Browsing by Subject "Dynamical systems"

Now showing 1 - 5 of 5
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    Effects of a Rogue Star on Earth's Climate
    (2020-08) Chandramouli, Harini
    The details of the way in which the Earth orbits the Sun can have profound effects on Earth’s climate. Elements such as the Earth’s tilt or how tight the orbit is can affect temperature distribution or glacial formation. One event that could lead to such changes is if a star passes near our solar system close enough to disturb Earth’s orbit. Over the Earth's 4.5 billion year history, many stars could have approached the Solar System. Disturbances that are forced due to such an event could have effects on the orbit of planets in the Solar System. Even small changes to the orbits of the planets can have large effects on the climate a planet experiences as they could disrupt the Milankovitch cycles. These effects would be visible in the Earth's sediment core data, which can be used to help model the Earth's position as it orbits the Sun. This work focuses on modeling the effects of a star passing within the Solar System on the eccentricity, semi-major axis, and argument of the periapsis. The changes to the Earth's climate due to the changes on those orbital elements are also considered. Here the focus is on what the system would look like if the star were to pass within Kuiper Belt and the Oort Cloud. Passage within the Kuiper Belt has can change the equilibrium temperature of the Earth up to $2^\circ$ C without even taking into consideration the interactions of the planets with each other on the climate. The position of a planet as the star passes has an effect on the system, giving different results for different initial conditions. This position is seen to lead to dramatic differences between the orbital elements for various solutions.
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    Efficient computational methods for uncertainty quantification of large systems.
    (2011-05) Gaurav, Gaurav
    The quest to design environment-friendly and sustainable engineering systems has witnessed more and more fervent efforts in recent years. With the growth of affordable large-capacity computing resources, predictive, science-based computational models have become instrumental in this pursuit. The present work develops efficient computational methods for the uncertainty analysis of large dynamical and mechanical systems with local nonlinearities and uncertainties. Two approaches have been utilized: (i) reduction of the size of the system, and (ii) use of parallel computing resources. The first approach utilizes the response of a nominal system to efficiently compute the response of related systems; three types of analysis methods have been developed. The first method can be utilized for efficient modal analysis of undamped linear systems with local stiffness uncertainties. The second method can perform efficient frequency domain analysis of linear systems with local damping and stiffness uncertainties. The third method can be utilized for efficient time domain analysis of systems with local nonlinearities and uncertainties. These methods provide gains in computational efficiency approaching three orders of magnitude for moderately-sized computational models compared to the corresponding conventional methods. The gains in computational efficiency are expected to be more pronounced as the dimensionality of the system is increased. The second approach to increase computational efficiency utilizes modern parallel computing devices, specifically, graphics processing units (GPUs) to perform uncertainty analysis of computational models. A variety of uncertainty quantification methods have been implemented on a GPU. The gains in computational efficiency compared to corresponding CPU-based implementations range from one to three orders of magnitude. These GPU implementations are expected to serve as initial bases for further developments in the use of GPUs in this field.
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    Generalizations for Insolation and Albedo to Adapt an Energy Balance Model to Other Planets
    (2019-05) Nadeau, Alice
    Interest in modeling the climates of other planets has been stimulated by observations of the Pluto-Charon system and seven Earth-sized planets orbiting the nearby star TRAPPIST-1. Furthermore, as of March 2019, over four thousand planets outside of our solar system have been discovered. Scientists are interested in what these planets might be like and if they could support life as we know it, but there is very little empirical information that they can collect in order to learn more about them. For this reason, scientists must rely on models to study climate on these planets. Because so little is known about our planetary neighbors compared to Earth, and even less is known about planets outside of our solar system, it is hard to faithfully model their climates using complex models such as such as the class of models referred to as General Circulation Models (GCMs). Instead, conceptual climate models may be preferred because the small number of state variables and parameters (relative to GCMs) make it easier to quantify possible behaviors of the system. Adapting well known conceptual models for Earth to extraterrestrial and extrasolar planets raises issues whose solutions draw from the fields of celestial mechanics, harmonic analysis and nonsmooth systems. This work focuses on a main component of conceptual climate models---incoming radiation absorbed by the planet---and the mathematical considerations for and implications of adapting this component to planets other than Earth. We generalize both the distribution of insolation and location of different albedos on the planet's surface. We find that the insolation distribution for slowly rotating planets approaches a rapid rotation distribution like the reciprocal of the rotation rate. Additionally, we show that it is possible to have stable, asymmetric configurations of ice in an energy balance model of Pluto.
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    Integrated dynamical models of down-the-hole percussive drilling
    (2014-07) Depouhon, Alexandre F.B.E.
    Due to the overall process complexity, studies about percussive drilling usually focus on a limited set of the (sub)processes underlying it, e.g., the hammer thermodynamics or the interaction between the bit and the rock. Following this paradigm, the assessment of the process performance is typically performed by considering a single percussive activation and a single interaction cycle between the bit and the rock, from arbitrary initial conditions.The need for an integrated approach to evaluate drilling performance, based on the dynamical interaction of the (sub)processes underlying drilling, is evident. Such an approach requires simplified models, however, as the computational cost associated with full scale models is simply unbearable. In this thesis, three dynamical integrated models are proposed and a preliminary analysis is conducted for a reference configuration and around it. The models couple three modules that represent: (i) the dynamics of the mechanical system, (ii) the interaction between the bit and the rock, and (iii) the activation of the mechanical system. For each module, simple representations are considered; of particular importance is the bit/rock interaction model which is a generalization to repeated interactions of experimental evidence observed for a single interaction.In the first model, the dynamics of a rigid bit is cast into a drifting oscillator and the activation modeled as a periodic impulsive force. The second and third models account for the dynamics of the piston and the activation results from the impact of the piston on the bit. They are respectively based on elastic and rigid representations of the two bodies. In the rigid model, analytical results of wave propagation in thin rods are used to represent the contact interaction between the piston and the bit. In the elastic model, wave propagation is resolved.Their preliminary analysis has revealed the occurrence of complex dynamical responses in the space of parameters. Expected trends are recovered around a reference configuration corresponding to a low-size hammer, with an increase of the rate of penetration with the feed force and the percussive frequency. An important sensitivity of the rate of penetration to the latter parameter is uncovered. Interestingly, our analyses show that when the activation period has the same order of magnitude as the timescale associated with the bit/rock interaction, a lower power consumption is observed, indicating a possible resonance phenomenon in the drilling system. Also, the predictions of the rigid model are shown to be in good agreement with the ones of the elastic model, in the explored range of parameters.Given the piecewise linear nature of the proposed models, dedicated numerical tools have been developed to conduct their analysis. As such, the thesis proposes a high-order time integration scheme for linear structural dynamics as well as a novel framework to evaluate the accuracy of such schemes, and a root-solving module to perform event-detection, for coupling with event-driven integration strategies. Specific to the framework is the account for both structural damping and external forcing in the evaluation of the scheme order of accuracy. Specific to the root-solving module is the forcing of event occurrence in the localization procedure.
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    Singular Perturbations of the Quadratic Map
    (2015) Maiers, Jordan