A transfer Bayesian learning methodology for SHM of monumental structures

A transfer Bayesian learning methodology for SHM of monumental structures Laura Ierimontia,∗, Nicola Cavalaglia, Ilaria Venanzia, Enrique Garc´ıa-Mac´ıasa, Filippo Ubertinia aDepartment of Civil and Environmental Engineering, University of Perugia. Via G. Duranti, 93 - 06125 Perugia, Italy. Abstract A critical aspect related to the damage detection strategies by using structural health monitoring (SHM) is the lack of diagnostic labels able to assign a damage class to the measured data. In this context, a semi-supervised learning methodology, designated as transfer Bayesian learning (TBL), is proposed with the main objective of labeling post-processed data by selecting a limited number of informative elements. The method suggested in this work allows to define multi-class labels by making use of a digital twin (DT) of the structure, as a function of specific damage-sensitive mechanical parameters. The methodology is applied in a monumental building, the Consoli Palace, located in Gubbio, central Italy. The structure is instrumented with several sensors in order to measure vibrations, temperature and possible variation of existing cracks’ amplitudes. Several nonlinear pushover analyses are carried out on a calibrated finite element (FE) model to use them in conjunction with Engineering judgment for the definition of the damage-sensitive areas. The DT is then used as class classifier by means of a sensitivity damage chart (SDC). Finally, a Bayesian model updating of the damage-dependent parameters allows the probabilistic ∗Corresponding author. Department of Civil and Environmental Engineering, University of Perugia. Via G. Duranti, 93 - 06125 Perugia, Italy. phone: +39 075 585 3908; fax: +39 075 585 3897. Email addresses: [email protected] (Laura Ierimonti ), [email protected] (Nicola Cavalagli), [email protected] (Ilaria Venanzi), [email protected] (Enrique Garc´ıa-Mac´ıas), [email protected] (Filippo Ubertini) Preprint submitted to Journal of L A T E X Templates March 10, 2021 damage detection and identification. Keywords: Transfer learning, Bayesian model updating, digital twin, damage classification, continuous monitoring Nomenclature αKriging approximation error βKriging regression parameters ˆyKriging predictor XDesign parameters xSSI-based state vector YObservation vector YVector collecting parameters of the FE model to be calibrated ySSI-based vector of output measurements MFE model SDesign sets FKriging regression model MMathematical model RKriging correlation matrix kjDamage threshold kj Φ Mode shape σStandard deviation ASSI-based system matrix 2 CSSI-based output matrix vSSI-based vector of noise of measurements wSSI-based vector of external input θKriging correlation parameters ζDamping coefficient cBayesian normalizing factor DTotal number of measured data dMeasured data fFrequency F(·) Standard Gaussian cumulative distribution function iIndex of natural vibration mode JObjective function jIndex of design parameters to be updated Jerr Bayesian error function kjmultiplying coefficients of xj MTotal number of vibration modes mIndex of measured data NTotal number of xjdesign parameters to be updated NsNumber of samples pProbability density function Pal jAlarm function Pdam jDamage probability 3 p1, p2Weights of objective function rKriging correlation components of R R2Coefficient of determination TDimension of training population tTime UTotal no. of FE uncertain parameters to be up Updtaed cov Covariance DT Digital Twin MF Modal Features ref Undamaged reference state SDC Sensitivity Damage Chart TBL Transfer Bayesian Learning TL Transfer Learning 1. Introduction Structural damages suffered by monumental structures during recent earthquakes have gained attention on SHM-based damage detection and localization methodologies, due to the low economic impact and non-destructive nature of SHM technology [1]. Indeed, continuous SHM can accomplish the complex task5 of ensuring risk reduction in regions characterized by a relevant cultural heritage like Italy, which has the greatest number of UNESCO world heritage sites in the world [2]. Different works in literature make use of SHM as a powerful tool for investigating the evolution with time of the structural modal parameters which are10 4 related to material’s damage and deterioration [3, 4, 5, 6] and for investigation of the dynamic behavior of masonry bell towers [7, 8, 9, 10, 11]. Since relevant variations in the dynamic response of an healthy structure can also be related to changes in the environmental conditions, many works are devoted to the study of proper techniques able to remove the effects of temperature and humidity15 [12, 13]. The use of surrogate models, i.e. DT, for handling uncertainties in SHM is explored in [14, 15]. Recently, the benefits of SHM were widely exploited in literature for bridges [16, 17, 18, 19] and a brief review on SHM for data-driven damage identification problems is given in [20]. SHM is also explored in literature for damage assessment by means of Bayesian20 model updating, in order to deal with different sources of uncertainty [21]. On the one hand, the use of an updated FE model to reach an accurate response prediction of the real structure is widely investigated in literature [22, 23, 24, 25, 26, 27], since a Bayesian statistical framework is able to handle the inherent ill-conditioning and possible non-uniqueness in model updating processes25 [28, 29]. On the other hand, only a few recent contributions deal with longterm SHM data and damage detection. In particular, Behmanesh et al. [30, 31] proposed a model updating technique based on hierarchical Bayesian modeling for identification of civil structural systems under changing environmental conditions accounting for different subgroups of measured data, while Sun et al.30 [32] adopted a hierarchical Bayesian framework with Laplace priors for updating the finite element model. An iterative procedure for damage detection and localization by using the Bayesian framework through the Transitional Markov Chain Monte Carlo (TMCMC) is proposed in [33]. A Bayesian model updating framework of an historic masonry tower is developed in [34], while in [35] a35 Bayesian-based technique is used for the updating of the mechanical properties associated to the base isolation system installed on a school building. As confirmed by the literature on the topic presented above, nowadays the most common procedure in civil SHM concerns the data acquisition and feature extraction (system eigenfrequencies and modal shapes) through signal process-40 ing [36, 37]. Then, a smart statistical classification is necessary to convert 5 monitoring data into damage information, improving the accuracy of predictive models. This task could be accomplished by exploiting machine learning (ML) techniques, which can be divided in three main categories: supervised learning, unsupervised learning, semi-supervised learning. In the supervised learning45 paradigm, monitoring data can be associated with known labeled features that define the meaning of data. Conversely, in the unsupervised learning paradigm the problem is characterized by a massive amount of unlabeled data. At the intermediate level, semi-supervised learning allows the use of data with and without descriptive labels, which is more feasible in the context of SHM dam-50 age identification. Indeed, damage identification problems can be recognized as a five-level hierarchical approach [36]: (i) detection; (ii) localization; (iii) classification ; (iv) assessment; (v) prediction. Conventional SHM-systems can be classified as unsupervised learning, since they typically allow to capture modifications of the global behavior (i) of the structure (novelty detection).55 Over the last decades, the concepts of ML have been approaching the SHM field [38, 39, 18]. A probabilistic framework for the classification, investigation and labelling of data is suggested in [40] and a semi-supervised Gaussian mixture model for a probabilistic damage-classification is presented in [41]. Considering the literature background, a big effort still needs to be done to60 develop an automated SHMand model-based damage detection framework considering at the same time: robustness, accuracy and long-term data. Since model-based approaches are typically ill-conditioned, some priority needs to be made to cleverly label the data. The process of prioritizing monitored data by using the numerical model fits into the transfer learning (TL) technique, a65 semi-supervised learning approach where the learning algorithm is allowed to build a labeled training set autonomously [42, 36, 43]. In this context, this paper presents a TBL methodology based on long-term monitoring data aimed at evaluating the possible damage of monumental structures, where TL concepts allow to mitigate the shortage of labeled structural70 data and a trained DT is used as a formal prior belief for damage assessment. The idea is that it is possible to learn relationships from data. In the context of 6 SHM, this means that it is possible to assign a damage state or class by using a trained numerical model in order to detect damage at the earliest possible time and in an automatic manner. Then, a Bayesian-based procedure allows to trace75 over time the probability of occurrence of possible damage scenarios. The case study is the Consoli palace located in Gubbio, near Perugia, in Italy, a complex historical masonry buildings. The structure has been monitored by the Authors since 2015 and the SHM sensors’ network has been enhanced in 2020. The continuous monitoring data are used for the Bayesian-based updating of80 specific mechanical properties of some portions of the structure, identified as the most damage-sensitive ones by means of non-linear static analyisis (NLSA) and Engineering judgment. The prior knowledge of the uncertain parameters is sequentially updated on the basis of a trained DT consisting of a surrogate model. Probabilistic damage identification is performed by using the SDC, i.e.,85 a graphical chart able to associate to the frequencies decay a stiffness reduction, which is a possible sign of damage. The effectiveness of the proposed methodology is demonstrated by the numerical simulation of a feasible damage scenario. The results show the advantage of having long-term monitoring data on the correct estimation of the uncertain parameters’ distribution. The main advan-90 tages and innovations of the proposed approach can be summarized as follows: i) the availability of a big amount of long-term monitoring data allows to timely estimate the trend of uncertain parameters, i.e., distinguishing variations due to changes in environmental conditions (which can be removed by means of regression models) from damage/deterioration over time; ii) the subsequent updating95 of the damage-sensitive uncertain parameters triggers the robust identification of possible damage scenarios; iii) the use of a digital twin allows to overcome the shortage of labeled structural data and to transfer knowledge between numerical models and online monitoring data; iv) the results are periodically updated in terms of damage probabilities, which allow an essential support for decision100 making; v) once the model is trained, the methodology is computationally efficient, allowing the rapid diagnosis and damage localization. The rest of the paper is organized as follows. Section 2 presents the proposed 7 methodology for continuous SHM and Bayesian model updating. Section 3 describes the monumental building selected as case study, the continuous mon-105 itoring system and the FE model. Section 4 illustrates the results and, finally, Section 5 concludes the paper. 2. The proposed TBL methodology The proposed methodology, schematically represented in Fig. 1, embraces different consequential steps which can be summarized as follows:110 1. Data acquisition from the monitoring system which typically consists of acceleration data, temperature information and static measurements, such as crack amplitudes. 2. Continuous post-processing over time in order to estimate the modal fea-115 tures (MF), i.e., natural frequencies fexp i, mode shapes Φexp iand damping coefficients ζexp iassociated with the ith natural vibration mode from operational vibration measurements. 3. If t=1; tdenoting discrete time (days):120 i) Preliminary FE modeling and evaluation of damage-sensitive portions on the basis of NLSA and Engineering judgment. ii) Calibration of the digital twin as a function of the uncertain parameters to be updated associated to each damage-sensitive portion. iii) Damage classification in order to construct the SDC, i.e., a graphical125 chart able to associate a stiffness reduction to the frequencies decay for the selected damage scenarios. 4. if t >1: i) Novelty detection on a daily basis t, i.e. daily average values of MF.130 If a novelty is detected, increase the frequency of the modal tests on a hourly basis t=t∗, i.e., MF(t∗). 8 ii) Perform the Bayesian model updating of the uncertain parameters by means of the SDC. iii) Assign a damage probability for damage identification.135 Figure 1: The proposed TBL methodology. The main advantage of the proposed methodology is the possibility to evaluate in a timely and continuous manner any MF variation in the structure which is commonly considered as a sign of possible damage [4]. Then, the use of the SDC, numerically reconstructed from the DT, allows to detect damage with a 9 where: efi=fexp i−ˆ fi(xj)) (13) eΦi=Φexp i−aiˆ Φi(xj)TΦexp i−aiˆ Φi(xj),(14) with ai=ΦT i,exp ˆ Φi(xj)/ˆ ΦT i(xj)ˆ Φi(xj)mode shape scaling factor; σfi,σΦi 235 being the standard deviations associated to the ith natural frequency and mode shape which can be arbitrarily assumed or evaluated on the basis of the statistical measures [47]. For the sake of clarity, in the present application σfiand σΦi are directly evaluated on the basis of statistical measures on the available data with the main objective of reducing the computational effort. It is worthwhile240 to underline that a better quantification of the likelihood function might lead to a more accurate estimation of the posterior distribution, which is an aspect that will deserve a special attention in future developments of this work. 2.5. Probabilistic damage identification245 The Bayesian model updating is extended to be used for the probabilistic damage identification. In particular, according to [30, 48], the probability Pdam j that the updated jth parameter kup jin a possibly damaged state is reduced from the undamaged state kref jcan be written as: Pdam j=Pkup j⩽(1 −kj)kref j|dref,ddam=F  (1 −k)kref j−kup j q(1 −k)2σ2 kref j +σ2 kup j  (15) where F(·) is the standard Gaussian cumulative distribution function, kj∈[0.1] is a damage threshold selected for the jth damage scenario and kref j= 1 is the reference undamaged state. Changes in Pdam jare studied as a sign of possible damage. The case of kup j= (1 −kj)kref jleads to Pdam j= 0.5 which is defined as alarm function Pal j, meaning that the the damage threshold hypothesized250 for the jth damage scenario is reached. The case of kup j>(1 −kj)kref jleads to Pdam j<0.5, denoting that the damage threshold is not reached. Finally, 16 the case of kup j<(1 −kj)kref jleads to Pdam j>0.5, revealing that the damage threshold is exceeded. Additionally, in order to clearly define a possible damage state, a damage factor255 DF can be introduced as: DF =kref j−kup j kj (16) From Eq. (16) it is possible to obtain DF=0 if kup j=kref jand DF=1 if kref j− kup j=kj. If DF>1, it means that kup jis strongly reduced with respect to the threshold kj. DF increases as kjtends to zero, meaning that the modal modification extracted from the data probably doesn’t correspond to the jth260 damage scenario (possible false alarm). 3. The case study The Consoli Palace is a medieval building located in Gubbio, Umbria, central Italy, which dates back to the 14th century. The complexity of the building lies in the articulated internal distribution of volumes and materials. Globally,265 the structure has a rectangular plan and it is arranged on a series of floors above (about 60 meters) and under (about 10 meters) the square level. With reference to Fig. 3, three structural components can be identified: a central body; a loggia, connected to the main structure along the south wall; a bell tower. The architectural style of each fa¸cade (East and West side), is characterized by270 round arched windows and merlons in the rooftop. The load-bearing walls have a thickness of about 1.2 m and they are connected through horizontal masonry vaults. The Palace is built in calcareous stone masonry with a regular and homogeneous texture. 275 3.1. The continuous SHM system The monitoring system (Fig. 3) was installed by the Department of Civil and Environmental Engineering of University of Perugia in July 2020 within the 17 framework of the PRIN2017 detect aging project (cfr. Acknowledgments) and it is characterized by:280 Figure 3: Continuous monitoring configuration.differenziare rispetto a figura CSHM. NICOLA •a data acquisition system; •twelve unidirectional accelerometers A1-A12, model PCB393B12 (measurement range ±0.5 g, frequency range 0.15-1000 Hz, broadband resolution 8 µg, resonant frequency ≥10 kHz) installed at the 3rd, 5th and roof level;285 •four S-series linear variable transducers (LVDTs), denoted as C1-C4 (mea18 surement range 0-0.5 mm, resolution 0.31 m); •four K-type thermocouples; •a wiring system connecting the sensors to a NI CompactDAQ-9132 data acquisition system equipped with a NI 9234 acquisition module for ac-290 celerometers (24-bit resolution, 102-dB dynamic range, and anti-aliasing filters) and a NI 9219 acquisition module (24-bit resolution, +-60 V range, 100 S/s) for LVDTs and thermocouples. The monitoring system was activated on July 18th 2020. Table 1 summarizes the SHM sensors’ network. Acceleration data are stored in separate files con-295 taining 30 min-long recordings with a sampling frequency of 100 Hz, while crack amplitudes and temperature values are sampled at 0.1 Hz. Data are transferred and contained in a cloud storage, and can be accessed by system administrators and visualized in a web-based platform. The frequency tracking during the time period July 18th 2020 - December 20th 300 2021 is depicted in Fig. 4 a) with the indication of the selected training period. Fig. 4 b) shows the tracking of the MAC value and Fig. 4 c) of the Modal Phase Collinearity (MPC) index, which expresses the linear functional relationship between the real and the imaginary parts of the unscaled mode shape vectors. The high values of the MPC index confirm the presence of real normal modes.305 3.2. FE model In order to reproduce the structural dynamic behaviour (local and global vibration modes) identified throught Ambient Vibration Tets (AVT), a 3D FE model of the structure has been built and calibrated within the Abaqus environment [49], as detailed in [13]. The mesh of the masonry is composed by310 three-dimensional tetrahedral and hexahedral first-order elements. An isotropic material is assigned to the FE model with invariant properties under rotation of each axis. Since masonry can be considered as a quasi-brittle material whose mechanical performance deteriorates (softens) under monotonic or cyclic loading, the non-linear behavior of the material is reproduced by using the well315 19 Channel Level Measure Direction A1 rooftop acceleration y A2 rooftop acceleration x A3 rooftop acceleration y A4 5 acceleration y A5 5 acceleration x A6 5 acceleration y A7 3 acceleration −y A8 3 acceleration x A9 3 acceleration −x A10 rooftop acceleration −y A11 rooftop acceleration x A12 rooftop acceleration −x C1-C3-C4 5 crack’s length − C2 6 crack’s length − T2 6 temperature − T1-T3-T4 5 temperature − Table 1: SHM sensors’ network within the building. known concrete damage plasticity (CDP) model, introduced by [50, 51]. Indeed, the CDP model is particularly suitable for masonry structures in which the material is particularly prone to damage [52]. The non-linear constitutive law takes into consideration the degradation of the elastic stiffness induced by plastic straining both in tension and compression. According to [53], in order to320 describe the inelastic multi-dimensional behavior, masonry is modeled through the Drucker-Prager strength criterion with Kc= 0.667, which represents the shape of the yield surface in plane stress (Kc= 1 stands for a circular surface); a value equal to 0.1 is adopted for the eccentricity term; the dilatancy angle is assumed equal to 10◦; the ratio between the ultimate compression strength in325 biaxial stress states and in uniaxial conditions is set equal to 1.16; the viscosity 20 Figure 4: MF tracking during the time period July 18th 2020 - December 20th 2021: a) Frequency tracking; b) MAC tracking; c) MPC tracking. parameter is assumed equal to 0.002. As reported in Fig. 5, four vibration modes are selected for the numerical simulations: 21 Figure 5: Principal vibration modes: a) Fx1; b) Fy1; c) L1; d) T1. •Fx1: global flexural mode along the East-West direction (f1=2.35 Hz and330 MAC=0.98); •Fy1: global flexural mode along the North-South direction direction (f2=3.05 Hz and MAC=0.76); •L1: local mode which pertains to the bell tower (f3=3.46 Hz and MAC=0.64) •T1: global torsional mode (f4=4.17 Hz MAC=0.97).335 The relatively low agreement between experimental and numerical higher order mode shapes was already discussed in [13, 54]. 3.3. Selection of damage scenarios NLSA along the two main directions of the building are carried out in order to identify potential cracking patterns that can be activated by an earthquake,340 and associate such patterns to damage-prone regions. As a matter of fact, some of the cracks already existing in the palace (e.g. the crack vertically oriented along the south wall) agree with the numerically predicted cracking patterns, indicating that some of the damaging mechanisms are already activated. More in detail, NLSA allows to determine two damage scenarios (Fig. 6b-c):345 D2 which represents the crack pattern resulting from NLSA along xdirection 22 capable of reproducing the existing pattern, especially along the south wall Fig. 6b) and the internal horizontal vaults; D3 which refers to the crack pattern resulting from NLSA along the ydirection 6c). Furthermore, on the basis of the Engineering judgment, additional damageFigure 6: Selected damage sensitive areas with reference to both FE model and real structure: a) R1, the loggia; b) D2, crack pattern y; c) D3, crack pattern x, d) R4, the bell tower; e) R5, the staircase; f) R6, the fa¸cade degradation. 350 prone regions are selected: R1, which represents the poor connection exerted between the loggia and the central body of the Consoli Palace; R4, which refers to the bell tower (Fig. 6d); R5, which represents damage to the principal 23 staircase (Fig.6e); R6, which represent degradation of the exterior texture of the main fa¸cade of the building, i.e., West fa¸cade (Fig.6c). In such areas, the355 Young’s modulus of the isotropic material is assumed as uncertain. Hence, the vector collecting the uncertain parameters is X={k1, .., k6}. 4. Analysis results 4.1. Simulation-based damage scenarios In order to calibrate the surrogate model, a total number of 500 samples360 (Ns) are randomly simulated for the uncertain parameters collected in vector X. Figs. 7 a)-d) illustrate the correspondence of both frequencies (R2) and Figure 7: FEM estimate vs Kriging estimate for the selected principal vibration modes: a) samples of mode Fx1 ; f) samples of mode Fy1 ; g) samples of mode L1 ; h) samples of mode T1. mode shapes JMAC = 1/NsPi(1 −MACi)evaluated from the FE model and the Kriging model. From the figure it can be noted that the surrogate predicted frequencies and mode shapes (Kriging estimate) well fit the corresponding FE365 values. The DT-based sensitivity damage chart SDC relating frequencies and uncertain parameters is depicted in Fig. 8, which enables to associate the expected value of kj, i.e. the reduction of the jth stiffness of the masonry, to a defined frequency decay.370 A graphical representation of the frequencies which mostly affect the damage 24 Figure 8: SDC for frequencies decays: a) D1; b) D2; c) D3; d) D4; e) D5; f) D6. 25 Figure 13: DF versus Pdam by simulating damage scenarios D4 and D6: a)-f) damage scenarios D1-D6 by simulating D4; g)-n) damage scenarios D1-D6 by simulating D6. and L1, reaches Pal after a certain number of updates (see the multiple points in damaged area). This fact is due to the higher sensitivity of D5 to small variations associated to Fx1. This sensitivity can be reduced by means of an efficient removal of environmental effects that will be reached after 1 year of 32 training period. Moreover, the damage scenario D4 perfectly fits the simulated450 one. Indeed, it is clearly visible the occurrence of the damaged state, since Pdam reaches 0.5 and DF reaches 1 immediately at the first step of model updating. Finally, Figs. 13 g)-n) demonstrate that a damage scenario (D6) associated with a slight frequency decay, allows to keep out D1-D2-D3-D4-D5, and suggests to pay specific attention to those damage scenarios associated with455 smaller frequencies variations. 5. Conclusions The present paper has presented a transfer learning Bayesian methodology for structural health monitoring of monumental buildings. The proposed method enters in the challenging context of integrated SHM, consisting of a per-460 manent network of sensors installed in historical buildings capable of assessing continuously over time the structural condition. The main advantages and innovations of the proposed approach concern the use of real-time long-term monitoring data, the robust identification of possible damage scenarios by using the TL concepts, where a digital twin allows to465 transfer knowledge between numerical models and monitoring data and the use of damage probabilities, which allow an essential support for decision making. The case study is the Consoli Palace, located in Umbria (Italy), which has been monitored since 2015 with an improvement of the SHM sensors’ network in 2020. The data stored between July 18th 2020 and January 14th 2021 have been used470 for the analysis. A FE model able to reproduce the identified structural dynamic behavior has been built and a series of NLSA, in conjunction with Engineering judgment, enabled to select damage-sensitive portions of the structure. A DT of the structure has been calibrated by using the Kriging model in order to define the SDC as a prior knowledge of possible damage. 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