The quasi-classical model of the spherical configuration in general relativity

We consider the quasi-classical model of the spin-free configuration on the basis of the self-gravitating spherical dust shell in General Relativity. For determination of the energy spectrum of the stationary states on the basis of quasi-classical quantization rules it is required to carry out some regularization of the system. It is realized by an embedding of the initial system in the extended system with rotation. Then, the stationary states of the spherical shells are S-states of the system with the intrinsic momentum. The quasi-classical treatment of a stability of the configuration is associated with the Langer modification of a square of the quantum mechanical intrinsic momentum. It gives value of critical bare mass of the shell determining threshold of stability. For the shell with the bare mass smaller or equal to the Planck's mass, the energy spectra of bound states are found. We obtain the expression for tunneling probability of the shell and construct the quasi-classical model of the pair creation and annihilation of the shells.Comment: 22 pages, sprocl.sty, 3 figure

Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of the hybrid dynamic models including continuous vehicle dynamics, the discrete models of computations and communications, and the size of the problem. We will overview recent advances in methodology and tools to model, analyze, and design robust autonomous aerospace systems operating in uncertain environment, with stress on efficient uncertainty quantification and robust design using the case studies of the mission including model-based target tracking and search, and trajectory planning in uncertain urban environment. To show that the methodology is generally applicable to uncertain dynamical systems, we will also show examples of application of the new methods to efficient uncertainty quantification of energy usage in buildings, and stability assessment of interconnected power networks

Nonlinear Coherent Modes of Trapped Bose-Einstein Condensates

Nonlinear coherent modes are the collective states of trapped Bose atoms, corresponding to different energy levels. These modes can be created starting from the ground state condensate that can be excited by means of a resonant alternating field. A thorough theory for the resonant excitation of the coherent modes is presented. The necessary and sufficient conditions for the feasibility of this process are found. Temporal behaviour of fractional populations and of relative phases exhibits dynamic critical phenomena on a critical line of the parametric manifold. The origin of these critical phenomena is elucidated by analyzing the structure of the phase space. An atomic cloud, containing the coherent modes, possesses several interesting features, such as interference patterns, interference current, spin squeezing, and massive entanglement. The developed theory suggests a generalization of resonant effects in optics to nonlinear systems of Bose-condensed atoms.Comment: 26 pages, Revtex, no figure

Long term nonlinear propagation of uncertainties in perturbed geocentric dynamics using automatic domain splitting

Current approaches to uncertainty propagation in astrodynamics mainly refer tolinearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationallyintensive. Differential algebra has already proven to be an efficient compromiseby replacing thousands of pointwise integrations of Monte Carlo runs with thefast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails in highly nonlinear dynamics or long term propagation. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial of the current state is split in two polynomials when its accuracy reaches a given threshold. The resulting polynomials accurately track uncertainties, even in highly nonlinear dynamics and long term propagations. Furthermore, valuable additional information about the dynamical system is available from the pattern in which those automatic splits occur. From this pattern it is immediately visible where the system behaves chaotically and where its evolution is smooth. Furthermore, it is possible to deduce the behavior of the system for each region, yielding further insight into the dynamics. In this work, the method is applied to the analysis of an end-of-life disposal trajectory of the INTEGRAL spacecraft

Reliability-based design optimization of shells with uncertain geometry using adaptive Kriging metamodels

Optimal design under uncertainty has gained much attention in the past ten years due to the ever increasing need for manufacturers to build robust systems at the lowest cost. Reliability-based design optimization (RBDO) allows the analyst to minimize some cost function while ensuring some minimal performances cast as admissible failure probabilities for a set of performance functions. In order to address real-world engineering problems in which the performance is assessed through computational models (e.g., finite element models in structural mechanics) metamodeling techniques have been developed in the past decade. This paper introduces adaptive Kriging surrogate models to solve the RBDO problem. The latter is cast in an augmented space that "sums up" the range of the design space and the aleatory uncertainty in the design parameters and the environmental conditions. The surrogate model is used (i) for evaluating robust estimates of the failure probabilities (and for enhancing the computational experimental design by adaptive sampling) in order to achieve the requested accuracy and (ii) for applying a gradient-based optimization algorithm to get optimal values of the design parameters. The approach is applied to the optimal design of ring-stiffened cylindrical shells used in submarine engineering under uncertain geometric imperfections. For this application the performance of the structure is related to buckling which is addressed here by means of a finite element solution based on the asymptotic numerical method

Stochastic System Design and Applications to Stochastically Robust Structural Control

The knowledge about a planned system in engineering design applications is never complete. Often, a probabilistic quantification of the uncertainty arising from this missing information is warranted in order to efficiently incorporate our partial knowledge about the system and its environment into their respective models. In this framework, the design objective is typically related to the expected value of a system performance measure, such as reliability or expected life-cycle cost. This system design process is called stochastic system design and the associated design optimization problem stochastic optimization. In this thesis general stochastic system design problems are discussed. Application of this design approach to the specific field of structural control is considered for developing a robust-to-uncertainties nonlinear controller synthesis methodology. Initially problems that involve relatively simple models are discussed. Analytical approximations, motivated by the simplicity of the models adopted, are discussed for evaluating the system performance and efficiently performing the stochastic optimization. Special focus is given in this setting on the design of control laws for linear structural systems with probabilistic model uncertainty, under stationary stochastic excitation. The analysis then shifts to complex systems, involving nonlinear models with high-dimensional uncertainties. To address this complexity in the model description stochastic simulation is suggested for evaluating the performance objectives. This simulation-based approach addresses adequately all important characteristics of the system but makes the associated design optimization challenging. A novel algorithm, called Stochastic Subset Optimization (SSO), is developed for efficiently exploring the sensitivity of the objective function to the design variables and iteratively identifying a subset of the original design space that has v i high plausibility of containing the optimal design variables. An efficient two-stage framework for the stochastic optimization is then discussed combining SSO with some other stochastic search algorithm. Topics related to the combination of the two different stages for overall enhanced efficiency of the optimization process are discussed. Applications to general structural design problems as well as structural control problems are finally considered. The design objectives in these problems are the reliability of the system and the life-cycle cost. For the latter case, instead of approximating the damages from future earthquakes in terms of the reliability of the structure, as typically performed in past research efforts, an accurate methodology is presented for estimating this cost; this methodology uses the nonlinear response of the structure under a given excitation to estimate the damages in a detailed, component level