Aug 2026· e-Journal of Nondestructive Testing· Vol 31· 0 citations
Abstract
Virtual sensing techniques, or soft sensing, aim at reconstructing physical quantities at locations where no direct measurements are available, enabling a complete representation of a system from sparse sensor information. These methods rely on the synergy between the measurement data, physics- or knowledge-based models, and statistical characterization of uncertainties. In recent years, virtual sensing has been consolidated into a broad set of approaches that include inverse finite-element formulations, data-driven methods, and control theory-based estimators. Among the latter, proportional observers (PO) and their multi-resolution extensions (MRPO) offer a practical alternative to Kalman-filters and “natural observers”. Their simple formulation, combined with frequency-band tuning, balances computational burden, noise robustness, and reconstruction accuracy.
This paper presents an observer-based virtual sensing (OB-VS) framework for reconstructing strain fields in ship structures using a limited number of physical sensors. The approach builds on PO and MRPO, exploiting linear relations between the measurement vector and the estimated state-space variables. The method leverages the mechanical transfer function of the structure and a statistical description of uncertainties in the frequency domain, including measurement noise, modelling inaccuracies, and unknown excitation. A key advantage of the PO formulation is that the prediction-error covariance matrix depends quadratically on the observer gain(s), ensuring the existence of a unique global minimum. The MRPO enhances this approach by defining multiple POs, each one tailored for a specific frequency band into which the signal is decomposed via wavelet multi-resolution analysis.
The methodology is applied to the experimental dataset relative to an elastic scaled ship-model campaign, performed within the “Digital Ship Structural Health Monitoring project” (dTHOR), granted by the European Defence Fund, aimed at developing a system based on innovative utilization of extensive on-board measurements, a comprehensive digital framework, and hybrid analysis and modelling. Measurements from 13 strain-gauges were utilized. Stepping forward with respect to previous 1D models, the present analysis employs a full 3D finite-element (FE) representation of the segmented hull. The observer is trained by extracting modal quantities from the FE solution (modal strains, modal mass, damping, and stiffness matrices) and constructing the corresponding transfer function. Because the FE model represents the dry hull, a correction is introduced to ensure consistency between numerical and experimental modal characteristics. The external (wave) excitation is estimated by combining acceleration, strain and rigid-body motion data.
Performance is evaluated by computing the errors in signal reconstruction both in time (Time Response Assurance Criterion) and frequency (Frequency Response Assurance Criterion). Both PO and MRPO achieve a high fidelity in predicting the strains measured by sensors not involved in the virtual sensing process. An a-posteriori comparison of several sensor layouts highlights the possibility of achieving an optimal trade-off between instrumentation effort and reconstruction accuracy based on end-user needs. The OB-VS approach keeps accurate in signal tracking even under incorrect assumptions on the external excitation, thus demonstrating its robustness.
Overall, this study highlights the potential of multi-resolution analysis for improving robustness to noise and spectral representation for real-time observer-based virtual sensing, enabling predictive digital-twin capabilities not only limited to on-board monitoring of ship structures.
Accurate structural dynamics analysis depends on the reliability of numerical models, which are commonly validated through modal testing. In practice, however, sensor placement during testing is often limited by spatial and environmental constraints, leading to incomplete measurement
data. To address these limitations, virtual sensing has emerged as an effective solution for estimating structural responses at unmeasured locations. This study proposes a physicsbased virtual sensing framework that integrates Kalman filter algorithm with a Newmarkbeta implicit time integration
to reconstruct full-field response. A modal reducedorder model is employed to improve computational efficiency. Unlike virtual sensing approaches that focus on inverse force identification, this framework reconstructs unmeasured acceleration responses under measured excitation, which is more
relevant to test and validation processes. The proposed method enables accurate identification of dominant natural frequencies using limited sensor data, without requiring extensive test-simulation correlation. The framework is validated through shaker tests on an automotive chassis crossmember,
where the reconstructed responses show agreement with experimental results in the frequency domain. The proposed framework provides a practical solution for reliable response estimation while reducing the number of physical sensors and correlation processes required in structural dynamic testing.
Jihwan Jang, Jongwha Lee, Sangwon Park et al.· INTER-NOISE and NOISE-CON Co...· 0 citations
Structural Health Monitoring (SHM) is increasingly being considered for predictive maintenance of aircrafts. This study investigates the use of strain and displacement measurements obtained from a limited number of sensors as input to a calibration matrix (CM) algorithm to compute the shapes of structures undergoing geometric nonlinear (GNL) structural deformation. To evaluate CM, GNL benchmark problems from the literature were expanded with sensors at specific locations on these structures. Two problems used for GNL inverse finite element methods (iFEM) in the literature were implemented as well, for comparison. Each problem was modeled using ABAQUS and subjected to significant deformations, resulting in strain and displacement measurements from simulated sensors. A novel GNL CM variant is presented that converts virtual sensor data into accurately reconstructed strain/displacement distributions and applied loads, achieving superior accuracy for the two state-of-the-art GNL iFEM problems. The accuracy was assessed for each problem based on the number and type of sensors employed. Under noiseless conditions, the presented method achieves a force reconstruction error of 1% or less for all problems utilizing 96 strain sensors and of 2% or less for all but one problem when employing 96 displacement sensors or a combination of strain and displacement sensors. A displacement error of 0.34% or less is achieved for all benchmark problems, using just 32 sensors of either type. When subjected to noise, the accuracy is comparable or better than the iFEM results with 96 sensors of either type for all but one of the benchmark problems, and superior when using a greater number of sensors, reaching median errors of less than 1%. A median force error of 2.5% is reached with 720 strain sensors when subject to 10% noise. This level of accuracy is achieved for both isotropic and composite laminate materials, and for structures undergoing buckling deformation.
Cornelis de Mooij, Marcias Martinez· Journal of Strain Analysis f...· 0 citations
Reliable deformation reconstruction supports structural health monitoring when a target sensor is unavailable, unreliable, or requires independent verification. The proposed framework contributes an interpretable correlation-driven representation for virtual sensing rather than new X-means, Gaussian process, LSSVR, or MK-LSSVR algorithms. BIC-guided X-means adaptively identifies monitoring-point groups with statistically similar standardized deformation histories. A fixed-covariance Gaussian process equipped witha squared exponential kernel, calibrated exclusively from training-period measurements and used solely as a probabilistic representation of inter-point dependence without explicit residual extraction, transforms the associated-point measurements into the time-varying synergistic expectation μsyn(t). This quantity is combined with conventional water pressure, temperature, and time-effect factors as the dynamic M3 input. The same kernel formulation defines fixed synergistic-variance and mutual-information diagnostics for configuration-level interpretation; these quantities are neither dynamic regressors nor evidence used to establish the reported performance conclusions. Controlled M1-M2-M3 comparisons under identical LSSVR and MK-LSSVR backbones isolate the effects of conventional factors, raw associated-point measurements, and the correlation-derived representation. The framework was evaluated at M-α/C2, M-μ/C1, and M-ρ/C3 using horizontal-displacement records from 20 monitoring points in a hydraulic concrete structure in Sichuan Province, China. For M-α, M3 + MK-LSSVR achieved an RMSE of 0.1016 mm, an MAE of 0.0388 mm, and an R2 of 0.9890; relative to M2 + MK-LSSVR, RMSE and MAE decreased by 68.4% and 69.3%, respectively. The same configuration gave the lowest RMSE and MAE and the highest R2 for M-μ and M-ρ, reproducing the within-target ranking in all three X-means groups. The framework is intended for virtual sensing, sensor-replacement support, data verification, and monitoring–consistency checking, not as a direct damage or safety indicator.
Integrating sensor measurements with complex numerical models for structural health monitoring holds great promise because both sources provide complementary and potentially high-fidelity sources of information. However, effectively assimilating data into computationally intensive models presents substantial challenges. Among these are noisy and potentially sparse data, simplifications and uncertainties in the computational model and strong nonlinearities that pose challenges to traditional assimilation approaches. While Bayesian updating offers a principled approach to data assimilation under uncertainty, it requires considerable numerical effort that is often not acceptable, in particular, if frequent and fast updates are required.
The present work takes [1] as a starting point, where a statistical version of the Finite Element method (statFEM) has been introduced. The main ingredients of statFEM are a propagation of uncertainty through a Finite Element model and (empirical) Bayesian updating that explicitly accounts for model misspecification. Subsequent work also considers Ensemble Kalman filtering as an alternative to linear Bayesian updating [2] and demonstrated its applicability in structural health monitoring contexts [3].
Here, we investigate the limits of updating methods, as presented in [1] or [4], and present a comparative computational analysis against alternative strategies such as Ensemble Kalman filtering for standard benchmark problems. In order to accelerate the prior computations, we employ surrogate models and further integrate adaptive sampling techniques. As a case study, we model aging material properties as spatially correlated random fields and assimilate noisy simulated measurements of structural responses. This allows us to demonstrate the efficient computation of stochastic priors and their assimilation via both the original statFEM and ensemble-based methods. Finally, we discuss the propagation of cracks and its implications for load-bearing capacity assessment within the data assimilation framework.
[1] Girolami, M., Febrianto, E., Yin, G., & Cirak, F. (2021). The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions. Computer Methods in Applied Mechanics and Engineering, 375, 113533.
[2] Duffin, C., Cripps, E., Stemler, T., & Girolami, M. (2021). Statistical finite elements for misspecified models. Proceedings of the National Academy of Sciences, 118(2), e2015006118.
[3] Muralidhar, N. K., Gräßle, C., Rauter, N., Mikhaylenko, A., Lammering, R. & Lorenz, D. A. (2023). Damage identification in fiber metal laminates using Bayesian analysis with model order reduction, Computer Methods in Applied Mechanics and Engineering, 403, 115737.
[4] Narouie, V., Wessels, H., Cirak, F., & Römer, U. (2025). Mechanical state estimation with a Polynomial-Chaos-Based Statistical Finite Element Method. Computer Methods in Applied Mechanics and Engineering, 441, 117970.
Lucas Hermann, Saddam Hijazi, C. Gräßle et al.· e-Journal of Nondestructive...· 0 citations
Computer-vision-based structural health monitoring (CVSHM) enables contactless displacement measurement at multiple locations on the vibrating structure. Additionally, such a measurement can be realized from a certain distance from the monitored infrastructure. It provides a possibility of reducing the costs of the CVSHM system. However, computer-vision-based (CV) vibration measurement can be significantly contaminated by measurement errors at higher vibration frequencies and low amplitudes.
A framework for CVSHM is proposed, which allows for robust detection, localization, and assessment of the damage [1]. As illustrated in the figure, the framework consists of: CV measurement of structural displacements, modal data extraction, and modal sensitivity-based model updating by changing the stiffness of monitored structural members. Displacement measurement is realized with the template matching technique, maximizing the zero-normalized cross-correlation function. Later, modal parameters are identified using the data-driven stochastic subspace identification method (SSI-DATA). Degrees of freedom suspected to be source of gross errors are removed. The last framework component is inspired by the augmented inverse estimate described in [2]. The proposed method additionally employs: (a) weighting of errors between the identified and model modal parameters, and (b) constraints imposed on the model updating procedure that structural stiffness parameters can only decrease with respect to the initial values. Such a method of damage assessment is termed weighted negative least square inverse estimate (WNLSIE). This method is physics-based and avoids the problems typical of machine-learning approaches, such as the need for the collection of training data or generalization of the trained model.
The proposed framework is tested using realistic synthetic videos representing vibrating truss structure. These videos are generated with physics-based graphical models (PBGM) and allow for employing the displacement ground truth data for comparison purposes [3]. Displacements of 19 truss nodes are measured and 29 truss members are monitored. Sampling frequency is 120 frames per second (fps). Excitations are unknown. The proposed framework allows for prediction of damaged truss member and its damage level, when error of measured displacement is at the level of 50 %.
The proposed framework for CVSHM is easy to implement and allows for detection, localization, and assessment of structural damage even for highly contaminated displacement data.
Bibliography
[1] M. Ostrowski, B. Blachowski, B. Wójcik, M. Żarski, P. Tauzowski, Ł. Jankowski, A framework for computer vision-based health monitoring of a truss structure subjected to unknown excitations, Earthquake Engineering and Engineering Vibration 22 (2023) 1–17. https://doi.org/10.1007/s11803-023-2154-3.
[2] X. Peng, F.J. Qin, Q.W. Yang, H. Chen, A Robust Estimate Method for Damage Detection of Concrete Structures Using Contaminated Data, Advances in Civil Engineering 2021 (2021) 6669958. https://doi.org/https://doi.org/10.1155/2021/6669958.
[3] Y. Narazaki, F. Gomez, V. Hoskere, M.D. Smith, B.F. Spencer, Efficient development of vision-based dense three-dimensional displacement measurement algorithms using physics-based graphics models, Structural Health Monitoring 20 (2021) 1841–1863. https://doi.org/https://doi.org/10.1177/1475921720939522.
M. Ostrowski, B. Błachowski, M. Żarski et al.· e-Journal of Nondestructive...· 0 citations
Optimal Sensor Placement (OSP) is a critical component of vibration-based Structural Health Monitoring (SHM), yet its effectiveness is often compromised by the deterministic nature of standard Finite Element (FE) models. This study evaluates two advanced OSP frameworks that explicitly account for epistemic and aleatory uncertainties: a variance-based method that utilizes hierarchical clustering to manage modal sensitivity variance, and a likelihood-maximization method designed to maximize the probability of achieving specific SHM objectives under measurement noise. Using a laboratory-tested glulam timber beam as a benchmark, the research investigates the impact of uncertain material properties and support stiffness on sensor performance. Experimental results from ambient and impact vibration tests serve as the ground truth for validation. The findings reveal that boundary condition uncertainties can lead to significant modal discrepancies, especially for higher modes. Both probabilistic methods identified robust sensor configurations that significantly outperformed random layouts, with the likelihood-maximization method achieving a 65.9% success rate for high-fidelity mode shape reconstruction (MAC ≥ 0.95). By bridging the gap between theoretical optimization and practical structural variability, these frameworks provide a reliable methodology for designing SHM systems in complex, real-world infrastructure.
Semih Gönen, E. Erduran, Oguzhan Gumus et al.· e-Journal of Nondestructive...· 0 citations
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