Browsing by Author "Hager, Corinna"

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Robust numerical algorithms for dynamic frictional contact problems with different time and space scales
(2010) Hager, Corinna; Wohlmuth, Barbara (Prof. Dr.)
In many technical and engineering applications, numerical simulation is becoming more and more important for the design of products or the optimization of industrial production lines. However, the simulation of complex processes like the forming of sheet metal or the rolling of a car tire is still a very challenging task, as nonlinear elastic or elastoplastic material behaviour needs to be combined with frictional contact and dynamic effects. In addition, these processes often feature a small mobile contact zone which needs to be resolved very accurately to get a good picture of the evolution of the contact stress. In order to be able to perform an accurate simulation of such intricate systems, there is a huge demand for a robust numerical scheme that combines a suitable multiscale discretization of the geometry with an efficient solution algorithm capable of dealing with the material and contact nonlinearities. The aim of this thesis is to design such an algorithm by combining several different methods which are described in the following. Our main field of application is structural mechanics. Here, we base the implementation on finite element methods in space and implicit finite difference schemes in time. The conditions for both plasticity and frictional contact are given in terms of a set of local inequality constraints which are formulated by introducing additional inner or dual degrees of freedom. As the meshes are generally non-matching at the contact interface, we employ mortar techniques to incorporate the contact constraints in a variationally consistent way. By using biorthogonal basis functions for the discrete multiplier space, the contact conditions can be enforced node-wise, and a two-body contact problem can be solved in the same way as a one-body problem. The next step in the construction of an efficient solution algorithm is to reformulate the local inequality conditions for plasticity and contact in terms of nondifferentiable equalities. These nonlinear complementarity functions can be combined with the equations for the bulk material to form a set of nonlinear semismooth equations which are then solved by means of a generalized form of the Newton method. Due to the local structure of the inequality constraints, this iterative scheme can be implemented as an active set strategy where the active sets are updated in each Newton iteration. Further, the additional dual degrees of freedom can easily be eliminated using local static condensation. We remark that the well-known radial return method is a special case of this general framework if the plastic hardening laws are linear. However, the convergence properties of the Newton iteration strongly depend on the choice of the NCP function. In this context, we show that the function corresponding to the radial return method is not optimal, and we present a family of modified NCP functions which allow for better convergence results. Another important issue for the robust simulation of dynamic contact problems is related to the inertia terms. If standard time discretization schemes like the trapezoidal rule are used, the contact stress often shows spurious oscillations in time that become worse when the time step is refined. In order to avoid this effect, we employ a modified mass matrix where no mass is associated with the contact nodes. By this, the original semi-discrete system decouples into an algebraic equation in time for the contact nodes and an ordinary differential equation in time for the other nodes. This in turn leads to much smoother results for the contact stress. We present an efficient way of obtaining the modified mass matrix by means of non-standard quadrature formulas used only for the elements near the contact boundary. Furthermore, we prove optimal a priori error estimates for the modified semi-discrete as well as for the fully discrete system, provided that the contact stress is given and that the solution is sufficiently regular. In the last part of the thesis, we deal with the situation that the body features fine local structures near the contact zone by incorporating the multiscale aspect into the discretization. For this, the domain is decomposed into several overlapping subdomains which have different grid spacing; one global mesh that does not resolve the details and overlapping local patches with a fine triangulation. Based on a surface coupling by means of the mortar method, we construct an iterative solution scheme for the coupled problem whose convergence rate is bounded independently of the mesh size or the Lame parameters. Finally, we employ the subdomain decomposition for introducing a finer time step size on the patch. We present suitable interface conditions with no numerical dissipation and prove a priori error estimates with respect to time for the resulting coupled energy-conserving system. The latter can efficiently be solved by the iterative procedure presented before.