Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Civil Engineering, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 121-123. / by Gary E.B. Choot. / M.S.

When structural optimization is performed via an iterative solution technique, it is possible to integrate the analysis and design iterations, in an integrated analysis and design procedure. The present work seeks to apply an integrated analysis and design approach in reliability based optimization, when a safety index approach is used. Two variants of the new approach are presented. Both of them are based on partially converged solution of the optimization procedure. The safety index approach employed allows us to use semi-analytical formulas to calculate the sensitivity derivatives of the safety constraints. The new approach is applied to the design of a simple structure. Both methods are robust to a satisfactory degree. The results are compared to those obtained by the safety index approach without integrating the analysis and design processes. The new methods substantially reduce the computational cost of optimization, which indicates that integrated analysis and design has the potential of removing a major obstacle, which is the excessive computational cost, in applying stochastic optimization to real life structural design. / Master of Science

The nonlinear stochastic vibrations of a beam with a varying cross-section are investigated. The nonlinearity is caused by midplane stretching and cubic in nature, and the forcing function is wide band white noise. The analysis is carried out by expanding the deflection curve in terms of the undamped linear modes. Substituting this expansion into the partial differential equation yields a set of ordinary differential equations in terms of the modal response functions, which are coupled through the nonlinear terms. The normal modes are found by the finite element method. The differential equations are then converted to a set of Ito's equations, from which a set of first-order differential equations for the response joint moments is found using the Fokker-Planck equation. These equations form an infinite hierarchy which is closed by the quasi-moment method. The solution is investigated near an internal resonance condition and the effects of higher order cumulants in the closure scheme and of additional modes to the expansion arc considered. It is shown that the second order solution is inadequate in the presence of internal resonances, but the fourth order solution proves to be adequate. The one mode approximation underestimates the nonlinear stiffening, and a multiple mode approach is necessary. It is also shown that the effect of an internal resonance of the stochastic vibration is to transfer of energy from the higher modes involved to the lower modes involved. / M.S.

We develop a theory for the analysis of chemical reactions in "isolated" containers. The main tool for this analysis consists of Boltzmann maps, which are discrete time dynamical systems that describe the time evolution of the normalized concentrations of the chemicals in the reactions. Moreover, the use of these maps allows us to draw conclusions about the continuous dynamical systems that the law of mass action associates with the different reactions. The theorems we prove show that entropy is a strict Liapunov function and that no complex evolution is expected out of the discrete dynamical systems. In fact, we prove convergence to a fixed point for most of the possible cases, and we give solid arguments for the convergence of the remaining ones. The analysis of the continuous systems is more complicated, and fewer results have been proven. However, the conclusions we draw are similar to those relative to the Boltzmann maps. Therefore, we suggest that no chaos is to be found in systems that do not exchange energy nor matter with the outer environment, both for the discrete and for the continuous cases. Such a phenomenon is more likely to occur in "closed" or in "open" reactors. Finally, we argue that the discrete dynamical systems have more physical content than the continuous ones, and that Boltzmann maps may be useful in the analysis of the non chaotic regions of many other kinds of finite dimensional maps. / Ph. D.

No description available.

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

A theory is presented which accounts for a particular aspect of nonlinearity caused by the deviation of plume "particles" from their mean trajectory in three-dimensional, statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities. Quasilinear expressions for the time-dependent nonlocal dispersivity and spatial covariance tensors of ensemble mean concentration are derived, as a function of time, variance σᵧ² of log hydraulic conductivity, degree of anisotropy, and flow direction. One important difference between existing linear theories and the new quasilinear theory is that in the former transverse nonlocal dispersivities tend asymptotically to zero whereas in the latter they tend to nonzero Fickian asymptotes. Another important difference is that while all existing theories are nominally limited to situations where σᵧ² is less than 1, the quasilinear theory is expected to be less prone to error when this restriction is violated because it deals with the above nonlinearity without formally limiting σᵧ². The theory predicts a significant drop in dimensionless longitudinal dispersivity when σᵧ² is large as compared to the case where σᵧ² is small. As a consequence of this drop the real asymptotic longitudinal dispersivity, which varies in proportion to σᵧ² when σᵧ² is small, is predicted to vary as σᵧ when σᵧ² is large. The dimensionless transverse dispersivity also drops significantly at early dimensionless time when σᵧ² is large. At late time this dispersivity attains a maximum near σᵧ² = 1, varies asymptotically at a rate proportional to σᵧ² when σᵧ² is small, and appears inversely proportional to σᵧ when σᵧ² is large. The actual asymptotic transverse dispersivity varies in proportion to σᵧ⁴ when σᵧ² is small and appears proportional to σᵧ when σᵧ² is large. One of the most interesting findings is that when the mean seepage velocity vector μ is at an angle to the principal axes of statistical anisotropy, the orientation of longitudinal spread is generally offset from μ toward the direction of largest log hydraulic conductivity correlation scale. When local dispersion is active, a plume starts elongating parallel to μ. With time the long axis of the plume rotates toward the direction of largest correlation scale, then rotates back toward μ, and finally stabilizes asymptotically at a relatively small angle of deflection. Application of the theory to depth-averaged concentration data from the recent tracer experiment at Borden, Ontario, yields a consistent and improved fit without any need for parameter adjustment.

Transmissivity, specific capacity, and steady state hydraulic head data collected from the Cortaro aquifer in Southern Arizona are analyzed statistically by means of regression and Kriging techniques. The statistics obtained in this manner are used to develop a stochastic model of the aquifer based on the finite element and Monte Carlo simulation methods. Three stages of generated head uncertainties are considered; (1) non-conditional, (2) conditional on transmissivity data and (3) conditional on both transmissivity and initial hydraulic head data (or inverse method). We found that simulated head values in stage 1 and 2 are associated with high variance amounting to 144.0 ft². When the statistics obtained from regression and Kriging in stage 2 are processed by means of the statistical inverse method of Neuman (1980), the result is a drastic reduction in the input head variance amounting to 75 percent reduction in the input head variance (i.e., 144 ft²). From these results, one may conclude that in order to minimize the variance of outputs generated by stochastic aquifer models, the input into such models must be created with the aid of appropriate statistical inverse procedure.

It is well known that the general geometric L´evy market models are incomplete, except for the geometric Brownian and the geometric Poissonian, but such a market can be completed by enlarging it with power-jump assets as Corcuera and Nualart [12] did on their paper. With the knowledge that an incomplete market due to jumps can be completed, we look at other cases of incompleteness. We will consider incompleteness due to more sources of randomness than tradable assets, transactions costs and stochastic volatility. We will show that such markets are incomplete and propose a way to complete them. By doing this we show that such markets can be completed. In the case of incompleteness due to more randomness than tradable assets, we will enlarge the market using the market’s underlying quadratic variation assets. By doing this we show that the market can be completed. Looking at a market paying transactional costs, which is also an incomplete market model due to indifference between the buyers and sellers price, we will show that a market paying transactional costs as the one given by, Cvitanic and Karatzas [13] can be completed. Empirical findings have shown that the Black and Scholes assumption of constant volatility is inaccurate (see Tompkins [40] for empirical evidence). Volatility is in some sense stochastic, and is divided into two broad classes. The first class being single-factor models, which have only one source of randomness, and are complete markets models. The other class being the multi-factor models in which other random elements are introduced, hence are an incomplete markets models. In this project we look at some commonly used multi-factor models and attempt to complete one of them. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2011.

Recent years have seen progress in the analysis of the heat kernel for certain reversible random walks in random environments. In particular the work of Barlow(2004) showed that the heat kernel for the random walk on the infinite component of supercritical bond percolation behaves in a Gaussian fashion. This heat kernel control was then used to prove a quenched functional central limit theorem. Following this work several examples have been analysed with anomalous heat kernel behaviour and, in some cases, anomalous scaling limits. We begin by generalizing the first result - looking for sufficient conditions on the geometry of the environment that ensure standard heat kernel upper bounds hold. We prove that these conditions are satisfied with probability one in the case of the random walk on continuum percolation and use the heat kernel bounds to prove an invariance principle. The random walk on dynamic environment is then considered. It is proven that if the environment evolves ergodically and is, in a certain sense, geometrically d-dimensional then standard on diagonal heat kernel bounds hold. Anomalous lower bounds on the heat kernel are also proven - in particular the random conductance model is shown to be "more anomalous" in the dynamic case than the static. Finally, the reflected random walk amongst random conductances is considered. It is shown in one dimension that under the usual scaling, this walk converges to reflected Brownian motion.