Efficient computation of current collection in bare electrodynamic tethers in and beyond OML regime

One key issue in the simulation of bare electrodynamic tethers (EDTs) is the accurate and fast computation of the collected current, an ambient dependent operation necessary to determine the Lorentz force for each time step. This paper introduces a novel semianalytical solution that allows researchers to compute the current distribution along the tether efficient and effectively under orbital-motion-limited (OML) and beyond OML conditions, i.e., if tether radius is greater than a certain ambient dependent threshold. The method reduces the original boundary value problem to a couple of nonlinear equations. If certain dimensionless variables are used, the beyond OML effect just makes the tether characteristic length L ∗ larger and it is decoupled from the current determination problem. A validation of the results and a comparison of the performance in terms of the time consumed is provided, with respect to a previous ad hoc solution and a conventional shooting method.

El Problema de Lambert en el Contexto de Optimización de Trayectorias Espaciales

Esta tesis se basa en el estudio de la trayectoria que pasa por dos puntos en el problema de los dos cuerpos, inicialmente desarrollado por Lambert, del que toma su nombre. En el pasado, el Problema de Lambert se ha utilizado para la determinación de órbitas a partir de observaciones astronómicas de los cuerpos celestes. Actualmente, se utiliza continuamente en determinación de órbitas, misiones planetaria e interplanetarias, encuentro espacial e interceptación, o incluso en corrección de orbitas. Dada su gran importancia, se decide investigar especialmente sobre su solución y las aplicaciones en las misiones espaciales actuales. El campo de investigación abierto, es muy amplio, así que, es necesario determinar unos objetivos específicos realistas, en el contexto de ejecución de una Tesis, pero que sirvan para mostrar con suficiente claridad el potencial de los resultados aportados en este trabajo, e incluso poder extenderlos a otros campos de aplicación. Como resultado de este análisis, el objetivo principal de la Tesis se enfoca en el desarrollo de algoritmos para resolver el Problema de Lambert, que puedan ser aplicados de forma muy eficiente en las misiones reales donde aparece. En todos los desarrollos, se ha considerado especialmente la eficiencia del cálculo computacional necesario en comparación con los métodos existentes en la actualidad, destacando la forma de evitar la pérdida de precisión inherente a este tipo de algoritmos y la posibilidad de aplicar cualquier método iterativo que implique el uso de derivadas de cualquier orden. En busca de estos objetivos, se desarrollan varias soluciones para resolver el Problema de Lambert, todas ellas basadas en la resolución de ecuaciones transcendentes, con las cuales, se alcanzan las siguientes aportaciones principales de este trabajo: • Una forma genérica completamente diferente de obtener las diversas ecuaciones para resolver el Problema de Lambert, mediante desarrollo analítico, desde cero, a partir de las ecuaciones elementales conocidas de las cónicas (geométricas y temporal), proporcionando en todas ellas fórmulas para el cálculo de derivadas de cualquier orden. • Proporcionar una visión unificada de las ecuaciones más relevantes existentes, mostrando la equivalencia con variantes de las ecuaciones aquí desarrolladas. • Deducción de una nueva variante de ecuación, el mayor logro de esta Tesis, que destaca en eficiencia sobre todas las demás (tanto en coste como en precisión). • Estudio de la sensibilidad de la solución ante variación de los datos iniciales, y como aplicar los resultados a casos reales de optimización de trayectorias. • También, a partir de los resultados, es posible deducir muchas propiedades utilizadas en la literatura para simplificar el problema, en particular la propiedad de invariancia, que conduce al Problema Transformado Simplificado. ABSTRACT This thesis is based on the study of the two-body, two-point boundary-value problem, initially developed by Lambert, from who it takes its name. Since the past, Lambert's Problem has been used for orbit determination from astronomical observations of celestial bodies. Currently, it is continuously used in orbit determinations, for planetary and interplanetary missions, space rendezvous, and interception, or even in orbit corrections. Given its great importance, it is decided to investigate their solution and applications in the current space missions. The open research field is very wide, it is necessary to determine specific and realistic objectives in the execution context of a Thesis, but that these serve to show clearly enough the potential of the results provided in this work, and even to extended them to other areas of application. As a result of this analysis, the main aim of the thesis focuses on the development of algorithms to solve the Lambert’s Problem which can be applied very efficiently in real missions where it appears. In all these developments, it has been specially considered the efficiency of the required computational calculation compared to currently existing methods, highlighting how to avoid the loss of precision inherent in such algorithms and the possibility to apply any iterative method involving the use of derivatives of any order. Looking to meet these objectives, a number of solutions to solve the Lambert’s Problem are developed, all based on the resolution of transcendental equations, with which the following main contributions of this work are reached: • A completely different generic way to get the various equations to solve the Lambert’s Problem by analytical development, from scratch, from the known elementary conic equations (geometrics and temporal), by providing, in all cases, the calculation of derivatives of any order. • Provide a unified view of most existing relevant equations, showing the equivalence with variants of the equations developed here. • Deduction of a new variant of equation, the goal of this Thesis, which emphasizes efficiency (both computational cost and accuracy) over all other. • Estudio de la sensibilidad de la solución ante la variación de las condiciones iniciales, mostrando cómo aprovechar los resultados a casos reales de optimización de trayectorias. • Study of the sensitivity of the solution to the variation of the initial data, and how to use the results to real cases of trajectories’ optimization. • Additionally, from results, it is possible to deduce many properties used in literature to simplify the problem, in particular the invariance property, which leads to a simplified transformed problem.

Modelo computacional para avaliação do desempenho hidrodinâmico de embarcações de planeio em águas calmas.

Em geral, uma embarcação de planeio é projetada para atingir elevados níveis de velocidade. Esse atributo de desempenho está diretamente relacionado ao porte da embarcação e à potência instalada em sua planta propulsiva. Tradicionalmente, durante o projeto de uma embarcação, as análises de desempenho são realizadas através de resultados de embarcações já existentes, retirados de séries sistemáticas ou de embarcações já desenvolvidas pelo estaleiro e/ou projetista. Além disso, a determinação dos atributos de desempenho pode ser feita através de métodos empíricos e/ou estatísticos, onde a embarcação é representada através de seus parâmetros geométricos principais; ou a partir de testes em modelos em escala reduzida ou protótipos. No caso específico de embarcações de planeio, o custo dos testes em escala reduzida é muito elevado em relação ao custo de projeto. Isso faz com que a maioria dos projetistas não opte por ensaios experimentais das novas embarcações em desenvolvimento. Ao longo dos últimos anos, o método de Savitsky foi largamente utilizado para se realizar estimativas de potência instalada de uma embarcação de planeio. Esse método utiliza um conjunto de equações semi-empíricas para determinar os esforços atuantes na embarcação, a partir dos quais é possível determinar a posição de equilíbrio de operação e a força propulsora necessária para navegar em uma dada velocidade. O método de Savitsky é muito utilizado nas fases iniciais de projeto, onde a geometria do casco ainda não foi totalmente definida, pois utiliza apenas as características geométricas principais da embarcação para realização das estimativas de esforços. À medida que se avança nas etapas de projeto, aumenta o detalhamento necessário das estimativas de desempenho. Para a realização, por exemplo, do projeto estrutural é necessária uma estimativa do campo de pressão atuante no fundo do casco, o qual não pode ser determinado pelo método de Savitsky. O método computacional implementado nesta dissertação, tem o objetivo de determinar as características do escoamento e o campo de pressão atuante no casco de uma embarcação de planeio navegando em águas calmas. O escoamento é determinado através de um problema de valor de contorno, no qual a superfície molhada no casco é considerada um corpo esbelto. Devido ao uso da teoria de corpo esbelto o problema pode ser tratado, separadamente, em cada seção, onde as condições de contorno são forçadamente respeitadas através de uma distribuição de vórtices.

A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches

The Boussinesq equation appears as the zeroth-order term in the shallow water flow expansion of the non-linear equation describing the flow of fluid in an unconfined aquifer. One-dimensional models based on the Boussinesq equation have been used to analyse tide-induced water table fluctuations in coastal aquifers. Previous analytical solutions for a sloping beach are based on the perturbation parameter, epsilon(N) = alphaepsilon cot beta (in which beta is the beach slope, alpha is the amplitude parameter and epsilon is the shallow water parameter) and are limited to tan(-1) (alphaepsilon) much less than beta less than or equal to pi/2. In this paper, a new higher-order solution to the non-linear boundary value problem is derived. The results demonstrate the significant influence of the higher-order components and beach slope on the water table fluctuations. The relative difference between the linear solution and the present solution increases as 6 and a increase, and reaches 7% of the linear solution. (C) 2003 Elsevier Ltd. All rights reserved.

Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

Operational Methods in the Environment of a Computer Algebra System

This article presents the principal results of the doctoral thesis “Direct Operational Methods in the Environment of a Computer Algebra System” by Margarita Spiridonova (Institute of mathematics and Informatics, BAS), successfully defended before the Specialised Academic Council for Informatics and Mathematical Modelling on 23 March, 2009.

Quasi-Regular Solutions for 3D Equations of Tricomi and Keldish Types

Цветан Д. Христов, Недю Ив. Попиванов, Манфред Шнайдер - Изучени са някои тримерни гранични задачи за уравнения от смесен тип. За уравнения от типа на Трикоми те са формулирани от М. Протер през 1952, като тримерни аналози на задачите на Дарбу или Коши–Гурса в равнината. Добре известно е, че новите задачи са некоректни. Ние формулираме нова гранична задача за уравнения от типа на Келдиш и даваме понятие за квазиругулярно решение на тази задача и на eдна от задачите на Протер. Намерени са достатъчни условия за единственост на такива решения.

Problems for P-Monge-Ampere Equations

2010 Mathematics Subject Classification: 35A23, 35B51, 35J96, 35P30, 47J20, 52A40.

Monte Carlo Algorithms for Linear Problems

MSC Subject Classification: 65C05, 65U05.

Time-domain simulation of wave-induced ship motions by a Rankine panel method

In this thesis, a numerical program has been developed to simulate the wave-induced ship motions in the time domain. Wave-body interactions have been studied for various ships and floating bodies through forced motion and free motion simulations in a wide range of wave frequencies. A three-dimensional Rankine panel method is applied to solve the boundary value problem for the wave-body interactions. The velocity potentials and normal velocities on the boundaries are obtained in the time domain by solving the mixed boundary integral equations in relation to the source and dipole distributions. The hydrodynamic forces are calculated by the integration of the instantaneous hydrodynamic pressures over the body surface. The equations of ship motion are solved simultaneously with the boundary value problem for each time step. The wave elevation is computed by applying the linear free surface conditions. A numerical damping zone is adopted to absorb the outgoing waves in order to satisfy the radiation condition for the truncated free surface. A numerical filter is applied on the free surface for the smoothing of the wave elevation. Good convergence has been reached for both forced motion simulations and free motion simulations. The computed added-mass and damping coefficients, wave exciting forces, and motion responses for ships and floating bodies are in good agreement with the numerical results from other programs and experimental data.

Comprehensive Characterization of Measurement Data Gathered by the Pressure Tube to Calandria Tube Gap Probe

Multi-frequency eddy current measurements are employed in estimating pressure tube (PT) to calandria tube (CT) gap in CANDU fuel channels, a critical inspection activity required to ensure fitness for service of fuel channels. In this thesis, a comprehensive characterization of eddy current gap data is laid out, in order to extract further information on fuel channel condition, and to identify generalized applications for multi-frequency eddy current data. A surface profiling technique, generalizable to multiple probe and conductive material configurations has been developed. This technique has allowed for identification of various pressure tube artefacts, has been independently validated (using ultrasonic measurements), and has been deployed and commissioned at Ontario Power Generation. Dodd and Deeds solutions to the electromagnetic boundary value problem associated with the PT to CT gap probe configuration were experimentally validated for amplitude response to changes in gap. Using the validated Dodd and Deeds solutions, principal components analysis (PCA) has been employed to identify independence and redundancies in multi-frequency eddy current data. This has allowed for an enhanced visualization of factors affecting gap measurement. Results of the PCA of simulation data are consistent with the skin depth equation, and are validated against PCA of physical experiments. Finally, compressed data acquisition has been realized, allowing faster data acquisition for multi-frequency eddy current systems with hardware limitations, and is generalizable to other applications where real time acquisition of large data sets is prohibitive.

Properties of a composite material with mixed imperfect contact conditions

We present an analytical solution of a mixed boundary value problem for an unbounded 2D doubly periodic domain which is a model of a composite material with mixed imperfect interface conditions. We find the effective conductivity of the composite material with mixed imperfect interface conditions, and also give numerical analysis of several of their properties such as temperature and flux.

Riemann-Hilbert problems for poly-Hardy space on the unit ball

In this paper, we focus on a Riemann–Hilbert boundary value problem (BVP) with a constant coefficients for the poly-Hardy space on the real unit ball in higher dimensions. We first discuss the boundary behaviour of functions in the poly-Hardy class. Then we construct the Schwarz kernel and the higher order Schwarz operator to study Riemann–Hilbert BVPs over the unit ball for the poly- Hardy class. Finally, we obtain explicit integral expressions for their solutions. As a special case, monogenic signals as elements in the Hardy space over the unit sphere will be reconstructed in the case of boundary data given in terms of functions having values in a Clifford subalgebra. Such monogenic signals represent the generalization of analytic signals as elements of the Hardy space over the unit circle of the complex plane.

A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.

Multiple solutions and numerical analysis to the dynamic and stationary models coupling a delayed energy balance model involving latent heat and discontinuous albedo with a deep ocean

We study a climatologically important interaction of two of the main components of the geophysical system by adding an energy balance model for the averaged atmospheric temperature as dynamic boundary condition to a diagnostic ocean model having an additional spatial dimension. In this work, we give deeper insight than previous papers in the literature, mainly with respect to the 1990 pioneering model by Watts and Morantine. We are taking into consideration the latent heat for the two phase ocean as well as a possible delayed term. Non-uniqueness for the initial boundary value problem, uniqueness under a non-degeneracy condition and the existence of multiple stationary solutions are proved here. These multiplicity results suggest that an S-shaped bifurcation diagram should be expected to occur in this class of models generalizing previous energy balance models. The numerical method applied to the model is based on a finite volume scheme with nonlinear weighted essentially non-oscillatory reconstruction and Runge–Kutta total variation diminishing for time integration.