18,958 research outputs found
Computing parametrized solutions for plasmonic nanogap structures
The interaction of electromagnetic waves with metallic nanostructures
generates resonant oscillations of the conduction-band electrons at the metal
surface. These resonances can lead to large enhancements of the incident field
and to the confinement of light to small regions, typically several orders of
magnitude smaller than the incident wavelength. The accurate prediction of
these resonances entails several challenges. Small geometric variations in the
plasmonic structure may lead to large variations in the electromagnetic field
responses. Furthermore, the material parameters that characterize the optical
behavior of metals at the nanoscale need to be determined experimentally and
are consequently subject to measurement errors. It then becomes essential that
any predictive tool for the simulation and design of plasmonic structures
accounts for fabrication tolerances and measurement uncertainties.
In this paper, we develop a reduced order modeling framework that is capable
of real-time accurate electromagnetic responses of plasmonic nanogap structures
for a wide range of geometry and material parameters. The main ingredients of
the proposed method are: (i) the hybridizable discontinuous Galerkin method to
numerically solve the equations governing electromagnetic wave propagation in
dielectric and metallic media, (ii) a reference domain formulation of the
time-harmonic Maxwell's equations to account for geometry variations; and (iii)
proper orthogonal decomposition and empirical interpolation techniques to
construct an efficient reduced model. To demonstrate effectiveness of the
models developed, we analyze geometry sensitivities and explore optimal designs
of a 3D periodic annular nanogap structure.Comment: 28 pages, 9 figures, 4 tables, 2 appendice
A robust error estimator and a residual-free error indicator for reduced basis methods
The Reduced Basis Method (RBM) is a rigorous model reduction approach for
solving parametrized partial differential equations. It identifies a
low-dimensional subspace for approximation of the parametric solution manifold
that is embedded in high-dimensional space. A reduced order model is
subsequently constructed in this subspace. RBM relies on residual-based error
indicators or {\em a posteriori} error bounds to guide construction of the
reduced solution subspace, to serve as a stopping criteria, and to certify the
resulting surrogate solutions. Unfortunately, it is well-known that the
standard algorithm for residual norm computation suffers from premature
stagnation at the level of the square root of machine precision.
In this paper, we develop two alternatives to the standard offline phase of
reduced basis algorithms. First, we design a robust strategy for computation of
residual error indicators that allows RBM algorithms to enrich the solution
subspace with accuracy beyond root machine precision. Secondly, we propose a
new error indicator based on the Lebesgue function in interpolation theory.
This error indicator does not require computation of residual norms, and
instead only requires the ability to compute the RBM solution. This
residual-free indicator is rigorous in that it bounds the error committed by
the RBM approximation, but up to an uncomputable multiplicative constant.
Because of this, the residual-free indicator is effective in choosing snapshots
during the offline RBM phase, but cannot currently be used to certify error
that the approximation commits. However, it circumvents the need for \textit{a
posteriori} analysis of numerical methods, and therefore can be effective on
problems where such a rigorous estimate is hard to derive
On the degree of the polynomial defining a planar algebraic curves of constant width
In this paper, we consider a family of closed planar algebraic curves
which are given in parametrization form via a trigonometric
polynomial . When is the boundary of a compact convex set, the
polynomial represents the support function of this set. Our aim is to
examine properties of the degree of the defining polynomial of this family of
curves in terms of the degree of . Thanks to the theory of elimination, we
compute the total degree and the partial degrees of this polynomial, and we
solve in addition a question raised by Rabinowitz in \cite{Rabi} on the lowest
degree polynomial whose graph is a non-circular curve of constant width.
Computations of partial degrees of the defining polynomial of algebraic
surfaces of constant width are also provided in the same way.Comment: 13 page
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