Shape sensing of 3D frame structures using an inverse Finite Element Method

A robust and efficient computational method for reconstructing the elastodynamic structural response of truss, beam, and frame structures, using measured surface-strain data, is presented. Known as “shape sensing”, this inverse problem has important implications for real-time actuation and control of smart structures, and for monitoring of structural integrity. The present formulation, based on the inverse Finite Element Method (iFEM), uses a least-squares variational principle involving section strains (also known as strain measures) of Timoshenko theory for stretching, torsion, bending, and transverse shear. The present iFEM methodology is based on strain–displacement relations only, without invoking force equilibrium. Consequently, both static and time-varying displacement fields can be reconstructed without the knowledge of material properties, applied loading, or damping characteristics. Two finite elements capable of modeling frame structures are derived using interdependent interpolations, in which interior degrees of freedom are condensed out at the element level. In addition, relationships between the order of kinematic-element interpolations and the number of required strain gauges are established. Several example problems involving cantilevered beams and three-dimensional frame structures undergoing static and dynamic response are discussed. To simulate experimentally measured strains and to establish reference displacements, high-fidelity MSC/NASTRAN finite element analyses are performed. Furthermore, numerically simulated measurement errors, based on Gaussian distribution, are also considered in order to verify the stability and robustness of the methodology. The iFEM solution accuracy is examined with respect to various levels of discretization and the number of strain gauges.