A primal-dual active set strategy for unilateral non-linear dynamic contact problems of thin-walled structures

The efficient modeling of 3D contact problems is still a challenge in non-linear implicit structural analysis. Most of the existing contact algorithms use penalty methods to satisfy the contact constraints, which necessitates a user defined penalty parameter. As it is well known, the choice of this additional parameter is somehow arbitrary, problem dependent and influences the accuracy of the analysis. We use a primal-dual active set strategy [

1

], based on dual Lagrange multipliers [

4

] to handle the nonlinearity 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 get 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 our 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 to apply the contact conditions directly to nodes lying on the contact surface.

The discretization in time is done with the implicit Generalized Energy-Momentum Method [

2

]. To conserve the total energy within our contact framework, we follow an approach from Laursen and Love [

3

], who introduce 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.