292 K. Worden et al. Fig. 34.3 Ecastré beam as sum of two cantilevers In a general theory, one would hope to prove theorems that were general, perhaps across particular classes of joint models; consider for example the reasonable conjecture: Suppose that given models MCi i (i =1, 2) are i-mirrors for structures Si in contexts Ci, then MA =MC 1 1 ⊕MJ MC 2 2 is an -mirror for the structure S1 ⊕SJ S2 in the context C1 ⊕C2 with ≥max( 1, 2) (where δ = −max( 1, 2) ≥0 is defined as the joining deficit). Finally, it is important to mention another use of the idea of joining models. One might simply wish to represent a complex structure in terms of substructures, even if there is no physical joining process involved (a situation that arises in hybrid testing [18]). A simple example will suffice. Suppose one wished to model a fixed-fixed beam, and to validate the model. However, suppose that one had no validation data for the beam, but one did possess a validated model for a cantilever beam; in fact the cantilever model had been established as an -mirror. Clearly, one can regard the fixed-fixed beam as two cantilevers joined perfectly at their tips. One could now attempt to answer the question above, as to whether joining two copies of the cantilever beam is an A-mirror for the fixed-fixed beam. In this case, one might assume that the joint model MJ is perfect; in practice a perfect joint when joining two FE models would be accomplished by seamlessly merging the meshes at the joint so that material continuity is as good at the joint as anywhere else. Perfect or idealised joints of this nature will be denoted by the symbol ⊕ P . Even in the case of a perfect joint, one should be aware of a caveat, and this relates tocontext. Suppose that the cantilever model was linear and had been validated on test data showing small or moderate deflections of the cantilever tip. When the cantilevers are joined, and the cantilever tips become the mid-point of the beam, the response of the real beam will become nonlinear for much smaller values of mid-point displacement than the values measured at the cantilever tip (Fig. 34.3). Many of the ideas discussed here are covered by the multilevel framework discussed in [5], and it may be that the ideas of reliability and relevance applied in that framework can be adopted in order to prove hypotheses like those pointed out in the current paper. 34.3.3 An Example Concerning Structural Health Monitoring One of the major problems with data-based Structural Health Monitoring (SHM) is that data from damaged structures is scarce. Although damage detection is possible even if one only have data from the normal condition of the structure of interest, using unsupervised learning [19]; higher-level diagnostics like locating damage or assessing its type or severity can only be accomplished if one has data from all the damage states of interest. It is inconceivable that one might carry out a test programme that systematically involved damaging numbers of high-value structures, so one has to turn towards modelling as a means of providing the necessary data. The context responses in an SHM problem are usually going to be features for machine learning. Given the importance of the specific context, new notation will be introduced; the SHM context will be denoted F.
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