Multinormal integrals by importance sampling for series system reliability

A structural system with multi-failure modes can be modeled as a series system if it fails whenever any of the failure mode occurs. Applying FORM, failure probability of a series system can be expressed using a complementary standard multinormal integral. However, the integral is increasingly more difficult as the dimension increases. Importance sampling method can be used to deal with such multi-fold integration. Considering the fact that the optimal importance sampling function can be determined for a linear limit state function in a uncorrelated standard normal space, this paper proposes an importance sampling function for multinormal integral as a linear combination of such optimal sampling functions. The accuracy and applicability of the method are investigated using numerical examples.     

Evaluation of accuracy and efficiency of some simulation and sampling methods in structural reliability analysis

Numerous simulation and sampling methods can be used to estimate reliability index or failure probability. Some point sampling methods require only a fraction of the computational effort of direct simulation methods. For many of these methods, however, it is not clear what trade-offs in terms of accuracy, precision, and computational effort can be expected, nor for which types of functions they are most suited. This study uses nine procedures to estimate failure probability and reliability index of approximately 200 limit state functions with characteristics common in structural reliability problems. The effects of function linearity, type of random variable distribution, variance, number of random variables, and target reliability index are investigated. It was found that some methods have the potential to save tremendous computational effort for certain types of limit state functions. Recommendations are made regarding the suitability of particular methods to evaluate particular types of problems.     

Adaptive radial-based importance sampling method for structural reliability

In this paper an adaptive radial-based importance sampling (ARBIS) method is presented. The radial-based importance sampling (RBIS) method, excluding a β-sphere from the sampling domain, is extended with an efficient adaptive scheme to determine the optimal radius β of the sphere. The adaptive scheme is based on directional simulation. The underlying basic methods are presented briefly. Several numerical examples demonstrate the efficiency, accuracy and robustness of the scheme. As such, the ARBIS method can be applied as a black-box and is of particular interest in applications with a low probability of failure, for example in structural reliability, in combination with a small number of stochastic variables.     

Reliability of uncertain dynamical systems with multiple design points

Asymptotic approximations and importance sampling methods are presented for evaluating a class of probability integrals with multiple design points that may arise in the calculation of the reliability of uncertain dynamical systems. An approximation based on asymptotics is used as a first step to provide a computationally efficient estimate of the probability integral. The importance sampling method utilizes information of the integrand at the design points to substantially accelerate the convergence of available importance sampling methods that use information from one design point only. Implementation issues related to the choice of importance sampling density and sample generation for reducing the variance of the estimate are addressed. The computational efficiency and improved accuracy of the proposed methods is demonstrated by investigating the reliability of structures equipped with a tuned mass damper for which multiple design points are shown to contribute significantly to the value of the reliability integral.     

Computational methods for accounting of structural uncertainties, applications to dynamic behavior prediction of piping systems

Computational probabilistic methods enable us to incorporate and propagate uncertainties in mechanical models. However, in some cases, classical methods, such as FORM/SORM methods and Monte-Carlo methods, can be computationally expensive or inaccurate. An efficient importance sampling method is then suggested to yield sufficiently accurate results with acceptable computational cost in an industrial context. The method is an importance sampling method based on a second order asymptotic approximation combined with the HyperCube Latin method. A clustering method is used to solve the global optimization problem which arises to find the points of maximum likelihood. The efficiency of the method compared to classical methods is illustrated with several examples. Considerable reduction of the statistical error of the estimated failure probability can be achieved. The interest of the method is assured provided the points of local maximum likelihood are not too numerous and uniformly distributed. The paper presents two vibratory test cases, the second one is an industrial piping system.     

Analysis of approximations for multinormal integration in system reliability computation

In the context of first-order reliability analysis, the computation of multivariate normal integrals is a key step in the analysis of the system probability of failure. Several approximate methods for multinormal integration have been developed, since the direct numerical integration in large dimensions (30–50) is not feasible. The product of conditional marginal (PCM) method was proposed as a simple and effective method for system reliability computation. Although PCM is fairly accurate in computing parallel system reliability, it can result in a large overestimation of the failure probability of series systems with highly reliable elements. The paper presents an improved version, referred to as I-PCM, to eliminate this shortcoming of the original method. The I-PCM method employs a simple modification of bivariate integrals based on the additive law of probability. Detailed error analyses and numerical examples are presented in the paper to illustrate the improved accuracy and efficiency of the proposed I-PCM method.     

Search-based importance sampling

Importance sampling as a special technique in Monte Carlo probability integration has been shown to be a highly efficient and rather unrestricted method. Non-Gaussian and dependent random variables and nonlinear limit functions can be treated relatively easily and with reasonable rates of convergence. A major draw-back, however, is the need to identify so-called “interesting” or “important” regions for integration. Reference to first-order second-moment (FOSM) methods may help, as well as numerical maximization routines applied. Each involves certain difficulties. An alternative procedure, based on directing and correcting the importance sampling function as sampling is carried out, is presented herein. In particular it is possible to have a multi-modal sampling function.     

Application of line sampling simulation method to reliability benchmark problems

A procedure denoted as Line Sampling (LS) has been developed for estimating the reliability of static and dynamical systems. The efficiency and accuracy of the method is shown by application to the subset of the entire spectrum of the posed benchmark problems [Schuëller GI, Pradlwarter HJ, Koutsourelakis PS. Benchmark study on reliability estimation in higher dimensions of structural systems. In URL: http://www.uibk.ac.at/mechanik/Publications/benchmark.html. Institute of Engineering Mechanics, Leopold-Franzens University, Innsbruck, Austria, 2004], i.e. in particular linear systems with random properties. The notion of design point excitation for non-linear systems is discussed and its use extended for reliability estimations of conservative non-linear MDOF systems considering critical conditional excitation.For solving the hysteretic MDOF system with uncertain structural parameters subjected to general Gaussian excitation, however, the general applicable subset procedure [Au SK, Beck JL. Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 2001;16:263–277] has been used combined with Importance Sampling.     

Time-dependent system reliability analysis by adaptive importance sampling

A method for evaluating time-dependent reliability of a structural system subjected to stochastic loads is presented. Structural deterioration due to environmental stressors is also taken into account. An adaptive Monte Carlo simulation procedure combined with conditional expectation is proposed. The optimum common ratio of the standard deviation of an importance sampling variables to that of the corresponding original valuables is estimated iteratively as well as their mean values. Unlike systems evaluated by simple Monte Carlo simulations, the accuracy of the failure probability evaluated by adaptive importance sampling is relatively insensitive to the magnitude of the probability.     

计算结构可靠指标的子域抽样法

提出了基于Monte Carlo法的用于结构可靠度计算的子域抽样法,其基本思想是以有界域模拟替代无界失效域。介绍了服从某种概率分布的变量在给定子区间内产生随机数的方法,并按照这种方法在选定的有界子区域内进行抽样模拟计算失效概率,并编制了仿真程序将抽样计算过程可视化。经算例验证,该算法抽样效率很高,且具有较高的模拟计算精度。     

An effective approximation to evaluate multinormal integrals

In structural system reliability theory, the evaluation of multivariate normal distributions is an important problem. Numerical integration of multinormal distributions with high accuracy and efficiency is known to be impractical when the number of distribution dimensions is large, typically greater than five. The paper presents a practical and effective approach to approximate a multinormal integral by a product of one-dimensional normal integrals, which are easy to evaluate. Examples considered in the paper illustrate a remarkable accuracy of the approximation in comparison with exact integration. Unlike a first-order multinormal approximation widely used in the literature, this method does not involve any iterative linearization, minimization or integration. Computational simplicity with high accuracy is the major advantage of the proposed method, which also highlights its potential for estimating reliability of structural systems.     

Response surface augmented moment method for efficient reliability analysis

The reliability analysis procedure based on design of experiment (DOE) is combined with the response surface method (RSM) for numerical efficiency. Instead of using the inefficient full factorial DOE, a response surface is constructed initially based on the data on the axial experimental points and updated successively by adding one more experimental point selected using an influence index, until the probability is converged. It is calculated using the Pearson system and the four statistical moments obtained from the experimental data complemented by the response surface. During the update of a response surface, cross product terms can be added into the formulation. The number of updating steps is finite, since the points to be added are selected among the set of points of full factorial design. The performance of the proposed method is tested with several examples containing various types of distributions. It is shown that the probability converges early in the process and thus the amount of calculation is only a small fraction of that of a full factorial design. This is comparable or even better than any other methods including FORM for a moderate number of random variables tested.     

基于整体承载极限状态的钢结构可靠度设计方法及其在门式钢刚架设计中的应用

本文提出了基于整体承载极限状态的钢结构可靠度设计思路。这种思路建立在钢结构整体非线性分析和验算的基础上,并确保结构整体而非构件的可靠度水平,使结构整体的实际可靠度水平尽可能地接近于设计的目标值。本文针对门式钢刚架结构建立了一套实现这种设计的方法,包括结构整体非线性分析、结构体系可靠度计算以及实用的设计表达式。通过三个门式钢刚架结构设计实例比较了传统的构件设计方法,不考虑体系可靠度的整体承载极限状态设计方法和本文考虑体系可靠度的整体承载极限状态设计方法的设计结果,说明本文所提出设计思想的先进性。     

基于可靠度理论的地铁施工项目风险控制研究

总结了地铁施工风险现状,提出将可靠度理论应用于地铁施工领域的思想,在此基础上,建立了地铁施工风险控制系统并串联模型。根据系统可靠度目标的要求,结合可靠度分配理论,确定组成并串联模型的各子系统的可靠度目标值,针对构成子系统的各基本事件,采取等分配法求解各基本事件可靠度,并给出了该方法的应用算例。研究表明,可靠度理论可以成功的应用到地铁施工领域,在给定系统安全目标值的条件下达到整个施工过程的安全优化。     

Ductile structural system reliability analysis using adaptive importance sampling

This paper presents a hybrid technique for efficient system reliability estimation of large ductile framed structures. The proposed procedure starts with a simple enumeration scheme, but quickly changes to an adaptive importance sampling scheme to make the process more efficient and easier to implement. The method solves the problem of including the effect of multiple failure sequences in an importance sampling scheme, for the system reliability estimation of large structures. The enumeration method is used to identify the first complete failure sequence. This failure sequence defines the initial failure domain for starting the adaptive sampling process. A weighted multi-modal sampling density is used to account for the contribution of different regions in the sampling domain to the system failure probability. As the simulation progresses, the failure domain is gradually modified to include the effect of other significant failure sequences and arrive at an accurate estimate of the system failure probability.     

Applications of diffusion models to reliability analysis of Daniels systems

Failure time is estimated for Daniels systems subject to quasistatic narrow band and dynamic broad band Gaussian load processes. Daniels systems are parallel systems with components (fibers) of independent indentically distributed resistances. The analysis is based on an approximate representation of response by diffusion models and probabilistic characteristics of the first passage time of these models. These characteristics include the mean and the probability of the first passage time and can be obtained respectively from a solution of an ordinary differential equation and by the path integral method. Results are given for simple Daniels system with two and three fibers.     

概率Pushover分析方法及其在结构体系抗震可靠度评估中的应用

本文结合我国现行的抗震规范 (GBJ 11— 89)和地震作用统计参数 ,提出了结构抗震分析的概率Pushover分析方法。根据结构体系可靠度的特点 ,提出了基于概率Pushover分析的结构体系抗震可靠度评估方法 ,并用重要抽样法检验了所提方法的计算精度。理论分析和算例结果表明 :概率Pushover分析方法是评估结构体系抗震可靠度的简便、实用方法。     

The equivalent extreme-value event and evaluation of the structural system reliability

The idea of equivalent extreme-value event and accordingly a new approach to evaluate the structural system reliability are elaborated. For any type of compound random event as combination of a set of random events represented by inequalities, an equivalent extreme-value event is defined. Elaborated investigations show that correlative information among the component random events is inherent in the equivalent extreme-value event. Since the probability density function of the equivalent extreme value could be obtained through the probability density evolution method, the idea of equivalent extreme-value event leads to a new uniform approach to evaluate the structural system reliability for both static and dynamic problems. Particularly, the investigation points out that computation of the dynamic reliability essentially involves dealing with infinite-dimensional correlation information and that is why the widely-used out-crossing process theory could be only an approximate and somewhat empirical reliability evaluation rather than an exact approach. The proposed approach is discussed in detail on how to construct the equivalent extreme-value event and then implement the procedure numerically. Two examples, of which one deals with static problem comparing the results with exact solution, the other deals with a nonlinear frame structure subjected to stochastic ground motions, are illustrated to validate the proposed method. The investigations show that the proposed approach is of satisfactory accuracy and applicable to the structural reliability analysis of various structures.     

Seismic system reliability analysis of bridges using the multiplicative dimensional reduction method

Abstract

A combined method of finite element reliability analysis and multiplicative dimensional reduction method (M-DRM) is proposed for systems reliability analysis of practical bridge structures. The probability distribution function of a structural response is derived based on the maximum entropy principle. To illustrate the accuracy and efficiency of the proposed approach, a simply supported bridge structure is adopted and the failure probability obtained are compared with the Monte Carlo simulation method. The validated method is then applied for the system reliability analysis for a practical high-pier rigid frame railway bridge located at the seismic-prone region. The finite element model of the bridge is developed using OpenSees and the M-DRM method is used to analyse the structural system reliability under earthquake loading.