The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space

Abstract

A two-scale Fourier transform for periodic homogenization in Fourier space is introduced. The transform connects the various existing techniques for periodic homogenization, i.e., two-scale convergence, periodic unfolding and the Floquet-Bloch expansion approach to homogenization. It turns out that the two-scale compactness results are easily obtained by the use of the two-scale Fourier transform. Moreover, the Floquet-Bloch eigenvalue problems for differential operators is recovered in a natural and straight forward way by the use of this transform. The transform is generalized to the (N + 1)-scale case. N. Wellander / Periodic homogenization in Fourier spaceNguetseng [24] and further developed in [1,2] and many other papers thereafter, e.g. in [19] and [20]. The Floquet-Bloch expansion [6,16] and the corresponding Bloch-wave homogenization method [3,5,[9][10][11] is a high-frequency method which recently has been further developed. It can be used to find the usual homogenization results, but more importantly, it provides dispersion relations for wave propagation in periodic structures [25], e.g. for photonic band gap structures. In [4] a two-scale transform was defined and used to homogenize fluid flow in porous media, see also [7,8,17] and [23] (in [8] and [17] it is called periodic unfolding). The two-scale transform simplifies the existing two-scale convergence proofs. The idea with the two-scale transform is to map bounded sequences of functions defined on L 2 (Ω) to sequences defined in the product space L 2 (Ω × T n ) and then taking the weak limit in L 2 (Ω × T n ). It gives the same limit as in the two-scale convergence method but when proving corrector results we do not have to assume any regularity of the solution to the cell problem, which was noted also in [9] using Bloch analysis. In [18] a similar approach was used to map nonperiodic coefficients to functions defined on Ω × T n .In this paper we give an explicit example of a two-scale transform operator by means of a two-scale Fourier transform, presented earlier in [36] and [37]. It turns out that the standard two-scale convergence results can be obtained at once in the frequency space, as well as we recover the classical Floquet-Bloch eigenvalue problems by the use of this transform.The paper is organized in the following way. In Section 2 we give some basic definitions, mainly to do with two-scale convergence. In Section 3 we define and explore the two-scale Fourier transform and its application to homogenization of PDEs. In Section 4 we define the two-scale transform operator by use of the two-scale Fourier transform. It is characterized and applied to the standard periodic elliptic PDE, proving convergence and correctors. Section 5 is devoted to the Floquet-Bloch homogenization method. We demonstrate how the corresponding eigenvalue problems are obtained by the use of the two-scale Fourier transform. In Section 6 we generalize to the (N + 1)-scale Fourier transform. Finally, in Section 7 we close the paper by some concluding remarks. Prelimi...

Two‐scale cut‐and‐projection convergence; homogenization of quasiperiodic structures

“…A similar method is the periodic unfolding approach , in which one first maps the original sequence of functions to a sequence that is defined on , and then takes the usual weak limit in suitable function spaces, using this extended domain. This is similar to the Fourier transform approach proposed in .…”

Homogenization of quasiperiodic structures and two-scale cut-and-projection convergence

“…Proof. The proof follows the line of the proof of Proposition 7 in [26], or by using Lemma 3 similarly as in Proposition 4, as follows. The limit (28) follows due to the a priori estimate and standard arguments, using admissible test functions φ(x, y) and ηφ(x, y) for the sequences {u η } and {curl u η }, respectively.…”

“…A similar method is the periodic unfolding approach [11], in which one first maps the original sequence of functions to a sequence that is defined on R n ×]0, 1[ n , and then takes the usual weak limit in suitable function spaces, using this extended domain. This approach is similar to the approach proposed in [26].…”

Homogenization of Nonlocal Electrostatic Problems by Means of the Two-Scale Fourier Transform

“…In this paper we return to a two-scale Fourier transform, which belongs to the class of two-scale transforms, presented in Wellander (2004;2007;2009). The transform is applied to nonlocal constitutive relations in electrostatic applications for periodic composites.…”

“…We define the two-scale Fourier transform, which is nothing but the standard Fourier transform evaluated at ξ + ε −1 m where ξ is restricted to a cube in R n with sidelength 1/ε, Wellander (2009).…”