Meshing strategies in the absolute nodal coordinate formulation-based Euler-Bernoulli beam elements

The absolute nodal coordinate formulation (ANCF) is a finite-element-based formulation developed for large deformation analysis in multibody applications. During the course of progress of the ANCF, a number of beam, plate and shell elements are introduced. Numerous different kinematic and strain energy definitions have been considered. However, the convergence behavior of ANCF-based beam elements has been investigated in the past primarily only with uniform mesh refinement. The objective of this paper is to study numerically the effects of different mesh refinement strategies within an Euler-Bernoulli beam-type problem solved with two-dimensional ANCF-based beam elements. To this end, h- and p-refinement meshing strategies are implemented for two-dimensional ANCF beam elements. The p-refinement is achieved by adding internal degrees of freedom to the elements. In addition, another p-refinement strategy is studied that employs higher-order derivatives as nodal coordinates. The rates of convergence for the different meshing strategies are calculated based on the known iteratively solved extensible elastica analytical solution. The numerical results indicate that the p-refinement strategy, with increasing number of higher-order derivatives at the element interfaces, in conjunction with h-refinement improves the rate of convergence. For the studied numerical example, this mesh strategy leads to better absolute error of displacement convergence than the uniform h-refinement or the p-refinement with internal nodes. The improvement in rate of convergence results from the use of higher-order derivatives that impose continuity to the bending and axial deformations across the elements, which may smooth the oscillations of bending and the axial strain fields.