A non-consistent start-up procedure for contact problems with large load-steps

Following up to a previous investigation, this paper proposes a strategy to deal with frictionless contact problems involving large penetrations, in the context of the node-to-segment formulation and of the pen- alty method. The rationale is based on two main considerations: the first one is that, within an iteration scheme, the use of consistent linearization is only convenient when the field of the unknowns is suffi- ciently close to the solution point; the second one is that, if the order of magnitude of the maximum con- tact pressure can be estimated a priori, this information can be exploited to approach the solution in a faster and more reliable way. Beside being very effective in case of large initial penetrations, where con- vergence using the standard algorithms would either not be achieved at all, or achieved with a large num- ber of iterations, the strategy can also be used to increase the convergence rate in contact problems with penetrations of usual magnitude. The proposed strategy is based on a check of the nodal contact pressure, to select the technique that has to be used to perform each iteration. If the contact pressure is smaller than a predefined limit, the prob- lem is solved in the standard way, using consistent linearization and Newton’s method. When the contact pressure exceeds the limit, a modified method is used. This one is based on the enforcement of a contact pressure limit and on the use of a simplified secant stiffness, where the geometric stiffness term is dis- regarded. The strategy has to be integrated with a specific ‘‘safeguard algorithm’’ to guarantee conver- gence to the correct solution also in cases where the maximum contact pressure has been underestimated. Two alternative procedures for this purpose are proposed. The solution strategy can be used with any contact formulation. Here, the efficiency and stability of the strategy is illustrated within the framework of the well-known node-to-segment formulation with a 2D implementation. Several example problems are presented