Advanced Markov Chain Monte Carlo Approach for Finite Element Calibration under Uncertainty

Uncertainty involved in the experiment data prohibits the wide applications of the finite element (FE) model updating technique into engineering practices. In this article, the Markov Chain Monte Carlo approach with a Delayed Rejection Adaptive Metropolis algorithm is investigated to perform the Bayesian framework for FE updating under uncertainty. A major advantage of this algorithm is that it adopts global and local adaptive strategies, which makes the FE model updating be robust to uncertainty. Another merit of the studied method is that it not only quantitatively predicts structural responses, but also calculates their statistical parameters such as the confidence interval. Impact test data of a grid structure are investigated to demonstrate the effectiveness of the presented FE model updating technique, in which the uncertainty parameters include the vertical and longitudinal spring stiffness that simulate the boundary
conditions, the end‐fixity factor for modeling semi‐rigid connections, and the elastic modulus for simulating the uncertainty associated with material property.     

Calculation of Posterior Probabilities for Bayesian Model Class Assessment and Averaging from Posterior Samples Based on Dynamic System Data

Abstract: In recent years, Bayesian model updating techniques based on dynamic data have been applied in system identification and structural health monitoring. Because of modeling uncertainty, a set of competing candidate model classes may be available to represent a system and it is then desirable to assess the plausibility of each model class based on system data. Bayesian model class assessment may then be used, which is based on the posterior probability of the different candidates for representing the system. If more than one model class has significant posterior probability, then Bayesian model class averaging provides a coherent mechanism to incorporate all of these model classes in making probabilistic predictions for the system response. This Bayesian model assessment and averaging requires calculation of the evidence for each model class based on the system data, which requires the evaluation of a multi‐dimensional integral involving the product of the likelihood and prior defined by the model class. In this article, a general method for calculating the evidence is proposed based on using posterior samples from any Markov Chain Monte Carlo algorithm. The effectiveness of the proposed method is illustrated by Bayesian model updating and assessment using simulated earthquake data from a ten‐story nonclassically damped building responding linearly and a four‐story building responding inelastically.     

Bayesian data analysis to quantify the uncertainty of intact rock strength

One of the main difficulties in the geotechnical design process lies in dealing with uncertainty. Uncertainty is associated with natural variation of properties, and the imprecision and unpredictability caused by insufficient information on parameters or models. Probabilistic methods are normally used to quantify uncertainty. However, the frequentist approach commonly used for this purpose has some drawbacks.First, it lacks a formal framework for incorporating knowledge not represented by data. Second, it has limitations in providing a proper measure of the confidence of parameters inferred from data. The Bayesian approach offers a better framework for treating uncertainty in geotechnical design. The advantages of the Bayesian approach for uncertainty quantification are highlighted in this paper with the Bayesian regression analysis of laboratory test data to infer the intact rock strength parameters σ_(ci) and m_i used in the Hoek-Brown strength criterion. Two case examples are used to illustrate different aspects of the Bayesian methodology and to contrast the approach with a frequentist approach represented by the nonlinear least squares(NLLS) method. The paper discusses the use of a Student's t-distribution versus a normal distribution to handle outliers, the consideration of absolute versus relative residuals, and the comparison of quality of fitting results based on standard errors and Bayes factors. Uncertainty quantification with confidence and prediction intervals of the frequentist approach is compared with that based on scatter plots and bands of fitted envelopes of the Bayesian approach. Finally, the Bayesian method is extended to consider two improvements of the fitting analysis. The first is the case in which the Hoek-Brown parameter, a, is treated as a variable to improve the fitting in the triaxial region. The second is the incorporation of the uncertainty in the estimation of the direct tensile strength from Brazilian test results within the overall evaluation of the intact rock strength.     

Bayesian analysis of uncertainty for structural engineering applications

There has been recent interest in differentiating aleatory and epistemic uncertainties within the structural engineering context. Aleatory uncertainty, which is related to the inherent physical randomness of a system, has substantially different effects on the analysis and design of structures as compared with epistemic uncertainty, which is knowledge based. Bayesian techniques provide powerful tools for integrating, in a rigorous manner, the two types of uncertainties. In a purely probabilistic viewpoint, the uncertainties merge, resulting in widened probability densities. From the viewpoint of design or experimentation, however, the two types of uncertainties have widely different effects. The purpose of this paper is to develop insight into these effects, using Bayesian-based analytical expressions for the aleatory and epistemic uncertainties. The paper goes beyond standard Bayesian conjugate distributions by incorporating the effects of model uncertainty, where the applicability of two or more analytical models are used to describe the structure of interest. The influence of multiple model uncertainties is explored for two problems: the Bayesian updating process as data is acquired, and the design of simple parallel systems.     

基于静力触探的土层自动划分方法与不确定性表征

提出了一种基于I_c数据自动划分土层的贝叶斯方法,所提方法不仅能够在考虑I_c的空间变异性的条件下自动划分土层,识别最可能的土层界面,而且能够定量地表征土层界面的不确定性,为制定下一步勘探方案和岩土工程设计提供参考依据。本文采用基于子集模拟的贝叶斯更新方法(CBUS)求解贝叶斯方程,产生土层厚度的后验分布样本,并计算每个可能的土层数目对应的模型证据,确定最可能土层数和最可能的土层界面深度,计算界面深度的标准差作为土层界面不确定性的量化指标。最后,通过上海市轨道交通10号线伊犁站基坑开挖现场的I_c数据和模拟I_c数据说明了所提方法的有效性和正确性。结果表明:所提方法划分的土层合理地反映了不同土层I_c的统计特性。相邻土层I_c的统计特性差异越大,界面深度的标准差越小,识别出的土层界面越可靠,反之亦然。     

Evidence-based framework for real-time life-cycle management of fatigue-critical details of structures

Ship structures are subjected to repetitive wave loads that can lead to fatigue damage. Accumulation of fatigue damage may eventually result in structural failure. In order to mitigate fatigue hazard to ship structures, timely inspections and maintenance actions must be planned during their life-span. As delays in the process of decision-making could result in the increase of the risk of failure, decision-supportive frameworks working in a real-time fashion are extremely important. In this paper, a life-cycle management framework for fatigue-critical details is proposed on the basis of Dynamic Bayesian Network (DBN) and Influence Diagram (ID). The proposed framework, established based on a stochastic fatigue crack growth model, leverages exact inference and discretisation of continuous random variables to enable real-time Bayesian updating and utility-based decision-making. Compared with previous life-cycle management studies, the new framework is capable of making decisions regarding different inspection techniques and appropriate repair actions in a holistic manner. The relatively simple structure of DBN and ID also makes the proposed framework especially attractive to stakeholders and practitioners, who may have limited knowledge of statistics, Bayesian updating and utility theories.     

Sensitivity of uncertainties in performance prediction of deteriorating concrete structures

Deterioration models for the condition and reliability prediction of civil infrastructure facilities involve numerous assumptions and simplifications. Furthermore, input parameters of these models are fraught with uncertainties. A Bayesian methodology has been developed by the authors, which uses information obtained through health monitoring to improve the quality of prediction. The sensitivity of prior and posterior predicted performance to different input parameters of the deterioration models, and the effect of instrument and measurement uncertainty, is investigated in this paper. The results quantify the influence of these uncertainties and highlight the efficacy of the updating methodology based on integrating monitoring data. It has been found that the probabilistic posterior performance predictions are significantly less sensitive to most of the input uncertainties. Furthermore, updating the performance distribution based on ‘event’ outcomes is likely to be more beneficial than monitoring and updating of the input parameters on an individual basis.     

基于检测数据更新的混凝土碳化深度预测

混凝土碳化深度预测模型及模型参数的选择均存在不可忽略的主观不确定性和随机性,应用于实际工程时存在显著的误差,而实际检测数据往往样本数量少、缺乏足够的完备性而不能用于实际工程中混凝土碳化深度的预测。以几个碳化深度预测模型计算结果的加权平均值来预测混凝土碳化深度,用贝叶斯方法结合检测信息和先验预测模型,更新预测模型权重的概率分布和相应模型分布参数的概率分布,采用更新后的模型权重和参数后验分布,可以更加准确地对结构的碳化规律进行评估和预测。以一个10 a期自然碳化试验结果为例,验证了本方法的有效性。     

Model selection, updating, and averaging for probabilistic fatigue damage prognosis

This paper presents a method for fatigue damage propagation model selection, updating, and averaging using reversible jump Markov chain Monte Carlo simulations. Uncertainties from model choice, model parameter, and measurement are explicitly included using probabilistic modeling. Response measurement data are used to perform Bayesian updating to reduce the uncertainty of fatigue damage prognostics. All the variables of interest, including the Bayes factors for model selection, the posterior distributions of model parameters, and the averaged results of system responses are obtained by one reversible jump Markov chain Monte Carlo simulation. The overall procedure is demonstrated by a numerical example and a practical fatigue problem involving two fatigue crack growth models. Experimental data are used to validate the performance of the method.     

基于集合卡尔曼滤波的岩土力学参数动态估计

 针对非确定性过程，引入集合卡尔曼滤波(EnKF)理论，视岩土变形体为一个随机动态系统，将位移观测值作为系统的输出，用集合卡尔曼滤波模型来描述系统的状态；进一步耦合数值分析方法实现岩土力学参数的随机动态估计，在有效地获得待估参数的同时还给出估计值的不确定性。通过数值算例表明，集合卡尔曼滤波可以有效地对含噪声的量测数据进行处理，能够跟踪岩土力学行为的动态变化。对比于常用最优化算法，集合卡尔曼滤波同时给出反演结果和先验知识的后验分布，显示出更好的实时性和可靠性。     

Slope reliability and back analysis of failure with geotechnical parameters estimated using Bayesian inference

A Bayesian approach is proposed for the inference of the geotechnical parameters used in slope design. The methodology involves the construction of posterior probability distributions that combine prior information on the parameter values with typical data from laboratory tests and site investigations used in design. The posterior distributions are often complex, multidimensional functions whose analysis requires the use of Markov chain Monte Carlo (MCMC) methods. These procedures are used to draw representative samples of the parameters investigated, providing information on their best estimate values, variability and correlations. The paper describes the methodology to define the posterior distributions of the input parameters for slope design and the use of these results for evaluation of the reliability of a slope with the first order reliability method (FORM). The reliability analysis corresponds to a forward stability analysis of the slope where the factor of safety (FS) is calculated with a surrogate model from the more likely values of the input parameters. The Bayesian model is also used to update the estimation of the input parameters based on the back analysis of slope failure. In this case, the condition FS = 1 is treated as a data point that is compared with the model prediction of FS. The analysis requires a sufficient number of observations of failure to outbalance the effect of the initial input parameters. The parameters are updated according to their uncertainty, which is determined by the amount of data supporting them. The methodology is illustrated with an example of a rock slope characterised with a Hoek-Brown rock mass strength. The example is used to highlight the advantages of using Bayesian methods for the slope reliability analysis and to show the effects of data support on the results of the updating process from back analysis of failure.     

Fatigue life prognosis study of welded tubular joints in signal support structures

Welded tubular joints have exhibited vulnerability to fatigue cracks. This paper presents the research on fatigue life prognosis study of steel welded tubular joints from traffic signal support structures. Fatigue testing results of six full-scale steel welded tubular joint specimens of signal support structures are first discussed. In order to predict the fatigue behavior for welded tubular structure and estimate its fatigue life, a fatigue life prognosis procedure based on Bayesian updating of the stochastic coefficients of the fatigue growth model is employed. Bayesian updating allows the use of dynamic diagnostic information with prior knowledge for improved prognosis. Surface fatigue crack growth data are recorded during fatigue tests of three specimens and the experimental data is used to demonstrate the fatigue life prognosis procedure for welded tubular structures. The prognosis results of this study provide insight into how fatigue hazard evolves in welded tubular joints of signal support structures.     

Evaluation of calibration efficacy under different levels of uncertainty

This paper examines how calibration performs under different levels of uncertainty in model input data. It specifically assesses the efficacy of Bayesian calibration to enhance the reliability of EnergyPlus model predictions. A Bayesian approach can be used to update uncertain values of parameters, given measured energy-use data, and to quantify the associated uncertainty. We assess the efficacy of Bayesian calibration under a controlled virtual-reality setup, which enables rigorous validation of the accuracy of calibration results in terms of both calibrated parameter values and model predictions. Case studies demonstrate the performance of Bayesian calibration of base models developed from audit data with differing levels of detail in building design, usage, and operation.     

A sensor-driven structural health prognosis procedure considering sensor performance degradation

This article presents a general framework for sensor-driven structural health prognosis and its application to probabilistic maintenance scheduling. Continuously collected sensor data is used to update the parameters of the stochastic structural degradation model. Uncertainty in sensor data (i.e. measurement error) is explicitly modelled as an evolving stochastic process. The proposed framework utilises Bayesian theorem and Markov Chain Monte Carlo (MCMC) sampling to calculate the posterior distributions of stochastic parameters of the structural degradation model. Bayesian updating allows the use of dynamic diagnostic information with prior knowledge for improved prognosis including risk analysis and remaining useful life (RUL) estimation. Although the proposed sensor-driven structural health prognosis procedure is illustrated with a fatigue-related example, it is applicable to more general applications such as corrosion and pavement cracking. A case study of the fatigue details found in a prototype steelgirder bridge has been conducted to demonstrate the proposed prognosis and maintenance scheduling procedure.     

Structural Model Updating and Health Monitoring with Incomplete Modal Data Using Gibbs Sampler

Abstract:   A new Bayesian model updating approach is presented for linear structural models. It is based on the Gibbs sampler, a stochastic simulation method that decomposes the uncertain model parameters into three groups, so that the direct sampling from any one group is possible when conditional on the other groups and the incomplete modal data. This means that even if the number of uncertain parameters is large, the effective dimension for the Gibbs sampler is always three and so high-dimensional parameter spaces that are fatal to most sampling techniques are handled by the method, making it more practical for health monitoring of real structures. The approach also inherits the advantages of Bayesian techniques: it not only updates the optimal estimate of the structural parameters but also updates the associated uncertainties. The approach is illustrated by applying it to two examples of structural health monitoring problems, in which the goal is to detect and quantify any damage using incomplete modal data obtained from small-amplitude vibrations measured before and after a severe loading event, such as an earthquake or explosion.     

Evaluation and Bayesian dynamic prediction of deterioration of structural performance

The deterioration of structural performance is a time-variant process with a large amount of uncertainties and incompleteness of load and environmental effects. It seems inevitable that the prediction of structural deterioration should be based on a philosophy of information updating. In the present paper, a new model system for structural performance prediction is introduced based on Bayesian dynamic linear model (DLM) theory. The system can implement Bayesian updating on the prediction of the time series process of structural deterioration. It can effectively incorporate useful information through the deterioration process of structures to update the prediction. Intervention and monitoring techniques are also designed to ensure the stability of the model system. This paper also defines an indicator of the so-called ‘condition index’ to evaluate the structural performance. Using condition indexes, the qualitative condition rating of structural performance based on visual inspection in current engineering practice can be integrated into the Bayesian DLMs. Case studies validate the advantages of Bayesian DLMs. The information updating has a favourable effect on reliability analysis and life prediction. The condition indexes are simple and convenient used in practice.     

Active learning structural model updating of a multisensory system based on Kriging method and Bayesian inference

Model updating techniques are often applied to calibrate the numerical models of bridges using structural health monitoring data. The updated models can facilitate damage assessment and prediction of responses under extreme loading conditions. Some researchers have adopted surrogate models, for example, Kriging approach, to reduce the computations, while others have quantified uncertainties with Bayesian inference. It is desirable to further improve the efficiency and robustness of the Kriging-based model updating approach and analytically evaluate its uncertainties. An active learning structural model updating method is proposed based on the Kriging method. The expected feasibility learning function is extended for model updating using a Bayesian objective function. The uncertainties can be quantified through a derived likelihood function. The case study for verification involves a multisensory vehicle-bridge system comprising only two sensors, with one installed on a vehicle parked temporarily on the bridge and another mounted directly on the bridge. The proposed algorithm is utilized for damage detection of two beams numerically and an aluminum model beam experimentally. The proposed method can achieve satisfactory accuracy in identifying damage with much less data, compared with the general Kriging model updating technique. Both the computation and instrumentation can be reduced for structural health monitoring and model updating.     

Structural parameter identification and damage detection for a steel structure using a two-stage finite element model updating method

A two-stage eigensensitivity-based finite element (FE) model updating procedure is developed for structural parameter identification and damage detection for the IASC-ASCE structural health monitoring benchmark steel structure on the basis of ambient vibration measurements. In the first stage, both the weighted least squares and Bayesian estimation methods are adopted for the identification of the connection stiffness of beam-column joints and Young’s modulus of the structure; then the damage detection is conducted via the FE model updating procedure for detecting damaged braces with different damage patterns of the structure. Comparisons between the FE model updated results and the experimental data show that the eigensensitivity-based FE model updating procedure is an effective tool for structural parameter identification and damage detection for steel frame structures.     

基于全局敏感性的模型更新混合模拟方法

针对混合模拟中数值结构本构参数不确定性,提出了基于Sobol全局敏感性分析的模型更新混合模拟方法,以试验子结构作为参数更新的对比目标.建立常见的钢框架与钢筋混凝土框架模型,推导出其独立本构参数作为更新对象,获得各个参数在地震作用全过程中全局灵敏度系数变化关系,据此进行了参数筛选,制定了核心参数更新与全参数更新两种更新策...