Ellipsometry is an optical technique used for studies of thin films and surfaces. The technique is based on measurement and analysis of the changes in the state of polarization that occur when polarized light is reflected on a sample surface. The multichannel Mueller matrix ellipsometer is a new system that is about to enter the commercial market. It can measure the full 4x4 Mueller matrix of a sample. The Laboratory of Applied Optics at the Department of Physics, Chemistry and Biology at Linköpings University has purchased one such multichannel Mueller matrix ellipsometer, called RC2, from J.A. Woollam Co., Inc. This project has the objective to investigate potentials and limitations of this new ellipsometer. This is done by comparing measurements carried out on RC2 with similar measurements made on a different, well known ellipsometer system, the VASE ellipsometer. A study of the theoretical background of ellipsometry has been made including a description of the Jones formalism of describing optical properties as well as the Stokes/ Mueller formalism. A short theoretical description of the RC2 principles, in order to better understand the new ellipsometer is also given. Measurements have been made on samples of varying complexity, including isotropic and anisotropic samples with in-plane anisotropy and out-of-plane anisotropy. On samples with no depolarization there should be no difference between the two ellipsometers. As can be seen in the experimental results there is some difference, but very little. Both for the isotropic samples, where Δ and ψ have been measured, and anisotropic, where the full Mueller matrix has been measured, we find a good match between VASE and RC2. When the samples are depolarizing, we expect to see a difference in the Mueller matrix. We do notice a significant difference in the measurements on tilted nanometer rods of TiO2, which has an advanced nano-structure in the upper layer. We can thus conclude that samples with depolarizing properties reveal this when measured with the new RC2 ellipsometer.

Fluid phenomena dominated by viscous effects can, in many cases, be modeled by the Stokes equations. The boundary integral form of the Stokes equations reduces the number of degrees of freedom in a numerical discretization by reformulating the three-dimensional problem to two-dimensional integral equations to be discretized over the boundaries of the domain. Hence for the study of objects immersed in a fluid, such as drops or elastic/solid particles, integral equations are to be discretized over the surfaces of these objects only. As outer boundaries or confinements are added these must also be included in the formulation. An inherent difficulty in the numerical treatment of boundary integrals for Stokes flow is the integration of the singular fundamental solution of the Stokes equations, e.g. the so called Stokeslet. To alleviate this problem we developed a set of high-order quadrature rules for the numerical integration of the Stokeslet over a flat surface. Such a quadrature rule was first designed for singularities of the type <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?1/%7C%5Cmathbf%7Bx%7D%7C" />. To assess the convergence properties of this quadrature rule a theoretical analysis has been performed. The slightly more complicated singularity of the Stokeslet required certain modifications of the integration rule developed for <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?1/%7C%5Cmathbf%7Bx%7D%7C" />. An extension of this type of quadrature rule to a cylindrical surface is also developed. These quadrature rules are tested also on physical problems that have an analytic solution in the literature. Another difficulty associated with boundary integral problems is introduced by periodic boundary conditions. For a set of particles in a periodic domain periodicity is imposed by requiring that the motion of each particle has an added contribution from all periodic images of all particles all the way up to infinity. This leads to an infinite sum which is not absolutely convergent, and an additional physical constraint which removes the divergence needs to be imposed. The sum is decomposed into two fast converging sums, one that handles the short range interactions in real space and the other that sums up the long range interactions in Fourier space. Such decompositions are already available in the literature for kernels that are commonly used in boundary integral formulations. Here a decomposition in faster decaying sums than the ones present in the literature is derived for the periodic kernel of the stress tensor. However the computational complexity of the sums, regardless of the decomposition they stem from, is <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathcal%7BO%7D(N%5E%7B2%7D)" />. This complexity can be lowered using a fast summation method as we introduced here for simulating a sedimenting fiber suspension. The fast summation method was initially designed for point particles, which could be used for fibers discretized numerically almost without any changes. However, when two fibers are very close to each other, analytical integration is used to eliminate numerical inaccuracies due to the nearly singular behavior of the kernel and the real space part in the fast summation method was modified to allow for this analytical treatment. The method we have developed for sedimenting fiber suspensions allows for simulations in large periodic domains and we have performed a set of such simulations at a larger scale (larger domain/more fibers) than previously feasible. / <p>QC 20121122</p>

University of Macau / Faculty of Science and Technology / Department of Electromechanical Engineering

The long time energy decay of vortices for several different initial flow scenarios is investigated both theoretically and numerically. The theoretical analysis is based on the energy method. Numerical calculations are done by solving the compressible Navier-Stokes equations using a high order stable finite difference method. The simulations verify the theoretical conclusion that vortices decay at a slow rate compared to other types of flows. Several Reynolds numbers and grid sizes in both two and three dimensions are considered.

No description available.

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The objective of the study is to investigate the capability of PANS (Partially Averaged Navier-Stokes Simulation) model over a wide range of Reynolds numbers and flow physics. In this regard, numerical simulations of turbulent flow past a circular cylinder are performed at ReD 140,000 and ReD 3900 using the PANS model. The high Reynolds number PANS results are compared with experimental results from Cantwell and Coles, Large Eddy Simulation results from Breuer, and Detached Eddy Simulation results from Travin et al. Low Reynolds number PANS results are compared with experimental results from Ong and Wallace and Large Eddy Simulation results from Breuer. The effects of the various PANS parameters (fk, f&#949;, &#963;ku, &#963;&#949;u) on the ability to capture turbulence physics at various Reynolds numbers are studied. It is confirmed, as previously predicted from theoretical considerations that: (i) for high Reynolds number flow f&#949; = 1 and &#963;ku = &#963;k × fk2 / f&#949; are most appropriate; and (ii) for low Reynolds number flow f&#949; = fk and &#963;ku = &#963;k are most suitable. These choices for the parameters stem from the fact that there is no clear separation of scales between the energy scales and the dissipation scales at low Reynolds number unlike in the high Reynolds number where there is a clear separation of scales between the energy containing scales and the dissipation scales. Also, in both cases it is found that decreasing fk leads to improved accuracy in predicting the flow statistics.

<p>Ellipsometry is an optical technique used for studies of thin films and surfaces. The technique is based on measurement and analysis of the changes in the state of polarization that occur when polarized light is reflected on a sample surface.</p><p>The multichannel Mueller matrix ellipsometer is a new system that is about to enter the commercial market. It can measure the full 4x4 Mueller matrix of a sample.</p><p>The Laboratory of Applied Optics at the Department of Physics, Chemistry and Biology at Linköpings University has purchased one such multichannel Mueller matrix ellipsometer, called RC2, from J.A. Woollam Co., Inc.</p><p>This project has the objective to investigate potentials and limitations of this new ellipsometer. This is done by comparing measurements carried out on RC2 with similar measurements made on a different, well known ellipsometer system, the VASE ellipsometer.</p><p>A study of the theoretical background of ellipsometry has been made including a description of the Jones formalism of describing optical properties as well as the Stokes/ Mueller formalism.</p><p>A short theoretical description of the RC2 principles, in order to better understand the new ellipsometer is also given.</p><p>Measurements have been made on samples of varying complexity, including isotropic and anisotropic samples with in-plane anisotropy and out-of-plane anisotropy.</p><p>On samples with no depolarization there should be no difference between the two ellipsometers. As can be seen in the experimental results there is some difference, but very little. Both for the isotropic samples, where Δ and ψ have been measured, and anisotropic, where the full Mueller matrix has been measured, we find a good match between VASE and RC2.</p><p>When the samples are depolarizing, we expect to see a difference in the Mueller matrix. We do notice a significant difference in the measurements on tilted nanometer rods of TiO2, which has an advanced nano-structure in the upper layer. We can thus conclude that samples with depolarizing properties reveal this when measured with the new RC2 ellipsometer.</p>

Dans une première partie, on introduit des estimateurs d'erreur a posteriori pour l'équation de la chaleur<br />dans R^d, d=2,3 via une méthode d'éléments finis non conformes en espace et un schéma d'Euler implicite en temps. Pour cette discrétisation, on élabore un indicateur d'erreur résiduel spatial basé sur les sauts des dérivées normales et tangentielles de notre approximation, ainsi qu'un indicateur résiduel temporel basé sur le saut du gradient à chaque pas de temps. Les bornes inférieures et supérieures de la norme de l'erreur forment les résultats principaux de cette étude. En outre, on montre que ces estimateurs sont fiables et efficaces. Dans une seconde partie, on traite le problème de Stokes dynamique. L'élaboration des estimateurs a posteriori est également basée sur des estimateurs spatiaux et temporels. Une preuve de leur fiabilité et de leur efficacité est donnée. Finalement, les tests numériques et un algorithme adaptatif confirment les prévisions théoriques et le bon comportement de ces estimateurs.

The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.