Title: Inverse FEM for Full-Field Reconstruction of Elastic Deformations in Shear Deformable Plates and Shells Authors: Alexander Tessler

The inverse problem of real-time reconstruction of full-field structural displacements is addressed through the application of a new variational formulation leading to versatile, robust and computationally efficient inverse shell finite element analysis. Utilizing surface strain measurements from strain sensors mounted on the load-carrying structural components, the methodology enables accurate computations of the three-dimensional displacement field. This high fidelity computational technology is essential for providing feedback to the actuation and control systems of the next generation of aerospace vehicles. INTRODUCTION Real-time reconstruction of full-field structural displacements is seen as enabling technology for providing feedback to the actuation and control systems of the next generation of aerospace vehicles with morphed-wing architecture. To facilitate such capabilities, load-carrying structural members will be instrumented with a network of strain sensors, e.g., fiber optic sensors with Bragg gratings. Reconstruction of a displacement vector at every material point of the structure from a set of discrete strain measurements constitutes an inverse mathematical problem. Inverse problems are ill posed in the sense that they do not necessarily satisfy conditions of existence, uniqueness, and stability. For this class of mathematical problems that use experimentally measured data known only approximately and containing random error, Tikhonov and Arsenin [1] formulated a general method for constructing approximate solutions that are stable under small changes in the measured data. Their approach is based upon the fundamental concept of a regularizing operator. Recently, Shkarayev et al. [2-3], Bogert et al. [4], and Tessler and Spangler [5], using different least-squares approaches, focused _____________ Alexander Tessler, NASA Langley Research Center, Hampton, Virginia 23681, U.S.A. [email protected] Jan L. Spangler, Lockheed Martin Aeronautics Company, NASA Langley Research Center, Hampton, Virginia 23681, U.S.A. on the inverse problem of reconstructing the three-dimensional displacements in plate and shell aerospace structures from in-situ strain-sensor measurements. This paper presents an inverse finite element method (iFEM) based on the TesslerSpangler [5] least-squares variational formulation. The error functional uses the leastsquares-difference terms comprised of the Mindlin-theory strain measures that are expressed in terms of the displacements and the corresponding strain measures computed from the experimental strains. A penalty-parameter controlled regularization term enforces appropriate constraint conditions on the transverse shear strains. By virtue of these assumptions, all strain compatibility relations are explicitly satisfied. The inverse formulation does not use elastic or inertial material properties. A three-node, inverse-shell element is developed having six conventional degreesof-freedom at each node, i.e., three displacements and three rotations. The kinematic variables are interpolated using the lowest-order anisoparametric € C0 -continuous functions, i.e., linear in-plane displacements and bending rotations, and a constrainedtype quadratic deflection. These functions were adopted from an earlier Mindlin shell element formulation [6-7]. A computational example is presented for a statically loaded cantilevered plate for which experimentally measured strains have been obtained in a structures laboratory. Application of iFEM is demonstrated on this problem, and comparisons with the measured deflections and those obtained using the direct FEM are discussed. THEORETICAL FOUNDATION The deformations of an inverse, three-node flat shell element, herein denoted as iMIN3, are fully defined by the three components of the displacement vector, in accordance with Mindlin theory (refer to Figure 1): Figure 1. Three-node, inverse shell element, iMIN3.