A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables
Introduction
During the last decades, the room available in car vehicles (e.g. in engine compartment) has plummeted because of the rapid development of on-board electronics. As a result, a need for very accurate numerical tools for design has appeared in automotive industry. In the meantime, a fast computation is necessary so that design duration remains suitable for industry. In this context, flexible pieces represent an outstanding challenge since, unlike most of car pieces, they cannot be modeled as rigid body solids in CAD software. This paper focuses on a specific type of flexible piece, namely electrical cables. Cables have a complex structure. A wire is made up of copper filaments wrapped in an elastomer duct. These wires are most of the time gathered in bundles which are themselves surrounded by various protections such as tape, PVC tube … Moreover, the full cable is often constituted of several drifted cable pieces forming a system with a complex geometry.
Due to its slender shape and its flexibility, one can consider a simple cable as a beam undergoing large displacements and large rotations. There exist several theories accounting for nonlinear beams, see Refs. [1], [2], [3]. The most widely used theory is the geometrically exact beam model whose founding principles were established by Reissner [4], [5] and further generalized by Simo [6]. Various finite element formulations (FEM) have been presented to solve these equations numerically. The notable works of [7], [8], [9], [10], [11] and more recently [12], [13] can, among others, be listed.
At the heart of all these formulations, the rotation parameterization is of paramount importance in the numerical models. The 3D rotations modeling indeed is not an easy task especially when computational efficiency is sought. The rotational vector-like parameterization used by many authors features only 3 parameters (minimum set in 3D). However, as no parameterization of less than 4 parameters can be singularity-free [14], this choice poses several numerical limitations and lacks robustness. A powerful alternative consists in using quaternions, a set of 4 singularity-free parameters. Firstly used only for storage in the numerical models, Zupan et al. [12] have recently shown their utility when used as primary variables. They also have developed a model without rotation matrices exploiting the high potential of quaternion algebra, and very efficient for numerical purposes [15]. In addition, quaternions offer an original description of rotations since they substitute the usual trigonometric functions by algebraic variables and lead to polynomial equilibrium equations.
The finite element method, very adapted to the assembly of complex geometries such as electrical cables ones, is applied to the continuous equilibrium equations. The algebraic system obtained is then generally solved by a classical predictor-corrector method (PCM) [16]. Even if the appearance of arc-length methods [17] has considerably enhanced the robustness of this type of method, they often require to choose a step size which may be very tricky for the user. A sufficiently small step size allows to compute even highly nonlinear part of equilibrium branches but in return may impractically increase the computation time, while a larger step size may spoil the convergence. Taking advantage of the polynomial form of the system of equations obtained when using quaternion parameters an alternative consists in replacing the PCM by the asymptotic numerical method (ANM) firstly presented by Damil and Potier-Ferry [18] and by Cochelin [19]. The ANM is a very powerful solver for quadratic problems and it overcomes all the drawbacks of the PCM. This technique indeed is very robust, does not require any tuning parameters and is thus well suited for an industrial use. In addition, Cochelin and Medale [20] have equipped the method with a bifurcation detector and improved its efficiency in the vicinity of bifurcation points.
Combining quaternions with the ANM has already been set up on a rod model discretized with a finite difference scheme by Lazarus et al. [21], which have got very promising results. We propose here to set up the technique on the finite-element based geometrically exact beam model, in what constitutes the main originality of this paper. Validations and illustrations of the method on very intricate problems are provided and discussed. A critical evaluation and future researches are presented by way of conclusion.
Section snippets
Governing equations
In this section, the classical quasi-static formulation of the geometrically exact beam model, based on the rotation vector, is firstly presented. It enables to explain all the main ingredients of the model and to discuss their physical meaning. Secondly, the equations are modified by using quaternions instead of the rotation vector. This leads to the formulation which serves our numerical model, presented in part 3.
Discrete form of equations
The finite element method is now applied to the weak form of governing equations (35) to obtain the discrete equilibrium equations. The beam is cut in Ne elements, which for sake of simplicity are supposed of equal length Le = L∕Ne. Each element is composed of n + 2 equally-spaced nodes: n internal nodes and one node at each extremity of the element. In each element e, the unknown field value at node is denoted .
The virtual work principle (35) is discretized using a standard
Asymptotic numerical method
To the best knowledge of the authors, up to now, solving the nonlinear systems (51) or (52) has always been carried out using classical predictor-corrector methods. In this part, the Asymptotic Numerical Method (ANM) introduced by Cochelin [19] and Damil and Potier-Ferry [18] is presented. It is a high order perturbation technique that allows to compute a large part of a nonlinear branch with only one matrix inversion. The method principle is, starting from a solution point, to seek the branch
Validation
The method presented above has been implemented in ManLab [39], an open-source Matlab package. In this section, we show the results of our code on several reference numerical experiments for which an analytical solution is known or which have been tested by other authors. These examples serve to demonstrate the validity and the accuracy of our model as well as its abilities. In these tests, the centerline is chosen such that it passes through the centroid of all the sections (no additional
Application on a practical example
The examples of the previous section give a good overview of the robustness and accuracy of the herein presented method. To show the potential of the method in an industrial environment, we henceforth present an application to a real example. In this example pictured on Fig. 14, a cable harness made of a main fixed cable which first divides itself in two parts themselves fixed and then in 3 movable pieces is studied. The mounting operation consists in plugging these 3 cables to 3 connectors of
Conclusion
In this paper, a finite element formulation of the geometrically exact beam model using rotational quaternions is presented. Their singularity-free property, their numerical efficiency and their algebraic nature are the main arguments for this choice. In particular, the latter trait leads to a polynomial formulation of the discrete equilibrium equations that we exploit through the use of the asymptotic numerical method. This method is a powerful solver for systems of quadratic nonlinear
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