Shape reconstruction of plane beam with finite deformation based on absolute nodal coordinate formulation

Most of the existing researches on deformation reconstruction of flexible structures with finite deformation are only based on the geometric relationship between curvature and strain, which ignores the longitudinal deformation and the coupling effect of the longitudinal deformation and the bending deformation. In order to construct a more accurate deformation reconstruction method which can be extended with the help of existing mechanical tools, this paper takes the plane beam as the object, partially inherits inverse finite element method developed by Tessler A, and regards the deformation reconstruction problem of plane beam as a kind of numerical optimization problem. Firstly, by introducing the absolute nodal coordinate formulation (ANCF) into the description of mapping relationship between strain and displacement, an inverse gradient reduced ANCF plane beam element is derived. Secondly, the inverse ANCF element is modified to simplify the degree of freedom of nodes and ensure the C2 continuity at nodes by introducing the penalty function, which not only ensures the problem is well-posed, but also improves the accuracy of the final result. Finally, based on the inverse ANCF element, the Newton method is used to develop two types of algorithms for deformation reconstruction under different working conditions, one is the element-by-element algorithm and the other is the multi-element algorithm. The numerical simulation results show that the reconstruction relative error of this method is less than 1% under the condition of large deformation, and it still maintains high accuracy under the condition of few measuring points. The convergence and computational efficiency of the method are verified by numerical simulation example. © 2021, Chinese Journal of Theoretical and Applied Mechanics Press. All right reserved.