The inverse Finite Element Method (iFEM) is a variationally-based technique proposed for solving the inverse problem of reconstructing the three-dimensional deformations of a beam or shell structure from experimentally measured surface strains (called shape sensing). The formulation is based on the minimization of a least-squares error functional that uses the complete set of strain measures consistent with linear, first-order shear-deformation theory. The main benefit of iFEM is that it is well-suited for C0-continuous displacement finite element discretizations, thus enabling the development of robust algorithms for application to complex civil and aeronautical structures. Existing iFEM works can be broadly classified into 1D iFEM for beams or frame structures, and 2D iFEM for plate or shell structures.
1D iFEM Formulation
The 1D formulation is based on the kinematic assumptions of classical Euler-Bernoulli beam theory or the first-order Timoshenko theory. It determines the displacement field by solving a least-squares error functional defined between certain ‘analytical’ and ‘experimental’ sectional strains representing the beam’s axial, bending, transverse shear and torsional deformation. Research on 1D iFEM has led to the development of various inverse beam elements, some of which have been experimentally for the displacement reconstruction of a thin-walled beam (as shown in the images).
2D iFEM Formulation
The 2D formulation is based on the kinematic assumptions of Mindlin theory, where the strain measures that make up the least-square error functional represent the membrane, curvature, and transverse shear strains defined in the shell midplane. A wide repository of inverse shell elements have been developed, like the three-node triangle, four-quadrilateral, and eight-node curved shell elements. An experimental application of the four-node shell for the displacement reconstruction of a stiffened aluminium panel is shown in the images below.
Hybrid Formulation
Latest research efforts are aimed at combining both 1D and 2D approaches using a hybrid scheme to improve accuracy, reduce the number of strain sensors required, and lower computation effort. The images below are of a composite stiffened panel designed and manufactured specifically to experimentally test this new scheme. Work is in progress and new updates and results should follow soon.
References
[1] A. Tessler, J. L. Spangler. A least-squares variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells. Comput. Methods Appl. Mech. Engrg. 194 (2005) 327–339.
[2] Gherlone, M.; Cerracchio, P.; Mattone, M.; Di Sciuva, M.; Tessler, A. Shape sensing of 3D frame structures using an inverse Finite Element Method. International Journal of Solids and Structures 2012, 49, 3100–3112.