# Contact dynamics with large deformations

#### Initial Situation

Contact problems with nonlinear material laws and Coulomb friction play important role in many engineering and technical applications. The main key is the construction of efficient algorithms for this highly nonlinear problem. To simulate such type of problems, nonconforming discretization techniques for the linear problem statement based on the Lagrange multipliers with dual shape functions were developed within the leaked Collaborative Research Center 404 at the University of Stuttgart In the bygone years. Optimal a priori error estimates were established for the resulting weak variational formulations. Furthermore, efficient algorithms to solve the arising nonlinear algebaic system based on semi-smooth Newton methods were developed. For the frictionless case these methods can be equivalently interpreted as primal-dual active set strategies. The main focus was the handling of the frictional problem with Coulomb's friction law in the three-dimensional setting. A semi-smooth Newton algorithm with a superlinear convergence behavior was developed for this highly nonlinear situation.

#### Current Results

We extend the above mentioned algorithms to contact dynamics in combination with nonlinear materials as neo-Hookean or more general Mooney-Rivlin laws. Since the contact constraints are treated by a semi-smooth Newton method it is quite natural to handle both nonlinearities, the contact and the material ones, in one iteration loop. We show, that for the formulation of the Newton system with respect to the new constrained basis, the same modifications as in the linear case have to be applied to the original system with respect to the standard basis. Thus, existing finite element codes can be easily generalized to our algorithms. For the dynamical prozess focus on energy-conserving time-discretization algorithms. We use the method by Gonzales as an algorithmic energy-conserving scheme. This methods conserves not only the energy but also the linear and the angular momentum. To avoid spurious osciallations in the Lagrange multiplier we use a sligthly modified mass matrix. This matrix can be interpreted as lumped version of the original one whereas the mass lumping is only applied at the possible contact region.

 Dynamical contact problem with neo-Hookean material and Coulomb friction

 Dynamical contact problem with neo-Hookean material: without friction (left) and Coulomb friction (right)