Inverse methods for health monitoring and shape sensing of aerospace structures

Shape sensing, i.e., reconstruction of the displacement field of a structure from surface-measured strains, has relevant implications for Structural Health Monitoring (SHM), as well as for control and actuation of smart structures. The knowledge of the full-field displacements implies that other essential response quantities such as stresses can be assessed, thus enabling real-time damage predictions by means of appropriate failure criteria. The inverse Finite Element Method (iFEM) is a shape-sensing methodology shown to be fast, accurate, and robust. In the present thesis, the general framework of iFEM, i.e., least-square variational statement and displacement-based finite element approximation, has been adopted to develop efficient and robust shape-sensing techniques, with focus on thin-walled structures, beam and frame structures and multilayered, composite and sandwich structures. The theoretical framework of the iFEM and its original formulation for plates and shell structures, developed on the basis of the First-order Shear Deformation Theory (FSDT), are firstly summarized. Then, the variational principle for three-dimensional frame structures, based on Timoshenko beam theory, is reviewed. This is followed by a discussion of two C0-continuous, beam inverse elements, a 0th-order element, having constant shear-section strain along the element length, and a newly formulated 1st-order element, having linear shear section strain. The formulation, originally proposed for circular cross-section beams, is extended to rectangular beams. In addition, relationships between the order of kinematic-element interpolations and the number of required strain gauges are established. A new formulation for the shape sensing of multilayered composite and sandwich structures possessing a high degree of anisotropy and heterogeneity is herein presented. The new formulation employs the iFEM as a general framework and the Refined Zigzag Theory (RZT) as the underlying plate theory. A three-node inverse plate finite element is formulated, enabling robust and efficient modeling of arbitraty plate structures. A methodology to infer applied loads from iFEM-predicted displacements is proposed. The evaluated loads can then be used within a direct finite element analysis to evaluate high-fidelity finite element stress solution. Several example problems involving thin-walled structures, three-dimensional frame structures and multilayered, composite and sandwich structures, undergoing static and dynamic response, as well as thermal deformation, are discussed. To simulate experimentally measured strains and to establish reference displacements, high-fidelity MSC/NASTRAN finite element analyses are performed. For the sandwich plate problems, exact elasticity solution or direct RZT solution are employed to the same purpose. To enable the analysis of partially instrumented structures considered in the present example problems, a method to select optimal weights to be used in the weighted least-square formulation for plates with sparse strain data is proposed. Numerically simulated measurement errors, based on Gaussian distribution, are also considered in order to verify the stability and robustness of the iFEM methodology. Furthermore, a comparative study involving iFEM and other shape sensing methodologies existing in the literature is discussed. An experimental test campaign is presented aiming to demonstrate that iFEM for beam and plate structures is reliable when experimentally measured strains are used as input data. The accuracy and robustness of iFEM with respect to unavoidable measurement errors, due to strain sensor locations, measurement systems, and geometry imperfections, are demonstrated for both static and dynamic loadings. The proposed methodology based on the iFEM is computationally efficient and accurate in reconstructing both static and time-varying displacement fields. Since only strain-displacement relationships are used, all type of structural deformation can be modeled without the knowledge of material properties, applied loading, or damping characteristics. For this reasons, the present shape sensing techniques are suitable for real-time Structural Health Monitoring.