Three-dimensional pseudo-dynamic reliability analysis of seismic shield tunnel faces combined with sparse polynomial chaos expansion

To address the seismic face stability challenges encountered in urban and subsea tunnel construction,an efficient probabilistic analysis framework for shield tunnel faces under seismic conditions is proposed.Based on the upper-bound theory of limit analysis,an improved three-dimensional discrete deterministic mechanism,accounting for the heterogeneous nature of soil media,is formulated to evaluate seismic face stability.The metamodel of failure probabilistic assessments for seismic tunnel faces is constructed by integrating the sparse polynomial chaos expansion method(SPCE)with the modified pseudo-dynamic approach(MPD).The improved deterministic model is validated by comparing with published literature and numerical simulations results,and the SPCE-MPD metamodel is examined with the traditional MCS method.Based on the SPCE-MPD metamodels,the seismic effects on face failure probability and reliability index are presented and the global sensitivity analysis(GSA)is involved to reflect the influence order of seismic action parameters.Finally,the proposed approach is tested to be effective by a engineering case of the Chengdu outer ring tunnel.The results show that higher uncertainty of seismic response on face stability should be noticed in areas with intense earthquakes and variation of seismic wave velocity has the most profound influence on tunnel face stability.

Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a generalized polynomial chaos expansion(GPCE)for statistical moment and reliability analyses associated with the stochastic output and a static reanalysis method to generate the input-output data set.In the reanalysis,we employ substructuring for a structure to isolate its local regions that vary due to random inputs.This allows for avoiding repeated computations of invariant substructures while generating the input-output data set.Combining substructuring with static condensation further improves the computational efficiency of the reanalysis without losing accuracy.Consequently,the GPCE with the static reanalysis method can achieve significant computational saving,thus mitigating the curse of dimensionality to some degree for UQ under high-dimensional inputs.The numerical results obtained from a simple structure indicate that the proposed method for UQ produces accurate solutions more efficiently than the GPCE using full finite element analyses(FEAs).We also demonstrate the efficiency and scalability of the proposed method by executing UQ for a large-scale wing-box structure under ten-dimensional(all-dependent)random inputs.

Sensitivity Analysis of Electromagnetic Scattering from Dielectric Targets with Polynomial Chaos Expansion and Method of Moments

In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.

Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.

Data-driven sparse polynomial chaos expansion for models with dependent inputs

Polynomial chaos expansions(PCEs)have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of the inputs.PCEs for models with independent inputs have been extensively explored in the literature.Recently,different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications.Typical approaches include building PCEs based on the Gram–Schmidt algorithm or transforming the dependent inputs into independent inputs.However,the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions,respectively.In this paper,we propose a data-driven approach to build sparse PCEs for models with dependent inputs without any distributional assumptions.The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output.The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency.Four numerical examples are implemented to validate the proposed algorithm.The source code is made publicly available for reproducibility.

A polynomial chaos expansion method for the uncertain acoustic field in shallow water

To obtain a universal model solving the uncertain acoustic field in shallow water, a non-intrusive model coupled polynomial chaos expansion （PCE） method with Helmholtz equa- tion is established, in which the polynomial coefficients are solved by probabilistic collocation method （PCM）. For the cases of Pekeris waveguide which have uncertainties in depth of water column, in both sound speed profile and depth of water column, and for the case of thermocline with lower limit depth uncertain, probability density functions （PDF） of transmission loss （TL） are calculated. The results show that the proposed model is universal for the acoustic propa- gation codes with high computational efficiency and accuracy, and can be applied to study the uncertainty of acoustic propagation in the shallow water en^-ironment with multiple parameters uncertain.

A novel robust aerodynamic optimization technique coupled with adjoint solvers and polynomial chaos expansion

Uncertainty is common in the life cycle of an aircraft, and Robust Aerodynamic Optimization(RAO) that considers uncertainty is important in aircraft design. To avoid the curse of dimensionality in surrogate-based optimization, this study proposes an adjoint RAO technique called “R-Opt”. Polynomial Chaos Expansion(PCE) is coupled with the R-Opt technique to quantify uncertainty in the responses of the target(including its mean and standard deviation). Only one process of PCE model construction is required in each iteration, and the gradients of uncertainty can be inferred via chain rules. The proposed method is more efficient than prevalent methods,and avoids the problem of a disagreement over the best PCE basis from among a number of PCE models(especially in case of sparse PCE). It also supports the application of sparse PCE.Two benchmark tests and two airfoil cases were used to verify R-Opt, and the optimal solutions were deemed to be robust. It improved the mean aerodynamic performance and reduced the standard deviation of the target.

基于PCE法和最大熵法的船舶不确定性优化设计

[目的]船舶不确定性优化设计(UDO)的关键在于不确定性的量化(UQ),传统的蒙特卡洛方法较为耗时且计算成本较高,故提出一种基于多项式混沌展开法(PCE)和最大熵法(MEM)的船舶UDO方法。[方法]首先,选取较低计算成本的PCE法来量化多个不确定参数作用下输出的随机性质;然后,根据正交多项式的特点,采用基于线性无关原则的改进概率配点法(IPCM)求解PCE系数;最后,利用推导的约束前4阶矩,结合最大熵法求解约束的概率密度函数(PDF),进而在失效域上积分得到UDO关注的约束失效概率。[结果]研究结果表明:改进概率配点法可以给出最优的概率配点数目,并显著减少样本点数量;基于PCE法和MEM法求解约束失效概率时,与蒙特卡洛法结果对比,其在不额外增加计算量的前提下亦可满足精度要求;散货船工程算例的优化结果验证了PCE法较常用的蒙特卡洛法在精度和效率上更具工程应用优势。[结论]该不确定性优化设计方法可以高效准确地实现船舶设计方案的稳健性和可靠性。

An Efficient Sampling Method for Regression-Based Polynomial Chaos Expansion

The polynomial chaos expansion(PCE)is an efficient numerical method for performing a reliability analysis.It relates the output of a nonlinear system with the uncertainty in its input parameters using a multidimensional polynomial approximation(the so-called PCE).Numerically,such an approximation can be obtained by using a regression method with a suitable design of experiments.The cost of this approximation depends on the size of the design of experiments.If the design of experiments is large and the system is modeled with a computationally expensive FEA(Finite Element Analysis)model,the PCE approximation becomes unfeasible.The aim of this work is to propose an algorithm that generates efficiently a design of experiments of a size defined by the user,in order to make the PCE approximation computationally feasible.It is an optimization algorithm that seeks to find the best design of experiments in the D-optimal sense for the PCE.This algorithm is a coupling between genetic algorithms and the Fedorov exchange algorithm.The efficiency of our approach in terms of accuracy and computational time reduction is compared with other existing methods in the case of analytical functions and finite element based functions.

A Polynomial Chaos Expansion Trust Region Method for Robust Optimization

Robust optimization is an approach for the design of a mechanical structure which takes into account the uncertainties of the design variables.It requires at each iteration the evaluation of some robust measures of the objective function and the constraints.In a previous work,the authors have proposed a method which efficiently generates a design of experiments with respect to the design variable uncertainties to compute the robust measures using the polynomial chaos expansion.This paper extends the proposed method to the case of the robust optimization.The generated design of experiments is used to build a surrogate model for the robust measures over a certain trust region.This leads to a trust region optimization method which only requires one evaluation of the design of experiments per iteration(single loop method).Unlike other single loop methods which are only based on a first order approximation of robust measure of the constraints and which does not handle a robust measure for the objective function,the proposed method can handle any approximation order and any choice for the robust measures.Some numerical experiments based on finite element functions are performed to show the efficiency of the method.

Polynomial chaos surrogate and bayesian learning for coupled hydro-mechanical behavior of soil slope

As rainfall infiltrates into soil slopes,the hydraulic and mechanical behaviors of soils are interacted.In this study,an efficient probabilistic parameter estimation method for coupled hydro-mechanical behavior in soil slope is proposed.This method integrates the Polynomial Chaos Expansion(PCE)method,the coupled hydro-mechanical modeling,and the Bayesian learning method.A coupled hydro-mechanical numerical model is established for the simulation of behaviors of unsaturated soil slope under rainfall infiltration,following by training a cheap-to-run PCE surrogate to replace it.Probabilistic estimation of soil parameters is conducted based on the Bayesian learning technique with the Markov Chain Monte Carlo(MCMC)simulation.A numerical example of an unsaturated slope under rainfall infiltration is presented to illustrate the proposed method.The effects of measurement durations and response types on parameter estimation are addressed.The result shows that with the increase of measurement duration,the uncertainties of soil parameters are significantly reduced.The uncertainties of hydraulic properties are reduced significantly using the pore water pressure data,while the uncertainties of soil strength parameters are reduced greatly using the measured displacement data.

基于多项式混沌展开的电力系统概率可用输电能力评估

大规模开发和利用风能有利于实现电力系统清洁低碳转型,是实现国家“碳达峰、碳中和”战略目标的重要技术手段,但风电出力的强不确定性对电力系统区域间可用输电能力(available transfer capability,ATC)评估带来了全新的挑战,传统用于求解计及风电出力不确定性的概率ATC评估模型在计算效率和计算精度方面均存在一定的不足。为此,该文提出一种基于多项式混沌展开(polynomialchaos expansion,PCE)的电力系统概率ATC评估方法,该方法首先构建基于机会约束的电力系统概率ATC评估模型;然后,根据风电出力预测误差的概率分布特征,选择对应的正交多项式为基函数以近似风电出力预测误差及电力网络中与之相关联的其他随机变量;进一步,借助Galerkin投影和基于一阶矩、二阶矩的机会约束转化方法,将所构建的机会约束模型的概率约束转化为确定性约束,实现基于机会约束的概率ATC评估模型向易于求解的确定性优化模型的转化;进而,将概率ATC评估模型的求解问题转化为ATC的最优多项式逼近系数的求解问题,根据求得的最优多项式逼近系数和选取的基函数计算电力系统ATC的概率分布特征;最后,通过修改后的PJM-5节点测试系统、IEEE-118节点测试系统及吉林西部电网实际算例验证了所提基于多项式混沌展开的电力系统概率ATC评估方法的准确性和有效性。

基于混合蒙特卡洛/多项式混沌展开方法的多参数随机等离子体不确定性分析

多项式混沌展开(polynomial chaos expansion, PCE)方法对于分析随机等离子体电磁传播不确定性已经展示出了相当大的潜力。然而,由于构建多项式混沌模型的计算量随着不确定性输入维数的增加呈指数增长,数值模拟耗时长,导致“维数灾难”问题。因此,本文基于非侵入式多项式混沌(non-intrusive polynomial chaos,NIPC)方法,采用混合蒙特卡洛(Monte Carlo, MC)/PCE方法研究了多层等离子体平板电子密度不确定性对透射系数的影响,并验证了所提出方法的实用性。与MC方法相比,本文方法可以大大加快仿真的速度,有效缓解了多项式展开项的数量随着随机变量维数的增加而快速增加的缺点,同时大大降低了MC方法的仿真次数,有利于电磁模型的多参数不确定性分析。

基于多项式混沌的机床几何误差灵敏度分析

为解决目前灵敏度分析方法普遍存在的样本需求量大且计算效率不高的问题,提出了一种基于多项式混沌展开的全局灵敏度分析方法。首先,以AC型双转台五轴数控机床为研究对象,根据旋量理论建立了完备的空间误差模型。其次,构建了机床几何误差的多项式混沌展开模型,采用正交匹配追踪实现模型的稀疏化,并给出了基于该方法的Sobol灵敏度指数。进而,对五轴数控机床几何误差进行了实例分析,测量并统计出41项误差的近似概率分布,分析了影响各方向位姿误差分量的关键几何误差。通过与蒙特卡洛法和拉丁超立方法进行对比,多项式混沌展开方法的正确性得到验证,且在不降低计算精度的前提下,样本量从1×10^(5)降低到1×10^(3),计算时间分别减少96.8%和98.1%,计算效率显著提高。

翼吊式飞机跨声速气动特性的不确定性分析

针对随机不确定性可能带来翼吊式飞机严重气动性能波动的问题,提出了一种基于主动学习加点策略的Gauss过程回归(Gaussian process regression,GPR)代理模型方法用于不确定性分析,该主动学习加点策略能够有效地降低模型不确定性,提高不确定预测的精度。关注来流不确定性输入,分别使用Smolyak稀疏网格多项式混沌展开(polynomial chaos expansion,PCE)方法和基于主动学习加点策略的GPR代理模型方法,结合Sobol灵敏度分析对翼-身-短舱-挂架几何进行了不确定性分析。结果表明,在跨声速条件下,攻角和Mach数的不确定性会引起翼吊式飞机升力系数和阻力系数的剧烈波动,其中升力系数的波动同时受攻角和Mach数的影响,阻力系数的波动主要由Mach数决定。

二阶混沌多项式的概率密度精确计算及其结构可靠度应用

混沌多项式展开是一种广泛使用的方法,用于建立功能函数的代理模型,以方便对随机结构进行不确定性量化及可靠度分析。然而,在某些可靠度分析问题中经常需要混沌多项式展开模型的概率密度函数作为随机变量的完整表达,但在一般情况下难以准确计算混沌多项式展开模型的概率密度函数。为研究二阶混沌多项式展开模型的概率密度函数计算方法,通过正交变换消除模型中的交叉项,推导出二阶混沌多项式展开其特征函数的显式表达式,然后利用快速傅里叶变换求得二阶混沌多项式展开的概率密度函数,并通过数值算例验证了所提方法在结构可靠度应用中的准确性和适用性。研究结果表明:所提方法能获得二阶混沌多项式模型的概率密度函数与累积分布函数,计算结果与理论精确解吻合,获得的非中心卡方分布的累积分布函数尾部可与精确值在10^(-8)水平上保持一致,且适用于高维情形。同时,所提方法能高效准确地给出不同响应阈值下的结构失效概率,即使是在10^(-8)水平上的小失效概率情形。相较于前四阶矩方法,所提方法计算精度更高,对于输出变量具有强非高斯性的情况依然适用。此外,由于二阶混沌多项式展开模型代理强非线性功能函数存在一定误差,因此所提方法对于强非线性问题存在一定局限性。

基于概率模型和数据驱动的动力总成悬置系统可靠性优化

针对电动汽车动力总成悬置系统(PMS)一部分参数为概率变量,一部分参数为离散数据的复杂不确定情形,开展了基于概率模型和数据驱动的电动汽车PMS可靠性优化设计研究。首先,基于任意多项式混沌(APC)展开和广义最大熵原理推导了一种求解该复杂不确定情形下PMS响应不确定性和可靠性的高效方法;然后,基于蒙特卡洛抽样,给出了该复杂不确定情形下求解PMS响应不确定性和可靠性的参考方法;接着,提出了一种基于APC展开法的灵敏度分析方法,进一步提出了一种考虑响应不确定性和可靠性的PMS优化设计方法;最后,通过应用算例验证方法的有效性,并对系统进行了灵敏度分析和可靠性优化。结果表明,所提出的方法可有效地处理电动汽车PMS一部分参数为概率变量、一部分参数为离散数据的复杂不确定情形,并能可靠地优化该情形下的系统固有特性,且方法具有较高的计算精度和计算效率。

基于稀疏多项式混沌展开模型的钢筋混凝土结构长期挠度预测

钢筋混凝土受弯结构的长期挠度预测对于评价其全寿命周期的可服役性和安全性具有重要意义。文章传统的经验法难以考虑所有的影响因素,为了能够准确预测钢筋混凝土结构的长期挠度,本文使用稀疏多项式混沌展开(polynomial chaos expansion,PCE)模型预测钢筋混凝土结构的长期挠度,并对影响结构挠度的参数进行全局灵敏度分析。使用实验数据集建立和评估稀疏PCE模型,与常见的代理模型(RBF、SVR和Kriging)和常见的机器学习模型(BP神经网络)进行比较,采用十折交叉验证算法对模型进行训练和检验。结果表明,稀疏PCE模型在预测钢筋混凝土结构长期挠度方面均优于常见的代理模型和机器学习模型,其相关系数R^(2)、相对平均绝对误差(RAAE)、相对最大绝对误差(RMAE)和均方根误差(RMSE)分别为0.970、0.108、0.537和0.062。稀疏PCE模型的RMSE值远远优于经验方法。最后,基于稀疏PCE的全局灵敏度分析结果对影响结构挠度的参数进行了重要性排序,其中,瞬时或即时测量的挠度a(i)、跨深比l/h、龄期t’混凝土强度fc’重要性程度较高,且依次递减。稀疏PCE模型可用于钢筋混凝土结构长期挠度预测,并且可量化评估影响钢筋混凝土结构长期适用性的关键因素。

基于多项式混沌展开法的涡流无损检测高效元模型辅助探测概率的分析

探测概率对于量化涡流无损检测系统的检测能力非常重要。模型辅助探测概率的参数需要大量试验或仿真数据才可以确定,往往难以实现。应用基于退化核函数加速的边界元法的数值模型,提出使用基于普通最小二乘法的多项式混沌展开算法的元模型提升三维涡流无损检测问题探测概率的效率。通过有限截面线圈检测金属板面槽的算例,引入线圈位置和提离距离为不确定传播的参数,测试结果表明,该元模型预测的模型参数与基于退化核函数加速的边界元法物理模型计算的模型参数相对误差在1%以内,极大降低了所需的计算开销。

高比例新能源电力系统可靠性高效评估和薄弱环节辨识方法

新能源渗透率的提高可能诱发供电中断的风险,使电力系统的运行可靠性降低,传统的中长期可靠性评估方法难以满足运行可靠性评估的时间需求。本文提出了一种电力系统运行可靠性高效评估算法,揭示了电力系统运行可靠性指标与风电出力不确定性因素的解析函数关系,避免不确定性因素变化时可靠性的重复计算。首先,基于隐马尔可夫模型对风电出力的分布特性进行建模;然后,通过状态枚举-混沌多项式展开方法建立可靠性指标与风电出力间的解析函数;最后,基于解析函数实现对实时风电出力下的新能源电力系统运行可靠性的高效评估及薄弱环节辨识。以修改的IEEE-RTS79系统为例进行分析计算,验证了所提方法的有效性。