## 3 Numerical Studies
### iFEM Studies Involving Uniaxial Strain Data  
Fiber optic sensors are ideal candidates for strain field reconstruction using iFEM as they can provide a dense set of strain measurements over the fiber length. A typical strain rosette measurement system comprises sensors attached via a cable network to a Data Acquisition (DAQ) system. The measurements are processed into viable information and are subsequently stored and analyzed for decision making. In comparison, the cable network of a fiber optic system is composed entirely of the fiber with strain measurements all along the fiber length. The instrumentation systems used for fiber optic sensors depend on the specific sensing technology used [10]. A distributed fiber optic system uses an interrogator to transmit optical pulses into the fiber and capture the backscattered light. The backscattered light is subsequently analyzed to obtain strain measurements along the fiber. In contrast to strain rosettes, fiber optic strain measurements are uniaxial, i.e., along the local fiber direction. Uniaxial strain measurements cannot provide all the components of in-plane strain (\(\varepsilon_{x},\ \varepsilon_{y},\gamma_{xy}\)) at a point, leading to inaccuracies in the iFEM reconstruction. The number and position of strain sensors used also affects the iFEM solution. Sensor configurations with an insufficient number of boundary elements with available strain data can lead to a singular system matrix and a breakdown of the iFEM solution. Similar is the case for configurations with discontinuous sensor patterns. Any proposed sensor configuration should avoid these conditions to ensure accurate iFEM results. In what follows, we describe efforts to reconstruct the strain field in a damaged plate using spatially distributed uniaxial fiber-optic strain sensors. The results are compared with the benchmark iFEM results of Section 3.1. As the location and orientation of the strain measurements depend on the fiber arrangement, different fiber patterns are investigated. The plate is assumed to be instrumented with one or two continuous fibers to recreate the sensor pattern of the benchmark model (Figure 3). The fiber arrangement should also maximize the quality and quantity of strain measurements, minimize fiber length, and avoid any self-intersections. Based on these requirements, two fiber arrangements are proposed (see Figure 8),  
* Configuration-1: a continuous non-intersecting fiber arranged as a wave on the plate. The fiber arrangement within an element is referred to as 'Unitcell-1' (Figure 9a), having the same pattern repeated across multiple elements. Strain measurements corresponding to Unitcell-1 are along the directions: \(\{0,\pm 90\}\), which is a mixture of \(\varepsilon_{x}\) and \(\varepsilon_{y}\) strain data at different 3 \(\times\) 3 Gauss integration points within an element. At most, three strain measurements are made per element. * Configuration-2: two continuous non-intersecting fibers are used to create a strain rosette within each element [35]. The fiber arrangement within an element is referred to as 'Unitcell-2' (Figure 9b), and the pattern is repeated across multiple elements. Strain measurements corresponding to Unitcell-2 are along directions: \(\{0,\pm 60\}\). These three measurements formed a strain rosette and are used to calculate the \(\varepsilon_{x},\ \varepsilon_{y}\) and \(\gamma_{xy}\) components of strain at the centroid of each element (Figure 9b). Strain measurements corresponding to Unicells-1 and 2 suffered from certain limitations. The fiber arrangements of Figure 8 are unable to replicate the benchmark pattern of Figure 3, resulting in certain boundary elements of the plate without any strain data (see Figure 10), potentially leading to a singular system matrix, **K**. This issue could be readily resolved by a pre-processing procedure that would enable these elements to be populated with strain data. For this purpose, a one-dimensional (1D) version of the Smoothing Element Analysis (SEA) [36] is used herein.