26 research outputs found
Discontinuous Galerkin Methods in Nanophotonics
In this thesis I present discontinuous Galerkin methods for Maxwell\u27s equations in both time- and frequency-domain. The method\u27s computational capabilities are extended by perfectly matched layers, dispersive and anisotropic materials, and sources. These techniques are applied to a study of coupling effects in split-ring resonator arrays
International Workshop on Finite Elements for Microwave Engineering
When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
Solution of the Schrodinger equation for quasi-one-dimensional materials using helical waves
We formulate and implement a spectral method for solving the Schrodinger
equation, as it applies to quasi-one-dimensional materials and structures. This
allows for computation of the electronic structure of important technological
materials such as nanotubes (of arbitrary chirality), nanowires, nanoribbons,
chiral nanoassemblies, nanosprings and nanocoils, in an accurate, efficient and
systematic manner. Our work is motivated by the observation that one of the
most successful methods for carrying out electronic structure calculations of
bulk/crystalline systems -- the plane-wave method -- is a spectral method based
on eigenfunction expansion. Our scheme avoids computationally onerous
approximations involving periodic supercells often employed in conventional
plane-wave calculations of quasi-one-dimensional materials, and also overcomes
several limitations of other discretization strategies, e.g., those based on
finite differences and atomic orbitals. We describe the setup of fast
transforms to carry out discretization of the governing equations using our
basis set, and the use of matrix-free iterative diagonalization to obtain the
electronic eigenstates. Miscellaneous computational details, including the
choice of eigensolvers, use of a preconditioning scheme, evaluation of
oscillatory radial integrals and the imposition of a kinetic energy cutoff are
discussed. We have implemented these strategies into a computational package
called HelicES (Helical Electronic Structure). We demonstrate the utility of
our method in carrying out systematic electronic structure calculations of
various quasi-one-dimensional materials through numerous examples involving
nanotubes, nanoribbons and nanowires. We also explore the convergence, accuracy
and efficiency of our method. We anticipate that our method will find numerous
applications in computational nanomechanics and materials science
Locally Implicit Time Integration for Linear Maxwell\u27s Equations
This thesis is concerned with the full discretization of Maxwell\u27s equations in cases where the spatial discretization has to be carried out with a locally refined grid. In such situations locally implicit time integrators are an appealing choice for the time discretization since they overcome the grid-induced stiffness of these problems. We analyze such a locally implicit time integrator in the case where the space discretization stems from a central fluxes discontinuous Galerkin method. In fact, we prove its stability under a CFL condition which solely depends on the coarse part of the spatial grid and give a rigorous error analysis showing that the integrator is second order convergent. Moreover, we extend this time integrator so that it can be applied to an upwind fluxes discontinuous Galerkin space discretization. We show that this novel integrator preserves the second order temporal convergence and that it inherits the improved properties of an upwind fluxes discretization (better stability, higher spatial convergence rate) compared to the central fluxes case