Keywords: domain decomposition, time and space multiscale, LATIN, parallel computing.

The behavior of large structures often results from complex phenomena taking place at a fine scale, with small length of variations both in space and time. For instance local cracking or local buckling have a global effect on large structures. In order to predict the behavior of such structures, sophisticated models have been developed to describe the material at a very fine scale compared to the scale of the structure. Corresponding numerical simulations require the use of fine discretizations over both space and time domains, which often leads to systems with a very large number of degrees of freedom whose calculation cost is generally prohibitive.

Recently, a new multiscale computational strategy has been proposed for the analysis of such problems [1,2,3]. This strategy is based on the LArge Time INcrement method (LATIN method [4]), which enables one to work globally over the time-space domain. It can be seen as a parallel mixed domain decomposition strategy in space, including automatic space and time homogenization, and for which no periodicity condition is needed, in opposition with standard homogenization techniques.

The first idea consists in splitting the structure into substructures and interfaces, and dividing the studied time interval into subintervals. Interface fields are separated into a macro part and a micro complement. Beacuse of the Saint-Venant's principle, an adapted choice of macro quantities, combined with the resolution of a global macro problem, ensures that micro quantities have only local effects. This choice provides to the method the numerical scalability with respect to space variables. A technique to extend this property to time aspects is in progress. The second idea consists in using an iterative technique involving the resolution of a set of independant problems on the refined "micro" scale and the resolution of a number of problems on the homogenized "macro" scale. A new approximation technique for the resolution of micro problems is presented as an improvement over the version introduced in [1]. It leads to drastic saving in terms of computation cost.

A three-dimensional numerical example, with (visco)plastic material and unilateral contact with friction is provided to show the performance of the strategy combined with the new approximation technique, and to highlight the parallel properties.

1

A. Nouy and P. Ladevèze, "Multiscale computational strategy with time and space homogenization: a radial-type approximation technique for solving microproblems", International Journal for Multiscale Computational Engineering, 2, 557-574, 2004. doi:10.1615/IntJMultCompEng.v2.i4.40

2

P. Ladevèze, D. Néron and P. Gosselet, "On a mixed and multiscale domain decomposition method", Computer Methods in Applied Mechanics and Engineering, Vol 196, Num 8, Pages 1526-1540, 2007. doi:10.1016/j.cma.2006.05.014

3

P. Ladevèze, D. Néron and J.-C. Passieux, "On multiscale computational mechanics with time-space homogenization", Textbook: Bridging the scales in Science and Engineering, Ed. J. Fish, to appear 2008.

4

P. Ladevèze, "Nonlinear Computational Structural Mechanics--New Approaches and Non-Incremental Methods of Calculation", Springer Verlag, 1999.

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