Probabilistic analysis of tunnel face seismic stability in layered rock masses using Polynomial Chaos Kriging metamodel

Face stability is an essential issue in tunnel design and construction.Layered rock masses are typical and ubiquitous;uncertainties in rock properties always exist.In view of this,a comprehensive method,which combines the Upper bound Limit analysis of Tunnel face stability,the Polynomial Chaos Kriging,the Monte-Carlo Simulation and Analysis of Covariance method(ULT-PCK-MA),is proposed to investigate the seismic stability of tunnel faces.A two-dimensional analytical model of ULT is developed to evaluate the virtual support force based on the upper bound limit analysis.An efficient probabilistic analysis method PCK-MA based on the adaptive Polynomial Chaos Kriging metamodel is then implemented to investigate the parameter uncertainty effects.Ten input parameters,including geological strength indices,uniaxial compressive strengths and constants for three rock formations,and the horizontal seismic coefficients,are treated as random variables.The effects of these parameter uncertainties on the failure probability and sensitivity indices are discussed.In addition,the effects of weak layer position,the middle layer thickness and quality,the tunnel diameter,the parameters correlation,and the seismic loadings are investigated,respectively.The results show that the layer distributions significantly influence the tunnel face probabilistic stability,particularly when the weak rock is present in the bottom layer.The efficiency of the proposed ULT-PCK-MA is validated,which is expected to facilitate the engineering design and construction.

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.

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.

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.

Analysis of stochastic bifurcation and chaos in stochastic Duffing-van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.

Uncertainty Analysis and Optimization of Quasi-Zero Stifness Air Suspension Based on Polynomial Chaos Method

To improve the vibration isolation performance of suspensions,various new structural forms of suspensions have been proposed.However,there is uncertainty in these new structure suspensions,so the deterministic research cannot refect the performance of the suspension under actual operating conditions.In this paper,a quasi-zero stifness isolator is used in automotive suspensions to form a new suspension−quasi-zero stifness air suspension(QZSAS).Due to the strong nonlinearity and structural complexity of quasi-zero stifness suspensions,changes in structural parameters may cause dramatic changes in suspension performance,so it is of practical importance to study the efect of structural parameter uncertainty on the suspension performance.In order to solve this problem,three suspension structural parameters d_(0),L_(0) and Pc_(0) are selected as random variables,and the polynomial chaos expansion(PCE)theory is used to solve the suspension performance parameters.The sensitivity of the performance parameters to diferent structural parameters was discussed and analyzed in the frequency domain.Furthermore,a multi-objective optimization of the structural parameters d_(0),L_(0) and Pc_(0) of QZSAS was performed with the mean and variance of the root-mean-square(RMS)acceleration values as the optimization objectives.The optimization results show that there is an improvement of about 8%−1_(0)%in the mean value and about 4_(0)%−55%in the standard deviation of acceleration(RMS)values.This paper verifes the feasibility of the PCE method for solving the uncertainty problem of complex nonlinear systems,which provide a reference for the future structural design and optimization of such suspension systems.

Uncertainty Quantification of Numerical Simulation of Flows around a Cylinder Using Non-intrusive Polynomial Chaos

The uncertainty quantification of flows around a cylinder is studied by the non-intrusive polynomial chaos method. Based on the validation with benchmark results, discussions are mainly focused on the statistic properties of the peak lift and drag coefficients and base pressure drop over the cylinder with the uncertainties of viscosity coefficient and inflow boundary velocity. As for the numerical results of flows around a cylinder, influence of the inflow boundary velocity uncertainty is larger than that of viscosity. The results indeed demonstrate that a five-order degree of polynomial chaos expansion is enough to represent the solution of flow in this study.

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.

Uncertainty through Polynomial Chaos： A Sensor Sensitivity and Correlation Analysis in EEG Problems

The author studies the effect of uncertain conductivity on the electroencephalography （EEG） forward problem. A three-layer spherical head model with different and random layer conductivities is considered. Polynomial Chaos （PC） is used to model the randomness. The author performs a sensitivity and correlation analysis of EEG sensors influenced by uncertain conductivity. The author addressed the sensitivity analysis at three stages： dipole location and moment averaged out, only the dipole moment averaged out, and both fixed. On average, the author observes the least influenced electrodes along the great longitudinal fissure. Also, sensors located closer to a dipole source, are of greater influence to a change in conductivity. The highly influenced sensors were on average located temporal. This was also the case in the correlation analysis. Sensors in the temporal parts of the brain are highly correlated. Whereas the sensors in the occipital and lower frontal region, though they are close together, are not so highly correlated as in the temporal regions. This study clearly shows that intrinsic sensor correlation exists, and therefore cannot be discarded, especially in the inverse problem. In the latter it makes it possible not to specify the conductivities. It also offers an easy but rigorous modeling of the stochastic propagation of uncertain conductivity to sensorial potentials （e.g., making it suited for research on optimal placing of these sensors）.

Stochastic Chaos with Its Control and Synchronization

The discovery of chaos in the sixties of last century was a breakthrough in concept,revealing the truth that some disorder behavior,called chaos,could happen even in a deterministic nonlinear system under barely deterministic disturbance.After a series of serious studies,people begin to acknowledge that chaos is a specific type of steady state motion other than the conventional periodic and quasi-periodic ones,featuring a sensitive dependence on initial conditions,resulting from the intrinsic randomness of a nonlinear system itself.In fact,chaos is a collective phenomenon consisting of massive individual chaotic responses,corresponding to different initial conditions in phase space.Any two adjacent individual chaotic responses repel each other,thus causing not only the sensitive dependence on initial conditions but also the existence of at least one positive top Lyapunov exponent(TLE) for chaos.Meanwhile,all the sample responses share one common invariant set on the Poincaré map,called chaotic attractor,which every sample response visits from time to time ergodically.So far,the existence of at least one positive TLE is a commonly acknowledged remarkable feature of chaos.We know that there are various forms of uncertainties in the real world.In theoretical studies,people often use stochastic models to describe these uncertainties,such as random variables or random processes.Systems with random variables as their parameters or with random processes as their excitations are often called stochastic systems.No doubt,chaotic phenomena also exist in stochastic systems,which we call stochastic chaos to distinguish it from deterministic chaos in the deterministic system.Stochastic chaos reflects not only the intrinsic randomness of the nonlinear system but also the external random effects of the random parameter or the random excitation.Hence,stochastic chaos is also a collective massive phenomenon,corresponding not only to different initial conditions but also to different samples of the random parameter or the random excitation.Thus,the unique common feature of deterministic chaos and stochastic chaos is that they all have at least one positive top Lyapunov exponent for their chaotic motion.For analysis of random phenomena,one used to look for the PDFs(Probability Density Functions) of the ensemble random responses.However,it is a pity that PDF information is not favorable to studying repellency of the neighboring chaotic responses nor to calculating the related TLE,so we would rather study stochastic chaos through its sample responses.Moreover,since any sample of stochastic chaos is a deterministic one,we need not supplement any additional definition on stochastic chaos,just mentioning that every sample of stochastic chaos should be deterministic chaos.We are mainly concerned with the following two basic kinds of nonlinear stochastic systems,i.e.one with random variables as its parameters and one with ergodical random processes as its excitations.To solve the stochastic chaos problems of these two kinds of systems,we first transform the original stochastic system into their equivalent deterministic ones.Namely,we can transform the former stochastic system into an equivalent deterministic system in the sense of mean square approximation with respect to the random parameter space by the orthogonal polynomial approximation,and transform the latter one simply through replacing its ergodical random excitations by their representative deterministic samples.Having transformed the original stochastic chaos problem into the deterministic chaos problem of equivalent systems,we can use all the available effective methods for further chaos analysis.In this paper,we aim to review the state of art of studying stochastic chaos with its control and synchronization by the above-mentioned strategy.

NON-INTEGRABILITY AND CHAOS OF A CONSERVATIVE COMPOUND PENDULUM

By using a series of canonical transformations (Birkhoff's series), an approximate integral of a conservative compound pendulum is evaluated. Level lines of this approximate integral are compared with the numerical simulation results. It is seen clearly that with a raised energy level, the nearly integrable system becomes non-integrable, i.e. the regular motion pattern changes to the chaotic one. Experiments with such a pendulum device display the behavior mentioned above.

Chaos Synchronization in Lorenz System

In this paper, we analyze chaotic dynamics of nonlinear systems and study chaos synchronization of Lorenz system. We extend our study by discussing other methods available in literature. We propose a theorem followed by a lemma in general and another one for a particular case of Lorenz system. Numerical simulations are given to verify the proposed theorems.

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。

基于多项式混沌展开的交直流系统全纯嵌入概率潮流计算方法

为快速、准确量化分析风电出力不确定性对交直流电力系统潮流分布的影响,提出一种基于多项式混沌展开(PCE)的交直流电力系统全纯嵌入概率潮流计算方法。该方法首先根据风电出力的概率分布特征选择最优正交基函数,构造近似风电出力概率分布特征的PCE表达式;其次,将该PCE表达式引入交直流电力系统的全纯嵌入潮流方程中,构建基于PCE的交直流电力系统全纯嵌入概率潮流计算模型;再次,通过Galerkin投影将所构建的全纯嵌入概率潮流计算模型转化为高维确定性全纯嵌入潮流计算模型;然后,借助确定性全纯嵌入潮流模型求解方法,实现对所转化的高维确定性全纯嵌入潮流模型的求解,并根据所得PCE逼近系数计算交直流电力系统潮流的概率分布特征;最后,通过修改的PJM 5节点、IEEE 30节点和IEEE 118节点交直流测试系统算例验证所提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。

来流变化导致飞行器气动力/热不确定度量化分析

为评估及量化来流条件变化导致高速飞行器气动力/热特性的不确定性程度,选择具有飞机前机身与座舱罩组合部件基本特征的双椭球模型为研究模型,采用数值模拟方法,获得了其流动结构和壁面热流、压力分布等特征,并通过与实验数据进行对比,验证了预测方法的可靠性。在此基础上,选取来流速度、来流温度、来流密度和壁面温度这4个来流参数作为不确定性变量,采用拉丁超立方试验设计与非嵌入式多项式混沌相结合的不确定度量化方法,开展了气动力/热不确定度量化分析和敏感性分析。结果表明,来流条件的不确定性对模型的升力、阻力及驻点热流的值均有较大影响,其中来流速度和来流温度的变化对壁面压强分布影响较大,来流速度、来流密度和壁面温度的变化对壁面热流的预测有着重要影响。

离心泵进口来流速度扰动不确定性对水力性能及流场的影响

离心泵运行过程中其进口来流速度随时间的波动对泵的水力性能具有不可忽略的影响,该研究采用非嵌入式多项式混沌(non-intrusive polynomial chaos, NIPC)方法对进口来流速度扰动的不确定性进行量化分析,探究其对离心泵水力性能的影响。结果表明:随机来流速度对泵的水力性能具有较大影响,且不确定度越大,其影响越大;进口来流速度扰动会引起泵叶片压力面尾缘与吸力面上压力分布及叶轮流道内流场的变化,从而造成泵的扬程及效率的波动且波动范围较大;同时,来流不确定性的影响在叶轮内流场中的传播是非对称且非均匀的。不同工况下来流速度随时间的扰动对泵性能的影响有所差异,大部分工况下来流速度扰动会造成泵性能的下降,其中不确定度为5%时,扬程最大可下降0.4 m,效率下降3%。对不同工况下泵的进口流速不确定性进行量化分析,能够对离心泵在整个运行工况下的稳健性进行综合评定,为泵稳健性设计提供一定的基础。

水下拖曳系统拖缆末端不确定性量化分析

在多变的海洋环境中,水下拖曳系统拖缆优化设计与拖体精确控制的关键在于拖缆末端不确定性的量化。针对传统不确定性量化方法蒙特卡罗(MC)法计算成本高、精度低的问题,提出一种基于多项式混沌(PC)的拖缆末端不确定性量化方法。利用拉丁超立方采样获取拖缆参数的样本集,并代入集中质量法模型求得拖缆末端位置坐标。通过PC方法生成拖缆末端响应的代理模型,根据正交多项式的特点量化拖缆末端的不确定性,同时与MC方法进行对比。结果表明:相比于MC方法,PC方法的计算结果关于样本数量的收敛速度更快,精度更高;运动响应不确定性与拖缆轴向长度近似正比例关系;缆长增大将导致末端的不确定性增大,且增大趋势逐渐平缓;拖缆参数不确定性一定时,增大母船航速有助于提高拖体在高度上的稳定性。PC方法的准确性和高效性得到验证。同时,拖缆末端不确定性量化分析结果可为相关工程问题提供指导。

低可探测DSI进气道几何敏感性分析

低可探测DSI进气道(蚌式进气道)设计受多个几何参数的影响,单几何参数分析难以全面反映外形对进气道性能的影响。本文采用非嵌入式多项式混沌方法对某低可探测DSI进气道开展研究,分析亚声速大攻角、最大飞行速度时,进气道性能对喉道面积、喉道位置、鼓包马赫数、鼓包前移量、唇罩前伸量、唇口前缘半径以及进气道收缩量的敏感程度,并开展了选型设计。结果表明,进气道性能对唇罩前伸量最敏感,鼓包马赫数和收缩量对亚声速大攻角下的进气道性能也有所贡献,并存在最佳的唇罩前伸量与进气道收缩量。选型设计后,进气道在亚声速大攻角时总压恢复系数提升4.6%,稳态总压畸变指数下降10.9%;在最大飞行速度时总压恢复系数提升3.1%,稳态总压畸变指数降低35.5%。合理地选择几何参数,减弱或消除进气道唇口处以及激波后的分离能够有效提升进气道性能。