Self-adapting time integration management in crash-worthiness and sheet metal forming computations

Variable step strategies are especially well suited to deal with problems characterized by high non-linearity and contact/impact, and resolved with an implicit scheme. Both phenomena are typical of dynamic simulations of contact-impact problems, as well as sheet metal forming. Constant step size strategies do not give a satisfactory answer for this kind of problem, 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 adaptation 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 (after contact, or during sheet metal forming) to very short (during contact-impact). thus ensuring precision while keeping the computation cost to a minimum. Furthermore, we will see that this strategy can be used in explicit schemes. Additionally, the proposed algorithm automatically takes decisions regarding the necessity of updating the tangent matrix or stopping the iterations, further reducing the computational costs especially, when the Augmented Lagrangian method is used. As an illustration of the capabilities of this algorithm, several numerical simulations (shock absorber devices for vehicle crash-worthiness or sheet metal forming) problems will be presented. Other simulations pertaining to the sheet metal forming for vehicle structures will also be investigated, thus demonstrating the versatility, the capabilities and the efficiency of the proposed strategy.