Self-adapting time integration strategies for non-linear structural dynamics

Adaptive strategies are specially well suited to deal with problems characterized by high non-linearity and contact/impact. Constant step size strategies do not give a satisfactory answer for this kind of problems, since it is very difficult, if not impossible, for the user to find an appropriate time step that does not lead to divergence nor generate extremely costly computations. An automatic time stepping algorithm is proposed, which takes into account the recent history of accelerations in the (deformable) bodies under consideration. More precisely, the adaption algorithm is based on estimators of the integration error of the differential dynamic balance equations. This allows for adaptation of the step size to capture correctly the transient phenomena, with characteristic times which can range from relatively long (in regime) to very short (impact), thus ensuring precision while keeping the computation cost to a minimum. This algorithm is applicable both to implicit and explicit time integration schemes. Additionally, in the case of implicit schemes, the proposed algorithm automatically takes decisions regarding the necessity of updating the tangent matrix or stopping the iterations, further reducing the computational cost. This is specially true when the selective updating scheme is combined with the augmented Lagrangian procedure for the treatment of contact. As an illustration, numerical simulations of the buckling of an automobile stringer under crash impact are presented, demonstrating the versatility, the capabilities and the efficiency of the proposed strategy.