## 3 Numerical Studies
### iFEM Studies Involving Uniaxial Strain Data
Figure 8: Fiber arrangements investigated: (**a**) Configuration-1 (with a wave-like fiber pattern, referred to as ‘Unitcell-1’, repeated across multiple elements), and (**b**) Configuration-2 (with a strain rosette fiber pattern, referred to as ‘Unitcell-2’, repeated across multiple elements). Figure 10: In-situ strain measurements and their locations corresponding to: (**a**) Unitcell-1, and (**b**) Unitcell-2. Figure 9: Unitcells corresponding to the fiber arrangements: (**a**) Unitcell-1 (single fiber arrangement measuring a mixture of \(\varepsilon_{x}\) and \(\varepsilon_{y}\) strains within each element), and (**b**) Unitcell-2 (dual fiber arrangement measuring \(\varepsilon_{x},\ \varepsilon_{y}\) and \(\gamma_{xy}\) strains at the centroid of each element). The SEA is a finite element (FE) based method that has been used as an FE post-processing tool for stress or strain'recovery' (improved prediction) and error estimation. It is a variational based approach where the structural domain is discretized using smoothing elements. The problem is solved by minimizing a discrete least-squares error functional enforcing the continuity of the strains and their derivatives. Thus, the SEA recovers C\({}^{1}\)-continuous strains with C\({}^{0}\)-continuous derivatives. As the present work focused on a 1D formulation of the SEA, a two-node linear smoothing element with quadratic interpolation of strains and linear interpolation of the strain derivatives is used. The accuracy of the smoothed, SEA-generated strain data depended on the interpolation order of the smoothing element used and the complexity of the plate's strain field. The SEA is expected to provide high accuracy results for an undamaged plate under uniform biaxial loading as the in-plane normal strains had uniform distributions, and the contribution of in-plane shear strain is negligible. For a damaged plate, recovering the strain field due to the crack is more challenging. In the presented context, however, the role of the SEA-based strain smoothing is to provide relatively accurate strain fields along the boundary of the plate. In addition to the in-situ measured strains, these smoothed strains would become input strains in the iFEM analysis. Figure 11 shows the use of the SEA on the strain data set of Unitcell-2. Along the grid lines, the SEA is used to interpolate existing FE strain data. Each grid line is discretized into a series of 1D smoothing elements, with at least one FE strain data point per element. The 1D SEA is used to smooth each in-plane strain component (\(\varepsilon_{x}\), \(\varepsilon_{y}\) and \(\gamma_{xy}\)) individually. At the end of the smoothing procedure, each element of the grid would have tri-axial strain data at the element centroid. A similar SEA approach is used for Unitcell-1 as well. The use of fiber optics enabled strain measurements along long one-dimensional sensor paths, making it suitable for the 1D SEA methodology. However, different smoothing strategies, such as using 2D SEA elements or an alternate smoothing strategy, may also be used depending on the problem. Note that Unitcell-1 lacked shear strain measurements, potentially leading to an inaccurate iFEM reconstruction. This issue is resolved by assuming a constant value of in-plane shear strain for all elements along the grid lines. A few additional strain rosettes are introduced in Configuration-1 to calculate shear strains at some discrete locations, and their average is used to represent the constant shear strain value for all elements along  
Figure 11: Use of Smoothing Element Analysis (SEA) for the strain data set of Unitcell-2; the position of the smoothing lines and the smoothed strain data generated have been shown. the grid lines. Although theoretically incorrect, this assumption is used to improve the quality of strain data available for the iFEM reconstruction. As the presented problem explored the plate cases under uniform biaxial loading, this assumption is expected to be moderately successful.