Robust Formulations for Beam-to-Beam Contact

Sammanfattning: Contact between beam elements is a specific category of contact problems which was introduced by Wriggers and Zavarise in 1997 for normal contact and later extended by Zavarise and Wriggers to include tangential and frictional contact. In these works, beam elements are assumed to have rigid circular cross-sections and each pair of elements cannot have more than one contact point. The method proposed in the early papers is based on introducing a gap function and calculating the incremental change of that gap function and its variation in terms of incremental change of the nodal displacement vector and its variation. Due to complexity of derivations, specially for tangential contact, it is assumed that beam elements have linear shape functions. Furthermore, moments at the contact point are ignored. In the work presented in this licentiate thesis, we mostly adress the questions of simplicity and robustness of implementations, which become critical once the number of contact is large.In the first paper, we have proposed a robust formulation for normal and tangential contact of beams in 3D space to be used with a penalty stiffness method. This formulation is based on the assumption that contact normal, tangents, and location are constant (independent of displacements) in each iteration, while they are updated between iterations. On the other hand, we have no restrictions on the shape functions of the underlying beam elements. This leads to a mathematically simpler derivation and equations, as the linearization of the variation of the gap function vanishes. The results from this formulation are verified and benchmarked through comparison with the results from the previous algorithms. The proposed method shows better convergence rates allowing for selecting larger loadsteps or broader ranges for penalty stiffness. The performance and robustness of the formulation is demonstrated through numerical examples.In the second paper, we have suggested two alternative methods to handle in-plane rotational contact between beam elements. The first method follows the method of linearizing the variation of gap function, originally proposed by Wriggers and Zavarise. To be able to do the calculations, we have assumed a linear shape function for the underlying beam elements. This method can be used with both penalty stiffness and Lagrange multiplier methods. In the second method, we have followed the same method that we used in our first paper, that is, using the assumption that the contact normal is independent of nodal displacements at each iteration, while it is updated between iterations. This method yields simpler equations and it has no limitations on the shape functions to be used for the beam elements, however, it is limited to penalty stiffness methods. Both methods show comparable convergence rates, performance and stability which is demonstrated through numerical examples.