Keywords: domain decomposition method, non-linear problems, damage.

Nowadays, many computational strategies exist to solve nonlinear multiscale problems. Two groups of methods can be distinguished: the ones that really treat multiscale aspect of the nonlinear problems and the ones that use a multiscale strategy to solve the linear problems arising from the linearization of the nonlinear problem. Only few methods can really be classified in the first group.

In the second group, one of the most popular and general family of methods is the so-called "Newton-Krylov-Schur" (NKS) family (see [1] for a review). Among these methods, we take an interest in the FETI method [2]. These methods perform well, but they may loose their efficiency when they encounter pathological phenomena such as local non-linearities. Because the convergence of Newton-type algorithms is linked to the strongest non-linearity in the domain, local non-linear phenomena may penalize the convergence of the global algorithm. In [3], the convergence of the NKS strategy is dramatically slowed down by local buckling, hence a strategy which introduces non-linear relocalization steps, inspired from the principles of the LaTIn method [4], is designed and assessed. Our approach, introduced in [5], is based on a different point of view.

Stating that standard NKS solvers do not exploit the domain decomposition in the non-linear context, we propose to introduce the domain decomposition directly in the non-linear formulation of the problem instead of only using it to solve linearized problems. First, the global non-linear problem, is decomposed into local problems under a global constraint imposed via a Lagrange multiplier. Second, a condensation step formulates the non-linear problem in terms of the unknown Lagrange multiplier. Third, a Newton-type algorithm is chosen to solve this non-linear interface problem. An iteration of our algorithm requires the solution of a global linear interface problem and the solution of a set of independent non-linear local problems (via another Newton-type solver) with Neumann interface conditions which may imply specific handling of infinitesimal rigid body motions. Consequently, treating non-linearity at its own local scale leads to a modification of the classical domain decomposition methods by the addition of a set of local non-linear iterations which are parallel and not expensive. These non-linear iterations decrease the number of global resolutions and result in significant CPU speedup.

1

P. Gosselet, C. Rey. "Non-overlapping domain decomposition methods in structural mechanics", Archives of Computational Methods in Engineering, 13(4), 515-572, 2007. doi:10.1007/BF02905857

2

C. Farhat, F.-X. Roux. "A method of finite element tearing and interconnecting and its parallel solution algorithm", International Journal for Numerical Methods in Engineering, 32, 1205-1227, 1991. doi:10.1002/nme.1620320604

3

P. Cresta, O. Allix, C. Rey, S. Guinard. "Nonlinear localization strategies for domain decomposition methods: application to post-buckling analysis", Computer Methods in Applied Mechanics and Engineering, 196, 1436-1446, 2007. doi:10.1016/j.cma.2006.03.013

4

P. Ladevèze, D. Néron, P.  Gosselet. "On a mixed and multiscale domain decomposition method", Computer Methods in Applied Mechanics and Engineering, 196(8), 1526-1540, 2006. doi:10.1016/j.cma.2006.05.014

5

J. Pebrel, C. Rey, P. Gosselet " A nonlinear dual domain decomposition method: application to structural problems with damage", International Journal for Multiscale Computational Engineering, (submitted).

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