## 3 Numerical Studies
### Benchmark iFEM Results  
Before investigating the effects of uniaxial strain data, iFEM reconstruction using strain data measured by a dense grid of strain rosettes is used to establish a set of benchmark iFEM results. These results corresponded to a high accuracy iFEM reconstruction and are used as reference results for forthcoming comparisons. As the plate is under biaxial loading, in-plane strains are uniform across the plate thickness. Hence, only strain measurements made either on the top or bottom surface of the plate are required for calculating the experimental strain measures. In the present work, strain measurements made on the top surface of the plate are used. Membrane strain measures, \(\mathbf{e}^{\epsilon}\) could be calculated using  
Figure 3: Sensor configuration of the benchmark inverse Finite Element Method (iFEM) model: **(a)** strain rosette grid (each point represents a strain rosette and the sensors divide the plate into nine square cells), and **(b)** magnified view of boundary elements with a maximum of five, and a minimum of three rosettes per element. Equation (6)), while the bending strain measures, \(\mathbf{k}^{\epsilon}\) are equal to zero (Equation (7)). The sensor configuration used for the benchmark results is shown in Figure 2(a), where each point designates a strain rosette, and a total of 440 strain rosettes are used. Such a sensor grid divided the plate into nine square cells, where strain measurements are performed only along the cell boundaries. At each sensor position, the \(\varepsilon_{x}\), \(\varepsilon_{y}\) and \(\gamma_{xy}\) components of strain are measured. Figure 2(b) shows a magnified view of the sensor positions within some boundary elements. Depending on element position, strain measurements are available in at least three and at most five points within each element. For this analysis, the integrals of Equation (12) are integrated numerically using the 3 \(\times\) 3 Gauss scheme. Initially, the benchmark iFEM model is used to reconstruct the strain field of an undamaged plate under biaxial loading. The FE and benchmark iFEM results for an undamaged plate are shown in Figure 4. As the plate is under uniform biaxial loading, the in-plane normal strains are uniform, and the shear strain is negligible. Hence, \(\varepsilon_{p}\) is a constant over the plate, as evident from the FE results (see Figure 3(a)). The contour plot of iFEM results also showed a constant value of \(\varepsilon_{p}\), with minor variations of the order of \(10^{-8}\) (which are negligible). Accurate reconstruction of the undamaged far-field strains is essential for the damage detection strategy presented, as it acted as the baseline strain field of the structure. The present results clearly demonstrated the accuracy of the benchmark iFEM model in reconstructing the undamaged strain field. Next, the reconstruction accuracy of a damaged strain field is investigated. The FE and benchmark iFEM results for Damage Case-1 are shown in Figure 5. Both plots showed a strain concentration in the vicinity of the crack, with the iFEM results having a more diffuse distribution and a lower magnitude of \(\varepsilon_{p}\). This is due to the absence of strain measurements close to the crack tip. The iFEM reconstruction utilized strains measured by the grid of sensors around the crack (see Figure 3), and the sensors measured a lower strain magnitude than at the crack tip. This resulted in a lower magnitude and a more diffuse strain distribution in the iFEM results. Nevertheless, for damage detection, the iFEM results of Figure 2(b) are promising as they successfully reconstructed a strain concentration at the damage site. The FE and benchmark iFEM results for Damage Case-2 are shown in Figure 6. In this case, the magnitude of \(\varepsilon_{p}\) is smaller due to the smaller damage size. The FE results are highly localized, and the iFEM results showed a more diffuse \(\varepsilon_{p}\) distribution at the damage site.