The Periodic Unfolding Method in Homogenization

Abstract

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

Variational approach to coarse-graining of generalized gradient flows

“…This second form of the rate functional shows clearly how I γ (ρ) = 0 is equivalent to the property that ρ solves the VFP Eq. (17). It also shows that if I γ (ρ) > 0, then ρ is an approximate solution in the sense that it satisfies the VFP equation up to some error −γ div p (ρ t h t ) whose norm is controlled by the rate functional.…”

“…All proofs of singular limits hinge on using certain special structure of the equations; well-known examples are compensated compactness [55,72], the theories of viscosity solutions [19] and entropy solutions [46,69], and the methods of periodic unfolding [16,17] and two-scale convergence [5]. Variational-evolution structure, such as in the case of gradient flows and variational rate-independent systems, also facilitates limits [28,51,53,54,67,70,71].…”

Approximation of bi-variate functions: singular value decomposition versus sparse grids

“…After unfolding (Cioranescu et al, 2008), it gives rise to the product of the macroscopic physical domain and the periodic microscopic domain of the cell problem; see Matache (2002). For multiple scales, a general product appears here which still can be written as the product of two domains, one containing, for example, the macroscopic scale and the other consisting of the product of the domains of the different microscales (Hoang & Schwab, 2005).…”

A symbolic transformation language and its application to a multiscale method

“…The statement of the approximate model (3) derived with the two-scale transform was announced in (Lenczner, 1997). Then several proofs have been published, in (Lenczner and Senouci-Bereksi, 1999), (Casado-Díaz, 2000), (Cioranescu et al, 2002), (Lenczner and Smith, 2007), and (Cioranescu et al, 2008). Here we follow the proof in (Lenczner and Smith, 2007) where an effort has been made to formulate proofs in an algebraic way and to avoid any abstract reasoning in the sequences of formal transformations.…”