## 2 Materials and Methods
### Synthetic Training Data  
A prerequisite for the generation of synthetic training data is an existing FE model of the structure which is to be monitored. The FE model is utilized to simulate different load cases, from the results of which strain data is then extracted for its intended use as training data. In order to acquire strain data which covers a broad range of load cases, i.e., to achieve high variance training data, a large number of unique simulations is necessary. In the presented work this computational effort is reduced by utilizing a framework for physics-driven feature generation of strain data for the training of ML-based SHM methods, developed by [PERSON] et al. [21]. It uses the FE model and the simulation results of a single, specific load case to generate a large amount of strain data representing new hypothetical, but physically probable, load cases. Therefore, these load cases do not require a simulation, but are instead derived from this objective load case. The framework, illustrated in Figure 5 as a process flow, is closely linked to the FE model of the structure. In its current state, it requires that the area of the sensor grid is modeled as a submodel using the sub-structuring technique [21; 24]. The FE model of the complete structure represents the global model in this context. When the global model is simulated with an objective load case, the resulting displacements at the submodel edges become available. These then serve as boundary conditions for the simulation of only the submodel, but with refined geometry and mesh settings. The framework uses these objective load case displacements \(\mathbf{U}^{O}\) and the stiffness matrix \(\mathbf{K}\) of the submodel as inputs. The stiffness matrix is further reduced in size through a static condensation procedure. For this, it is assumed that no external forces are applied to the submodel, i.e., the relevant displacements result from the boundary conditions at the master nodes, introduced by the loaded global model [21]. The corresponding sub-matrices and sub-vectors are denoted by the subscript \(m\). In a following step, the eigenvectors \(\mathbf{\phi}_{m,i}\) of the resulting reduced stiffness matrix \(\mathbf{K}_{red}\) are calculated, where \(i=1\ldots N_{m}\) and \(N_{m}\) is the size of the quadratic matrix \(\mathbf{K}_{red}\). Those eigenvectors which best approximate the objective load case displacements \(\mathbf{U}_{m}^{O}\) are selected for further processing steps. The set of best  
Figure 4: Location of the different sized damages introduced to the spoiler demonstrator in relation to the sensor grid. The position and size of both damages are illustrated by the red, dash-lined circles. fitting eigenvectors is denoted as \(E_{B}\). A linear combination of all eigenvectors with the corresponding optimal coefficients \(x_{i}^{O}\), where \(i\in E_{B}\), is able to closely replicate the objective load case displacements. The coefficients \(x_{i}^{O}\) are determined for the selected eigenvectors through a least squares optimization with respect to the objective load case displacement:  
\[\min_{x}\ \left\|\ \mathbf{U}_{m}^{O}-\sum_{i\in E_{B}}\mathbf{\phi}_{m,i}x_{i}\right\| \tag{1}\]  
By applying the elements of the individual eigenvectors as displacements at the sub-model edges (instead of the objective load case displacements), strain data \(\epsilon_{i}\) acquired using each eigenvector is obtained. The superposition of this eigenvector strain data, additionally scaled with the optimal coefficients, again results in approximately the same strain data as it would result from a simulation with the objective load case displacements at the submodel edges.