Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering

There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering

A Three-Dimensional Recovery-Based Discontinuous Galerkin Method for Turbulence Simulations

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106503/1/AIAA2013-515.pd

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 $\omega$ and $2\omega$ is paramount to excite the second harmonic response.Comment: 31 pages, 7 figure

Analysis of enhanced mixing by natural and forced convection with application to chemical reactor design

We looked for new designs to enhance mixing on catalytic chemical reactors. Two models were proposed. The first one, used forced convection as the mixing enhancer and consisted of a stack of corotating disks with the catalytic coating on their surface and which were enclosed in a cilyndrical cavity. Two operation modes were studied for this type of reactor, discontinuous and semicontinuous one. In the semicontinuous mode a flux is fed to the reactor through the external wall. This configuration showed good efficiency compared to other type of reactors. The second model proposed used natural convection as the mixing enhancer. In this case a cubical cavity which had the catalytic surface on the bottom wall, was heated from below which activated the convection in the system leading to the mixing of the fluid in it. It was seen efficiency is determined by flow and mass transfer within boundary layers regions.Se buscaron nuevos diseños que aumentaran el mezclado en reactores catalíticos. Dos modelos fueron propuestos. El primero, usó convección forzada como el impulsor del mezclado y consistía de una pila de discos corotatorios con el catalizador en sus superficies y los cuales se encontraban dentro de una cavidad cilíndrica. Dos modos de operación fueron estudiados para este tipo de reactor, discontinuo y semicontinuo. En el modo semicontinuo se alimentó un flujo al reactor mediante la pared exterior. Esta configuración mostró una buena eficiencia comparado con otro tipo de reactores. El segundo modelo propuesto usó la convección natural como el impulsor del mezclado. En este caso una cavidad cúbica que tenía el catalizador en su pared inferior, fue calentada por debajo, lo que activaba la convección en el sistema llevando al mezclado del fluido en el mismo. La eficiencia es determinada por la transferencia de flujo y masa dentro de las regiones de capa límite