Absolute Nodal Coordinate Formulation-Based Shape Sensing Approach for Large Deformation: Plane Beam

Abstract

The inverse finite element method (IFEM) is currently one of the most studied methods in the field of shape sensing, in other words, the reconstruction of the displacement field of a structure from discrete strain measures. The current research is still insufficient in applying IFEM to flexible structures undergoing large deformation that are in increasing demand, especially in terms of computational efficiency. Hence, an element-by-element IFEM approach based on absolute nodal coordinate formulation (ANCF) is developed in the paper. Taking the plane beam as the object, a class of gradient-deficient ANCF plane beam element is introduced to provide a concise nonlinear nodal displacement/strain relationship. Similar to IFEM, the inverse ANCF (IANCF) plane beam element is obtained in the form of least-square formulation, which means IANCF describes the deformation reconstruction problem as a nonlinear optimization problem. Because the computational complexity of solving nonlinear optimization problems increases rapidly with the increase of the number of decision variables, an element-by-element solution algorithm that solves each element relatively independently is adopted, and the explicit iterative formula is given by the Newton method. Besides, a curvature continuity constraint is introduced to improve the well-posed-ness of this problem and the smoothness of the reconstructed shape. Through numerical analysis, IANCF exhibits remarkable accuracy in various deformation degrees and its insensitivity to the weight factors inherited from IFEM. In the experiment conducted with surface-mounted distributed optical fiber sensors, the effectiveness of IANCF for practical structures is verified.