Asymptotic behavior of acoustic waves scattered by very small obstacles

Barucq, Hélène; Diaz, Julien; Mattesi, Vanessa; Tordeux, Sebastien

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S705-S731

The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain. The use of asymptotic models is very efficient and the purpose of this work is to provide a rigorous justification of a new asymptotic model for low-cost numerical simulations. This model is based on asymptotic near-field and far-field developments that are then matched by a key procedure that we describe and demonstrate. We show that it is enough to focus on the regular part of the wave field to rigorously establish the complete asymptotic expansion. For that purpose, we provide an error estimate which is set in the whole space, including the transition region separating the near-field from the far-field area. The proof of convergence is established through Kondratiev’s seminal work on the Laplace equation and involves the Mellin transform. Numerical experiments including multiple scattering illustrate the efficiency of the resulting numerical method by delivering some comparisons with solutions computed with a finite element software.

Classification : 35C20, 35L05, 74J20
Keywords: Acoustic wave propagation, matched asymptotic expansion method, scattering problem, Mellin transform, singularity theory

[1] S. Acosta, High order surface radiation conditions for time-harmonic waves in exterior domains. Comput. Methods Appl. Mech. Eng. 322 (2017) 296–310. | MR

[2] M. Ainsworth, P. Monk and W. Muniz, Dispersive and dissipative properties of discontinuous galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27 (2006) 5–40. | MR | Zbl

[3] H. Alzubaidi, X. Antoine and C. Chniti, Formulation and accuracy of on-surface radiation conditions for acoustic multiple scattering problems. Appl. Math. Comput. 277 (2016) 82–100. | MR

[4] C. Baldassari, H. Barucq, H. Calandra and J. Diaz, Numerical performances of a hybrid local-time stepping strategy applied to the reverse time migration. Geophys. Prospect. 59 (2011) 907–919.

[5] C. Baldassari, H. Barucq, H. Calandra, B. Denel and J. Diaz, Performance analysis of a high-order discontinuous galerkin method application to the reverse time migration. Commun. Comput. Phys. 11 (2012) 660–673. | MR

[6] H. Barucq, R. Djellouli and E. Estecahandy, Efficient dg-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems. Int. J. Numer. Methods Eng. 98 (2014) 747–780. | MR

[7] H. Barucq, V. Mattesi and S. Tordeux, The mellin transform. Technical Report RR-8743, INRIA Bordeaux Sud Ouest (2015).

[8] H. Barucq, F. Faucher and H. Pham, Localization of small obstacles from back-scattered data at limited incident angles with full-waveform inversion. J. Comput. Phys. 370 (2018) 1–24. | MR

[9] A. Bendali, P.-H. Cocquet and S. Tordeux, Approximation by multipoles of the multiple acoustic scattering by small obstacles in three dimensions and application to the foldy theory of isotropic scattering. Arch. Ration. Mech. Anal. 219 (2016) 1017–1059. | MR

[10] N. Burk and G. Lebeau, Annales Scientifiques de I’École Normale Supérieure – Injections de Sobolev Probabilistes et Applications 4 (2013). | MR | Zbl | Numdam

[11] M. Cassier and C. Hazard, Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the foldy–lax model. Wave Motion 50 (2013) 18–28. | MR

[12] D. P. Challa and M. Sini, On the justification of the foldy–lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes. Multiscale Model. Simul. 12 (2014) 55–108. | MR

[13] X. Claeys, Analyse asymptotique et numérique de la diffraction d’ondes par des fils minces. Ph.D. thesis, Université de Versailles Saint-Quentin-en-Yvelines (2008).

[14] M. Costabel and M. Dauge, Les problèmes à coins en 10 leçons.

[15] J. D. De Basabe, M. K. Sen and M. F. Wheeler, The interior penalty discontinuous galerkin method for elastic wave propagation: grid dispersion. Geophys. J. Int. 175 (2008) 83–93.

[16] J. Diaz and M. J. Grote, Multi-level explicit local time-stepping methods for second-order wave equations. Comput. Methods Appl. Mech. Eng. 291 (2015) 240–265. | MR

[17] V. M. Dikasov, An inverse problem for the wave equation in a ball. Funct. Anal. App. 25 (1991) 56–58. | MR | Zbl

[18] L. L. Foldy, The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Phys. Rev. 67 (1945) 107. | MR | Zbl

[19] R. F. B. Frank, W. J. Olver, D. W. Lozier and C. W. Clark, NIST Handbook of Mathematical Functions. NIST and Cambridge University Press (2010). | Zbl

[20] M. J. Grote, A. Schneebeli and D. Schötzau, Discontinuous galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44 (2006) 2408–2431. | MR | Zbl | DOI

[21] A. Gumerov and R. Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions. Elsevier (2004). | MR

[22] C. Gundlach, J. Martín-García and D. Garfinkle, Summation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions. Class. Quantum Grav. 30 (2013) 145003. | MR | Zbl | DOI

[23] A. M. Il’In, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. American Mathematical Society (1991).

[24] V. A. Kondratiev, Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967) 209–292. | MR | Zbl

[25] J. Labat, Modélisation multi-échelle de la diffraction des ondes électromagnétiques par de petits obstacles. Ph.D. thesis, Pau (2019).

[26] J. Labat, V. Péron and S. Tordeux, Equivalent multipolar point-source modeling of small spheres for fast and accurate electromagnetic wave scattering computations. Wave Motion 92 (2019) 102409. | MR | DOI

[27] N. N. Lebedev, Spherical Functions & Their Applications. Dover Publication (1975).

[28] J. Li, H. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers. SIAM J. Appl. Math. 73 (2013) 1721–1746. | MR | Zbl | DOI

[29] J. Li, H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement. SIAM J. Imaging Sci. 6 (2013) 2285–2309. | MR | Zbl | DOI

[30] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers. Multiscale Model. Simul. 12 (2014) 927–952. | MR | DOI

[31] P. A. Martin, Multiple Scattering: Interaction of Time-harmonic Waves with $N$ Obstacles. In Vol. 107 of Cambridge University Press (2006). | MR | Zbl

[32] V. Mattesi, Propagation des ondes dans un milieu comportant des petites hétérogénéités: analyse asymptotique et calcul numérique. Ph.D. thesis, Université de Pau et des Pays de l’Adour (2014).

[33] V. Mattesi and S. Tordeux, Equivalent source modelling of small heterogeneities in the context of 3D time-domain wave propagation equation. In: Waves 2013. Gammarth, Tunisia (2013).

[34] M. N’Diaye, Étude et développement de méthodes numériques d’ordre élevé pour la résolution des équations différentielles ordinaires (EDO): Applications à la résolution des équations d’ondes acoustiques et électromagnétiques. Ph.D. thesis, 2017. Thèse de doctorat dirigée par Barucq, Hélène et Duruflé, Marc Mathématiques appliquées Pau (2017).

[35] J.-C. Nédélec, Acoustic and electromagnetic equations. In: Vol. 144 of Applied Mathematical Sciences. Integral Representations for Harmonic Problems. Springer-Verlag, New York (2001). | MR | Zbl

[36] J. Novak, J.-L. Cornou and N. Vasset, A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere. J. Comput. Phys. 229 (2010) 399–414. | MR | Zbl | DOI

[37] A. D. Poularikas, Transforms and Applications Handbook. CRC Press, 2010. | MR | Zbl | DOI

[38] M. Rietmann, M. Grote, D. Peter and O. Schenk, Newmark local time stepping on high-performance computing architectures. J. Comput. Phys. 334 (2017) 308–326. | MR | DOI

[39] M. E. Taylor, Partial differential equations I. Basic theory, 2nd edition. In: Vol. 115 of Applied Mathematical Sciences. Springer, New York (2011). | MR | Zbl

[40] S. Tordeux, Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes. Ph.D. thesis, Université de Versailles Saint-Quentin-en-Yvelines (2004).

[41] V. Villamizar, S. Acosta and B. Dastrup, High order local absorbing boundary conditions for acoustic waves in terms of farfield expansions. J. Comput. Phys. 333 (2017) 331–351. | MR | DOI

[42] S. N. Vladimir Maz’Ya and B.1 Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhäuser (2000). | MR