Stable time step estimates for NURBS-based explicit dynamics

Abstract

Automobile crashworthiness is a complex application for numerical methods in dynamics of structures which includes many high non-linearities. Explicit techniques are widely used for structural dynamics dealing with difficult and large problems that prevent the use of implicit methods. We propose, in this paper, a deep study of the stable time step, which guarantees the stability of the method, and its estimates, for one-dimensional and two-dimensional problems. Element and nodal time steps are presented and adapted to highly regular B-spline and NURBS functions, in the context of isogeometric analysis. The size of the proposed stable time estimates benefits from the properties of regularity and extended support of the basis. Their performance is assessed and compared in several examples, with an arbitrary mesh, uniform or non-uniform, and considering polynomial orders from one to five. The smoothness and order of the polynomials have a significant effect on the stable time step and its estimates. Several lumping schemes of the mass matrix are presented and their accuracy is assessed.