Model order reduction for the fatigue life prediction of wire ropes in tension and bending

Helically wounded steel wires in tension and bending are widely encountered, especially in the energy and power transmission fields, e.g. mooring lines for floating offshore wind turbines (FOWT) or overhead electrical conductors. The life prediction of such structures involving fretting fatigue phenomena between their constituent wires is crucial. However, wire-scale simulations for complex and multiple loads requires effective computational and modeling strategies. Examples include the use of wire models and beam-to-beam contact algorithms [1] or the use of homogenization techniques [2]. Model reduction techniques and multi-scale domain decomposition methods (DDM) are investigated here in order to appreciate their ability to deal with this particular class of problem. Model reduction for frictional contact problems is a challenging issue mainly due to the non-linear and non-regular character of frictional phenomena. It is indeed difficult to represent rapid changes in contact status and complex propagative phenomena of sliding/sticking zones as well as their strong multiscale content. Few works exist in the literature. Among them, one can cite a posteriori approaches with reduced basis enrichment such as the non-negative matrix factorization method [3] or the cone-projected greedy algorithm [4]. Among a priori methods that build the reduced basis on the fly throughout the computation by a progressive Proper Generalized Decomposition (PGD), one can cite the works developed in the framework of the non-incremental nonlinear solver called the LATIN method [5] and its application to contact problems [6, 7]. In this case, the resulting algorithm shares similar features with augmented Lagrangian methods known for their robustness in dealing with frictional contact problems [8]. The main difference with the LATIN method comes from the fact that, unlike classical incremental nonlinear solvers, an iterate of the solution is generated at each iteration over the entire space-time domain, which makes this method particularly suitable for a space-time separate representation of the solution for PGD model order reduction. In this presentation, an SVD analysis of simulation results from [9] for a metric length of wire rope belonging to a FOWT mooring line is first proposed to illustrate its reduction potential. The analysis of contact quantities performed layer by layer for the spiral strand wire rope, shows that quantities for the outer layers where slip occurs mainly due to bending are more difficult to represent with a reduced order model. It is also more difficult to accurately represent frictional contact forces than normal contact forces and slip. The multiscale content of the problem (global modes corresponding to the wire rope tension and bending and local modes corresponding to contact and friction between wires) suggests that the use of multiscale methods may be beneficial. In a second step, a model reduction method by PGD combined with the nonlinear solver LATIN is used on a simple one-dimensional problem which is a simplification of the loading seen by a constituent wire of a wire rope. It is shown that the reduction potential depends closely on the importance of the sliding front propagation (see Figure 1). Sorting the reduced basis and controlling its size using an appropriate downsizing algorithm is also crucial to recover optimality with respect to SVD [7]. Finally, the LATIN-based multiscale mixed DDM with PGD [6] is applied to a two-dimensional problem representative of the targeted application. It is shown to what extent the DDM coarse problem and the introduction of the PGD, which builds reduced basis per subdomain, are able to effectively represent the multiscale content of the problem.