iFEM benchmark problems for solid elements

Structural health monitoring (SHM) is a growing field of research, as it has the potential to simultaneously improve the reliability of structures and reduce their maintenance cost. SHM requires accurate stress and strain information, preferably for the entire structure. Unfortunately, it is often infeasible to instrument every part of the structure, making it necessary to estimate the stress and strain fields based on data from a limited number of sensors. One promising technique for making this estimate is the inverse finite element method (iFEM), which can be applied to any combination of geometry and loading conditions. In addition, it can also process several different types of sensor data. In this study, benchmark problems based on the MacNeal and Harder linear elastic problem set for FEM algorithms were extended to test the accuracy of iFEM algorithms. As the benchmarks use linear elastic materials, small displacements and strains, the iFEM implementation was also limited to these conditions. Accurate iFEM estimates can be obtained for the benchmark problems for which accurate FEM solutions can be obtained with solid elements, specifically 3-dimensional 20 node hexahedral elements with reduced integration (C3D20R), based on either displacement sensors, strain sensors, or both combined, and provided that a sufficient number of sensors is used. The iFEM algorithms generally produce more accurate estimates of displacements than of strains. The addition of Tikhonov regularization does not result in a significant increase in accuracy for either the displacement or strain distribution estimates and can even deteriorate the results in certain cases.