A nonsmooth local frame approach to beam-to-beam contact

Highly slender structures undergoing large displacements, such as cables and hoses are integral parts of most modern high performance engineering systems and interest from the industry in the virtual simulation of such systems has strongly increased in recent years. We focus on distributed line-to-line contact interactions, which are most appropriate for modeling closely packed slender structures that remain in contact over extended regions of their length. The modeling of frictional contact interactions, one of the key aspects for accurate prediction of mechanical behaviour, is particularly challenging and still a matter of scientific research.

In this thesis we adopt a geometric framework for flexible multibody systems which is characterized by a rigorous mathematical handling of frame transformations. The geometrically exact beam finite element framework on the special Euclidean group is selected for modeling slender structures. The equations of motion are interpreted as being expressed in the local frame attached to the body resulting in interesting frame invariance properties.

An essential ingredient of the selected contact model is the formulation of non-penetration conditions as a unilateral restriction of relative motions between material particles and the treatment of these conditions in a discrete setting. This work establishes the contact kinematics, including the normal gap and tangential slip, between geometrically exact beams building on the special Euclidean group formalism. The resulting formulation expresses the contact forces in the local frame and inherits the frame invariance properties from the chosen geometric framework.

These unilateral constraints may cause nonsmooth effects such as discontinuous spatial distribution of contact reaction forces, dynamic impacts and frictional stick-slip transitions which require advanced numerical techniques. First, a quasi-static methodology is developed using an Augmented Lagrangian technique combined with a nonsmooth Newton solver. The elaboration of an alternative Gauß-Seidel dual solver is also explored. Then, the dynamic problem is tackled using the nonsmooth generalized-α time integration scheme. For quasi-static and dynamic cases we implemented two beam-to-beam contact elements. The first one is a simple collocation technique. The second one is the mortar method, where the non-penetration constraints are enforced in a weak sense. All the developments have been implemented into the open source software Odin connecting our work to a wide library of finite elements.

Various numerical experiments were performed for the quasi-static and the dynamic case. These include the static frictionless contact of a cantilever with a rigid wall for which a novel analytic solution was derived. The examples show the potential of combining all the previously mentioned concepts to form an appropriate framework for handling geometric non-linearities, discontinuities, and complex frictionless and frictional contact configurations exhibited by cable assemblies.