Automatic time stepping algorithms for implicit numerical simulations of blade/casing interactions.

An automatic time stepping algorithm for non-linear problems, solved by implicit schemes, is presented. The time step computation is based on the estimation of an integration error calculated from the acceleration difference. It is normalised according to the size of the problem and the integration parameters. This time step control algorithm modifies the time step size only if the problem has a long time physical change. Additionally, the Hessian matrix can be kept constant for several iterations, even though the problem is non-linear. A criterion selecting if the Hessian matrix must be calculated or not is developed. Finally, a criterion of iterations divergence is also proposed. It avoids the determination, by the user, of a maximal iteration number. This minimises the total number of iterations, and thus the computation cost. Industrial numerical examples are presented that demonstrate the performances (precision and computational cost) of the algorithms.