270 D. J. Wagg et al. 30.2.3 Data-Augmented Modelling Computer models, regardless of the level of fidelity, are typically not able to capture all possible physics exhibited by an engineering system. As a consequence, a digital twin will augment the outputs from computer models with data to get closer to providing ultra-realistic predictions. One way to begin to quantify this is to define the model discrepancy. This is simply the mismatch between the computer model output and the measured process from the physical twin (assuming for simplicity there is no observational uncertainty). Two points are worthy of note here. First, model discrepancy it usually quite straightforward to measure (or estimate in the presence of observational uncertainty) even if the physical twin and/or computer model(s) are very complex. Second, even when the parameters are treated as deterministic and considered to be “truly” known, there will typically still be a mismatch, and hence some level of model discrepancy. Therefore, based on the fact that computer modelling alone will be inadequate, models will be augmented by information from physically recorded data in order to create a digital twin. In fact this augmentation process is one of the core attributes of a digital twin, and for the purpose of demonstrating the concept it will be assumed that the digital twin has just a single computer model. Then, following the approach of Kennedy and O’Hagan [16], the computer model in the digital twin will be represented as z(x) =y(x) +e =η(x, θ) +δ(x) +e, (30.1) where z(x) and y(x) are respectively the observational and bias (or model discrepancy)-corrected computer model outputs based on the given inputs x. The bias-corrected computer model output is equal to the sum of the computer model η(x, θ) and the model discrepancy δ(x), where θ are parameters of the computer model. The observations are assumed to be uncertain, and this is represented in the model by the addition of error, e. The definitions in Eq. (30.1) allow us to build a digital twin in which firstly, data sets are used to quantify the model discrepancy, δ(x). Then secondly, this information is used to add a correction (i.e. calibrate) the computer model so that the augmented outputs, z(x), properly reflect the measured outputs from the physical twin. In the next section a numerical example of this process will be presented. 30.2.4 Numerical Example The importance of the model discrepancy term is demonstrated for a simple numerical example; a mass, tension wire system, shown schematically in Fig. 30.3. The objective is to predict the natural frequency of the systemf (in Hz), given different tensions T, where the mass mis unknown. To reflect the concept of model discrepancy it is assumed that the “true” system has an off-centred mass where, L=1ma =0.2 m (Eq. (30.2) and Fig. 30.3a) and that the “true” mass is 5.45 kg. However, the model of the system does not include the ability in incorporate an offset, instead modelling the system with a centred mass, representing a level of missing physics (Eq. (30.3) and Fig. 30.3b), we have ftrue = 1 2π T(a +b) mab 1 2 (30.2) fmodel = 1 π T mL 1 2 (30.3) (a) (b) Fig. 30.3 Mass, tensioned wire system schematic. Panel (a) shows the model; centred mass, tensioned wire and panel (b) the ‘true’ system; off-centred mass, tensioned wire (L=1ma =0.2m)
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