PSI - Issue 84

Alberto Barontini et al. / Procedia Structural Integrity 84 (2026) 352–359

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Both formulations are here used within the classic backward sequential EfI method. The underlying rationale of these approaches is that closely spaced points in the model exhibit a higher correlation in prediction error. Therefore, a more spatially distributed sensor layout should help avoid biased regions. Alternative approaches, by contrast, have focused on a stochastic framework to quantify uncertainties and their impact on the optimisation process. In this context, one possible solution consists in a probabilistic-based optimisation where optimal sensor locations are treated as stochastic variables. This is the case for the approach hereafter referred to as A3, in which a Monte Carlo simulation is performed using the estimated distributions of the model’s uncertain parameters to generate multiple sets of optimal sensor locations. Recurrent sensor locations across these sets are identified and counted. Locations that appear in all solutions, or in the majority of them (e.g. more than 65% of the solutions), are considered vital, while the remaining locations are selected based on their conditional probabilities, collected in a sensor affinity (SA) matrix (Castro-Triguero et al., 2013). Alternative methods have instead sought to reduce the computational cost by quantifying uncertainties prior to the optimisation, thereby enabling a single optimisation run. Among these, an early approach framed the problem as the maximisation of the info-gap robustness function. The resulting sensor network corresponds to the configuration that maximises the minimum distinguishability over a set of sampled cases for a specified number of sensors (Vinot et al., 2005). In that work, the authors employed an objective function based on the condition number of the mode shape matrix and adopted a heuristic forward sequential approach. In the present study, instead, a consistent methodology, hereafter called A4, based on backward sequential EfI optimisation is proposed. Accordingly, at each iteration, the sensor location to be discarded is the one that contributes least to the effective independence distribution across all generated samples of mode shapes corresponding to the uncertain model parameters. An alternative approach, hereafter referred to as A5, adopts a recent reformulation of the fractional eigenvalue distribution matrix of the original EfI method expressed as follows (Kim et al., 2018): = [ ]⨂[ ] − 1 + ( )[[ ]⨂[ ]] − 1 (5) which accounts for the variance induced by model uncertainties. Under this formulation, any random realisation of the mode shape matrix can be expressed as: = + (6) Finally, the last method investigated, hereafter referred to as A6, aims to maximise the mean value of the FIM determinant while simultaneously minimising its standard deviation. The authors originally proposed a metaheuristic multi-objective optimisation approach (H. An et al., 2022). However, for consistency, the underlying rationale is reformulated here within a heuristic backward sequential framework. Accordingly, at each iteration, the sensor location whose removal yields the minimum value of the following objective function is discarded: = ∗ +(1− ) ∗ (7) 3. Case study application The numerical case study investigated in this work consists of the finite element (FE) model of a truss bridge, inspired by benchmark cases originally adopted in the development of some of the approaches compared herein (Castro-Triguero et al., 2013; Kim et al., 2018). The truss comprises 36 elements and 16 nodes, each characterised by two translational degrees of freedom (DOFs). At each node n , the DOF aligned with the x-axis (horizontal) is numbered as 2( − 1), while the DOF aligned with the y-axis (vertical) is numbered as 2 . Thus, odd-numbered locations correspond to DOFs oriented horizontally, while even-numbered locations are oriented vertically. DOF 1, 2, and 30 are constrained. Following Castro-Triguero et al. (2013), the mean cross-sectional area is taken as 0.0025 m² (coefficient of Variation, CoV), the mean Young’s modulus as 70 GPa (8% CoV), and the mean material density as 2800 kg/m³ (4% CoV). These three parameters are assumed to be uncertain and normally distributed. To this end, 4000 samples are