular
microwave billiard that contains a photonic crystal, are discussed. Close to
the Dirac point states have been detected which are localized at the straight
edge of the photonic crystal corresponding to a zigzag edge in graphene.
We present measurements of transmission and reflection spectra of a microwave
photonic crystal composed of 874 metallic cylinders arranged in a triangular
lattice. The spectra show clear evidence of a Dirac point, a characteristic of
a spectrum of relativistic massless fermions. In fact, Dirac points are a
peculiar property of the electronic band structure of graphene, whose
properties consequently can be described by the relativistic Dirac equation. In
the vicinity of the Dirac point, the measured reflection spectra resemble those
obtained by conductance measurements in scanning tunneling microscopy of
graphene flakes.
Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrödinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective 'speed of light' c* approximately 10(6) m s(-1). Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m(c) of massless carriers in graphene is described by E = m(c)c*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.
When combined with Krylov projection methods, polynomial filtering can provide a powerful method for extracting extreme or interior eigenvalues of large sparse matrices. This general approach can be quite efficient in the situation when a large number of eigenvalues is sought. However, its competitiveness depends critically on a good implementation. This paper presents a technique based on such a combination to compute a group of extreme or interior eigenvalues of a real symmetric (or complex Hermitian) matrix. The technique harnesses the effectiveness of the Lanczos algorithm with partial reorthogonalization and the power of polynomial filtering. Numerical experiments indicate that the method can be far superior to competing algorithms when a large number of eigenvalues and eigenvectors is to be computed.
In this paper, we address a fast approach for an accurate eigenfrequency determination, based on a finite-element computation of electromagnetic fields for a superconducting cavity and further employment of the Lanczos method for the eigenvalue determination. The major challenges posed by this paper are: 1) the ability of the approach to tackle the large-scale eigenvalue problem and 2) the capability to extract many, i.e., order of thousands, eigenfrequencies for the considered problem. In addition to the need to ensure high precision of the calculated eigenfrequencies, we compare them side by side with the reference data available from analytical expressions and CEM3D eigenmode solver. Furthermore, the simulations have shown high accuracy of this technique and good agreement with the reference data. Finally, all of the results show that the suggested technique can be used for precise determination of many eigenfrequencies.
A discretization method is described for the solution of the
inhomogeneous, as well as the homogeneous, Maxwell equations in a
finite, three-dimensional, source-free region. The distribution of
material (permittivity, permeability, conductivity) can be an arbitrary
function of the position. The eigenvalue-calculation of dielectric
loaded waveguides serves as numerical example.
This is a revised edition of a book which appeared close to two decades ago. Someone scrutinizing how the field has evolved in these two decades will make two interesting observations. On the one hand the observer will be struck by the staggering number of new developments in numerical linear algebra during this period. The field has evolved in all directions: theory, algorithms, software, and novel applications. Two decades ago there was essentially no publically available software for large eigenvalue problems. Today one has a flurry to choose from, and the activity in software development does not seem to be abating. A number of new algorithms appeared in this period as well. I can mention at the outset the Jacobi-Davidson algorithm and the idea of implicit restarts, both discussed in this book, but there are a few others. The most interesting development to the numerical analyst may be the expansion of the realm of eigenvalue techniques into newer and more challenging applications. Or perhaps, the more correct observation is that these applications were always there, but they were not as widely appreciated or understood by numerical analysts, or were not fully developed due to lack of software.
It is often necessary to filter out an eigenspace of a given matrix A before performing certain computations with it. The eigenspace usually corresponds to undesired eigenvalues in the underlying application. One such application is in information retrieval, where the method of latent semantic indexing replaces the original matrix with a lower-rank one using tools based on the singular value decomposition. Here the low-rank approximation to the original matrix is used to analyze similarities with a given query vector. Filtering has the effect of yielding the most relevant part of the desired solution while discarding noise and redundancies in the underlying problem. Another common application is to compute an invariant subspace of a symmetric matrix associated with eigenvalues in a given interval. In this case, it is necessary to filter out eigenvalues that are not in the interval of the wanted eigenvalues. This paper presents a few conjugate gradient-like methods to provide solutions to these types of problems by iterative procedures which utilize only matrix-vector products.
We describe monocrystalline graphitic films, which are a few atoms thick but are nonetheless stable under ambient conditions,
metallic, and of remarkably high quality. The films are found to be a two-dimensional semimetal with a tiny overlap between
valence and conductance bands, and they exhibit a strong ambipolar electric field effect such that electrons and holes in
concentrations up to 1013 per square centimeter and with room-temperature mobilities of ∼10,000 square centimeters per volt-second can be induced by
applying gate voltage.
State of the art in the simulation of electromagnetic fields based on large scale finite element eigenanalysis
- W Ackermann
- G Benderskaya
- T Weiland
- W. Ackermann
Randzustände in einem supraleitenden Mikrowellen-Diracbillard. Bachelor’s thesis
- C Cuno
- C. Cuno
Simulation Technology AG: CST Microwave StudioⓇ
- Cst - Computer