Shape sensing techniques allow for the time-efficient reconstruction of displacements based on measured strain data. There are technical applications, where the structure of interest is deformed in the geometrically non-linear domain. In aeronautics, this is the case for high-aspect-ratio wings, which are more frequently found in future designs. Only shape sensing methods that specifically take the non-linearity into account, can deliver appropriate displacement estimates for such application. A shape sensing method based on the linear modal approach can be utilised incrementally to capture the geometric non-linearity; it has therefore been denoted incremental modal method (IMM). This paper presents analytical relations for the uncertainty propagation for the various input quantities of the method, specifically strain mode shapes, displacement mode shapes, and measured strain. Deterministic shape sensing and uncertainty propagation are demonstrated using data obtained with a finite element model of a high-aspect-ratio wing experiencing geometric non-linear deflections in flapwise bending. Virtual strain and acceleration sensors are assumed for this setup, imitating the instrumentation conceivable for experimental work. The results obtained by analytical propagation are compared to Monte Carlo simulations for the purpose of validation. The derived propagation formulas make it possible to follow the evolution of the uncertainties over the number of increments. Given that the variability of the input quantities is known, the number of increments that minimise uncertainties can be determined for a model-free application of the shape sensing. Together with the deterministic estimates provided by an FE model, it is possible to determine the ideal number of increments for a specific shape sensing application in the geometrically non-linear domain.

Shape sensing techniques enable the real-time reconstruction of wing displacements based on measured strain. As wing designs become more flexible, they may at some point exhibit geometric non-linear deformation. This eventually leads to erroneous displacement estimates if applied methods rely on the assumption of linear deformation. In this research, the Incremental Modal Method (IMM) is presented which accounts for the change of the employed mode shapes due to structural deformation. The method is applied on the finite element model of a high aspect ratio wing undergoing geometric non-linear deflections in flap-wise bending. In the process, a setup of virtual strain sensors is presumed which is representative as instrumentation in experiments. Along a reference line of the wing, the displacement estimates of IMM are compared to results obtained using the Modal Rotation Method (MRM), another shape sensing scheme recently developed for the non-linear regime. For the chosen segmentation and virtual instrumentation of the investigated wing, IMM proves to be a promising candidate for realtime displacement reconstruction in experiments, provided that mode shapes in intermediate deflected states can be determined.

The reconstruction of the deformations from strain measurements is usually called shape sensing and is of interest in the field of structural health monitoring. In this paper the problem is studied with a limited amount of strain data since for real applications and an on-line monitoring it might be unfeasible to employ a large amount of sensors. In this context the deformations are computed with the inverse Finite Element Method (iFEM) taking advantage from a preliminary strain pre-extrapolation done with the Smoothing Element Analysis (SEA). The results are then compared with another common shape sensing method, the so-called Modal Method (MM), using as study case a composite wing-like structure. Noise and uncertainties and their influences on the results are also assessed. The outcome obtained points out that iFEM is largely improved after the preliminary strain pre-extrapolation. However, for this study case, the Modal Method performs better despite the intrinsic disadvantage in requiring the material properties of the structure (compared to iFEM, where no constitutive relation is needed), which affects the results in case uncertainties are present in the system.