30 On Key Technologies for Realising Digital Twins for Structural Dynamics Applications 271 200 300 400 500 600 700 800 900 1000 200 400 600 800 1000 1 2 3 4 5 6 1 2 3 4 5 6 (a) (b) Calibrated Model ±3σ Calibrated Model μ Observations Fig. 30.4 Mass tension wire system example. Panel (a) indicates the model discrepancy between the model and “true” system when the “true” parameter value is used. Panel (b) presents the results of Bayesian calibration Clearly when the computer model uses the “true” value of mthere will be model discrepancy, as shown in Fig. 30.4a, where the computer model, true system and experimental observation (with e ∼ N(0, 0.01 2 )) are compared when m = 5.45 kg. If calibration is performed (here Bayesian calibration is utilised) without considering model discrepancy the estimated parameter value will be biased and there is no guarantee the functional form of the output will be correct. Figure 30.4b presents the outcome of Bayesian calibration for the model (with a prior M ∼ N(5.45,0.55 2 )) where the maximuma posteriori probability (MAP) estimate is M=5.01 kg. The result also demonstrates the difficulty in replicating the output correctly as model form errors are apparent (for further examples on the importance of model discrepancy see [17]). 30.2.5 Implications for Digital Twin Technology It should be noted that the example presented is highly simplified compared to the intended application for digital twin technology. However, the intention is to demonstrate the power of data augmentation applied to models containing unmodelled physics. A more general interpretation of the process is that of grey-box modelling. The grey box model is formed by combining a white box (the physics-based model) with a black box (a machine learning or statistical process) in order to capture model discrepancy. Without quantifying model discrepancy, parameters inferred during an uncertainty quantification process will typically be biased or potentially “over-confident”, leading to inaccurate predictions [17]. In a digital twin where biased parameters at a low-level model are then combined with other augmented models, this may lead to considerable errors at a full-system level. This affect could be compounded in a digital twin which includes multiple models, particularly with modelling issues such as mesh mismatches, which will result in several sources of model form errors, that if propagated to the next model/level will compound further. Trivially, bias will occur in calibrated parameters across the complete set of models, if discrepancy isn’t accounted for. As a result, in contrast to the case of a single validated model, it will be essential for digital twins attempting to join multiple models to incorporate mechanisms for inferring and compensating for model discrepancy. Once quantified model discrepancy should be used to inform model improvements. By interrogating where the largest sources of model discrepancy exist and the functional form of the bias, improvement to the physical models can be made. This aids building confidence in predictions by ultimately leading to a reduction in uncertainty, where the digital twin will systematically improve and evolve over the life-cycle of the structure.
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