Towards a nonsmooth multibody finite element framework for braiding simulation

Highly deformable slender structures such as cables and hoses are ubiquitous in engineering systems, where they interact with each other and rigid bodies through kinematic joints and contact conditions. The global dynamics of such assemblies can be captured using system-level simulations. However this is not straightforward, as rigid bodies demand a robust handling of large amplitude motions and constrained relative motions, while for flexible bodies, these overall motions are accompanied by local deformations. In addition, nonsmooth phenomena can be induced from impacts or stick-slip transitions. Such factors make the model susceptible to numerical ill-conditioning and stability issues; all of which pose as central problems in the targeted application of this thesis. This thesis proposes two original contributions. (i.) First, we address the simulation of the braiding process, which is widely used in the fabrication of preforms for composite production. On the one end, thin and flexible textile yarns flow from bobbin carriers in a controlled manner, driven by horn gears to form an interlocked mesh. On the other end, yarns undergo contact-friction interactions with the mandrel. As the process parameters strongly influence the final preform geometry, a system-level model is useful. We propose a multibody approach, where the carriers and the mandrel are modelled as rigid bodies, and the yarns are modelled using a geometrically exact beam finite element model. Absolute coordinates represent both large rigid body motions and elastic deflections of the yarns. A local frame representation of the equations of motion leads to reduced geometric nonlinearities. Phenomena such as tension compensation in the yarn are modelled using judicious choices of kinematic joints, and carrier motions using switching bilateral constraints. Furthermore, unilateral contact constraints and friction laws govern the yarn-to-mandrel interactions. We explore two possible contact formulations: a mortar line-to-line contact formulation, and a collocation node-to-surface method. A strict enforcement of constraints using Lagrange multipliers induces a nonsmooth response. Numerical discretization in space is performed using a Lie group $SE(3)$ interpolation scheme. To solve the semi-discrete system, we use the nonsmooth generalized-$\alpha$ method (NSGA) method. The choice of the time step solver in the NSGA is also addressed. An augmented Lagrangian formulation with a semi-smooth Newton method relies on the tangent operator, which becomes ill-posed when the density of contact points becomes large. Instead, we use a nonlinear block Gauss-Seidel strategy which considers a sequential resolution of each contact point. Finally, we present the preliminary simulation of an industrial multi-layer braiding machine. (ii.) Second, the novel approach of switching bilateral constraints for carrier motions provided motivation to address a more fundamental question: \textit{can we establish a general modelling procedure for multibody systems with switching bilateral constraints}? In this work, we address a special class of multibody systems, where switching functions trigger instantaneous changes in the algebraic constraint expressions, thus leading to a time-discontinuous response. These switching functions define switching surfaces, that partition the system dynamics into discrete regimes with nonsmooth transitions. Switching surfaces are thus instrumental in our modelling framework, as they orchestrate the geometry of the constraint space. The equations of motion can either be formulated as hybrid differential-algebraic equations or as an equality of differential measures with constraints. At the switching surface, an impact law based on an intermediate gradient is introduced. We propose to determine this intermediate gradient by interpolation between the pre- and post-switch gradients. Theoretical arguments and numerical results show that the choice of this intermediate gradient drives the energy behavior at switching. The numerical integration demands special attention, as classical DAE solvers fail to handle discontinuities at switching surfaces. Event-driven and time-stepping schemes from nonsmooth dynamics are well-suited for this class of problems. We use a benchmark test to compare the solutions obtained from classical and nonsmooth versions of the generalized-$\alpha$ method. Further, using the NSGA scheme, three examples of multibody systems with switching bilateral constraints are successfully simulated.