Redundant coordinates in harmonic force-field calculations

Redundancy relations between vibrational coordinates may be linear (as for rectilinear coordinates used in deriving a G matrix), or non-linear (as for curvilinear coordinates used in formulating model force fields). It is shown that geometrically defined internal coordinates are necessarily curvilinear. Hence it is shown that linear force constants can occur in model force field calculations involving redundant coordinates, in disagreement with the recent proposal of Gussoni and Zerbi.

Alternate treatments of jacobian singularities in polar coordinates within finite-difference schemes

Jacobian singularities of differential operators in curvilinear coordinates occur when the Jacobian determinant of the curvilinear-to-Cartesian mapping vanishes, thus leading to unbounded coefficients in partial differential equations. Within a finite-difference scheme, we treat the singularity at the pole of polar coordinates by setting up complementary equations. Such equations are obtained by either integral or smoothness conditions. They are assessed by application to analytically solvable steady-state heat-conduction problems.

The curvature of the conical intersection seam: an approximate second-order analysis

We present a method for analyzing the curvature (second derivatives) of the conical intersection hyperline at an optimized critical point. Our method uses the projected Hessians of the degenerate states after elimination of the two branching space coordinates, and is equivalent to a frequency calculation on a single Born-Oppenheimer potential-energy surface. Based on the projected Hessians, we develop an equation for the energy as a function of a set of curvilinear coordinates where the degeneracy is preserved to second order (i.e., the conical intersection hyperline). The curvature of the potential-energy surface in these coordinates is the curvature of the conical intersection hyperline itself, and thus determines whether one has a minimum or saddle point on the hyperline. The equation used to classify optimized conical intersection points depends in a simple way on the first- and second-order degeneracy splittings calculated at these points. As an example, for fulvene, we show that the two optimized conical intersection points of C2v symmetry are saddle points on the intersection hyperline. Accordingly, there are further intersection points of lower energy, and one of C2 symmetry - presented here for the first time - is found to be the global minimum in the intersection space

The curvature of the conical intersection seam: an approximate second-order analysis

We present a method for analyzing the curvature (second derivatives) of the conical intersection hyperline at an optimized critical point. Our method uses the projected Hessians of the degenerate states after elimination of the two branching space coordinates, and is equivalent to a frequency calculation on a single Born-Oppenheimer potential-energy surface. Based on the projected Hessians, we develop an equation for the energy as a function of a set of curvilinear coordinates where the degeneracy is preserved to second order (i.e., the conical intersection hyperline). The curvature of the potential-energy surface in these coordinates is the curvature of the conical intersection hyperline itself, and thus determines whether one has a minimum or saddle point on the hyperline. The equation used to classify optimized conical intersection points depends in a simple way on the first- and second-order degeneracy splittings calculated at these points. As an example, for fulvene, we show that the two optimized conical intersection points of C2v symmetry are saddle points on the intersection hyperline. Accordingly, there are further intersection points of lower energy, and one of C2 symmetry - presented here for the first time - is found to be the global minimum in the intersection space

A geometric mass-preserving redistancing scheme for the level set function

In this paper we describe and evaluate a geometric mass-preserving redistancing procedure for the level set function on general structured grids. The proposed algorithm is adapted from a recent finite element-based method and preserves the mass by means of a localized mass correction. A salient feature of the scheme is the absence of adjustable parameters. The algorithm is tested in two and three spatial dimensions and compared with the widely used partial differential equation (PDE)-based redistancing method using structured Cartesian grids. Through the use of quantitative error measures of interest in level set methods, we show that the overall performance of the proposed geometric procedure is better than PDE-based reinitialization schemes, since it is more robust with comparable accuracy. We also show that the algorithm is well-suited for the highly stretched curvilinear grids used in CFD simulations. Copyright (C) 2010 John Wiley & Sons, Ltd.

Quantum rings of arbitrary shape and non-uniform width in a threading magnetic field

The electronic states of quantum rings with centerlines of arbitrary shape and non-uniform width in a threading magnetic field are calculated. The solutions of the Schrodinger equation with Dirichlet boundary conditions are obtained by a variational separation of variables in curvilinear coordinates. We obtain a width profile that compensates for the main effects of the curvature variations in the centerline. Numerical results are shown for circular, elliptical, and limacon-shaped quantum rings. We also show that smooth and tiny variations in the width may strongly affect the Aharonov-Bohm oscillations.

Derived Bedrock Elevations, Strain Rates and Stresses from Measured Surface Elevations and Velocities - Jakobshavns-Isbrae, Greenland

Jakobshavns Isbrae (69 degrees 10'N, 49 degrees 5'W) drains about 6.5% of the Greenland ice sheet and is the fastest ice stream known. The Jakobshavns Isbrae basin of about 10 000 km(2) was mapped photogrammetrically from four sets of aerial photography, two taken in July 1985 and two in July 1986. Positions and elevations of several hundred natural features on the ice surface were determined for each epoch by photogrammetric block-aerial triangulation, and surface velocity vectors were computed from the positions. The two flights in 1985 yielded the best results and provided most common points (716) for velocity determinations and are therefore used in the modeling studies. The data from these irregularly spaced points were used to calculate ice elevations and velocity vectors at uniformly spaced grid paints 3 km apart by interpolation. The field of surface strain rates was then calculated from these gridded data and used to compute the field of surface deviatoric stresses, using the flow law of ice, for rectilinear coordinates, X, Y pointing eastward and northward. and curvilinear coordinates, L, T pointing longitudinally and transversely to the changing ice-flow direction. Ice-surface elevations and slopes were then used to calculate ice thicknesses and the fraction of the ice velocity due to basal sliding. Our calculated ice thicknesses are in fair agreement with an ice-thickness map based on seismic sounding and supplied to us by K. Echelmeyer. Ice thicknesses were subtracted from measured ice-surface elevations to map bed topography. Our calculation shows that basal sliding is significant only in the 10-15 km before Jakobshavns Isbrae becomes afloat in Jakobshavns IsfJord.

The CMCC Eddy-permitting Global Ocean Physical Reanalysis (C-GLORS v5, 1980-2014), links to NetCDF files

The CMCC Global Ocean Physical Reanalysis System (C-GLORS) is used to simulate the state of the ocean in the last decades. It consists of a variational data assimilation system (OceanVar), capable of assimilating all in-situ observations along with altimetry data, and a forecast step performed by the ocean model NEMO coupled with the LIM2 sea-ice model. KEY STRENGTHS: - Data are available for a large number of ocean parameters - An extensive validation has been conducted and is freely available - The reanalysis is performed at high resolution (1/4 degree) and spans the last 30 years KEY LIMITATIONS: - Quality may be discontinuos and depend on observation coverage - Uncertainty estimates are simply derived through verification skill scores

Computation of nonlinear streaky structures in boundary layer flow

En esta tesis se integran numéricamente las ecuaciones reducidas de Navier Stokes (RNS), que describen el flujo en una capa límite tridimensional que presenta también una escala característica espacial corta en el sentido transversal. La formulación RNS se usa para el cálculo de “streaks” no lineales de amplitud finita, y los resultados conseguidos coinciden con los existentes en la literatura, obtenidos típicamente utilizando simulación numérica directa (DNS) o nonlinear parabolized stability equations (PSE). El cálculo de los “streaks” integrando las RNS es mucho menos costoso que usando DNS, y no presenta los problemas de estabilidad que aparecen en la formulación PSE cuando la amplitud del “streak” deja de ser pequeña. El código de integración RNS se utiliza también para el cálculo de los “streaks” que aparecen de manera natural en el borde de ataque de una placa plana en ausencia de perturbaciones en la corriente uniforme exterior. Los resultados existentes hasta ahora calculaban estos “streaks” únicamente en el límite lineal (amplitud pequeña), y en esta tesis se lleva a cabo el cálculo de los mismos en el régimen completamente no lineal (amplitud finita). En la segunda parte de la tesis se generaliza el código RNS para incluir la posibilidad de tener una placa no plana, con curvatura en el sentido transversal que varía lentamente en el sentido de la corriente. Esto se consigue aplicando un cambio de coordenadas, que transforma el dominio físico en uno rectangular. La formulación RNS se integra también expresada en las correspondientes coordenadas curvilíneas. Este código generalizado RNS se utiliza finalmente para estudiar el flujo de capa límite sobre una placa con surcos que varían lentamente en el sentido de la corriente, y es usado para simular el flujo sobre surcos que crecen en tal sentido. Abstract In this thesis, the reduced Navier Stokes (RNS) equations are numerically integrated. This formulation describes the flow in a three-dimensional boundary layer that also presents a short characteristic space scale in the spanwise direction. RNS equations are used to calculate nonlinear finite amplitude “streaks”, and the results agree with those reported in the literature, typically obtained using direct numerical simulation (DNS) or nonlinear parabolized stability equations (PSE). “Streaks” simulations through the RNS integration are much cheaper than using DNS, and avoid stability problems that appear in the PSE when the amplitude of the “streak” is not small. The RNS integration code is also used to calculate the “streaks” that naturally emerge at the leading edge of a flat plate boundary layer in the absence of any free stream perturbations. Up to now, the existing results for these “streaks” have been only calculated in the linear limit (small amplitude), and in this thesis their calculation is carried out in the fully nonlinear regime (finite amplitude). In the second part of the thesis, the RNS code is generalized to include the possibility of having a non-flat plate, curved in the spanwise direction and slowly varying in the streamwise direction. This is achieved by applying a change of coordinates, which transforms the physical domain into a rectangular one. The RNS formulation expressed in the corresponding curvilinear coordinates is also numerically integrated. This generalized RNS code is finally used to study the boundary layer flow over a plate with grooves which vary slowly in the streamwise direction; and this code is used to simulate the flow over grooves that grow in the streamwise direction.

Einleitung in die allgemeine Theorie der krummen Flächen.

Available on demand as hard copy or computer file from Cornell University Library.

Excitation energies from ground-state density-functionals by means of generator coordinates

The generator-coordinate method is a flexible and powerful reformulation of the variational principle. Here we show that by introducing a generator coordinate in the Kohn-Sham equation of density-functional theory, excitation energies can be obtained from ground-state density functionals. As a viability test, the method is applied to ground-state energies and various types of excited-state energies of atoms and ions from the He and the Li isoelectronic series. Results are compared to a variety of alternative DFT-based approaches to excited states, in particular time-dependent density-functional theory with exact and approximate potentials.

Comparison of Preoperative Nd:YAG Laser Anterior Capsulotomy Versus Two-Stage Curvilinear Capsulorhexis in Phacoemulsification of White Intumescent Cataracts

BACKGROUND AND OBJECTIVE: To compare the results of preoperative Nd:YAG laser anterior capsulotomy versus two-stage continuous curvilinear capsulorhexis (CCC) in phacoemulsification of eyes with white intumescent cataracts and liquefied cortex. PATIENTS AND METHODS: Twenty-three eyes with white intumescent cataract were consecutively randomized for phacoemulsification with preoperative Nd:YAG laser anterior capsulotomy (group 1, n = 11) or two-stage CCC (group 2, n = 12) procedures. Intraoperative findings and postoperative outcomes were compared using the nonparametric tests. RESULTS: Postoperative Visual acuity, mean surgical time, mean effective phacoemulsification time, and frequency of complications were not significantly different between the two groups (P > .05). Two cases in each group were converted to the extracapsular technique. Excluding these four patients, surgical time was shorter In group 1 (P = .017). CONCLUSION: Preoperative Nd:YAG laser anterior capsulotomy is a safe technique in decompressing the capsular bag before phacoemulsification of white intumescent cataracts with liquefied cortex.

Propagation of nonstationary curved and stretched premixed flames with multistep reaction mechanisms

The propagation speed of a thin premixed flame disturbed by an unsteady fluid flow of a larger scale is considered. The flame may also have a general shape but the reaction zone is assumed to be thin compared to the flame thickness. Unlike in preceding publications, the presented asymptotic analysis is performed for a general multistep reaction mechanism and, at the same time, the flame front is curved by the fluid flow. The resulting equations define the propagation speed of disturbed flames in terms of the properties of undisturbed planar flames and the flame stretch. Special attention is paid to the near-equidiffusion limit. In this case, the flame propagation speed is shown to depend on the effective Zeldovich number Z(f) , and the flame stretch. Unlike the conventional Zeldovich number, the effective Zeldovich number is not necessarily linked directly to the activation energies of the reactions. Several examples of determining the effective Zeldovich number for reduced combustion mechanisms are given while, for realistic reactions, the effective Zeldovich number is determined from experiments. Another feature of the present approach is represented by the relatively simple asymptotic technique based on the adaptive generalized curvilinear system of coordinates attached to the flame (i.e., intrinsic disturbed flame equations [IDFE]).

Global and local coordinates

This chapter described the global and local coordinate systems utilized in the formulation of spatial multibody systems. Global coordinate system is considered in the present work to denote the inertia frame. Additionally, body-fixed coordinate systems, also called local coordinate systems, are utilized to describe local properties of points that belong to a particular body. Furthermore, the process of transforming local coordinates into global coordinates is characterized by considering a transformation matrix. In the present work, Cartesian coordinates are utilized to locate the center of mass of each rigid body, as well as the location of any point that belongs to a body.

Vector of coordinates, velocities and accelerations

This chapter describes the how the vector of coordinates are defined in the formulation of spatial multibody systems. For this purpose, the translational motion is described in terms of Cartesian coordinates, while rotational motion is specified using the technique of Euler parameters. This approach avoids the computational difficulties associated with the singularities in the case of using Euler angles or Bryant angles. Moreover, the formulation of the velocities vector and accelerations vector is presented and analyzed here. These two sets of vectors are defined in terms of translational and rotational coordinates.