Abstract
The present thesis is concerned with the numerical simulation of contact problems of thin-walled structures using the finite element method. A mortar-based contact formulation is presented and combined with suitable strategies for the discretization in space and time.
In view of a useful coupling with the element independent contact description, a trilinear surface oriented hybrid shell element is derived on the basis of the 7-parameter shell model by Büchter and Ramm (1992). Additionally, a trilinear geometric nonlinear brick element based on the principle of Hu-Washizu is devised. Numerical tests demonstrate the performance of both element formulations.
For the discretization in time, two implicit time integration algorithms are used. In addition to the existing "Generalized-α"-Method especially the "Generalized-Energy-Momentum-Method" is applied. The latter is proven to be unconditionally stable in all performed numerical analyses.
The essential part of this thesis is the extension of the mortar contact formulation presented by Hüeber and Wohlmuth (2005) to the geometrically nonlinear regime. Introducing continuously approximated Lagrange Multipliers, physically representing the contact pressure, the geometric impenetrability condition is formulated in a weak, integral sense. Using dual shape functions (Wohlmuth (2000)) for the interpolation of the Lagrange Multipliers allows for a nodal decoupling of the geometric constraints. The combination with an active set strategy results in an algorithm which allows for elimination of the discrete nodal values of the Lagrange Multipliers. They can be easily recovered from the displacements in a variational consistent way. In contrast to many other formulations, the resulting contact algorithm combines two main advantages: Only the discrete nodal displacements appear as primal unknowns, thereby the size of the system of equations to be solved remains constant; there is no need for any user defined paramters like a penalty parameter.
Detailed numerical analyses of dynamic contact problems illustrate the necessity of additional, algorithmic energy-conserving strategies. The "Velocity-Update"-method by Laursen and Love (2002) is characterized by the fact that it guarantees the exact conservation of energy while simultaneously satisfying the geometric impenetrability condition. This method is revised according to the presented contact formulation and generalized for combination with the "Generalized-Energy- Momentum-Method".
Numerical examples are investigated to analyze and judge the effectiveness of the proposed solution strategy.
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