Approximate Bayesian computation for structural identification of ancient tie-rods using noisy modal data基于噪声模态数据的古代拉杆结构识别的近似 Bayes 计算
![]()
Monchetti S, Pepi C, Viscardi C, Gioffrè M, 2024. Approximate Bayesian computation for structural identification of ancient tie-rods using noisy modal data. Probabilistic Engineering Mechanics, 77: 103674.DOI: 10.1016/j.probengmech.2024.103674
石拱门和拱顶是常见历史结构构件，由于地震作用或桥台沉降，其经常承受不对称荷载。过去几十年里，许多研究试图增进对这类构件结构行为的理解，以便进行预防性保护。对现有建筑结构性能的评估通常依赖未知输入参数 (包括几何形状、力学特性、物理特性和边界条件) 下的有效数值模型。这些参数可通过确定性优化函数估计，旨在最小化数值模型的输出与观测动力/静力结构响应的差异。然而，确定性方法忽略了输入参数和观测的不确定性。在此情形下，Bayes 方法有助于估计未知数值模型参数及其不确定性 (后验分布)。该方法根据当前观测更新模型参数的先验知识 (先验分布)，并通过似然函数考察影响观测量的所有不确定性源。然而，有两个重要挑战: 似然函数可能未知或过于复杂而无法评估，且近似后验分布的计算成本可能过高。本研究采用近似 Bayes 计算 (approximate Bayesian computation, ABC) 处理难以确定的似然函数，从而应对挑战。此外，采用精确代理模型，如混沌多项式展开 (polynomial chaos expansion, PCE) 和人工神经网络 (artificial neural network, ANN)，可降低计算成本。该研究主要为简单结构系统 (拉杆) 建立数值模型，并通过观测结构响应 (模态数据、应变、位移) 的 Bayes 更新推断未知输入参数，如力学性能和边界条件。本研究主要创新之处在于，一方面，提出了一种不同不确定性源下估计古代拉杆轴向力的可靠方法；另一方面，应用近似 Bayes 计算获得结构识别反问题的解。关键词: 近似 Bayes 计算, Bayes 模拟推断, 混沌多项式展开, Markov 链 蒙特卡罗方法, 结构识别, 拉杆Masonry arches and vaults are common historic structural elements that frequently experience asymmetric loading due to seismic action or abutment settlements. Over the past few decades, numerous studies have sought to enhance our understanding of the structural behavior of these elements for the purpose of preventive conservation. The assessment of the structural performance of existing constructions typically relies on effective numerical models guided by a set of unknown input parameters, including geometry, mechanical characteristics, physical properties, and boundary conditions. These parameters can be estimated through deterministic optimization functions aimed at minimizing the discrepancy between the output of a numerical model and the measured dynamic and/or static structural response. However, deterministic approaches overlook uncertainties associated with both input parameters and measurements. In this context, the Bayesian approach proves valuable for estimating unknown numerical model parameters and their associated uncertainties (posterior distributions). This involves updating prior knowledge of model parameters (prior distributions) based on current measurements and explicitly considering all sources of uncertainties affecting observed quantities through likelihood functions. However, two significant challenges arise: the likelihood function may be unknown or too complex to evaluate, and the computational costs for approximating the posterior distribution can be prohibitive. This study addresses these challenges by employing Approximate Bayesian Computation (ABC) to handle the intractable likelihood function. Additionally, the computational burden is mitigated through the use of accurate surrogate models such as Polynomial Chaos Expansions (PCE) and Artificial Neural Networks (ANN). The research focuses on setting up numerical models for simple structural systems (tie-rods) and inferring unknown input parameters, such as mechanical properties and boundary conditions, through Bayesian updating based on observed structural responses (modal data, strains, displacements). The main novelties of this research regard, on the one hand, the proposal of a methodology for obtaining a reliable estimate of the axial force in ancient tie-rods accounting for different sources of uncertainty and, on the other hand, the application of ABC to obtain the structural identification inverse problem solution.
Keywords: Approximate Bayesian computation; Bayesian simulated inference; Polynomial chaos expansion; Markov chain Monte Carlo methods; Structural identification; Tie-rods.![]()
Fig. 1. Santa Maria della Consolazione Temple (a) and tie-rods (b)
图 2: 激光测振仪记录的拉杆位移时变曲线与相关功率谱密度谱Fig. 2. Displacement time history recorded by laser vibrometer on the tie-rod (a) and associated power spectral density spectrum (b)
图 3: 模拟拉杆动力行为的计算模型与不确定模型参数Fig. 3. Computational model reproducing the dynamic behavior of the tie-rod and uncertain model parameters θ
Fig. 4. Natural frequencies variation due to variation in the rotational spring stiffnesses k_l and k_r
图 5: 采用近似 Bayes 计算、Markov 链蒙特卡罗与基于近似 Bayes 计算和Markov 链蒙特卡罗的混沌多项式展开更新参数的后验概率密度函数: (a) Young 模量; (b) 质量密度; (c) 截面横向尺寸; (d)轴向荷载Fig. 5. Posterior PDF of the updated parameters using ABC, MCMC and PCE based ABC and MCMC: (a) Young modulus E; (b) Mass density ρ; (c) Cross section transversal dimension a; (d) Axial load N
图 6: Kullback-Leibler 距离、后验均值比与后验变异系数比Fig. 6. Kullback-Leibler distances (a) posterior mean value ratio (b) and posterior CV ratio (c)
图 7: 采用近似 Bayes 计算与不同自振频率数更新参数的后验概率密度函数: (a) Young 模量; (b) 质量密度; (c) 截面横向尺寸; (d)轴向荷载Fig. 7. Posterior PDF of the updated parameters using ABC and different number of natural frequencies y_0, i = 1, ..., 5: (a) Young modulus E; (b) Mass density ρ; (c) Cross section transversal dimension a; (d) Axial load N
图 8: 采用不同自振频率数更新参数的近似 Bayes 计算后验均值与真实值之比以及 Bayes 平均绝对百分比误差Fig. 8. Ratio between the ABC posterior mean of the updated parameters using different numbers of natural frequencies, from 1 to 5, and the true value (a); BMAPE values (b)
作者信息 | Authors
意大利佛罗伦萨大学 (University of Florence) 土木与环境工程系
Chiara Pepi, 通讯作者 (Corresp.)意大利佩鲁贾大学 (University of Perugia) 土木与环境工程系Email: chiara.pepi@unipg.it
意大利佛罗伦萨大学 (University of Florence) 统计与计算机科学应用系
意大利佩鲁贾大学 (University of Perugia) 土木与环境工程系
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)
