34 On Digital Twins, Mirrors and Virtualisations 291 {y2,y4,y6}. In this situation, there are two simple ways to establish M : • The trivial approach is to simply change the output deck of MC, so that the model outputs the required variables (if it didn’t before). • One can add a numerical interpolation step to the process in order to estimate the variables in C from those in C. In the first case, it should be a fairly straightforward matter to establish that the model is an -mirror based on the existing theory of error estimates for FE models [16, 17], and one would expect that ≈ . In the second case, one should be able to use error estimates from the numerical analysis of interpolation, combined with some reasonable assumptions about the continuity of the beam profile. One could also bound the errors based on much coarser assumptions e.g. one could estimate how far y4 could get fromy3 and y5 before the induced stresses in the beam exceeded the yield stress. Although the latter approach would likely work, it would probably yield an , so conservative that one would find the value impractical in terms of model trust. In an exercise like this, the objective would be to find the lowest bound on possible. A more interesting problem arises in the case of the extended definition of context. Suppose C covered points 1, 3 and 5 at low levels of excitation, and C covered points 2, 4 and 6 at a higher level of excitation; there would be two different answers to this question, depending on whether MC was linear or nonlinear. 34.3.2 An Example Concerning Assembly This example concerns a very important objective of any programme of ‘virtualisation’. Suppose one could validate a model of a full-scale assembled structure using only test data acquired from substructure testing. The cost savings in the design/production cycle would be potentially very high. It is important that the ‘algebra’ of models being developed covers this situation, and this will entail an understanding of how to model joints and joining processes. For the sake of simplicity, consider the case of two substructures (but note that this is not a real restriction, as the substructure assembly can be considered recursively). The substructures, denoted S1 and S2, will be assumed to have individual contexts C1 and C2 respectively. It will be assumed that the substructures will be joined using some technology, which can itself be modelled; in the general case, one assumes that the joint may itself be a substructure SJ. With a small abuse of mathematical notation, the assembled structure SA will be denoted by, SA =S1 ⊕SJ S2 (34.8) For simplicity, it will be assumed that all the responses from the substructures can still be measured; in this case one can denote the new context by CA =C1 ⊕C2. (In this case, the ⊕is largely just a direct sum with some reordering of symbols and deletion of copies of symbols that appear in the environment context twice.) In general, one would have to allow for the fact that the joining process might eliminate a possible measurement point on the substructure, and thus change the context by removing a variable. It is assumed that each substructure Si has amodel MCi associated with it, and that the models have been validated using test data from the individual structures, and it has been established that MC i is an i-mirror in each case. Furthermore, assume that the joint/joining process has a model MJ, and that this model may or may not have been validated. The model of the assembled structure is denoted, MA =M1 ⊕MJ M2 (34.9) The key question is now, Given the assumptions stated, is it possible to show that there exists any A such that MA is an A-mirror for SA in the context CA, in the absence of any test data for the assemblySA? If so, then what is the smallest A for which this is true? Of course, one could also attempt to accommodate uncertainty, and frame the question in terms of α-mirrors. This is the most difficult question so far, but it also offers the highest returns, if it can be answered. The problem also depends on whether a validated model for MJ is available. For example, consider the case when the joint is a weld, and that coupon tests have established some of the material properties of the weld material (perhaps with a high degree of uncertainty). Even allowing for the fact that the issue is not just about material properties, one would expect Ato be a monotonically-increasing function of the weld parameter uncertainties. One might also model the weld as a hybrid model, given that the physics of the joint are not perfectly understood. From first principles, one might approach the problem from the same viewpoint as before; one could make reasonable/trusted assumptions about the real joint and the model joint, and try to determine how far they can diverge.
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