A high–order perturbation of surfaces (HOPS) approach to Fokas integral equations: Three–dimensional layered–media scattering

Nicholls, David

doi:10.1090/qam/1411

Author: David P. Nicholls
Journal: Quart. Appl. Math. 74 (2016), 61-87
MSC (2010): Primary 65R20, 65N35
DOI: https://doi.org/10.1090/qam/1411
Published electronically: December 3, 2015
MathSciNet review: 3472520
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Abstract: In this paper we discuss a novel High–Order Perturbation of Surfaces (HOPS) method for the simulation of linear acoustic waves in a three–dimensional layered, periodic structure. The model we consider is that of linear time–harmonic wave propagation which generates three–dimensional, quasiperiodic, outgoing solutions of Helmholtz equations, coupled across irregular layer interfaces. We significantly enhance and stabilize the approach of the author and D. Ambrose, which is a formulation of the problem utilizing the integral equation formalism of Fokas and collaborators. The method is phrased in terms of interfacial variables resulting in a method which has an order of magnitude fewer unknowns than a volumetric approach to this problem. In contrast to classical integral equation formulations, the current contribution does not require specialized quadratures or periodized fundamental solutions. Additionally, as a result of the HOPS philosophy, our new approach is not only faster and better conditioned than the algorithm of the author and Ambrose, it also delivers the scattering returns of an entire family of solutions with a single simulation rather than requiring a new approximation for each profile of interest. Detailed numerical simulations are presented which demonstrate the efficiency, fidelity, and spectral accuracy which can be realized with this new methodology.

References

M. J. Ablowitz, A. S. Fokas, and Z. H. Musslimani, On a new non-local formulation of water waves, J. Fluid Mech. 562 (2006), 313–343. MR 2263547, DOI 10.1017/S0022112006001091
David M. Ambrose and David P. Nicholls, Fokas integral equations for three dimensional layered-media scattering, J. Comput. Phys. 276 (2014), 1–25. MR 3252567, DOI 10.1016/j.jcp.2014.07.018
George A. Baker Jr. and Peter Graves-Morris, Padé approximants, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 59, Cambridge University Press, Cambridge, 1996. MR 1383091, DOI 10.1017/CBO9780511530074
Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR 538168
Florian Bleibinhaus and Stéphane Rondenay, Effects of surface scattering in full-waveform inversion, Geophysics 74(6):WCC69–WCC77, 2009.
L. M. Brekhovskikh and Yu. P. Lysanov, Fundamentals of ocean acoustics, 2nd ed., Springer Series on Wave Phenomena, vol. 8, Springer-Verlag, Berlin, 1991. MR 1100922, DOI 10.1007/978-3-662-07328-5
Oscar P. Bruno and Fernando Reitich, Numerical solution of diffraction problems: A method of variation of boundaries, J. Opt. Soc. Am. A 10(6):1168–1175, 1993.
Oscar P. Bruno and Fernando Reitich, Numerical solution of diffraction problems: A method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities, J. Opt. Soc. Am. A 10(11):2307–2316, 1993.
Oscar P. Bruno and Fernando Reitich, Numerical solution of diffraction problems: A method of variation of boundaries. III. Doubly periodic gratings, J. Opt. Soc. Am. A 10(12):2551–2562, 1993.
R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, An improved operator expansion algorithm for direct and inverse scattering computations, Waves Random Media 9 (1999), no. 3, 441–457. MR 1705850, DOI 10.1088/0959-7174/9/3/311
David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR 1635980, DOI 10.1007/978-3-662-03537-5
T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Extraordinary optical transmission through sub-wavelength hole arrays, Nature 391(6668):667–669, 1998.
Athanassios S. Fokas, A unified approach to boundary value problems, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 78, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. MR 2451953, DOI 10.1137/1.9780898717068
L. Geli, P. Y. Bard, and B. Jullien, The effect of topography on earthquake ground motion: a review and new results, Bulletin of the Seismological Society of America 78(1):42–63, 1988.
David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. MR 0520152
L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325–348. MR 918448, DOI 10.1016/0021-9991(87)90140-9
J. Homola, Surface plasmon resonance sensors for detection of chemical and biological species, Chemical Reviews 108(2):462–493, 2008.
D. Komatitsch and J. Tromp, Spectral-element simulations of global seismic wave propagation-I. Validation, Geophysical Journal International 149(2):390–412, 2002.
Harun Kurkcu and Fernando Reitich, Stable and efficient evaluation of periodized Green’s functions for the Helmholtz equation at high frequencies, J. Comput. Phys. 228 (2009), no. 1, 75–95. MR 2464068, DOI 10.1016/j.jcp.2008.08.021
Nathan C. Lindquist, Timothy W. Johnson, Jincy Jose, Lauren M. Otto, and Sang-Hyun Oh, Ultrasmooth metallic films with buried nanostructures for backside reflection-mode plasmonic biosensing, Annalen der Physik 524:687–696, 2012.
Alison Malcolm and David P. Nicholls, A boundary perturbation method for recovering interface shapes in layered media, Inverse Problems 27 (2011), no. 9, 095009, 18. MR 2831891, DOI 10.1088/0266-5611/27/9/095009
Alison Malcolm and David P. Nicholls, A field expansions method for scattering by periodic multilayered media, Journal of the Acoustical Society of America 129(4):1783–1793, 2011.
MATLAB, version 7.10.0 (R2010a), The MathWorks Inc., Natick, Massachusetts, 2010.
D. Michael Milder, An improved formalism for rough-surface scattering of acoustic and electromagnetic waves, in Proceedings of SPIE - The International Society for Optical Engineering (San Diego, 1991), volume 1558, Int. Soc. for Optical Engineering, Bellingham, WA, 1991, pp. 213–221.
D. Michael Milder, An improved formalism for wave scattering from rough surfaces, J. Acoust. Soc. Am. 89(2):529–541, 1991.
M. Moskovits, Surface–enhanced spectroscopy, Reviews of Modern Physics 57(3):783–826, 1985.
David P. Nicholls, Three-dimensional acoustic scattering by layered media: a novel surface formulation with operator expansions implementation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 (2012), no. 2139, 731–758. MR 2892310, DOI 10.1098/rspa.2011.0555
David P. Nicholls and Fernando Reitich, A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 6, 1411–1433. MR 1869643, DOI 10.1017/S0308210500001463
David P. Nicholls and Fernando Reitich, Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators, J. Comput. Phys. 170 (2001), no. 1, 276–298. MR 1843612, DOI 10.1006/jcph.2001.6737
David P. Nicholls and Fernando Reitich, Analytic continuation of Dirichlet-Neumann operators, Numer. Math. 94 (2003), no. 1, 107–146. MR 1971215, DOI 10.1007/s002110200399
D. P. Nicholls, F. Reitich, T. Johnson, and S.-H. Oh, Fast high–order perturbation of surfaces (HOPS) methods for simulation of multi–layer plasmonic devices and metamaterials, Journal of Optical Society of America, A, Volume 31, Issue 8, 1820–1831, 2014.
Roger Petit (ed.), Electromagnetic theory of gratings, Topics in Current Physics, vol. 22, Springer-Verlag, Berlin-New York, 1980. MR 609533
R. Gerhard Pratt, Frequency-domain elastic wave modeling by finite differences: A tool for crosshole seismic imaging, Geophysics 55(5):626–632, 1990.
F. Reitich, T. Johnson, S.-H. Oh, and G. Meyer, A fast and high-order accurate boundary perturbation method for characterization and design in nanoplasmonics, Journal of the Optical Society of America, A 30:2175, 2013.
F. Reitich and K. K. Tamma, State-of-the-art, trends, and directions in computational electromagnetics [Special issue on computational electromagnetics], CMES Comput. Model. Eng. Sci. 5 (2004), no. 4, 287–294. MR 2070390
F. J. Sanchez-Sesma, V. J. Palencia, and F. Luzon, Estimation of local site effects during earthquakes: An overview, in V. K. Gupta, editor, from Seismic Source to Structural Response, University of Southern California, Department of Civil Engineering, 2004, pp. 44–70.
F. J. Sanchez-Sesma, E. Perez-Rocha, and S. Chavez-Perez, Diffraction of elastic waves by three-dimensional surface irregularities. Part II, Bulletin of the Seismological Society of America 79(1):101–112, 1989.
Peter J. Shull, Nondestructive Evaluation: Theory, Techniques, and Applications, Marcel Dekker, 2002.
E. A. Spence and A. S. Fokas, A new transform method I: domain-dependent fundamental solutions and integral representations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), no. 2120, 2259–2281. MR 2659494, DOI 10.1098/rspa.2009.0512
E. A. Spence and A. S. Fokas, A new transform method II: the global relation and boundary-value problems in polar coordinates, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), no. 2120, 2283–2307. MR 2659495, DOI 10.1098/rspa.2009.0513
L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing, Wiley, New York, 1985.
J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics 74(6):WCC1–WCC26, 2009.
O. C. Zienkiewicz, The finite element method in engineering science, McGraw-Hill, London-New York-Düsseldorf, 1971. The second, expanded and revised, edition of The finite element method in structural and continuum mechanics. MR 0315970