Tato disertační práce se zabývá analýzou stability a konvergence klasických numerických metod pro řešení obyčejných diferenciálních rovnic. Jsou představeny klasické jednokrokové metody, jako je Eulerova metoda, Runge-Kuttovy metody a nepříliš známá, ale rychlá a přesná metoda Taylorovy řady. V práci uvažujeme zobecnění jednokrokových metod do vícekrokových metod, jako jsou Adamsovy metody, a jejich implementaci ve dvojicích prediktor-korektor. Dále uvádíme generalizaci do vícekrokových metod vyšších derivací, jako jsou např. Obreshkovovy metody. Dvojice prediktor-korektor jsou často implementovány v kombinacích modů, v práci uvažujeme tzv. módy PEC a PECE. Hlavním cílem a přínosem této práce je nová metoda čtvrtého řádu, která se skládá z dvoukrokového prediktoru a jednokrokového korektoru, jejichž formule využívají druhých derivací. V práci je diskutována Nordsieckova reprezentace, algoritmus pro výběr proměnlivého integračního kroku nebo odhad lokálních a globálních chyb. Navržený přístup je vhodně upraven pro použití proměnlivého integračního kroku s přístupe vyšších derivací. Uvádíme srovnání s klasickými metodami a provedené experimenty pro lineární a nelineární problémy.

<div>The orbits of interest for potential missions are stable or nearly stable to maintain long term presence for conducting scientific studies and to reduce the possibility of rapid departure. Near Rectilinear Halo Orbits (NRHOs) offer such stable or nearly stable orbits that are defined as part of the L1 and L2 halo orbit families in the circular restricted three-body problem. Within the Earth-Moon regime, the L1 and L2 NRHOs are proposed as long horizon trajectories for cislunar exploration missions, including NASA's upcoming Gateway mission. These stable or nearly stable orbits do not possess well-distinguished unstable and stable manifold structures. As a consequence, existing tools for stationkeeping and transfer trajectory design that exploit such underlying manifold structures are not reliable for orbits that are linearly stable. The current investigation focuses on leveraging stretching direction as an alternative for visualizing the flow of perturbations in the neighborhood of a reference trajectory. The information supplemented by the stretching directions are utilized to investigate the impact of maneuvers for two contrasting applications; the stationkeeping problem, where the goal is to maintain a spacecraft near a reference trajectory for a long period of time, and the transfer trajectory design application, where rapid departure and/or insertion is of concern.</div><div><br></div><div>Particularly, for the stationkeeping problem, a spacecraft incurs continuous deviations due to unmodeled forces and orbit determination errors in the complex multi-body dynamical regime. The flow dynamics in the region, using stretching directions, are utilized to identify appropriate maneuver and target locations to support a long lasting presence for the spacecraft near the desired path. The investigation reflects the impact of various factors on maneuver cost and boundedness. For orbits that are particularly sensitive to epoch time and possess distinct characteristics in the higher-fidelity ephemeris model compared to their CR3BP counterpart, an additional feedback control is applied for appropriate phasing. The effect of constraining maneuvers in a particular direction is also investigated for the 9:2 synodic resonant southern L2 NRHO, the current baseline for the Gateway mission. The stationkeeping strategy is applied to a range of L1 and L2 NRHOs, and validated in the higher-fidelity ephemeris model.</div><div><br></div><div>For missions with potential human presence, a rapid transfer between orbits of interest is a priority. The magnitude of the state variations along the maximum stretching direction is expected to grow rapidly and, therefore, offers information to depart from the orbit. Similarly, the maximum stretching in reverse time, enables arrival with a minimal maneuver magnitude. The impact of maneuvers in such sensitive directions is investigated. Further, enabling transfer design options to connect between two stable orbits. The transfer design strategy developed in this investigation is not restricted to a particular orbit but applicable to a broad range of stable and nearly stable orbits in the cislunar space, including the Distant Retrograde Orbit (DROs) and the Low Lunar Orbits (LLO) that are considered for potential missions. Examples for transfers linking a southern and a northern NRHO, a southern NRHO to a planar DRO, and a southern NRHO to a planar LLO are demonstrated.</div>

The thesis deals with speedup and accuracy of numerical computation, especially when differential equations are solved. Algorithms, which are fulling these conditions are named semi-analytical. One posibility how to accelerate computation of differential equation is paralelization. Presented paralelization is based on transformation numerical solution into residue number system, which is extended to floating point computation. A new algorithm for modulo multiplication is also proposed. As application applications in solution of differential calculus are the main goal it is discussed numeric integration with modified Euler, Runge - Kutta and Taylor series method in residue number system. Next possibilities and extension for implemented residue number system are mentioned at the end.

Disertační práce se zabývá simulací elektronických obvodů. Popisuje metodu kapacitorové substituce (CSM) pro převod elektronických obvodů na elektrické obvody, jež mohou být následně řešeny pomocí numerických metod, zejména Moderní metodou Taylorovy řady (MTSM). Tato metoda se odlišuje automatickým výběrem řádu, půlením kroku v případě potřeby a rozsáhlou oblastí stability podle zvoleného řádu. V rámci disertační práce bylo autorem disertace vytvořeno specializované programové vybavení pro řešení obyčejných diferenciálních rovnic pomocí MTSM, s mnoha vylepšeními v algoritmech (v porovnání s TKSL/386). Tyto algoritmy zahrnují zjednodušování obecných výrazů na polynomy, paralelizaci nezávislou na integrační metodě atp. Tento software běží na linuxovém serveru, který komunikuje pomocí protokolu TCP/IP. Toto vybavení bylo úspěšně použito pro simulaci VLSI obvodů, jejichž řešení pomocí CSM bylo značně rychlejší a spotřebovávalo méně paměti než state-of-the-art SPICE.

<p>Thorium dioxide (ThO₂) and metallic uranium (alpha-U) represent important alternative nuclear fuels. Investigating the behavior of defects introduced into these materials in an irradiation environment is critical for understanding microstructure evolution and property changes. The objective of this dissertation is to investigate the clustering of point defects in ThO₂ and alpha-U under irradiation, into voids and prismatic dislocation loops as a function of irradiation dose rate and temperature. To achieve this, we have developed a mean-field cluster dynamics (CD) model based on reaction rate theory to predict the evolution of self-interstitial atom (SIA) and vacancy loops in neutron-irradiated alpha-U. Detailed atomistic simulations have been carried out using molecular dynamics (MD) to study the configuration of such loops and compute their energetics, which are essential parameters of the CD model. Bond-boost hyper-MD simulations have been performed to compute the diffusivity of uranium SIA and vacancies, which govern the kinetics of the clustering phenomenon. Another CD model has been demonstrated for proton-irradiated ThO₂, considering the clustering of Th and O SIA and vacancies into SIA loops and voids, respectively, with varying sizes and stoichiometry. The compositions of all SIA loops and voids dictated by crystallography of ThO₂ in its fluorite structure have been presented in their respective cluster composition space (CCS). The CD model solves the density evolution of off-stoichiometric loops and voids, with irradiation, in their respective CCS. MD simulations have been performed to compute the energetics of different clusters in their CCS, as parameters of the CD model. Temperature-accelerated MD simulations have been performed to compute the diffusivity of Th and O point defects, that dictates the kinetics of defect clustering on irradiation. In alpha-U, the CD predictions show an accumulation of small sized vacancy loops and the growth of SIA loops with irradiation dose, which closely fits the reported size distribution of loops in neutron-irradiated alpha-U by Hudson and coworkers. The CD predicted density of defect clusters in proton-irradiated ThO₂, shows the evolution of near-stoichiometric SIA loops in their CCS. The size distribution of SIA loops at high irradiation doses closely corresponds to the transmission electron microscopy (TEM) observations reported in the literature. Also, the CD model did not predict the growth of voids and vacancy clusters, which is consistent with findings in literature. The model was further used to predict the density of sub-nanometric defect clusters and point defects, on low-dose irradiation, that significantly impairs the thermal conductivity of ThO₂. An extensive TEM and CD investigation has also been carried out to study the growth and coarsening of SIA loop and voids during post-irradiation isochronal annealing of ThO₂ at high temperatures.</p>

<p dir="ltr">In Chapter 1, we introduce geometric motions from the general perspective of gradient flows. Here we develop the basic framework in which to pose the two main results of this thesis.</p><p dir="ltr">In Chapter 2, we examine the pinch-off phenomenon for a tubular surface moving by surface diffusion. We prove the existence of a one parameter family of pinching profiles obeying a long wavelength approximation of the dynamics.</p><p dir="ltr">In Chapter 3, we study a diffusion-based numerical scheme for curve shortening flow. We prove that the scheme is one time-step consistent.</p>

<p dir="ltr">Neural operator learning is emerging as a prominent technique in scientific machine learn- ing for modeling complex nonlinear systems with multi-physics and multi-scale applications. A common drawback of such operators is that they are data-hungry and the results are highly dependent on the quality and quantity of the training data provided to the models. Moreover, obtaining high-quality data in sufficient quantity can be computationally prohibitive. Faster surrogate models are required to overcome this drawback which can be learned from datasets of variable fidelity and also quantify the uncertainty. In this work, we propose a Multi-Level Stacked Deep Operator Network (MLSDON) which can learn from datasets of different fidelity and is not dependent on the input function. Through various experiments, we demonstrate that the MLSDON can approximate the high-fidelity solution operator with better accuracy compared to a Vanilla DeepONet when sufficient high-fidelity data is unavailable. We also extend MLSDON to build robust confidence intervals by making conformalized predictions. This technique guarantees trajectory coverage of the predictions irrespective of the input distribution. Various numerical experiments are conducted to demonstrate the applicability of MLSDON to multi-fidelity, multi-scale, and multi-physics problems.</p>

No description available.

It is important to know the heat generated due to nuclear decay in the final repository for spent nuclear fuel. In Sweden, the heating powers generated in spent nuclear fuels are currently measured in the calorimeter System 251 at the Clab facility, Oskarshamn. In order to better measure, and increase understanding, of the temperature measurements in the calorimeter, a simple thermodynamic model of its temperature evolution was developed. The model was described as a system of ordinary differential equations, which were solved, and the solution was applied to calibration measurements of the calorimeter. How precise the model is, how its parameters affect the model, et cetera, are addressed. How the temperature evolution of the system changes as the values of parameters in the model are changed is addressed. The mass correction of the calorimeter could be estimated from this model, which validated the established mass correction of the calorimeter. How the measurement results from the calorimeter would be affected if the volume of the calorimeter was changed was also considered. Additionally, gamma radiation escape from the calorimeter without being detected as heat in the calorimeter. The gamma escape energy fraction was estimated by SERPENT simulations of the calorimeter, as a function of the initial photon energy. The gamma escape was also estimated for different values of the radius of System 251.

No description available.