Civil structures and infrastructures are critical for the life of the world population and play a strategic role for the global economy [1
]. Aging and ever-increasing extreme loading conditions threaten existing and new structural systems, stressing the need of real-time structural health monitoring (SHM) procedures to detect and identify any deviation from the damage-free baseline [2
Vibration-based SHM techniques investigate the structural health by recording and analyzing the vibration response, e.g., acceleration or displacement multivariate time series, of the monitored structure. Two competitive SHM approaches can be formally distinguished [3
]: the model-based one, e.g., [4
], and the data-based one, e.g., [6
]. The former is usually implemented through an updating strategy of a physics-based model on the basis of measured experimental data, which attempts to estimate the location and the extent of the occurred structural changes. The latter is based on a machine learning (ML) paradigm that, once trained, can be used as a black-box tool. ML systems automatically learn how the features, originated from the recorded data, are statistically correlated with the sought damage patterns [8
]. After the advent of deep learning (DL), which can incorporate the selection and extraction of optimized features into the end-to-end learning processes, the feature engineering stage has been progressively automatized.
This work proposes an output-only approach to the damage localization problem (see for instance [9
]), leveraging a synergic combination of multi-fidelity (MF) data-driven meta-modeling and Bayesian parameter identification. The probability distribution of the unknown damage parameters is approximated through a Markov chain Monte Carlo (MCMC) sampling algorithm.
MCMC has been applied in Bayesian model updating and model class selection in structural mechanics as well as in SHM, see, e.g., [11
]. In this work, MCMC is used to construct a Markov chain of the sought damage parameters, whose limit distribution is the target probability distribution. The probability distribution is sequentially updated by exploring the support of the damage parameters with a density of steps proportional to the unknown posterior distribution. The sampling acceptance is governed by the evidence of the current parameters to represent sparse dynamic response measurements, as provided by a sensors network, by means of a data-driven surrogate model.
Because handling finite element (FE) simulations within an MCMC analysis is computationally impractical, a FE model capable of simulating the effect of damage on the structural response is adopted only to build labelled datasets of vibration recordings for known damage positions, see for instance [13
]. A data-driven surrogate model is adopted instead to map operational and damage parameters to the associated vibration signals in place of the FE model. Such surrogate is based on a multi-fidelity deep neural network (MF-DNN) trained on synthetic data of multiple fidelities, a ML paradigm adopted and extended for instance in [14
]. Specifically, a limited amount of high fidelity (HF) data and a lot of cheaper low fidelity (LF) data are considered. This type of meta-modeling is useful to alleviate the high demand during training of HF data, potentially expensive to collect. Indeed, the LF data supply useful information on the trends of HF data, allowing the MF-DNN to enhance the prediction accuracy only leveraging few HF data in comparison to the single-fidelity method [16
3. Virtual Experiment
The proposed method is validated on the digital twin shown in Figure 1
. The HF model in Equation () is obtained from a FE discretization resulting in
dofs and integrated in time using the Newmark method. The structure is made of concrete, whose mechanical properties are: Young’s modulus
; Poisson’s ratio
. The structure is excited at the tip by a distributed load
, acting on an area of
, as depicted in Figure 1
. The load
varies in time according to
respectively denote the load amplitude and frequency, collected as
. Damage is introduced by reducing the material stiffness by
within the subdomain
, which is a box
as depicted in Figure 1
. The target position of this reduction is given by the coordinates of its center and can be identified with a single abscissa
running along the axis of the structure. Hence, the input parameters of the HF part are collected as
. Also the Rayleigh damping matrix, which account for a
damping ratio on the first 4 structural modes, is affected by the damage through the stiffness matrix. Synthetic displacement recordings
, are collected from
dofs, mimicking a monitoring system arranged as depicted in Figure 1
, for a time interval
The reduced-order model in Equation (2
), i.e., the LF model used to construct
, has been built performing a POD upon 40,000 snapshots in time, collected while exploring the parametric input space
. 14 POD-bases are selected and stored in matrix
, in place of the original 4659 dofs, after having fixed a suitable tolerance on the energy norm of the reconstruction error (
); for further details see, e.g., [9
For the training of the surrogate model in Equation (4
= 10,000 and
= 1000 instances have been collected from the LF and HF model, respectively. Concerning the compression of the LF data for the sake of prior dimensionality reduction, 104 POD-bases have been selected (
) and stored in matrix
, in place of 1600 data points.
The mean squared error and the mean absolute error have been used as loss functions for the training of
, respectively, together with the Adam optimization algorithm [21
]. The implementation has been carried out through the Tensorflow
], running on an Nvidia GeForce RTX 3080
An example of the reconstruction capabilities achieved by the surrogate model is shown in Figure 2
for the monitored gdl
, where the outcome of the regression over the POD-basis coefficients, ruled by the
, and the corresponding expanded LF signal are reported together with the signal enrichment, provided by the
. To quantify the accuracy of the predicted signals, the Pearson correlation coefficients (PCC) between predicted and ground truth HF signals are adopted as a measure of fitness. The PCC coefficients are evaluated with respect to 40 testing instances generated with the HF model while exploring the parametric input space
. The minimum PCC value over the 40 testing instances for each monitored channel is respectively
, which largely validate the performance of the surrogate model. The other way around, if the
is employed without being coupled with the
, the maximum PCC value drops to
, showing the utility of the MF setting that outperforms the single-fidelity based method.
In the absence of experimental data, the Bayesian estimation of the damage parameter
is simulated by considering pseudo-experimental instances, generated with the HF model, that have been corrupted by adding independent, identically distributed Gaussian noise featuring a signal-to-noise ratio equal to 80 to each vibration recording. Batches of
observations relative to the same damage condition but different operational conditions are processed during the evaluation of the likelihood in Equation (6
). The prior pdf
is taken as uniform, while, to account for the bounded domain in which
can fall, a truncated Gaussian centered on the last accepted state is considered for the proposal
. The adaptive Metropolis [23
] algorithm is adopted in order to ease the calibration of the proposal distribution, enabling its covariance to be tuned on the basis of past samples as the sampling evolves. The MCMC algorithm is run for 5000 samples, the first 500 of which are removed to get rid of the burn-in period. The obtained chain is ultimately thinned by discarding 3 samples over 4 to remove dependencies among consecutive samples.
Two examples of MCMC analyses are reported in Figure 3
, showing the generated Markov chains alongside the estimated posterior mean and credibility intervals. In both cases, the damage parameter
, here normalized between 0 and 1, is properly identified. It has to be noted that the larger uncertainty in the second case is somehow expected; indeed, given the structural layout and the placing of the sensors, the sensitivity of measures to damage positions far apart from the clamped side is smaller.
This paper has presented a stochastic approach for SHM, here applied to the problem of damage localization in case of slow damage progression. The presence of damage has been postulated as already detected, e.g., as identified by an early warning tool, and only the localization task has been analyzed. The Bayesian identification of damage parameters is achieved through an MCMC sampling algorithm, adopted to approximate their posterior distribution conditioned on a set of measurements. Few investigations are present in literature involving the use of MCMC for the health monitoring of civil structures, and this is the first one considering a MF-DNN surrogate model to accelerate the computation of the conditional likelihood. The surrogate model learns from simulated data of multiple fidelities, i.e., few HF data and several inexpensive LF data, such to alleviate the computational burden of the supervised training stage. The method has been assessed on a numerical case study, showing remarkable accuracy under the effect of measurement noise and varying operational conditions.
The method is suitable for structural typologies whose damage patterns can be represented by a stiffness reduction fixed within the time interval of interest. Since it enables a time scale separation between damage growth and damage assessment, this is a standard assumption for most practical scenarios in SHM. Such description of damage is consistent with the adopted vibration-based SHM approach, and allows the structure to be modeled as a linear system both in the presence and absence of damage. Moreover, as shown in [9
], even if the stiffness reduction takes place over domains of different size from that one adopted during the dataset construction, it is still possible to identify the correct position of damage.
Considering data-driven algorithms, damage localization is often addressed by exploiting a DL feature extractor followed by a classification or a regression module, e.g., as done in [9
]. However, due to the need of training in a simulated environment, the risk of losing generalization capabilities on real monitoring data is high. The proposed procedure tries to overcomes such generalization problems. Damage is located by seeking for those parameters of the surrogate model producing the closest output to the measured one, in terms of a suitable distance function measuring the signals similarity. For this reason and thanks to the fully stochastic framework here considered, which is suitable for dealing with noisy data and modeling inaccuracies, it is reasonable to expect a better ability of generalizing outside the training regime.
Besides the need of validating the proposed methodology within a suitable experimental setting, the next studies will extended the Bayesian identification also to the parameters controlling the operational conditions. Moreover, a usage monitoring tool powered by a suitable data-driven paradigm will be considered to provide useful prior knowledge as opposite to an informative flat prior. The analysis of dynamic effects resulting from localized damage mechanisms is also left for future investigations.