Stable time step estimates for NURBS-based explicit dynamics

• We propose a study of time step estimates for explicit dynamics.
• We examine the influence of the order and the regularity of the shape functions on the estimates.
• The accuracy of newly proposed estimates is assessed on 1D and 2D problems.
• Elements on the boundary penalize the time step with smooth B-splines.
• We propose to unclamp the knot vector to increase the size of the critical time step.

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.