On a mixed and multiscale domain decomposition method

Abstract

This paper presents a reexamination of a multiscale computational strategy with homogenization in space and time for the resolution of highly heterogeneous structural problems, focusing on its suitability for parallel computing. Spatially, this strategy can be viewed as a mixed, multilevel domain decomposition method (or, more accurately, as a "structure decomposition" method). Regarding time, a "parallel" property is also described. We also draw bridges between this and other current approaches.Key words: domain decomposition, multiscale, computational mechanics IntroductionIn the structural mechanics field, one can observe a surge of interest in the multiscale analysis of structures with complex microstructural geometry and/or complex behavior. When accurate solutions are required, calculations must be performed on a finely discretized model of the structure (defined on what is called the "micro" level) consistent with short lengths of variation, both in space and in time. A major application is the analysis of composite structures described on the microscale or on the mesoscale [1]; there are also other applications, such as in [2,3]. These types of situations lead to problems with very large numbers of degrees of freedom and computation costs which are prohibitive if one uses classical finite element codes. One of the main objectives * Corresponding author. E-mail: pierre.ladeveze@lmt.ens-cachan.fr Preprint of the last few decades has been to develop efficient and robust computational strategies suitable for these types of problems. One of these strategies uses the theory of the homogenization of periodic media [4][5][6]. Other developments and the associated computational approaches can be found in [7][8][9][10][11][12][13][14]. Besides periodicity, these strategies rely on the fundamental assumption that the ratio between the two scales is small. Thus, the boundary zones require specific treatment because in these zones the microstructure cannot be homogenized. Here, we follow a recently introduced multiscale computational strategy for nonlinear evolution problems. This strategy involves an automatic homogenization technique in space as well as in time [15,16] which is an extension of previous works limited to space alone [17,18]. This strategy, developed in a general framework, makes no a priori assumption regarding the form of the solution and, therefore, does not suffer from the limitations of standard homogenization techniques; moreover, it relies on an iterative algorithm. Until now, this strategy has been developed in the framework of small displacements of (visco)plastic structures under possible contact with or without friction. This paper is a reconsideration of this computational strategy in order to assess its adaptability to parallel computing. Therefore, we examine its three main characteristics. The first characteristic is that it can be viewed as a mixed, multilevel domain decomposition method or, more precisely as will explained further on, as a "structure decomposition" method. Ind...

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