A New Algorithm for the Evaluation of the Elastic Forces in the Absolute Nodal Coordinate Formulation

Abstract

The use of the Absolute Nodal Coordinate Formulation (ANCF) to model flexible multibody systems leads to a system of equations in which the mass matrix is constant. Among other advantages of the ANCF, as the simplicity of the formulation of joint constraint, the constancy of the mass matrix is specially useful since it implies that the mass matrix is calculated once before the analysis and stored as a invariant quantity. On the other hand, the evaluation of the elastic forces has been considered as a difficult task since they result in large nonlinear functions. Both linear and nonlinear formulations based on either Continuum Mechanics or Timoshenko/Euler-Bernoulli approaches have been supposed to lead to the same level of complexity of the elastic force function. The evaluation of the elastic forces when using a nonlinear Continuum Mechanics approach requires, as presented in the literature, the integration over the volume of the element. However, it can be shown that this integration can be avoided for every evaluation since there is a closed form of the elastic force function that can be obtained before the simulation. Finite elements based on ANCF have such a large number of nodal coordinate (24 and 48 for three-dimensional beams and plates, respectively) that the search for a closed form of the elastic force function becomes difficult and costly. This paper presents a set of constant matrices that are integrated in advance (before simulation) and allows the evaluation of the elastic forces without the integration over the volume of the element suggested in the literature. In addition, the amount of data that has to be stored in order to evaluate the function is reduced. These constant matrices together with the mass matrix complete the set of invariants of the ANCF that allows an acceleration of simulations. Numerical simulations comparing CPU time and number of arithmetic operations carried out when these invariant matrices are used with those carried out when they are not used, support the convenience of their use.