Uncertainty Propagation Analysis Of Structures Using Probability Box Theory

In practical engineering problems, uncertainties are common, and studying the theories and methods of uncertain analysis is important for reliability design of industrial products. Probability theory is traditional uncertain analysis methodology, in which a great amount of experimental samples is required to construct the precise probability distribution. Unfortunately, it often seems very difficult to obtain sufficient experimental samples because of the limitation of experimental condition and cost, and hence probability method will encounter some limitations in applicability. Interval theory is a relative convenient and effective uncertain analysis method, in which interval is used to model the uncertainty of a variable. However, interval can only define the range of a variable and cannot describe the probability distribution of a variable. Thus statistical information from experiment cannot be sufficiently applied in interval. Probability box(p-box) theory is regarded as the mixture of interval theory and probability theory, which covers the shortage of two theories. In addition, the existing interval, probability model, Dempster-Shafer structures can be transferred into p-box structures. However, it is still at its preliminary stage for the present p-box research. Some key technical difficulties remain, such as the low computational efficiency of p-box propagation methods, short of correlation modeling theory and analytical method, the shortage of application study etc.This dissertation conducts a systematical research for computational efficiency of uncertainty propagation of p-box, and aims at contributing some useful researches and trials on mathematical programming theories and practical algorithms. In the basic theory study of p-box, the mathematical property of some of p-boxes that numerical characters of p-box are monotonic with respect to their distribution parameters is investigated. Meanwhile, the similar monotonicity which exists in the monotonic function with p-boxes is proven. On the method study of p-box, utilizing the monotonic analysis searches the extreme point of parameter distributions to improve the computational efficiency of traditional double loop solving methods. Besides, the accept-reject sampling method and collaborative optimization method are used to reduce the repetitive computation. Based on the studies of two aspects, some practical and efficient methods are developed in under sections. The following studies are carried out in this dissertation:(1) For a linear uncertain problem, a method computing intervals of mean and variance is presented, and it is generalized to the solution of static finite element model. The monotonicity of mean and variance of commonly used cumulative distribution functions with respect to distribution parameter is discussed. Furthermore, mean and variance of the linear function are also monotonic with respect to the distribution parameter of input p-box. By the monotonicity analysis, the double loop solution of mean and variance can be transformed into the single loop solution of probability problem.(2) Condition that the first and second origin moment with respect to the distribution parameter is monotonic is demonstrated. The similar monotonicity exited in the first to fourth origin moment of function and distribution parameters is also proven. Monotonicity property of p-box is determined by the distribution type of p-box and the property of function. Based on the monotonicity analysis, a method solving the first to fourth origin moment is proposed, which avoids the optimized searching or random sampling and improves the computational efficiency.(3) Two double-loop sampling methods respectively are suggested to solve the probability bounds of structural response based on the accept-reject sampling technology and the monotonicity analysis. The existing sampling methods for p-boxes are commonly involved in the double loop sampling or nest sampling. Outer sampling is to acquire the random distribution parameters from interval, and inner sampling are to computes the response distribution by the sampling method. The accept-reject sampling method executes unitive sample for the same distribution density function by constructing the same envelope function, which can avoid the repetitive analysis of inner sampling and improve the calculation efficiency of double-loop sample method. The mathematical property that the cumulative distribution functions of monotonic function respect to distribution parameter of p-box are monotonic is investigated. The method based on the monotonicity analysis utilizes the property to reduce the outer sample of the double-loop sampling method.(4) An efficient interval sampling method based on a collaborative optimization strategy is suggested. P-box is discretized into a range of intervals by the interval sampling method. An envelope interval is construed which includes all of intervals from p-box. Selecting the proper original point, the Rosen projection method is used to solve the interval problem. The optimization path is saved for the following analysis. By referring to the search path of envelope interval, the appropriate initial iteration point of sub-intervals can be determined. Based on the initial point, the repetitive search of the overlapped interior region of the intervals is avoided. The collaborative optimization method requires much less search time comparing with the conventional optimization method.(5) Dependence in the p-boxes is investigated. Modeling the dependent p-boxes by copula from the original experiment data is proposed and an uncertain propagation analysis method for the dependent p-boxes is also presented. For three types of uncertain experimental data, p-box is used to quantify the epistemic uncertain and the correlation of experimental data is constructed by the copula. Based on the sample theory of copula, the dependent p-boxes are discretized into a range of correlative random intervals, and then surrogate-model and collaborative optimization are applied to solve the interval problem.