On the interest of a multiscale strategy for the model-order reduction of frictional contact problems

The simulation of architectured materials such as spiral strands wire rope or electrical conductors involving multiple frictional contact interactions is probably one of the most challenging problems in structural mechanics. Frictional contact problems are characterized by strong nonlinearities and non-smooth behaviors, which may require prohibitive computational costs, especially with time-dependent loadings and parametric studies. Acceleration or parallelization methods are therefore a necessity to address this issue such as multigrid methods or domain decomposition methods (DDM). Mixed DDM strategies also include those based on the LArge Time INcrement (LATIN) nonlinear solver applied to quasistatic contact problems. The LATIN method presents a robust treatment of contact conditions, sharing similarities with augmented Lagrangian approaches, and naturally leads to a mixed DDM. Another complementary way to decrease computational cost consists in using reduced-order model (ROM) techniques by seeking the solution of the given problem in a reduced-order basis (ROB). For contact problems, most of model reduction techniques rest on a posteriori methods, mainly for frictionless parametric problems with a small number of contact interfaces. Moreover, for problems involving a huge variety of loading conditions and nonlinear phenomena, such as frictional contact interactions, a predetermined ROB may not be able to efficiently capture non-regular and propagating multiscale phenomena that occur at contact interfaces: sliding, sticking and separation zones being difficult to represent. For this reason, an a priori approach based on the Proper Generalized Decomposition (PGD) coupled with the LATIN method is here used in order to tackle efficiently evolutions in frictional contact conditions. Frictional contact problems present a multiscale content, with global modes on the structural level and localized modes bringing corrections at the contact interface level, which suggests that proposing a model reduction method with a multiscale approach may be beneficial for problems of this type. In this presentation the efficiency of the multiscale LATIN-based mixed DDM with PGD is investigated in the case of frictional contact problems with large and multiple contact interfaces, where the contact conditions have an important effect at the global scale. Its robustness and effectiveness are tested on a two-dimensional frictional contact problem with more complex loadings. Guidelines are also given for choosing the parameters of the strategy, in particular those driving the construction of the reduced basis.