A Robust Method for Beam-to-Beam contact Problems Based on a Novel Tunneling Constraint

The need for better and more efficient computational methods to model the mechanics of contact has increasingly attracted the interest of researchers in this area over the last two decades. Contact problems--wherein two or more bodies meet through an interface--are extremely challenging from both mathematical and engineering points of view, because of the complex nonlinearities due to moving contact interfaces. Over the years, a significant effort has been devoted to address the resolution of continuum contact using the finite element method through several techniques including penetration functions, mortar methods, and domain decomposition. Nonetheless, a comparable effort to resolve beam contact problems is still lacking in the current literature. Motivated by a number of deficiencies encountered in existing models for beam contact, this dissertation concerns the finite element modeling of contact among structural beams of zero thickness undergoing large deformations in space. The resulting methods and algorithms provide an unprecedented range of applicability in simulating not only contact of thin structural beams, but also collisions of cables (e.g., knot tying and surgical thread simulations), and dislocation dynamics, among others. The prevailing contact modeling techniques rely on a gap penetration constraint that is enforced by a penalty method to resolve overlapping between two or more contacting beams. Early attempts to resolve this problem, however, have displayed two major drawbacks: not only would the said constraint function be not extendable to planar beam contact problems, but also it would break down when dealing with zero thickness structural elements (e.g., thin beams, cables, and ropes). It was also observed that the existing formulations are difficult to implement and, often, computationally inefficient. Moreover, the use of the penalty method, which dominates most of the existing literature available to date, leads to failures in preserving the exact resolution of contact and, sometimes, results in ill conditioned problems. Especially within a large scale system involving multiple contacting bodies, if the penalty approximation is too crude, then the simulations may predict non physical outcomes. In this dissertation, a robust approach that is based on a novel contact constraint is proposed. This approach allows a simple and unified treatment of possible beam contact scenarios in two and three dimensional problems involving large deformations; and is based on a Lagrange multiplier method to exactly enforce the contact constraint across the interfaces between beam element pairs. Simply based on the calculation of an oriented volume/area, the proposed approach possesses the ability to resolve any contact scenario exactly and efficiently, while being relatively easy to implement and more widely applicable than the few beam contact formulations available to date. The primary focus of this work is the inner workings of the novel beam contact formulation proposed and, then, the delineation of the details of its numerical implementation in a finite element method setting. Application of the new approach is demonstrated through several benchmark problems, where specific consideration is given to the assessment of its robustness. The verification and assessment studies indicate that the proposed model can successfully resolve beam to beam contact with high fidelity for a broad spectrum of scenarios, wherein existing methods usually fail. The lessons learned from this study may be applied to resolving line contact in a wide range of areas including structural mechanics, textile mechanics, and dislocation dynamics. They may serve as a useful tool for developing efficient and accurate numerical models to simulate anything from progressive collapse of structures under extreme event scenarios and crushing of foams, to modeling of knots or surgical threads, just to name a few. An incidental aim of this work has been to make the mechanics of beam contact a more accessible topic of learning