Spontaneous emission from radiative chiral nematic liquid crystals at the photonic band-gap edge: An investigation into the role of the density of photon states near resonance

In this article, we investigate the spontaneous emission properties of radiating molecules embedded in a chiral nematic liquid crystal, under the assumption that the electronic transition frequency is close to the photonic edge mode of the structure, i.e., at resonance. We take into account the transition broadening and the decay of electromagnetic field modes supported by the so-called "mirrorless"cavity. We employ the Jaynes-Cummings Hamiltonian to describe the electron interaction with the electromagnetic field, focusing on the mode with the diffracting polarization in the chiral nematic layer. As known in these structures, the density of photon states, calculated via the Wigner method, has distinct peaks on either side of the photonic band gap, which manifests itself as a considerable modification of the emission spectrum. We demonstrate that, near resonance, there are notable differences between the behavior of the density of states and the spontaneous emission profile of these structures. In addition, we examine in some detail the case of the logarithmic peak exhibited in the density of states in two-dimensional photonic structures and obtain analytic relations for the Lamb shift and the broadening of the atomic transition in the emission spectrum. The dynamical behavior of the atom-field system is described by a system of two first-order differential equations, solved using the Green's-function method and the Fourier transform. The emission spectra are then calculated and compared with experimental data. © 2013 American Physical Society.

Spontaneous emission from radiative chiral nematic liquid crystals at the photonic band-gap edge: an investigation into the role of the density of photon states near resonance.

In this article, we investigate the spontaneous emission properties of radiating molecules embedded in a chiral nematic liquid crystal, under the assumption that the electronic transition frequency is close to the photonic edge mode of the structure, i.e., at resonance. We take into account the transition broadening and the decay of electromagnetic field modes supported by the so-called "mirrorless"cavity. We employ the Jaynes-Cummings Hamiltonian to describe the electron interaction with the electromagnetic field, focusing on the mode with the diffracting polarization in the chiral nematic layer. As known in these structures, the density of photon states, calculated via the Wigner method, has distinct peaks on either side of the photonic band gap, which manifests itself as a considerable modification of the emission spectrum. We demonstrate that, near resonance, there are notable differences between the behavior of the density of states and the spontaneous emission profile of these structures. In addition, we examine in some detail the case of the logarithmic peak exhibited in the density of states in two-dimensional photonic structures and obtain analytic relations for the Lamb shift and the broadening of the atomic transition in the emission spectrum. The dynamical behavior of the atom-field system is described by a system of two first-order differential equations, solved using the Green's-function method and the Fourier transform. The emission spectra are then calculated and compared with experimental data.

On the application of differential geometry to MDO

Multidisciplinary Design Optimization (MDO) is a methodology for optimizing large coupled systems. Over the years, a number of different MDO decomposition strategies, known as architectures, have been developed, and various pieces of analytical work have been done on MDO and its architectures. However, MDO lacks an overarching paradigm which would unify the field and promote cumulative research. In this paper, we propose a differential geometry framework as such a paradigm: Differential geometry comes with its own set of analysis tools and a long history of use in theoretical physics. We begin by outlining some of the mathematics behind differential geometry and then translate MDO into that framework. This initial work gives new tools and techniques for studying MDO and its architectures while producing a naturally arising measure of design coupling. The framework also suggests several new areas for exploration into and analysis of MDO systems. At this point, analogies with particle dynamics and systems of differential equations look particularly promising for both the wealth of extant background theory that they have and the potential predictive and evaluative power that they hold. © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

On a classical problem of acoustic wave scattering by a free vortex: Numerical modelling

The scattering of sound from a point source by a Rankine vortex is investigated numerically by solving the Euler equations with the novel high-resolution CABARET method. For several Mach numbers of the vortex, the time-average amplitudes of the scattered field obtained from the numerical modeling are compared with the theoretical scaling laws' predictions. Copyright © 2009 by Sergey Karabasov.

Building flexible user interfaces for solving PDEs

FEniCS is a collection of software tools for the automated solution of differential equations by finite element methods. In this note, we describe how FEniCS can be used to solve a simple nonlinear model problem with varying levels of automation. At one extreme, FEniCS provides tools for the fully automated and adaptive solution of nonlinear partial differential equations. At the other extreme, FEniCS provides a range of tools that allow the computational scientist to experiment with novel solution algorithms. © 2010 American Institute of Physics.

A FEniCS-based programming framework for modeling turbulent flow by the Reynolds-averaged Navier-Stokes equations

Finding an appropriate turbulence model for a given flow case usually calls for extensive experimentation with both models and numerical solution methods. This work presents the design and implementation of a flexible, programmable software framework for assisting with numerical experiments in computational turbulence. The framework targets Reynolds-averaged Navier-Stokes models, discretized by finite element methods. The novel implementation makes use of Python and the FEniCS package, the combination of which leads to compact and reusable code, where model- and solver-specific code resemble closely the mathematical formulation of equations and algorithms. The presented ideas and programming techniques are also applicable to other fields that involve systems of nonlinear partial differential equations. We demonstrate the framework in two applications and investigate the impact of various linearizations on the convergence properties of nonlinear solvers for a Reynolds-averaged Navier-Stokes model. © 2011 Elsevier Ltd.

Optimal control with stochastic PDE constraints and uncertain controls

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby forgoing the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented. © 2011 Elsevier B.V.

Boundary integral technique for the numerical modelling of the air flow for the Aaberg exhaust system

A boundary integral technique has been developed for the numerical simulation of the air flow for the Aaberg exhaust system. For the steady, ideal, irrotational air flow induced by a jet, the air velocity is an analytical function. The solution of the problem is formulated in the form of a boundary integral equation by seeking the solution of a mixed boundary-value problem of an analytical function based on the Riemann-Hilbert technique. The boundary integral equation is numerically solved by converting it into a system of linear algebraic equations, which are solved by the process of the Gaussian elimination. The air velocity vector at any point in the solution domain is then computed from the air velocity on the boundary of the solution domains.

Seafloor interaction with steel catenary risers

This paper describes the key features of a seafloor-riser interaction model. The soil is represented in terms of non-linear load-deflection (P- y) relationships, which are also able to account for soil stiffness degradation due to cyclic loading. The analytical framework considers the riser-seafloor interaction problem in terms of a pipe resting on a bed of springs, and requires the iterative solution of a fourth-order ordinary differential equation. A series of simulations is used to illustrate the capabilities of the model. Thanks to the non-linear soil springs with stiffness degradation it is possible to simulate the trench formation process and estimate moments in a riser. Copyright © 2008 by The International Society of Offshore and Polar Engineers (ISOPE).

A differential lyapunov framework for contraction analysis

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves. © 2013 IEEE.