Solid Dynamics: Nonlinear Beams and Beam Contact
Christoph Meier, Alexander Popp and Wolfgang A. Wall
Standard beam-to-beam contact formulation available in the literature model the mechanical interaction of beams as point-to-point contact based on the solution of the closest-point problem between two space curves. While this procedure provides a very efficient, consistent and easy-to-implement contact algorithm, it fails whenever contact between two parallel beams has to be displayed. Furthermore, not only its numerical performance but also the applicability of this mechanical model in general, already decreases considerably for beams enclosing small angles. There exist only a few alternatives in the literature, mostly based on a collocation-type approach, which enable the treatment of small contact angles. However, these formulations typically require a beam-slenderness-dependent, very fine, and therefore expensive, contact discretization in order to detect and resolve beam-to-beam contact in the range of large contact angles. Consequently, the numerical effort increases dramatically with the slenderness ratio of the beams (e.g. when modelling thin fibers). The respective drawbacks of these two different types of beam-to-beam contact formulations lead to a dilemma, whenever applications require the resolution of arbitrary geometrical contact configurations spanning the whole range of possible contact angles (e.g. fibers in composite materials, filaments in biopolymer networks etc.)
Our approach is based on the idea of combining the advantages, while abstaining from the disadvantages, of these two types of formulations. Thus, we apply a standard point-to-point contact formulation in the range of large contact angles. The scope of small contact angles is covered by a line-to-line contact formulation that essentially preserves the advantages of the collocation-type approaches by additionally providing the desirable property of consistence in case of mesh-refinement. Within a prescribed angle interval, the model-transition between these two basic formulations is realized in a consistent manner without loosing essential properties such as conservation of linear momentum, angular momentum and energy. The gain in efficiency offered by this algorithm is twofold: On the one hand, a rougher discretization can be chosen for the more expensive small-angle contact formulation, since the critical case of large contact angles is treated by a different method. On the other hand, the total number of contact pairs evaluated with the small-angle formulation decreases considerably since for a sensibly chosen transition sector the predominant majority of contact pairs is treated by the large-angle contact formulation. While this algorithm already faces one essential challenge in the contact modelling of highly slender structures, i.e. the requirement of a sufficiently tight spatial resolution, a second pre-requisite in terms of a sensible fine time discretization has to be ensured by different means. In this context, we develop algorithms that allow for displacement increments per time step that exceed the order of magnitude of the beam radius, which is the typical limitation concerning the time step size of many standard beam-to-beam contact formulations.
