Homogenization of the Transmission Eigenvalue Problem for a Periodic Media

Transcription

1 Homogenization of the Transmission Eigenvalue Problem for a Periodic Media Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas iharris@math.tamu.edu Joint work with: F. Cakoni and H. Haddar Research Funded by: NSF Grant DMS and University of Delaware Graduate Student Fellowship SIAM CSE Conference 2017 Homogenization of the TE for Periodic Material 1 / 20

2 Problem Statement for Periodic Materials We consider the time harmonic Acoustic scattering in R 3 or Electromagnetic scattering in R 2 (TE-polarization case). u i D u s u = u R d \D Figure: An example of a periodic scatterer. Homogenization of the TE for Periodic Material 2 / 20

3 The Scattering Problem for Periodic Media The time harmonic scattering by a periodic media with scattered field u s ɛ H 1 loc (Rd ) and incident field u i = exp(ikx ŷ), where the total field u ɛ = u s ɛ + u i and scattered field satisfy u s ɛ + k 2 u s ɛ = 0 in R d \ D A(x/ɛ) u ɛ + k 2 n(x/ɛ)u ɛ = 0 in D u ɛ = uɛ s + u i and u ɛ = ν A ν (us ɛ + u i ) on D ( ) lim r m 1 u s 2 ɛ iku s r ɛ = 0. r Homogenization of the TE for Periodic Material 3 / 20

4 Convergence of the Coefficients The matrix A(y) L ( Y, R d d) is Y -periodic symmetric positive definite and the function n(y) L (Y ) is a positive Y -periodic function. We have that as ɛ 0 n ɛ := n(x/ɛ) n h := 1 Y Y n(y) dy weakly in L A ɛ := A(x/ɛ) A h in the sense of H-convergence (i.e. for u ɛ u in H 1 (D) then A ɛ u ɛ A h u in [L 2 (D)] m ) where A h is a constant symmetric positive definite matrix Homogenization of the TE for Periodic Material 4 / 20

5 Far-Field Operator It is known that the radiating scattered field which depends on the incident direction ŷ, has the following asymptotic expansion u s ɛ(x, ŷ; k) = eik x x m 1 2 { ( )} 1 u ɛ (ˆx, ŷ; k) + O x We now define the far field operator as L 2 (S) L 2 (S) (Fg)(ˆx) := u ɛ (ˆx, ŷ; k)g(ŷ) ds(ŷ) S as x where S = {x R m : x = 1} is the unit circle or sphere. Homogenization of the TE for Periodic Material 5 / 20

6 The Inverse Problem: Obtain information about the macro/micro structure of the periodic scattering object where the period is characterized by a small parameter ɛ 1 from the scattered field. F. Cakoni, H. Haddar and I. Harris Homogenization approach for the transmission eigenvalue problem for periodic media and application to the inverse problem, Inverse Problems and Imaging , Volume 9, Issue 4, 2015 (arxiv: ). Homogenization of the TE for Periodic Material 6 / 20

7 The TE-Problem for a Periodic Media Homogenization is used to study composite periodic media. We are interested in the limiting case as ɛ 0 for the TE-Problem. Find non-trivial { k ɛ, (w ɛ, v ɛ ) } R + X (D) such that: A(x/ɛ) w ɛ + k 2 ɛ n(x/ɛ)w ɛ = 0 in D v ɛ + k 2 ɛ v ɛ = 0 in D w ɛ = v ɛ and w ɛ = v ɛ ν A ν Note that the spaces for the solution (w ɛ, v ɛ ) will become precise later since they depend on whether A = I or A I. on D Homogenization of the TE for Periodic Material 7 / 20

8 Reconstructing the Real TEs Consider the Far-Field equation is given by (Fg z )(ˆx) = exp( ikz ˆx) for a z D. Let g z,δ be the regularized solution to the Far-Field equation if k ɛ R + is not a TE then g z,δ L 2 (S) is bounded as δ 0 if k ɛ R + is a TE then g z,δ L 2 (S) is unbounded as δ 0. Where we assume that lim δ 0 Fg z,δ exp( ikz ˆx) L 2 (S) = 0. Homogenization of the TE for Periodic Material 8 / 20

9 Numerical Examples i.e. plot k g α z Figure: A = Diag(5, 6) and n = 2 where the domain D is a 2 2 Square. I. Harris, F. Cakoni and J. Sun Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids. Inverse Problems 30 (2014) Homogenization of the TE for Periodic Material 9 / 20

10 Numerical Examples We now compute the TEs using the Far-Field Equation (FFE) and the FEM where we fix A = Diag(5, 6) and n = 2. Table: Comparison of FFE Computation v.s. FEM Calculations Method Domain 1st TE 2nd TE FFE square (2 2) FEM square (2 2) FFE circle (R = 1) FEM circle (R = 1) Homogenization of the TE for Periodic Material 10 / 20

11 Alternative Methods for computing TEs Inside-Out Duality Method Stefan Peters The Inside-Outside Duality in Inverse Scattering Theory. Ph.D. Thesis (2016). Generalized LSM Lorenzo Audibert Qualitative methods for heterogeneous media. Ph.D. Thesis (2015). Homogenization of the TE for Periodic Material 11 / 20

12 The case of an Isotropic media (i.e. A = I ) For this case (w ɛ, v ɛ ) L 2 (D) L 2 (D) with u ɛ = w ɛ v ɛ H 2 0 (D), where we have that the difference u ɛ satisfies for n ɛ := n(x/ɛ) 0 = ( + kɛ 2 ) 1 ( ) n ɛ + k 2 n ɛ 1 ɛ uɛ in D. The equivalent variational form is given by 1 ( uɛ + k 2 )( ɛ u ɛ ϕ + k 2 n ɛ 1 ɛ n ɛ ϕ ) dx = 0 for all ϕ H0 2 (D). D We can rewrite the variational form as A ɛ,kɛ (u ɛ, ϕ) k 2 ɛ B(u ɛ, ϕ) = 0 for all ϕ H 2 0 (D). Homogenization of the TE for Periodic Material 12 / 20

13 Lemma (Cakoni-Haddar-IH) Let n min n(y) n max, then there exists an infinite sequence of real transmission eigenvalues k ɛ, j for j N such that k j (n max, D) k ɛ, j < k j (n min, D) if n min > 1 k j (n min, D) k ɛ, j < k j (n max, D) if n max < 1. Bounded Eigenfunctions: We let u ɛ denote the eigenfunction corresponding to the eigenvalue k ɛ with u ɛ H 1 (D) = 1 for all ɛ > 0 α u ɛ 2 L 2 (D) A ɛ,k ɛ (u ɛ, u ɛ ) = k 2 ɛ B(u ɛ, u ɛ ) k 2 ɛ u ɛ 2 H 1 (D) provided that n min > 1 or n max < 1 = u ɛ are bounded in H 2 (D) Then appeal to Elliptic Regularity and Homogenization Theory. Homogenization of the TE for Periodic Material 13 / 20

14 Theorem (Cakoni-Haddar-IH) If n min > 1 or n max < 1, then there is a subsequence of { k ɛ, (w ɛ, v ɛ ) } R + L 2 (D) L 2 (D) that converges weakly to (v, w) L 2 (D) L 2 (D) and k ɛ k such that w + k 2 n h w = 0 and v + k 2 v = 0 in D w = v and w ν = v on D ν provided that k ɛ is bounded, where n h = 1 n(y) dy. Y Y Homogenization of the TE for Periodic Material 14 / 20

15 Reconstructing Material Properties A = I We can reconstruct the effective material property n h by finding a n 0 such that k 1 (n 0 ) = k 1 (n ɛ ) where w + k 2 n 0 w = 0 and v + k 2 v = 0 in D w = v and w ν = v ν on D. Now let n(x/ɛ) = sin 2 (2πx 1 /ɛ) + 2 Table: Reconstruction from scattering data ɛ k 1 (n ɛ ) n h n Homogenization of the TE for Periodic Material 15 / 20

16 Theorem (Cakoni-Haddar-IH) Assume that A(y) I and n(y) 1 have different sign in Y, or if n(y) = 1 and A(y) I is positive (or negative) definite. Then there is a subsequence of { kɛ, (w ɛ, v ɛ ) } R + H 1 (D) H 1 (D) that converges weakly to (w, v) H 1 (D) H 1 (D) and k ɛ k that satisfies A h w + k 2 n h w = 0 and v + k 2 v = 0 in D w w = v and = v on D ν ν Ah provided that k ɛ is bounded, where A(x/ɛ) A h in the sense of H-convergence as ɛ 0. Homogenization of the TE for Periodic Material 16 / 20

17 Reconstructing Material Properties n = 1 We can reconstruct the effective material property A h = a h I by finding an a 0 such that k 1 (a 0 ) = k 1 (A ɛ ) where a 0 w + k 2 w = 0 and v + k 2 v = 0 in D w w = v and a 0 ν = v on D. ν ( Now let A(x/ɛ) = 1 sin 2 ) (2πx 2 /ɛ) cos 2 (2πx 1 /ɛ) + 1 Table: Reconstruction from scattering data ɛ k 1 (A ɛ ) a h a Homogenization of the TE for Periodic Material 17 / 20

18 In the paper we have also considered: The convergence of the interior transmission problem for the cases where A = I and A I Construct a bulk corrector to prove strong convergence for A I Numerical test for the order of convergence Open Problem: Determining the corrector term in the asymptotic expansion for transmission eigenvalues to determine microstructure information. Homogenization of the TE for Periodic Material 18 / 20

19 Some References A. Bensoussan, J.L. Lions, G. Papanicolau Asymptotic Analysis for Periodic Structures Chelsea Publications, 1978 F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory Springer, Berlin I. Harris, Non-Destructive Testing of Anisotropic Materials University of Delaware, Ph.D. Thesis (2015). Homogenization of the TE for Periodic Material 19 / 20

20 Figure: Questions? Homogenization of the TE for Periodic Material 20 / 20