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

The 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 struct ures 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.