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.

Developing reliable health indicators (HIs) for aeronautical composite structures is challenging because damage evolution is complex, stochastic, and affected by uncertainties such as impact events and manufactured defects. In this context, passive and active structural health monitoring (SHM) techniques provide complementary information: acoustic emission (AE) captures temporally dense signatures of damage activity, whereas guided waves (GW) provide state-sensitive interrogation of the structure at discrete inspection times. This work therefore investigates heterogeneous sensor fusion for HI construction in composite T-stiffener panels subjected to run-to-failure compression-compression fatigue loading. Modality-specific AE- and GW-based HI frameworks are first developed independently using signal processing and AI. The resulting HIs are then synchronized in the fatigue-cycle domain and combined through inter-modality fusion to obtain a single fused HI. The study addresses key challenges including the absence of true HI labels, synchronization of passive and active measurements, and learning from limited multimodal overlap. The results show the potential of combining AE and GW to construct more informative and robust HIs for prognostics of composite aerostructures.

Structural Health Monitoring (SHM) has emerged as a candidate approach for cryogenic hydrogen systems, reusable launch vehicles, and space structures operating under extreme environmental and operational conditions (EOCs). Reliable SHM across such conditions requires both compensation of EOC influences and stable transducer behavior over the full temperature range. In guided wave (GW)-based SHM, this is particularly critical because of the strong temperature dependence of piezoceramic materials. This work presents a comparative study of hard (HRD-PZT), soft (SFT-PZT) and very-soft (VSF-PZT) Lead Zirconate-Titanate transducers from approximately 27 K to 353 K. Electromechanical impedance (EMI) measurements on unbonded discs in a helium cryostat and in a climate chamber were used to extract the temperature-dependent d₃₁ coefficient via the radial-mode procedure. Guided wave experiments were performed on an instrumented copper plate in the same cryostat (40–290 K) and on aluminum plates in a climate chamber (233–353 K), enabling a cross-substrate consistency check. At room temperature, the EMI measurements reproduce the expected VSF > SFT > HRD ordering, and the GW measurements show the same observed ordering. At cryogenic temperatures the EMI hierarchy is preserved, VSF-PZT retains the largest absolute d₃₁, but the GW amplitude hierarchy reverses: HRD-PZT delivers the largest A₀- and S₀-mode amplitudes below approximately 220 K. GW amplitudes are well described by a two-stage linear trend for HRD-PZT and by a sigmoidal trend for SFT- and VSF-PZT. In the climate-chamber range, a non-monotonic behaviour with a maximum near 310–320 K is consistent with softening of the structural adhesive near its glass-transition temperature. These results highlight that transducer selection for cryogenic SHM cannot rely on room-temperature datasheet values alone and must combine intrinsic piezoelectric properties with the temperature-dependent electromechanical coupling to the host structure

Recently, unsupervised constitutive model discovery has gained attention through frameworks based on the Virtual Fields Method (VFM), most prominently the EUCLID approach. However, the performance of VFM-based approaches, including EUCLID, is affected by measurement noise and data sparsity, which are unavoidable in practice. The statistical finite element method (statFEM) offers a complementary perspective by providing a Bayesian framework for assimilating noisy and sparse measurements to reconstruct the full-field displacement response, together with quantified uncertainty. While statFEM recovers displacement fields under uncertainty, it does not strictly enforce consistency with constitutive relations or aim to yield interpretable constitutive models. In this work, we couple statFEM with unsupervised constitutive model discovery in the EUCLID framework, yielding statFEM--EUCLID. The results show that this integration reduces sensitivity to noise and data sparsity, while ensuring that the reconstructed fields remain consistent with both equilibrium and constitutive laws. The framework is demonstrated for isotropic hyperelastic materials. Its computational efficiency and robustness make it appealing for further generalization, with the ultimate goal to discover hidden damage mechanisms from observational data. We expect the discovery of hidden damage mechanisms to be a game changer for reliable prognosis. The presented methodology embraces co-design of structures and sensors as it allows to update design-phase computational models given observational data.