Maximum likelihood estimation of probabilistically described loads in beam structures梁结构荷载概率描述的最大似然估计
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Tsiotas-Niachopetros A, Silionis NE, Anyfantis KN, 2024. Maximum likelihood estimation of probabilistically described loads in beam structures. Probabilistic Engineering Mechanics, 76: 103627.DOI: 10.1016/j.probengmech.2024.103627
摘要 | Abstract
近年来重点逐渐转向预测性维护，以提高服役结构的可靠性。运营过程中，通过处理现场传感器获取的结构响应数据，可以为这一方向提供额外见解。结构健康监测 (structural health monitoring, SHM) 方法非常适合这一任务；然而，考虑到随机结构荷载的影响对其鲁棒性至关重要。本工作提出了一个基于最大似然估计 (maximum likelihood estimation, MLE) 的框架，其目标是对随机结构荷载描述中的不可测量进在该框架下，以点荷载作用且荷载大小和作用点具有随机性的结构梁为例进行实现。关注点是推断基础概率分布函数 (probability distribution function, PDF) 的超参数。通过边缘最大似然估计目标进行 (荷载) 反识别过程，其中采用随机蒙特卡罗 (Monte Carlo, MC) 积分进行边缘化，并采用遗传算法 (genetic algorithm, GA) 进行优化。采用 Cramer-Rao (CR) 下界生成 95% 的置信区间 (confidence interval, CI) 以量化不确定性估计。关键词: 结构健康监测, 概率反问题, 边缘最大似然估计, 遗传算法In recent years, focus has been shifted towards predictive maintenance in an effort to improve the reliability of operating structures. Processing structural response data obtained from in-situ sensors during operation can provide added value towards this direction. Structural Health Monitoring (SHM) methods are uniquely suited for this task; however, accounting for the effect of stochastic structural loads is critical for their robustness. In this work, a framework based on Maximum Likelihood Estimation (MLE) is presented, whose goal is to obtain inferences on typically unobservable quantities that describe stochastic structural loading. A structural beam is employed as a demonstrative case study, that is subjected to point loads with stochastic magnitude and application points. The hyperparameters that govern their underlying probability distribution functions (pdf) are the quantities of inferential interest. The inverse (load) identification process is performed using a marginalized MLE objective, where stochastic Monte Carlo (MC) integration is employed to perform the marginalization and Genetic Algorithms (GAs) are used as the optimizer. The Cramer–Rao (CR) lower bound is used to produce 95 % Confidence Intervals (CIs) to quantify estimation uncertainty.Keywords: Structural Health Monitoring; Probabilistic inversion; Marginal-MLE; Genetic Algorithms![]()
图 1: (a) 固定未知荷载且荷载向量分量大小与坐标需进行推断的一般结构; (b) 随机荷载且大小与位置随机分布超参数需进行推断Fig. 1. A generic structure subjected to fixed yet unknown loads with the magnitudes and application coordinates of the load vector components being of inferential interest (a) or alternatively subjected to random loads with the distribution hyper-parameters of the random magnitude and location being of inferential interest (b)
图 2: 边缘最大似然估计中设置超参数估计问题的示意图Fig. 2. Schematic representation of the hyperparameter estimation problem cast in a Marginal Maximum Likelihood Estimation (M-MLE) setting
Fig. 3. Problem geometry
图 4: 采用 Cramer-Rao 下界与 Bayes 后验边际 (Nagel & Sudret 2016) 和目标值的边缘最大似然参数估计渐近分布Fig. 4. Asymptotic distributions of the M-MLE parameter estimates using the CR lower bound along with Bayesian posterior marginals (Nagel & Sudret 2016) and target values
图 5: 10 个样本下不同迭代的最佳与平均拟合度Fig. 5. Best and mean fitness over generations for a sample size n = 10
图 6: 研究案例: (a) 俯视图; (b) 侧视图; (c) 前视图Fig. 6. Top (a), side (b) and front (c) view of the considered test case
Fig. 7. Experimental set-up used to obtain strain response observations
Fig. 8. A linear plot demonstrating the accuracy of the surrogate model for the longitudinal strain component ɛ_xx
图 9: 边缘最大似然估计目标中随机积分计算的均值百分比误差随样本数的变化Fig. 9. Percentage error of the mean as a function of the number of samples for the stochastic integral calculation within the M-MLE objective
图 10: 采用重要性抽样的边缘最大似然估计目标中随机积分计算的均值百分比误差随样本数的变化Fig. 10. Percentage error of the mean as a function of the number of samples for the stochastic integral calculation within the M-MLE objective using IS
Fig. 11. Optimization results and target values in the form of barcharts, with error bars representing the bounds obtained using the CIs
图 12: 采用传统蒙特卡罗与重要性抽样的不同迭代最佳与平均拟合度Fig. 12. Maximum (left panel) and mean (right panel) fitness over generations using crude MC and IS
图 13: 目标与估计超参数值的荷载相关参数概率密度函数对比Fig. 13. Comparison between load-related parameter pdfs for target and estimated hyperparameter values
作者信息 | Authors
Andreas Tsiotas-Niachopetros 希腊雅典技术大学 (National Technical University of Athens) 船舶与海洋工程学院
希腊雅典技术大学 (National Technical University of Athens) 船舶与海洋工程学院
Konstantinos N. Anyfantis, 通讯作者 (Corresp.) 希腊雅典技术大学 (National Technical University of Athens) 船舶与海洋工程学院Email: kanyf@naval.ntua.gr
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
