34 On Digital Twins, Mirrors and Virtualisations 293 Meshed + MI Su Fig. 34.4 Insertion of a local damage model into an undamaged structure model Assume two ingredients: the first is a validated model of the undamaged structure of interest Su, denoted byMuF. Further assume a set of data{Du Tr ,Du T} which has been used to validate the data. Further assume that MuF is an u-mirror, according to some appropriate metric. The second ingredient is a local damage model Md, which has been validated in a context Cl using data from coupon tests. The model may have been updated on the basis of test data and may well be a hybrid (grey-box) model. Assume that under the circumstances MdCl is an d-mirror for the context Cl according to some appropriate metric. Finally, we assume that there are no validation data for the damaged structure Sd. The problem is essentially a joining problem; however, it is of a specific type and merits a little more new notation. An insertion model MI is defined as an algorithm or prescription for embedding the model MdCL in MuF, in such a way that the result is a model for Sd. This differs from the previous joint definitions in that there is no new physics associated with the join. MI could be a very simple process i.e. if the two component models are FE models, insertion will only really mean harmonising the two meshes along the boundary of the join. One can think of the process as a type of surgery9 i.e. one cuts out a healthy region of MuF and replaces in with MdCl , as in Fig. 34.4, and then harmonises the meshes at the boundary. 10 Clearly this means that there will need to be compatibility conditions which guarantee some degree of smoothness/continuity across the boundary.11 There is another compatibility condition required here by the theory; the models Mu andMd must exchange information in such a way that the dynamics evolves appropriately i.e. the response context of Cl must overlap with the environmental context of F i.e. Cl ∩F =φ. In fact, in a general assembly model MC1 ⊕ MI MC2, it will usually be necessary that C1∩C2 =φ and C1 ∩C2 =φ (where φ represents the empty set here). As a fairly simple example, consider the problem of modelling a crack in a pressure vessel (Fig. 34.4). The undamaged model MuF represents the vessel; the damage model MdCl , represents a through crack in a section of plate. By joining the two models, one can embed a crack of arbitrary location, length or orientation in the vessel (the process might require some care near the boundaries). A subtlety here is that the crack model might have been validated for flat specimens, in which case a modification might be needed for compatibility with the curved surface of the vessel. A more important issue is the following. The behaviour of the structure will usually be modelled using macroscopic physics, while the detailed crack model will require microscopic physics; this means that the features have to be chosen very carefully so that the behaviour of the crack is communicated over the boundary effectively. 9Surgery is a mathematical technique for building complicated topological spaces from simpler ones [20]. It may be that the technique can be applied in the context of joining models. 10This is similar to the situation in real-time hybrid testing where coordinate sets are defined in each domain, which need to be synchronised in order to form the join. Errors in the synchronisation process then give a measure of how imperfect the joint is. 11Note that this is rather perverse version of surgery, where undamaged tissue is replaced by damaged.
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