7 research outputs found
Multiscale nonconvex relaxation and application to thin films
-convergence techniques are used to give a characterization of the
behavior of a family of heterogeneous multiple scale integral functionals.
Periodicity, standard growth conditions and nonconvexity are assumed whereas a
stronger uniform continuity with respect to the macroscopic variable, normally
required in the existing literature, is avoided. An application to dimension
reduction problems in reiterated homogenization of thin films is presented.Comment: 40 pages, 5 figure
Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands
Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting.
It is shown by the reiteraded periodic two-scale convergence method that the
sequence of minimizers of a class of highly oscillatory minimizations problems
involving convex functionals, converges to the minimizers of a homogenized
problem with a suitable convex function
Heterogeneous thin films: Combining homogenization and dimension reduction with directors
We analyze the asymptotic behavior of a multiscale problem given by a
sequence of integral functionals subject to differential constraints conveyed
by a constant-rank operator with two characteristic length scales, namely the
film thickness and the period of oscillating microstructures, by means of
-convergence. On a technical level, this requires a subtile merging of
homogenization tools, such as multiscale convergence methods, with dimension
reduction techniques for functionals subject to differential constraints. One
observes that the results depend critically on the relative magnitude between
the two scales. Interestingly, this even regards the fundamental question of
locality of the limit model, and, in particular, leads to new findings also in
the gradient case.Comment: 28 page
The Periodic Unfolding Method in Homogenization
International audienceThe periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104] (with the basic proofs in [Proceedings of the Narvik Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakkotosho, Tokyo, 2006, pp. 119-136]). In the present paper we go into all the details of the method and include complete proofs, as well as several new extensions and developments. This approach is based on two distinct ideas, each leading to a new ingredient. The first idea is the change of scale, which is embodied in the unfolding operator. At the expense of doubling the dimension, this allows one to use standard weak or strong convergence theorems in L(p) spaces instead of more complicated tools (such as two-scale convergence, which is shown to be merely the weak convergence of the unfolding; cf. Remark 2.15). The second idea is the separation of scales, which is implemented as a macro-micro decomposition of functions and is especially suited for the weakly convergent sequences of Sobolev spaces. In the framework of this method, the proofs of most periodic homogenization results are elementary. The unfolding is particularly well-suited for multiscale problems (a simple backward iteration argument suffices) and for precise corrector results without extra regularity on the data. A list of the papers where these ideas appeared, at least in some preliminary form, is given with a discussion of their content. We also give a list of papers published since the publication [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104], and where the unfolding method has been successfully applied
An homogenization result in
Abstract An integral representation result is provided for the -limit of integral functionals arising in homogenization problems for the study of coherent thermochemical equilibria in multiphase solids