On the homogenization of the Helmholtz problem with thin perforated walls of finite length

Semin, Adrien; Delourme, Bérangère; Schmidt, Kersten

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 29-67.

In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in ℝ² with a thin and periodic layer of finite length. The layer may consists of a periodic pertubation of the material coefficients or it is a wall modelled by boundary conditions with an periodic array of small perforations. We consider the periodicity in the layer as the small variable δ and the thickness of the layer to be at the same order. Moreover we assume the thin layer to terminate at re-entrant corners leading to a singular behaviour in the asymptotic expansion of the solution representation. This singular behaviour becomes visible in the asymptotic expansion in powers of δ where the powers depend on the opening angle. We construct the asymptotic expansion order by order. It consists of a macroscopic representation away from the layer, a boundary layer corrector in the vicinity of the layer, and a near field corrector in the vicinity of the end-points. The boundary layer correctors and the near field correctors are obtained by the solution of canonical problems based, respectively, on the method of periodic surface homogenization and on the method of matched asymptotic expansions. This will lead to transmission conditions for the macroscopic part of the solution on an infinitely thin interface and corner conditions to fix the unbounded singular behaviour at its end-points. Finally, theoretical justifications of the second order expansion are given and illustrated by numerical experiments. The solution representation introduced in this article can be used to compute a highly accurate approximation of the solution with a computational effort independent of the small periodicity δ.

Classification : 32S05, 35C20, 35J05, 35J20, 41A60, 65D15
Mots clés : Helmholtz equation, thin periodic interface, method of matched asymptotic expansions, method of periodic surface homogenization, corner singularities

[1] T. Abboud and H. Ammari, Diffraction at a curved grating: TM and TE cases, homogenization. J. Math. Anal. Appl. 202 (1996) 995–1026 | DOI | MR | Zbl

[2] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol. 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964) | MR | Zbl

[3] Y. Achdou, Etude de la réflexion d’une onde électromagnétique par un métal recouvert d’un revêtement métallisé. Technical report, INRIA (1989)

[4] Y. Achdou, Effect of a thin metallized coating on the reflection of an electromagnetic wave. C. R. Acad. Sci. Paris, Ser. I 314 (1992) 217–222. | MR | Zbl

[5] Y. Achdou, O. Pironneau and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147 (1998) 187–218 | DOI | MR | Zbl

[6] M. Artola and M. Cessenat, Diffraction d’une onde électromagnétique par une couche composite mince accolée à un corps conducteur épais. I. Cas des inclusions fortement conductrices. C. R. Acad. Sci. Paris, Ser. I 313 (1991) 231–236 | MR | Zbl

[7] M. Artola and M. Cessenat, Scattering of an electromagnetic wave by a slender composite slab in contact with a thick perfect conductor. II. Inclusions (or coated material) with high conductivity and high permeability. C. R. Acad. Sci. Paris, Ser. I 313 (1991) 381–385 | MR | Zbl

[8] A. Bendali, A. Makhlouf and S. Tordeux, Field behavior near the edge of a microstrip antenna by the method of matched asymptotic expansions. Quart. Appl. Math. 69 (2011) 691–721 | DOI | MR | Zbl

[9] V. Bonnaillie-Noël, M. Dambrine, S. Tordeux and G. Vial, Interactions between moderately close inclusions for the Laplace equation. Math. Models Meth. Appl. Sci. 19 (2009) 1853–1882 | DOI | MR | Zbl

[10] A.-S. Bonnet-Ben Dhia, D. Drissi and N. Gmati, Mathematical analysis of the acoustic diffraction by a muffler containing perforated ducts. Math. Models Meth. Appl. Sci. 15 (2005) 1059–1090 | DOI | MR | Zbl

[11] D. Bresch and V. Milisic, High order multi-scale wall-laws, Part I: the periodic case. Quart. Appl. Math. 68 (2010) 229–253 | DOI | MR | Zbl

[12] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011) | MR | Zbl

[13] G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptot. Anal. 50 (2006) 121–173 | MR | Zbl

[14] I. S. Ciuperca, M. Jai and C. Poignard, Approximate transmission conditions through a rough thin layer: the case of periodic roughness. European J. Appl. Math. 21 (2010) 51–75 | DOI | MR | Zbl

[15] X. Claeys, On the theoretical justification of Pocklington’s equation. Math. Models Meth. Appl. Sci. 19 (2009) 1325–1355 | DOI | MR | Zbl

[16] X. Claeys and B. Delourme, High order asymptotics for wave propagation across thin periodic interfaces. Asymptot. Anal. 83 (2013) 35–82 | MR | Zbl

[17] ConceptsDevelopment Team. Webpage of Numerical C++ Library Concepts 2. http://www.concepts.math.ethz.ch (2016)

[18] M. Dauge, S. Tordeux and G. Vial, Selfsimilar perturbation near a corner: matching versus multiscale expansions for a model problem. In Around the research of Vladimir Maz’ya. II, vol. 12 of Int. Math. Ser. (N. Y.). Springer-Verlag, New York (2010) 95–134 | DOI | MR | Zbl

[19] B. Delourme, Modèles et asymptotiques des interfaces fines et périodiques en électromagnétisme Ph.D. thesis, Université Pierre et Marie Curie (2010)

[20] B. Delourme, K. Schmidt and A. Semin, When a thin periodic layer meets corners: asymptotic analysis of a singular poisson problem. Technicalreport 2015

[21] B. Delourme, K. Schmidt and A. Semin, On the homogenization of thin perforated walls of finite length. Asymptotic Anal. 97 (2016) 211–264 | DOI | MR | Zbl

[22] Ph. Frauenfelder and Ch. Lage, Concepts – an object-oriented software package for partial differential equations. ESAIM: M2AN 36 (2002) 937–951 | DOI | Numdam | MR | Zbl

[23] C.I. Goldstein, A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. Comput. 39 (1982) 309–324 | DOI | MR | Zbl

[24] A.M. Il'In, Matching of asymptotic expansions of solutions of boundary value problems, vol. 102 of Translations of Mathematical Monographs. Translated from the Russian by V. Minachin (V.V. Minakhin). American Mathematical Society, Providence, RI (1992) | MR | Zbl

[25] P. Joly and S. Tordeux, Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion. Multiscale Model. Simul. 5 (2006) 304–336 | DOI | MR | Zbl

[26] P. Joly and S. Tordeux, Matching of asymptotic expansions for waves propagation in media with thin slots. II. The error estimates. ESAIM: M2AN 42 (2008) 193–221 | DOI | Numdam | MR | Zbl

[27] P. Joly and A. Semin. Construction and analysis of improved kirchoff conditions for acoustic wave propagation in a junction of thin slots. ESAIM Proc. 25 (2008) 44–67 | DOI | MR | Zbl

[28] V.A. Kozlov, V.G. Mazya and J. Rossmann, Elliptic boundary value problems in domains with point singularities, vol. 52 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1997) | MR | Zbl

[29] A.L. Madureira and F. Valentin. Asymptotics of the Poisson problem in domains with curved rough boundaries. SIAM J. Math. Anal. 38 (2006/07) 1450–1473 | MR | Zbl

[30] A. Makhlouf, Justification et amélioration de modèles d’antenne patch par la méthode des développements asymptotiques raccordés. Ph.D. thesis, Institut National des Sciences Appliquées de Toulouse (2008)

[31] V. Maz’Ya, S. Nazarov and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. 111 of Operator Theory: Advances and Applications. Translated from the German by Georg Heinig and Christian Posthoff. Birkhäuser Verlag, Basel (2000) | MR | Zbl

[32] S.A. Nazarov. The Neumann problem in angular domains with periodic and parabolic perturbations of the boundary. Tr. Mosk. Mat. Obs. 69 (2008) 182–241 | MR

[33] S. Nicaise and A.-M. Sändig, General interface problems. I, II. Math. Methods Appl. Sci. 17 (1994) 395–429, 431–450 | DOI | MR | Zbl

[34] G. Panasenko, High order asymptotics of solutions of problems on the contact of periodic structures. Sbornik: Math. 38 (1981) 465–494 | Zbl

[35] G.A. Pavliotis and A.M. Stuart, Multiscale Methods: Averaging and Homogenization. Springer (2008) | MR | Zbl

[36] J.-R. Poirier, A. Bendali, and P. Borderies, Impedance boundary conditions for the scattering of time-harmonic waves by rapidly varying surfaces. IEEE Trans. Antennas and Propagation 54 (2006) 995–1005 | DOI

[37] J.-R. Poirier, A. Bendali, P. Borderies and S. Tournier, High order asymptotic expansion for the scattering of fast oscillating periodic surfaces. In Proc. 9th Int. Conf. on Mathematical and Numerical Aspects of Waves Propagation (Waves 2009), Pau, France 2009

[38] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains J. Functional Anal. 18 (1975) 27–59 | DOI | MR | Zbl

[39] E. Sánchez-Palencia, Nonhomogeneous media and vibration theory. Vol. 127 of Lecture Notes in Physics. Springer Verlag, Berlin (1980) | MR | Zbl

[40] E. Sánchez-Palencia, Un problème d’écoulement lent d’un fluide incompressible au travers d’une paroi finement perforée. In Homogenization methods: theory and applications in physics (Bréau-sans-Nappe 1983), vol. 57 of Collect. Dir. Études Rech. Élec. France. Eyrolles, Paris (1985) 371–400 | MR

[41] K. Schmidt and P. Kauf, Computation of the band structure of two-dimensional photonic crystals with hp finite elements. Comput. Methods Appl. Mech. Engrg. 198 (2009) 1249–1259 | DOI | MR | Zbl

[42] C. Schwab, p- and hp-finite element methods: Theory and applications in solid and fluid mechanics. Oxford University Press, Oxford, UK (1998) | MR | Zbl

[43] M. Van Dyke Perturbation methods in fluid mechanics. Applied Mathematics and Mechanics. Vol. 8. Academic Press, New York (1964) | MR | Zbl