PSI - Issue 84

358 Alberto Barontini et al. / Procedia Structural Integrity 84 (2026) 352–359 the formulation of the covariance matrix in A1 is proposed by the authors as one possible option, and its performance is known to depend strongly on the choice of the correlation distance , here assumed equal to ̅ . This parameter plays a crucial role in defining the correlation of the prediction error between the responses of sensors located in neighbouring regions (Papadimitriou & Lombaert, 2012) and may therefore have influenced the resulting placement. By contrast, the alternative formulation adopted in A2, which incorporates modal information (Vincenzi & Simonini, 2017), appears to provide a more robust solution. This result is particularly noteworthy given that A2 does not require any sampling of the uncertain parameters, namely it can be conducted on the reference mode shapes, thereby significantly reducing the computational burden while producing a placement that, for the problem considered, is comparable to and largely overlaps with those obtained using the other methods. Methods A4 and A6 entail an intermediate computational burden, as they perform a single optimisation run while accounting for the effect of sensor removal across the entire sample set. Thus, when compared to A3, they do not require the subsequent analysis of sensor occurrence. However, the repeated evaluation of the FIM determinant in method A6 may significantly increase the computational cost. Moreover, the solution obtained with A6 depends on the definition of the weighting parameter , which is here set to 0.5 to assign equal importance to the two competing objectives, namely maximising the mean and minimising the standard deviation of the determinant across the samples. It is worth noting that maximising the mean tends to favour central locations that are more strongly involved in the mode shapes, particularly the lower modes, whereas minimising the standard deviation penalises the same locations, as they are more strongly affected by variability induced by the sources of uncertainty. This trade-off is likely responsible for the comparatively poorer performance of A5. Although A5 has been shown to perform well when a larger number of modes is targeted and more sensors are available (Kim et al., 2018), in the present case it results in a clustered sensor configuration. This behaviour can be attributed to the fact that A5 favours locations exhibiting larger variance in the modal coordinates, with the aim of reducing uncertainty by instrumenting points characterised by higher variability. In the present case, mode 1 exhibits high variance at central locations, which likely promotes their selection in the final configuration. This penalises A5, which would otherwise have presented a lower computational cost while still accounting for uncertainties. 4. Conclusions The present work analysed and compared six existing methodologies that account for model error and sources of uncertainty in order to achieve a more robust optimisation of sensor placement based on preliminary information. Two of these methodologies (A1-A2) do not require explicit identification or characterisation of uncertainties, one treats both model parameters and sensor placement as stochastic (A3), while the remaining approaches (A4-A5-A6) sample the models on the basis of uncertain parameters and perform a single optimisation run. Two of the methods were originally formulated within the well-established backward sequential EfI framework (A3, A5). To ensure a fair comparison, the remaining methodologies were also reformulated within the same optimisation scheme. Four sensors were assumed to be available for placement, and four modes were selected as targets for the optimisation. Most approaches produced comparable optimised configurations, with spread distribution of vertical sensors and the inclusion of one horizontal sensor. Among the methods that account for errors through a correlated covariance matrix, A2 yields a superior solution. For the case examined, this approach produces a sensor placement comparable to that obtained using stochastic methods, in particular A3, A4 and A6, while reducing the computational burden. A2 was already proposed as an improvement of A1 formulation. Nonetheless, A1 could lead to better performance through a calibrated or informed definition of the correlation distance. A4 and A6 provide a well-distributed and potentially effective final placement, with A4 being particularly attractive, since A6 requires the iterative evaluation of the determinant of the FIM. As for A3, although potentially more computationally demanding than the other analysed alternatives due to the multiple optimisation runs and a postprocessing stage required, it may offer deeper insights into recurrent alternative sensor configurations by explicitly accounting for their affinities, this representing a valuable future improvement of the method. Finally, with regard to the A5 method, although it is computationally less demanding and capable of producing a well-distributed sensor network when a larger number of sensors and modes is considered, it resulted in a clustered configuration for the specific problem analysed, with the first mode shape appearing to dominate the weighting. A more detailed investigation of its dependence on the variance of the mode