【作者】 李龙；

【导师】 徐军； 康为江；

【作者基本信息】 湖南大学 ， 建筑与土木工程（专业学位）， 2022， 硕士

【摘要】 结构可靠度分析和可靠度设计优化是目前可靠度理论研究的主要内容。本论文围绕工程设计对高效、高精度的结构可靠度分析和可靠度设计优化方法的迫切需求,提出了两种高效的可靠度分析方法,并对概率积分上做出改进,同时将提出的可靠度方法用于可靠度设计优化中。主要研究内容如下:针对单个分布模型的问题和混合分布的适用性问题,提出了一种基于正态逆高斯分布的自适应混合分布方法。通过判断限制条件来选择混合分布模型中分量的个数。使用混合容积公式求解统计矩和拉普拉斯变换,通过匹配拉普拉斯变换的离散值建立非线性方程组,求解方程组得到混合分布模型的未知参数,然后重构功能函数的概率密度函数。五个不同类型的数值算例证明了所提方法可精确地计算功能函数的失效概率,尤其是小失效概率的计算。针对传统鞍点法中对累积生成函数的解析式的假设,提出了一种基于多项式展开的鞍点法。利用迭代的思想不断增加多项式展开的项数,逐渐逼近功能函数的累积生成函数。通过匹配累积生成函数的离散值计算多项式的未知参数。一旦获得功能函数的累积分布函数,通过求解鞍点得到功能函数的可靠度指标。当迭代过程中的可靠度指标变化率连续三次小于给定的阈值时,停止迭代,输出最终的可靠度指标。对于静力问题,采用上述提到的容积公式作为数值积分方法。对于动力问题,采用基于部分分层抽样的数论法用来生成高维空间下的样本。数值算例的结果表明:所提方法在静力和动力可靠度问题中均能准确地预测失效概率。基于可靠度问题中用到的数值积分方法,对概率(贝叶斯)积分进行了研究,提出了一种新的序列试验点设计方法。新的选点方法从贝叶斯公式预测的积分值出发,推导积分值的误差来源。同时根据新的样本点和训练样本之间的空间相关性,提出了衡量点到训练样本之间的距离的公式。根据上述研究,提出了一种基于置信区间和距离的序列选点方法。结果表明:相较于传统的确定性积分方法和抽样方法,所提方法能更好地兼顾效率和精度,同时能量化积分的不确定性。针对可靠度设计优化中计算效率低问题,提出了一种基于百分位数的序列解耦优化方法。在该方法中采用自适应混合分布模型计算设计优化中的可靠度评估。通过重构概率约束函数的累积分布函数,得到可靠度指标对应的约束值,然后平移极限状态面,用确定性的极限状态面代替概率约束面后,进行确定性的优化,得到优化值,然后判断前后两次优化目标变化值是否小于给定的阈值,如满足则输出结果,如不满足则继续解耦优化,重复上述操作,直到满足条件。结果说明所提方法相较经典可靠度设计优化方法更为高效。更多还原

【Abstract】 Nowadays the chief content of reliability theory includes structural reliability analysis and reliability-based design optimization.This paper presents two efficient reliability analysis methods,an active learning function based on probability integral and take the proposed reliability analysis method into reliability-based design optimization.The main research contents are following:An adaptive mixture distribution based on normal-inverse Gaussian distribution is put forward to deal with the problem of the applicability of single distribution model and mixture distribution.In this paper,a limited condition is proposed to decide to choose which distribution we will use,a single distribution model or mixture distribution model.The proposal calculates the unknown parameters of distribution by matching the discrete values of Laplace transform,then reconstructs the probability density function of performance function.Five examples demonstrate the proposal can accurately predict failure probability of performance function,especially in rare events.An improved saddlepoint approximation method based on polynomial expansion is presented to deal with the inflexible assumption about cumulative generating function.The proposed approach which stems from the idea of iteration approximates the cumulative generating function of performance function by keeping increasing the number of terms of polynomial expansion.The unknown parameters of cumulative generating function are determined by solving system of the nonlinear equations at discrete values in cumulative generating function,the failure probability can easily be obtained by saddlepoint approach after obtaining cumulative generating function.When the change rate of the reliability index in the iteration is lower than given threshold for three consecutive times,the iteration is stopped and the final reliability index is output.For static problems,this paper considers a mixture cubature formula as numerical integral method.For dynamic problems,a numerical sampling based partially stratified sampling is used to calculate related numerical quadrature.The results illustrate that the proposed method can predict small failure probability in both static problems and high-dimensional dynamic structure.A new sequential experimental design point method in Bayesian Quadrature is presented to tackle the problem of expensive computation in reliability analysis.The new method of selecting training sample point is made by two parts,the first part deduces the error source that affects the result by the formula of Bayesian Quadrature,the second part presents a distance function that can measure the distance between a new sample point and training pool.A new selecting point method is presented based on these two parts.The results demonstrate this approach has a good trade-off in accuracy and efficiency compared to deterministic integral methods and sampling method.Besides,it can quantify the uncertainty of the value of integral while other methods cannot do.A sequential reliability-based design optimization based on quantile is used to calculate the design problem and the above-mentioned adaptive mixture distribution is applied to reconstruct cumulative distribution function of the probability constraint function in optimization and the corresponding quantile,the probabilistic constraint surface is replaced by a deterministic constraint surface by moving the limit state surface.When the change value of the optimization target before and after the two times is less than the given threshold value,if so,output the result,if not,continue the decoupling optimization,repeat the above operations until the conditions are met.Three examples demonstrate the proposed method is more efficient than other classical reliability-based design optimization methods.更多还原