State of the Art of ANCF Elements Regarding Geometric Description, Interpolation Strategies, Definition of Elastic Forces, Validation and the Locking Phenomenon in Comparison with Proposed Beam Finite Elements

Abstract

The focus of the present article lies on new enhanced beam finite element formulations in the absolute nodal coordinate formulation (ANCF) and its embedding in the available formulations in the literature. The ANCF has been developed in the past for the modeling of large deformations in multibody dynamics problems. In contrast to the classical nonlinear beam finite elements in literature, the ANCF does not use rotational degrees of freedom, but slope vectors for the parameterization of the orientation of the cross section. This leads to several advantages compared to the classical formulations, e.g. ANCF elements do not necessarily suffer from singularities emerging from the parameterization of rotations. In the classical large rotation vector formulation, the mass matrix is not constant with respect to the generalized coordinates. In the case of ANCF elements, a constant mass matrix follows, which is advantageous in dynamic analysis. In the standard geometrically exact formulation, the parameterization of rotations leads to a nonlinear term containing quadratic velocities. The so-called quadratic velocity vector vanishes for ANCF elements which is advantageous in dynamic applications. In the present article, existing beam finite element formulations are analyzed and improved to derive formulations which are able to solve industrial applications with high performance, efficiency and accuracy. The interest lies especially on finite element formulations for multibody dynamics systems that are capable of large deformations in order to derive accurate solutions of nonlinear engineering and research problems. Existing shear deformable ANCF beam finite elements show an overly stiff behavior caused by the locking phenomenon. Existing locking problems are discussed in this article and avoided in the proposed elements in order to gain accurate solutions. The state of the art of ANCF elements in literature is reviewed including the basic description of the kinematics, interpolation strategies and definition of elastic forces, but also problems and known disadvantages arising in the existing elements, as e.g. the locking phenomenon. Regarding the description of the elastic forces, the present article shows the two standard approaches in literature as well as new enhanced formulations to avoid locking: a continuum mechanics based formulation for the elastic forces with elimination of Poisson and shear locking, and an extended hybrid structural mechanics based formulation for the elastic forces including a term for penalizing shear and cross section deformation. The definition of the element kinematics and interpolation strategies in existing elements are discussed and compared to the according descriptions of the proposed elements. A comparison of the solution of the proposed finite elements to analytical solutions in the literature and to the solution retrieved from commercial finite element software have shown high accuracy and high order of convergence. The speed of convergence is evaluated regarding different interpolation strategies and different formulations for the elastic forces. The investigations show that the proposed elements have high potential for simulation of geometrically nonlinear problems arising from real-life multibody applications and therefore are highly competitive with existing elements in commercial finite element codes. It has to be mentioned here that most of the studies on nonlinear elements based on the ANCF in literature use linear constitutive laws. Regarding many applications in which geometric and material nonlinearities arise, elastic material models are not sufficient to represent the real problem accurately. For this reason, an extension of the material model is necessary in order to fulfill the requirements of current but also of future materials arising in engineering or research. An overview of existing nonlinear material models in literature can be found in the “Appendix”.