Appropriate SCF basis sets for orbital studies of galaxies and a `quantum-mechanical' method to compute them

We address the question of an appropriate choice of basis functions for the self-consistent field (SCF) method of simulation of the N-body problem. Our criterion is based on a comparison of the orbits found in N-body realizations of analytical potential-density models of triaxial galaxies, in which the potential is fitted by the SCF method using a variety of basis sets, with those of the original models. Our tests refer to maximally triaxial Dehnen gamma-models for values of $\gamma$ in the range 0<=gamma<=1. When an N-body realization of a model is fitted by the SCF method, the choice of radial basis functions affects significantly the way the potential, forces, or derivatives of the forces are reproduced, especially in the central regions of the system. We find that this results in serious discrepancies in the relative amounts of chaotic versus regular orbits, or in the distributions of the Lyapunov characteristic exponents, as found by different basis sets. Numerical tests include the Clutton-Brock and the Hernquist-Ostriker (HO) basis sets, as well as a family of numerical basis sets which are `close' to the HO basis set. The family of numerical basis sets is parametrized in terms of a quantity $\epsilon$ which appears in the kernel functions of the Sturm-Liouville (SL) equation defining each basis set. The HO basis set is the $\epsilon=0$ member of the family. We demonstrate that grid solutions of the SL equation yielding numerical basis sets introduce large errors in the variational equations of motion. We propose a quantum-mechanical method of solution of the SL equation which overcomes these errors. We finally give criteria for a choice of optimal value of $\epsilon$ and calculate the latter as a function of the value of gamma.Comment: 22 pages, 13 figures, Accepted in MNRA

Forward diffraction modelling : analysis and application to grating reconstruction

The semiconductor industry uses lithography machines for manufacturing complex integrated circuits (also called ICs) onto wafers. Because an IC is built up layer by layer and feature sizes get smaller and smaller, tight control of the lithography process is required to guarantee a fast production of working ICs. Typically a lot of information on the lithography process can be obtained by measuring test structures or gratings which are scattered over the wafer. These gratings are tiny periodic structures much smaller than ICs. First these gratings are illuminated and its response (a scattered intensity) is measured. For certain applications like overlay metrology the asymmetry in this measured signal (due to an offset between two gratings) can be used to align the lithographic process. For other applications like critical dimension (CD) metrology one is interested in the shape of the grating lines that produced the measured signal. Since this information is not directly available but encrypted in the measurement, a reconstruction algorithm is used to extract it. The reconstructed values like height, width and sidewall angle can then be related to machine settings like dose and focus which control the lithographic process. In particular the CD metrology application requires rigorous mathematical models that solve optical diffraction problems for periodic gratings in combination with advanced reconstruction algorithms. This thesis focuses on the optical diffraction problem for 1D periodic gratings. Starting from Maxwell's equations a reduced model is derived by simplifying both the grating and the incident electromagnetic field. The former is approximated with an infinitely periodic layered structure with isotropic non-magnetic materials. The latter is approximated with a time-harmonic incident plane wave. The reduced model is discretised using two different mode expansion methods, Bloch and the Rigorous Coupled-Wave Analysis (RCWA). Bloch expands the electromagnetic field in each layer in terms of the exact eigenfunctions whereas RCWA only uses approximate eigenfunctions. After truncation of the involved series a transmission problem is derived by matching the fields at the layer interfaces. Having solved the resulting linear system, the scattered field can be computed easily. Both mode expansion methods solve a similar linear system containing a large but sparse block-structured coefficient matrix. However, special care needs to be taken when solving this system stably and efficiently. Therefore a stable condensation algorithm is derived based on Riccati transformations that decouples the exponentially growing and decaying terms that are present in the solution. This separation or decoupling is the key feature explaining the stability which is not always clear in alternative condensation algorithms. Furthermore the algorithm is optimised for speed by using a two-stage approach. Finally it is shown that the resulting stable recursions are identical to those used in the 'enhanced transmittance matrix approach" (a frequently used condensation algorithm), thereby confirming its stability as well. This thesis also examines and extends both mode expansions methods. The Bloch method is generalised to deal with multiple material transitions inside a grating layer covering a wider range of applications. However, lossy or fully asymmetric gratings are still hard to solve. On the other hand the Fourier discretisation used in RCWA is much more exible but only approximates the more exact discretisation of Bloch. Therefore two RCWA modifications have been investigated to improve the accuracy while keeping its exibility and relatively straightforward implementation. Adaptive Spatial Resolution applies an additional layer specific coordinate transformation before Fourier discretising the problem again. A good transformation not only refines near a material interface but also does this in a smooth way. A significant improvement in accuracy is observed that approaches and sometimes outperforms the results obtained with the Bloch method. The second modification removes the Fourier discretisation completely and uses a finite difference approximation in the periodic direction. Although this approach allows for a better discretisation near a material interface, the sparsity of the resulting matrices could not be exploited to make a competitive implementation within the standard RCWA framework. Finally the integration of the forward diffraction model in the CD reconstruction application is discussed. Either a library based or real-time regressions approach can be used for this reconstruction. Both approaches rely heavily on having an accurate and fast forward model. By exploiting additional symmetries and smart reuse of information, acceptable library fill times and real-time reconstructions are now feasible

The aeroacoustics of supersonic jets

This research project was a joint experimental/computational study of noise in supersonic jets. The experiments were performed in a low to moderate Reynolds number anechoic supersonic jet facility. Computations have focused on the modeling of the effect of an external shroud on the generation and radiation of jet noise. This report summarizes the results of the research program in the form of the Masters and Doctoral theses of those students who obtained their degrees with the assistance of this research grant. In addition, the presentations and publications made by the principal investigators and the research students is appended

NASA Workshop on Distributed Parameter Modeling and Control of Flexible Aerospace Systems

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach

Theoretical models for duct acoustic propagation and radiation

The development of computational methods in acoustics has led to the introduction of analysis and design procedures which model the turbofan inlet as a coupled system, simultaneously modeling propagation and radiation in the presence of realistic internal and external flows. Such models are generally large, require substantial computer speed and capacity, and can be expected to be used in the final design stages, with the simpler models being used in the early design iterations. Emphasis is given to practical modeling methods that have been applied to the acoustical design problem in turbofan engines. The mathematical model is established and the simplest case of propagation in a duct with hard walls is solved to introduce concepts and terminologies. An extensive overview is given of methods for the calculation of attenuation in uniform ducts with uniform flow and with shear flow. Subsequent sections deal with numerical techniques which provide an integrated representation of duct propagation and near- and far-field radiation for realistic geometries and flight conditions

12th International Conference on Vibrations in Rotating Machinery

Since 1976, the Vibrations in Rotating Machinery conferences have successfully brought industry and academia together to advance state-of-the-art research in dynamics of rotating machinery. 12th International Conference on Vibrations in Rotating Machinery contains contributions presented at the 12th edition of the conference, from industrial and academic experts from different countries. The book discusses the challenges in rotor-dynamics, rub, whirl, instability and more. The topics addressed include: - Active, smart vibration control - Rotor balancing, dynamics, and smart rotors - Bearings and seals - Noise vibration and harshness - Active and passive damping - Applications: wind turbines, steam turbines, gas turbines, compressors - Joints and couplings - Challenging performance boundaries of rotating machines - High power density machines - Electrical machines for aerospace - Management of extreme events - Active machines - Electric supercharging - Blades and bladed assemblies (forced response, flutter, mistuning) - Fault detection and condition monitoring - Rub, whirl and instability - Torsional vibration Providing the latest research and useful guidance, 12th International Conference on Vibrations in Rotating Machinery aims at those from industry or academia that are involved in transport, power, process, medical engineering, manufacturing or construction