Unilateral non‐linear dynamic contact of thin‐walled structures using a primal‐dual active set strategy

Abstract

AbstractThe efficient modelling of three‐dimensional contact problems is still a challenge in non‐linear implicit structural analysis. We use a primal‐dual active set strategy (SIAM J. Optim. 2003; 13:865–888), based on dual Lagrange multipliers (SIAM J. Numer. Anal. 2000; 38:989–1012) to handle the non‐linearity of the contact conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to achieve a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way.For the application to thin‐walled structures we adapt a three‐dimensional non‐linear shell formulation including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface‐oriented shell element, which allows the application of contact conditions directly to nodes lying on the contact surface. Shell typical locking phenomena are treated with the enhanced‐assumed‐strain‐method and the assumed‐natural‐strain‐method.The discretization in time is done with the implicit Generalized‐α method (J. Appl. Mech. 1993; 60:371–375) and the Generalized Energy–Momentum Method (Comp. Methods Appl. Mech. Eng. 1999; 178:343–366) to compare the development of energies within a frictionless contact description.In order to conserve the total energy within the discretized frictionless contact framework, we follow an approach from Laursen and Love (Int. J. Numer. Methods Eng. 2002; 53:245–274), who introduced a discrete contact velocity to update the velocity field in a post‐processing step.Various examples show the good performance of the primal‐dual active set strategy applied to the implicit dynamic analysis of thin‐walled structures. Copyright © 2006 John Wiley & Sons, Ltd.