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Khalil, Mohamed 
Deriving and Implementing a Contact Formulation in an Open-Source Finite Elements Framework 
Contact mechanics is a wide class of physical phenomena present in a variety of life situations, as well as engineering applications. This, in turn, triggers the motivation for continuously devel- oping a reliable mathematical model, and implementing a robust & efficient numerical simulation scheme for such phenomena. In the scope of this work, a formulation derivation and an imple- mentation of a segment-to-segment contact scheme based on collocation method for numerical integration and standard Lagrange multipliers was presented. The contribution of this work is proposing a reliable contact approach –the segment-to-segment approach –, which employs an exact contact constraint enforcement scheme –Lagrange multipliers –, and aiming at reducing the overhead of interface pre-processing for numerical integration by using the collocation inte- gration method. For the sake of completeness, a review of solid mechanics concepts has been first introduced to the reader, as well as discussions about the constraint enforcement meth- ods and different approaches for numerical integration adopted in contact mechanics numerical implementations. Moreover, the necessary linearizations related to the formulation, which are necessary for the Newton-Raphson solution method were appended in full detail. The formulation was implemented in a C++ open-source framework, Kratos Multiphysics, and different aspects are particularly tested through the common benchmarks in the literature. Additional engineering test cases serve as supplementary verification of implementation capabilities. The focus of the work was dedicated to investigating the implementation of 2D in full detail, nevertheless, an ex- tension to 3D using the proposed method has been also included. For the 2D case, the benchmarks has been fulfilled successfully, and the method has demonstrated notable robustness bearing in mind its innate simplification of the collocation integration method. In 3D, the influence of said simplifications required large number of integration points to fulfill the benchmark test studied, which reflected a notably expensive computational cost. Finally, the foreseen future outlook for this work has been proposed, based on the results and discussion. 
Bletzinger, K.-U. 
Technische Universität München 
TUM Institution:
Lehrstuhl für Statik