17 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 nested hybridizable discontinuous Galerkin method for computing second-harmonic generation in three-dimensional metallic nanostructures
In this paper, we develop a nested hybridizable discontinuous Galerkin (HDG)
method to numerically solve the Maxwell's equations coupled with the
hydrodynamic model for the conduction-band electrons in metals. By means of a
static condensation to eliminate the degrees of freedom of the approximate
solution defined in the elements, the HDG method yields a linear system in
terms of the degrees of freedom of the approximate trace defined on the element
boundaries. Furthermore, we propose to reorder these degrees of freedom so that
the linear system accommodates a second static condensation to eliminate a
large portion of the degrees of freedom of the approximate trace, thereby
yielding a much smaller linear system. For the particular metallic structures
considered in this paper, the resulting linear system obtained by means of
nested static condensations is a block tridiagonal system, which can be solved
efficiently. We apply the nested HDG method to compute the second harmonic
generation (SHG) on a triangular coaxial periodic nanogap structure. This
nonlinear optics phenomenon features rapid field variations and extreme
boundary-layer structures that span multiple length scales. Numerical results
show that the ability to identify structures which exhibit resonances at
and is paramount to excite the second harmonic response.Comment: 31 pages, 7 figure
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
Efficient and accurate nonlinear model reduction via first-order empirical interpolation
We present a model reduction approach that extends the original empirical
interpolation method to enable accurate and efficient reduced basis
approximation of parametrized nonlinear partial differential equations (PDEs).
In the presence of nonlinearity, the Galerkin reduced basis approximation
remains computationally expensive due to the high complexity of evaluating the
nonlinear terms, which depends on the dimension of the truth approximation. The
empirical interpolation method (EIM) was proposed as a nonlinear model
reduction technique to render the complexity of evaluating the nonlinear terms
independent of the dimension of the truth approximation. We introduce a
first-order empirical interpolation method (FOEIM) that makes use of the
partial derivative information to construct an inexpensive and stable
interpolation of the nonlinear terms. We propose two different FOEIM algorithms
to generate interpolation points and basis functions. We apply the FOEIM to
nonlinear elliptic PDEs and compare it to the Galerkin reduced basis
approximation and the EIM. Numerical results are presented to demonstrate the
performance of the three reduced basis approaches.Comment: 38 pages, 6 figures, 6 table
Simulation of the interaction of light with 3âD metallic nanostructures using a proper orthogonal decompositionâGalerkin reducedâorder discontinuous Galerkin timeâdomain method
International audienc
HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB
This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems
A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations
International audienceWe present a non-intrusive model order reduction (NIMOR) method with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level L (L â„ 1) over a target parameterized space. This method can deal with the so-called curse of dimensionality in high dimensional space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level L + 1, which includes the sparse grids from approximation level L. A nested proper orthogonal decomposition (POD) method is employed to extract time-and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time-and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. The performance of this NIMOR method is illustrated numerically by considering two classical test cases: the scattering of a plane wave by a 2-D dielectric disk and the scattering of a plane wave by a multi-layer heterogeneous medium. The prediction capabilities of the NIMOR method are evaluated by varying the relative permittivity. Numerical results indicate that the NIMOR method is a promising approach for simulating accurately and in fast way parameterized timedomain electromagnetic problems