A micro-macro computational strategy including homogenization for analysing heterogeneous structures
Résumé
Today, structural analysis is more and more concerned with material models which are described down to the micro-scale. This is the case, for instance, when dealing with composite materials. For linear analysis, the treatment of such two-level problems is currently performed using techniques which take into account homogenization coupled with a local reanalysis. The technique which is mastered the best is probably that initiated by Sanchez-Palencia for periodic media [1]. The macro-level solution yields the effective values of the unknowns; the micro-level solution results from a local reanalysis with a specific treatment of the boundary areas considered as distinct from the interior area. Of course, a constraint in the use of such a method lies in the fact that the ratio between the small scale and the large scale has to be small. The goal of the proposed micro-macro computational strategy introduced in [2] [3] is to avoid several of the limitations of the classical homogenization techniques, and to make the procedure suitable for the most powerful computer resources available today, i.e. parallel-architecture computers. Other low-cost computational methods, such as multigrid methods, make use of two scales or two grids as well, but in these the basis is essentially numerical and far removed from the "homogenization" background of the mechanical engineer. For the sake of simplicity, the proposed micro-macro computational strategy is presented here for linear elastic problems without introducing any discretisation. The first step is to decompose the structure into an assembly of simple components: substructures and interfaces. For instance, a substructure may include one or several cells of composite structures and an interface represents the behavior of the liaison between two substructures. Each of these components possesses its own variables and equations. An interface transfers both a distribution of displacements and a distribution of forces. Since each of the components possesses its own variables and equations, the unknowns are both the displacements and the forces along the interfaces. The resulting approach is therefore termed a "mixed" substructuring technique [4]. The originality of this method lies in the separation of the unknowns into macro quantities and an additional micro component. The second characteristic of the micro-macro strategy is the use of the so-called LATIN method on the problem to be solved, described as an assembly of substructures and interfaces. The LATIN method is a nonincremental iterative computational strategy. It deals with the entire time interval under investigation [4]. For linear problems, the strategy involves a numerical parameter which can be interpreted as an interface stiffness. At each iteration, one must solve a "macro” problem defined on the entire structure and a set of independent linear problems related to the substructures. The latter are the "micro" problems; they are expected to describe the short-wavelength effects of the solution. The "macro" problem is related to the homogenized structure. Only a few iterations are needed to reach a satisfactory solution and convergence has been proved mathematically.
This approach requires no specific treatment for boundary areas and the structures concerned are not required to be periodic media. The homogenization procedure is performed automatically within the algorithm. Furthermore, this approach is well-suited
to parallel-architecture computers, since the underlying method can be interpreted as a 2- level mixed domain decomposition method.
Several numerical examples in the case of highly heterogeneous elastic structures and perfect interfaces illustrate the possibilities of this method as well as its performance compared to other domain decomposition methods [5].