34 On Digital Twins, Mirrors and Virtualisations 287 point; however, the authors here would argue that the current proposal is more sympathetic to the needs of the digital twin concept, because of the explicit attention given to context and environment. There is no intention here to play down any previous works on general methodologies, the assumption is that the tools already proposed will play important roles. One example of a general framework for V&V is provided in [5]. That publication provides a methodology for estimating the uncertainty in system-level predictions, where system-level parameters are estimated in terms of lower-level experiments. The paper is largely concerned with calibration and uncertainty propagation, and introduces tools for estimating the reliability of models. Perhaps more importantly for the current discussion, the paper introduces a concept of ‘relevance’ which quantifies the relationship between the system-level model and lower-level models, and potentially allows a ‘confidence’ measure in terms of extrapolating from lower levels to the system level. The paper by Nagel and Sudret [6], proposes a Bayesian unified framework which provides a ‘. . . toolkit for statistical model building. It forms some kind of superstructure that embeds a variety of stochastic inverse problems as special cases’. (There are of course, many other papers one could cite; however, there is no intention here to provide a survey.) Another fair criticism of the current paper is that the new term ‘mirror’ is not needed either, it refers simply to a validated model; however, it is introduced here because it refers to a specific class of models and because, as discussed above, there is a need to distinguish the idea from the more overarching digital twin. The layout of the paper is as follows. The next section will make the main series of definitions of the important concepts in the framework: contexts, mirrors etc. The section will also define the concepts of environments and virtualisations which are central to the idea of a digital twin. Section 34.3 will discuss a number of example problems in which the idea of a mirror would be fruitful, assuming that the appropriate mathematical underpinnings of the theory can be provided. The paper finishes with some discussion and conclusions. 34.2 Mirrors 34.2.1 Basic Definitions To start with the simplest situation, the discussion will initially consider only physics-based models; data-based and hybrid models5 will be brought in later. One must begin with a structure (or system) S; this is the physical object of interest. It will be interpreted as having an objective reality independent of its surroundings i.e. it is possible to think of it in a vacuum remote from any other matter. Temporal changes in the confirmation and behaviour of the structure will be summarised in a state vector s(t) = {s1(t), . . . sNS(t)}, which consists of a set of NS instantaneous measurements (at time t) which completely characterise its state. Now, the environment of the structure could be considered as all physical reality exterior to it; however, that is too general. Considering the fact that the environment could also be characterised by a state vector; theenvironment Eof Swill be defined as the set of environmental variables that can actually affect S i.e. a change in a variable will evoke a change in the state s(t). With this in mind, one will have an environmental state vector e(t) ={e1(t), . . .eNE(t)}. Recognising that one will generally only wish to model some aspects of the behaviour of S, a context C for S will be defined as a set of variables C = {eC i ∈ E,sC j ∈ s;i,j}. The subset {eC i } will be referred to as the environmental context, and the subset {sC j } as the response or predictive context. Now, a schedule WC for the context C will be a set of time series {e C W (ti);i =1, . . .Nt ;ti ∈ [0,T]}. (In principle, the set {ti} could be continuous or discrete.) The response r C W (t), to a schedule WC is defined as the measurement sequence resulting from testing the structure and imposing the schedule as inputs. As the process will generally be dynamic, it will be denoted by the functional, rC W(t) =S[e C W(t) ≡WC] (34.1) One can now define the test TC W associated with the schedule WC in the context C, as the set T C W ={ eC W ,rC W}. In general, tests will be carried out for multiple purposes; for the moment, it will be observed that data are captured for training of models and for testing of models. For this reason, it is useful to divide data accordingly. Supposing that tests have been 5Hybrid models are also referred to in the literature as grey-box or data-augmented models. In the statistics literature, the addition of a data-based model in order to correct a physics-based model is commonly called model bias correction or model discrepancy correction; the most influential framework is probably that proposed in [7].
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