材料属性、外部载荷和几何尺寸等不确定性广泛存在于工程结构中。多个参数不确定性的耦合将导致结构响应产生无法忽略的偏差，因此精确且高效地量化参数的不确定性是非常有必要的。概率模型通常需要大量的实验样本信息得到精确的概率密度函数，这对于一些实际工程是十分困难的。而使用非概率凸集模型仅需要少量信息得到不确定参数的边界，便可描述参数的不确定性，十分适用于小样本、贫信息、高可靠性要求的机械结构。针对在满足精度和效率的前提下采用凸集模型控制不确定性这一问题，本文对非概率凸集模型及其应用进行了研究，主要研究内容如下：

基于改进多维平行六面体的非概率凸集模型。为更加高效地度量结构不确定性，提出一种改进多维平行六面体模型。通过重新定义区间变量的相关角和边缘区间，给出模型不确定域的显式表达式，进而给出依据实验样本点构建多维平行六面体模型的方法。算例分析表明，改进多维平行六面体模型能够包络更多的样本点且具备更小的体积，是一种比传统多维平行六面体模型的更为精确和紧凑的模型。

基于区间椭球交集的非概率凸集模型。本文提出了一种新的量化未知但有界不确定性的非概率凸集模型，定义为区间与椭球的交集。为了精确地分析所提模型的精度，给出了其体积的计算公式。与区间模型和椭球模型相比，区间椭球交集模型可以提供更小的不确定域体积，适合处理高维问题，从而在精度和效率之间取得了更好的平衡。通过三个算例分析表明所提模型的有效性。

基于区间椭球交集模型的半解析传播分析。本文将区间椭球交集模型应用于结构不确定性传播分析，提出了一种半解析法求解结构响应区间。通过将区间椭球交集模型响应区间的求解转化为求解至多2n-1 个椭球模型的响应区间，给出了区间椭球交集模型响应的半解析解，并给出了所提方法的求解步骤。通过三个数值算例验证了所提出的传播分析方法的有效性和可行性。

Uncertainties such as material properties, external loads and geometric dimensions widely exist in engineering structures. The coupling of multiple parameter uncertainties may result in non-negligible deviations in structural response. Consequently, it is of necessity to accurately and efficiently quantify the uncertainty of parameters. The probability model generally requires a large number of experimental samples to obtain accurate probability density functions. However, these samples are not always available in practical engineering structures. The non-probabilistic convex model only requires the boundary or scope of uncertain parameters, and therefore is more suitable for mechanical structures with limited samples, poor information and high reliability requirements. Aiming at the problem of accurately and efficiently quantifying the uncertainty of parameters, this paper studies the non-probabilistic convex model and its application. The main research contents are as follows:

(1) Non-probabilistic convex model based on improved multidimensional parallelepiped. An improved multidimensional parallelepiped model is presented to more reasonably and efficiently quantify structural uncertainties. An explicit expression for the uncertainty domain of the parallelepiped model is given by defining the correlation angle and marginal intervals of interval variables. A method is further formulated to construct a multidimensional parallelepiped model based on experimental samples. The results by three numerical examples show that the proposed model can envelop more sample points and has a smaller volume than the traditional multidimensional parallelepiped model.

(2) Non-probabilistic convex model based on intersection of interval and ellipsoidal. A non-probabilistic interval and ellipsoidal intersection model, namely the intersection of the interval model and the ellipsoidal model, is proposed for quantifying the unknown-but-bounded uncertainty. An analytical formula is given for the volume of the uncertainty domain, which is benefit for the quantitative evaluation of model accuracy. The proposed model can provide a smaller volume than either the interval model or the ellipsoidal model and is suitable for dealing with the high dimension problems, thus achieving a better balance with accuracy and efficiency. Three numerical examples are provided to demonstrate the effectiveness of the proposed modelling.

(3) A semi-analytical propagation analysis based on the interval and ellipsoidal intersection model. Based on the interval and ellipsoidal intersection model, a semi-analytical method is developed to obtain the bounds of the linear or near-linear response function. By transforming the solution of the response interval of the interval ellipsoidal intersection model into the solution of the response interval of up to 2n-1 ellipsoidal models, the semi-analytical solution of the response interval can be given. The implementation steps of the proposed method are also given. Three numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed propagation method.