A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

Neutron slowing down with inelastic scattering

The emphasis in reactor physics research has shifted toward investigations of fast reactors. The effects of high energy neutron processes have thus become fundamental to our understanding, and one of the most important of these processes is nuclear inelastic scattering. In this research we include inelastic scattering as a primary energy transfer mechanism, and study the resultant neutron energy spectrum in an infinite medium. We assume that the moderator material has a high mass number, so that in a laboratory coordinate system the energy loss of an inelastically scattered neutron may be taken as discrete. It is then consistent to treat elastic scattering with an age theory expansion. Mathematically these assumptions lead to balance equations of the differential-difference type.

The steady state problem is explored first by way of Laplace transformation of the energy variable. We then develop another steady state technique, valid for multiple inelastic level excitations, which depends on the level structure satisfying a physically reasonable constraint. In all cases the solutions we generate are compared with results obtained by modeling inelastic scattering with a separable, evaporative kernel.

The time dependent problem presents some new difficulties. By modeling the elastic scattering cross section in a particular way, we generate solutions to this more interesting problem. We conjecture the method of characteristics may be useful in analyzing time dependent problems with general cross sections. These ideas are briefly explored.

Pulsed neutron experiments in graphite

Detailed pulsed neutron measurements have been performed in graphite assemblies ranging in size from 30.48 cm x 38.10 cm x 38.10 cm to 91.44 cm x 66.67 cm x 66.67 cm. Results of the measurement have been compared to a modeled theoretical computation.

In the first set of experiments, we measured the effective decay constant of the neutron population in ten graphite stacks as a function of time after the source burst. We found the decay to be non-exponential in the six smallest assemblies, while in three larger assemblies the decay was exponential over a significant portion of the total measuring interval. The decay in the largest stack was exponential over the entire ten millisecond measuring interval. The non-exponential decay mode occurred when the effective decay constant exceeded 1600 sec^( -1).

In a second set of experiments, we measured the spatial dependence of the neutron population in four graphite stacks as a function of time after the source pulse. By doing an harmonic analysis of the spatial shape of the neutron distribution, we were able to compute the effective decay constants of the first two spatial modes. In addition, we were able to compute the time dependent effective wave number of neutron distribution in the stacks.

Finally, we used a Laplace transform technique and a simple modeled scattering kernel to solve a diffusion equation for the time and energy dependence of the neutron distribution in the graphite stacks. Comparison of these theoretical results with the results of the first set of experiments indicated that more exact theoretical analysis would be required to adequately describe the experiments.

The implications of our experimental results for the theory of pulsed neutron experiments in polycrystalline media are discussed in the last chapter.

Time-dependent monoenergetic neutron transport in two adjacent semi-infinite media

An exact solution to the monoenergetic Boltzmann equation is obtained for the case of a plane isotropic burst of neutrons introduced at the interface separating two adjacent, dissimilar, semi-infinite media. The method of solution used is to remove the time dependence by a Laplace transformation, solve the transformed equation by the normal mode expansion method, and then invert to recover the time dependence.

The general result is expressed as a sum of definite, multiple integrals, one of which contains the uncollided wave of neutrons originating at the source plane. It is possible to obtain a simplified form for the solution at the interface, and certain numerical calculations are made there.

The interface flux in two adjacent moderators is calculated and plotted as a function of time for several moderator materials. For each case it is found that the flux decay curve has an asymptotic slope given accurately by diffusion theory. Furthermore, the interface current is observed to change directions when the scattering and absorption cross sections of the two moderator materials are related in a certain manner. More specifically, the reflection process in two adjacent moderators appears to depend initially on the scattering properties and for long times on the absorption properties of the media.

This analysis contains both the single infinite and semi-infinite medium problems as special cases. The results in these two special cases provide a check on the accuracy of the general solution since they agree with solutions of these problems obtained by separate analyses.

Spectral density of first order Piecewise linear systems excited by white noise

The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form ẋ + f(x) = m/Ʃ/j = 1 h_j(x)n_j(t) where f and the h_j are piecewise linear functions (not necessarily continuous), and the n_j are stationary Gaussian white noise. For such systems, it is shown how the Laplace-transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32.11.

This method is applied to 4 subclasses: (1) m = 1, h₁ = const. (forcing function excitation); (2) m = 1, h₁ = f (parametric excitation); (3) m = 2, h₁ = const., h₂ = f, n₁ and n₂ correlated; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed.

Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steady-state density (the limit of the transition density as t → ∞).

4th European Conference on Energy Efficiency and Sustainability in Architecture and Planning : Donostia-San Sebastián, 1-3 Julio 2013

XXXII Cursos de Verano de la UPV/EHU

5th European Conference on Energy Efficiency and Sustainability in Architecture and Planning : Donostia-San Sebastián, 7-9 Julio 2014

XXXIII Cursos de Verano de la UPV/EHU

Prólogo

Homenaje a Georges Laplace, realizado en Vitoria-Gasteiz el 13,14 y 15 de noviembre de 2012. Edición a cargo de Aitor Calvo, Aitor Sánchez, Maite García-Rojas y Mónica Alonso-Eguíluz.

Francesc Gusi i Jener (1942-2012): in memoriam

Homenaje a Georges Laplace, realizado en Vitoria-Gasteiz el 13,14 y 15 de noviembre de 2012. Edición a cargo de Aitor Calvo, Aitor Sánchez, Maite García-Rojas y Mónica Alonso-Eguíluz.

León y caballo: Georges lLaplace. El yin y el yang de un prehistoriador inolvidable

Homenaje a Georges Laplace, realizado en Vitoria-Gasteiz el 13,14 y 15 de noviembre de 2012. Edición a cargo de Aitor Calvo, Aitor Sánchez, Maite García-Rojas y Mónica Alonso-Eguíluz.

Historia de una relación por venir: caracteres analíticos para el análisis funcional

Homenaje a Georges Laplace, realizado en Vitoria-Gasteiz el 13, 14 y 15 de noviembre de 2012. Edición a cargo de Aitor Calvo, Aitor Sánchez, Maite García-Rojas y Mónica Alonso-Eguíluz.

Tipología y tecnología. Una dependencia recíproca

Homenaje a Georges Laplace, realizado en Vitoria-Gasteiz el 13, 14 y 15 de noviembre de 2012. Edición a cargo de Aitor Calvo, Aitor Sánchez, Maite García-Rojas y Mónica Alonso-Eguíluz

La estratigrafía analítica: 25 años de ensayo y aprendizaje en el análisis e interpretación del registro arqueológico

Homenaje a Georges Laplace, realizado en Vitoria-Gasteiz el 13,14 y 15 de noviembre de 2012. Edición a cargo de Aitor Calvo, Aitor Sánchez, Maite García-Rojas y Mónica Alonso-Eguíluz.

Clasificación del formato tridimensional de materiales líticos desde una perspectiva geoarqueológica

Homenaje a Georges Laplace, realizado en Vitoria-Gasteiz el 13, 14 y 15 de noviembre de 2012. Edición a cargo de Aitor Calvo, Aitor Sánchez, Maite García-Rojas y Mónica Alonso-Eguíluz.

Los artefactos retocados del nivel C4d1j de Isturitz: estudio arqueopetrológico desde la Tipología Analítica

Homenaje a Georges Laplace, realizado en Vitoria-Gasteiz el 13, 14 y 15 de noviembre de 2012. Edición a cargo de Aitor Calvo, Aitor Sánchez, Maite García-Rojas y Mónica Alonso-Eguíluz.