A unified approach for beam-to-beam contact
Graphical abstract
Introduction
In countless fields of application, mechanical system performance is essentially determined by highly slender fiber- or rod-like components. Industrial webbing, high-tensile ropes and cables, fiber-reinforced composite materials or polymer materials, but also biological tissue or biopolymer networks (see e.g. [1]) can be identified as typical examples of such fiber-dominated systems. Geometrically nonlinear beam finite elements are an efficient and accurate tool for modeling and solving these systems numerically. In [2], different types of nonlinear beam element formulations have been evaluated and compared, and the so-called geometrically exact beam formulations (see e.g [3], [4], [5], [6], [7], [8], [9], [10], [11]) have been recommended in terms of model accuracy and computational efficiency. While all the mentioned finite element formulations are based on the Simo–Reissner beam theory, an alternative geometrically exact element formulation based on the Kirchhoff–Love theory of thin rods and incorporating the modes of axial tension, torsion and anisotropic bending has been proposed in the authors’ recent contributions [12], [13], [14]. These formulations are tailored for high beam slenderness ratios as considered in this work. Furthermore, they consist of a -continuous beam centerline representation which enables smooth beam-to-beam contact kinematics.
The applications mentioned above are characterized by an intensive mechanical contact interaction between individual fibers and by geometrically complex contact configurations. Some recent contributions focusing on the analytical modeling of contact interaction between thin fibers are, e.g., the investigation of ropes with single- and bi-helical fiber substructures [15], the theoretical treatment of knot-mechanics [16] or the analysis of optimal topologies and packing densities in filamentous materials based on an implicit consideration of contact [17]. The arguably most popular numerical contact formulation for slender continua [18] models mechanical beam-to-beam contact interaction by means of a discrete contact force acting at the closest point of the beam centerlines. This model, in the following denoted as point-to-point contact formulation, results in a rather compact and efficient numerical formulation, which subsequently has been extended to frictional problems considering friction forces [19] and friction torques [20], rectangular beam cross-sections [21], [22], smoothed centerline geometries [23], constraint enforcement via Lagrange multipliers instead of penalty methods [24] and adhesion effects [25]. Quite recently, it has also been applied to self-contact problems [26]. In the recent works [27] and [28], additional contact points located in the neighborhood of the actual closest point have been proposed in order to improve the accuracy of this purely point-based approach when applied in the regime of small contact angles. Nevertheless, this formulation still relies on the existence of a unique closest point projection between the beams. First investigations concerning existence and uniqueness of closest point projections as well as possible shortcomings of purely point-based procedures have been made in [29] and [20].
In [30], it has been shown analytically that the existence of the closest point solution cannot be guaranteed and consequently that point-to-point contact approaches cannot be applied in a considerable range of small contact angles. Since such configurations are very likely in complex fibrous systems, alternative beam contact models are required. One of the few existing alternatives is the formulation developed by Durville [31], [32], [33], [34], [35]. It is based on a collocation-point-to-segment formulation and the definition of proximity zones on an intermediate geometry. A second alternative proposed by Chamekh et al. [36], [37] is based on a Gauss-point-to-segment type approach and has mostly been applied to self-contact problems. In our earlier work [30], different beam contact formulations have been investigated and eventually a Gauss-point-to-segment formulation in combination with consistently linearized integration interval segmentation, a smooth contact force law and a -continuous beam element formulation has been suggested as model of choice for the contact interaction of slender beams. Since these alternative approaches consider contact forces that are distributed along the beams, they will be denoted as line-to-line contact formulations in the following.
Even though line-to-line approaches yield accurate and robust contact models in the entire range of possible contact angles, their computational efficiency decreases considerably with increasing beam slenderness ratio. In Section 5, it will be shown that especially in the range of large contact angles the number of Gauss or collocation points required by these approaches and the resulting computational effort is prohibitively high as compared to point-to-point contact formulations. Thus, on the one hand, the point-to-point contact formulation serves as sensible mechanical model and very efficient numerical algorithm in the range of intermediate and large contact angles while it is not applicable for small contact angles. On the other hand, the line-to-line contact formulation provides a very accurate and robust mechanical model in the small-angle regime whereas the computational efficiency dramatically decreases with increasing contact angles. These properties motivate the development of a novel all-angle beam contact (ABC) formulation that combines the advantages of two worlds: The formulation is based on a standard point-to-point contact formulation applied in the range of large contact angles while the scope of small contact angles is covered by the line-to-line contact formulation proposed in [30]. Two different variants of a smooth model-transition procedure between the regimes of point and line contact are investigated, a variationally consistent transition on penalty potential level and a simpler variant on contact force level. Both variants lead to exact conservation of linear and angular momentum, while only the variationally consistent variant enables exact energy conservation. Based on analytical investigations, recommendations are made concerning the optimal ratio between the two penalty parameters of the point and the line contact, the required number of line contact Gauss points and the choice of the model transition interval. All configuration-dependent quantities are consistently linearized allowing for an application within implicit time integration schemes. The resulting ABC formulation is supplemented by the contact contributions of the beam endpoints (see [30]).
The novel formulation successfully addresses one of the most essential challenges in the contact modeling of highly slender structures: To achieve a sufficiently fine spatial contact resolution at manageable computational costs. A second typical limitation for many standard beam contact algorithms in the range of high slenderness ratios is the requirement of an adequately small time step size. In that regard, we propose a step size control of the nonlinear solution scheme, which allows for displacement increments per time step that exceed the order of magnitude of the beam cross-section radius. Additionally, we propose a very efficient two-stage contact search algorithm based on dynamically adapted search segments for each finite element. This does not only result in a very tight set of potential contact pairs, but it also enables a subdivision into potential point-to-point and potential line-to-line contact pairs. The latter property is essential in order to fully exploit the efficiency potential of the proposed ABC formulation. All the presented algorithmic components are tailored for the most challenging case of arbitrary discretization orders and lengths that typically lead to high element slenderness ratios and deformations. The interplay of these individual constituents yields the first beam-to-beam contact formulation that combines a significant degree of robustness and universality in the treatment of complex contact scenarios and arbitrary beam-to-beam orientations with high computational efficiency, especially in the limit of extreme beam and element slenderness ratios.
The remainder of this paper is organized as follows. In Section 2, we sketch the main constituents of the beam element formulation proposed in [12], [13]. In Sections 3 Point-to-point contact formulation, 4 Line-to-line contact formulation, the basics of standard point contact models and of the line contact formulation derived in [30] are outlined. The novel all-angle beam contact formulation, the related theoretical investigations and the recommendations concerning optimal parameter choice are presented in Section 5, while Section 6 contains algorithmic aspects such as contact search, step size control, penalty force laws and treatment of endpoint contacts. Finally, detailed numerical verifications are presented in Section 7. While the first two examples in Sections 7.1 Example 1: Beam rotating on arc, 7.2 Example 2: Impact of free flying beams aim to investigate the accuracy and consistency of the all-angle beam contact formulation regarding contact force distributions and conservation properties, the remaining two examples in Sections 7.3 Example 3: Simulation of a biopolymer network, 7.4 Example 4: Dynamic failure of a rope are intended to bridge the gap towards challenging real-world applications. Therein, also the attainable efficiency gains of the proposed algorithm as compared to standard line-to-line contact formulations are quantified.
Section snippets
Variational problem statement and applied beam formulation
In the following section, the basics of a geometrically exact beam element formulation recently proposed by the authors and employed for the numerical examples in this work will be repeated. Although this specific beam element formulation can be regarded as very beneficial for the treatment of highly slender beams, the ABC formulation proposed in the present work is not restricted to this element formulation at all. On the contrary, it describes beam-to-beam contact interaction in a very
Point-to-point contact formulation
Within this section, we briefly review the main constituents of a standard point-to-point beam contact formulation as introduced in [18]. Thereto, we consider two arbitrarily curved beams with cross-section radii and , respectively. The beam centerlines are represented by two parametrized curves and with curve parameters and . Furthermore, and denote the tangents to these curves at positions and , respectively. In what follows, we assume that
Line-to-line contact formulation
Here, we will briefly repeat the most important aspects of the line-to-line contact formulation developed in [30]. In contrary to the point-to-point contact model, this formulation is based on a line constraint enforced along the entire beam length. The relevant kinematic quantities of this approach are illustrated in Fig. 2, Fig. 2(a).
Here, a distinction has to be made between a master beam (beam ) and a slave beam (beam ). The closest master point to a given slave point is determined
Limitations of existing beam-to-beam contact formulations
In the last two sections, we have presented two basic contact formulations that enable the mechanical modeling of beam-to-beam contact in the sense of a point-to-point and a line-to-line contact interaction. In the next two Sections 5.1.1 Limitations of point-to-point contact formulation, 5.1.2 Limitations of line-to-line contact formulation, practically relevant limitations of these basic formulations will be investigated. The results of this study will serve as foundation for the development
Algorithmic aspects
In the following Sections 6.1 Contact search algorithm, 6.2 Step size control, further information concerning the employed contact search algorithm and a step size control applied to the iterative displacement increments within the nonlinear solution scheme will be given. The latter method enables displacements per time step that are larger than the beam cross section radius, which is the typical time step size limitation of standard beam-to-beam contact algorithms. In Section 6.3, different
Numerical examples
The first two examples of this section aim at investigating the accuracy and consistency of the proposed ABC formulation. The first example focuses on the contact force evolutions in the model transition range, while the second example verifies the conservation properties already shown theoretically in Appendix B. Finally, we want to verify the robustness and efficiency of the proposed contact algorithm when applied to practically relevant applications. Thereto, we employ the force-based ABC
Conclusion
The aim of this work was the development of an efficient and robust beam-to-beam contact formulation capable of modeling complex contact scenarios with arbitrary geometrical configurations in unstructured systems of highly slender fibers. It has been shown that line contact formulations represent very accurate and robust mechanical models in the range of small contact angles, whereas their computational efficiency considerably decreases with increasing contact angles. This fact can be
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