A multiscale large time increment/FAS algorithm with time‐space model reduction for frictional contact problems

Abstract

A multiscale strategy using model reduction for frictional contact computation is presented. This new approach aims to improve computation time of finite element simulations involving frictional contact between linear and elastic bodies. This strategy is based on a combination between the LATIN (LArge Time INcrement) method and the FAS multigrid solver. The LATIN method is an iterative solver operating on the whole time-space domain. Applying an a posteriori analysis on solutions of different frictional contact problems shows a great potential as far as reducibility for frictional contact problems is concerned. Timespace vectors forming the so-called reduced basis depict particular scales of the problem. It becomes easy to make analogies with multigrid method to take full advantage of multiscale information. A. GIACOMA ET AL. in large-scale elasticity problems with contact and friction. Once again, because of the nonsmooth property of contact problems, convergence proof of generalized Newton's methods are hard to provide. Another method consists in casting the contact problems into a linear complementarity problem (LCP) and using specific solvers such as active-set methods, Lemke's algorithm, projected successive over relaxation. In [20], an LCP formulation is used to solve a frictional contact problem by faceting the Coulomb's cone. Nevertheless, this approximation is tough and not efficient (the problem to solve becomes larger). Nowadays, LCP formulations are suited to frictionless problems.Generally speaking, and even within a quasi-static context, all these nonlinear solvers can lead to prohibitive time of computations. Acceleration strategies based on multigrid methods were proposed [21,22]. Computational costs spur recent and intensive works on efficient model reduction techniques in various fields [23][24][25][26]. But because of the nondifferentiable nature of frictional contact, application of such methods seems to be heretic.The aim of this work is to propose a strategy accelerating solution for frictional contact problems in the finite element framework, embedding model reduction techniques also well-suited for parametric studies. Linear elastic, homogeneous, isotropic behavior, and quasi-static evolution are assumed for the bodies. Frictional contact laws involve a nonlinear and nonsmooth behavior at the boundary of the body. To solve this mechanical problem, the nonincremental LArge Time INcrement (LATIN) method is used. This method is well-known for its ability to solve difficult nonlinear large problems (nonlinear material, contact problems...) [27,28] with a global time-space approach. This method is closed to augmented Lagrangian methods. Its great advantages are non refactorization of matrices (stiffness matrix remains constant through LATIN iterations) and explicit subsequent iterations (no iterations to handle the nonlinear behavior solved only on the contacting boundary). So the cost of a LATIN iteration is low even if the number of iterations can be elevated as augmented Lagrangian method...

Toward an optimal a priori reduced basis strategy for frictional contact problems with LATIN solver

“…This innovative non-incremental approach is well known for its ability to solve efficiently various non-linear time dependent problems such as frictional contact problems [27,60,61], large displacement [62], non-linear materials [46,63,64], transient dynamics [65][66][67]. …”

“…In [27], analysis of RB provided by SVD is carried out for frictional contact problems and the LATIN/FAS strategy is proposed. It is exemplified that the RB has a multiscale content.…”

An efficient quasi-optimal space-time PGD application to frictional contact mechanics

“…This method, close to augmented Lagrangian methods, is well-known for its ability to solve difficult non-linear and time-dependent large problems with a global time-space approach (non-linear material [19], contact problems [4,15,20], large displacement [21], transient dynamics [22,23], fracture mechanics [24,25]...). The non-incremental LATIN method was proposed as a commitment of three principles, which are, for the elastic frictional contact problems: …”

“…S can be written accurately (up to a certain level) with a linear combination of a few vectors • Dominant trends (scale separability): some vectors of the basis (which are supposed to be the first ones) are highly contributory to generate S whereas other ones are less important. These vectors depict the different scales of the problem [4].…”

Mesoscale study of particle sedimentation with inertia effect using dissipative particle dynamics

“…Analytical theories with simplified assumptions and experimental approaches with semi-empirical formulas are usually difficult to describe the forces acting on particles, the momentum transfer of fluid-solid system and the change of inter-phase boundary (Giacoma et al 2014;Marco and Dimitris 2011;among others).…”