In this paper we propose an original approach for the simulation of the time-dependent response of a floating elastic plate using the so-called Singularity Expansion Method. This method consists in computing an asymptotic behaviour for large time obtained by means of the Laplace transform by using the analytic continuation of the resolvent of the problem. This leads to represent the solution as the sum of a discrete superposition of exponentially damped oscillating motions associated to the poles of the analytic continuation called resonances of the system, and a low frequency component associated to a branch point at frequency zero. We present the mathematical analysis of this method for the two-dimensional sea-keeping problem of a thin elastic plate (ice floe, floating runway, ...) and provide some numerical results to illustrate and discuss its efficiency.
@article{M2AN_2007__41_5_925_0,
author = {Hazard, Christophe and Loret, Fran\c cois},
title = {The singularity expansion method applied to the transient motions of a floating elastic plate},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {41},
year = {2007},
pages = {925-943},
doi = {10.1051/m2an:2007040},
mrnumber = {2363889},
zbl = {1140.74029},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2007__41_5_925_0}
}
Hazard, Christophe; Loret, François. The singularity expansion method applied to the transient motions of a floating elastic plate. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) pp. 925-943. doi : 10.1051/m2an:2007040. http://gdmltest.u-ga.fr/item/M2AN_2007__41_5_925_0/
[1] and, Handbook of mathematical functions. Dover Publications, New York, 9th edn. (1970).
[2] and, A class of analytic perturbations for one-body Schrödinger hamiltonians. Comm. Math. Phys. 22 (1971) 269-279. | Zbl 0219.47011
[3] and, Numerical experiments on the resonance poles associated to acoustic and elastic scattering by a plane crack, in Mathematical and Numerical Aspects of Wave Propagation, E. Bécache et al. Eds., SIAM (1995). | Zbl 0874.73077
[4] , The Neumann problem in the half-space. C. R. Acad. Sci. Paris Ser. I 335 (2002) 151-156. | Zbl 1019.35026
[5] and, Les résonances d'un trou noir de Schwarzschild. Ann. Henri. Poincarré 59 (1993) 280-294. | Numdam | Zbl 0793.53094
[6] and, Spectral properties of many body Schrödinger operators with dilation analytic interactions. Comm. Math. Phys. 22 (1971) 280-294. | Zbl 0219.47005
[7] , The Singularity Expansion Method, in Transient Electromagnetic Fields, L.B. Felsen Ed., Springer-Verlag, New York (1976).
[8] , Analyse fonctionnelle, Théorie et application. Masson, Paris (1983). | MR 697382 | Zbl 0511.46001
[9] and, Resonance expansions in semi-classical propagation. Comm. Math. Phys. 232 (2001) 1-12. | Zbl 1042.81582
[10] and, Solution of equations involving analytic functions. J. Comput. Phys. 45 (1982) 210-220. | Zbl 0484.65026
[11] and, A justification of the two-dimensional linear plate model. J. Mécanique 18 (1979) 315-344. | Zbl 0415.73072
[12] and, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 1, Dunod, Paris (1984). | MR 792484 | Zbl 0642.35001
[13] and, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 3, Dunod, Paris (1985). | MR 902802 | Zbl 0749.35005
[14] and, Hybrid ray mode analysis of transient scattering, in Low and Frequency Asymptotics, V.K. Varadan and V.V. Varadan Eds. (1986).
[15] and, Light fluid approximation for sound radiation and diffraction by thin elastic plates. J. Sound Vibration 213 (1998) 333-374.
[16] , The Singularity Expansion Method, in Fifth International Conference on Mathematical and Numerical Aspects of Wave propagation, SIAM (2000) 494-498. | Zbl 0960.76072
[17] and, Determination of scattering frequencies for an elastic floating body. SIAM J. Math. Anal. 24 (1993) 1458-1514. | Zbl 0784.76013
[18] and, Generalized eigenfunction expansions for conservative scattering problems with an application to water waves. Proceedings of the Royal Society of Edinburgh (2007) Accepted. | MR 2359924 | Zbl pre05235750
[19] , Perturbation theory for linear operators. Springer-Verlag, New York (1984). | Zbl 0531.47014
[20] and, Generic simplicity of resonances. Helv. Phys. Acta 8 (1995) 531-538. | Zbl 0844.47040
[21] , Theory of functions, Part II. Dover, New York (1947). | MR 19722
[22] and, Computing the zeros of analytic functions. Lect. Notes Math. 1727, Springer (2000). | MR 1754959 | Zbl 0945.65018
[23] , and, Linear Water Waves, a Mathematical Approach. Cambridge (2002). | Zbl 0996.76001
[24] , Problèmes inverses en diffraction d'ondes basés sur la notion de résonances. Ph.D. thesis, University of Paris IX, France (1997).
[25] and, Decaying modes for the wave equation in the exterior of an obstacle. Comm. Pure Appl. Math. 22 (1969) 737-787. | Zbl 0181.38201
[26] , and, Variational formulations for the determination of resonant states in scattering problems. SIAM J. Math. Anal. 23 (1992) 579-608. | Zbl 0801.35098
[27] , On locating all zeros of an analytic function within a bounded domain by a revised Delvess/Lyness method. SIAM J. Numer. Anal. 20 (1983). | MR 708463 | Zbl 0545.65031
[28] , Time-harmonic or resonant states decomposition for the simulation of the time-dependent solution of a sea-keeping problem. Ph.D. thesis, Centrale Paris school, France (2004).
[29] , and, Numerical computation of the scattering frequencies for acoustic wave equations. Comput. Phys. 75 (1988) 345-358. | Zbl 0643.65080
[30] and, The transient motion of a floating body. J. Fluid Mech 44 (1970) 303-313. | Zbl 0215.29003
[31] and, Transient acoustic diffraction and radiation by an axisymmetrical elastic shell: a new statement of the basic equations and a numerical method based on polynomial approximations. J. Sound Vibration 241 (2001) 459-483.
[32] , Spectral solution of time dependent shallow water hydroelasticity. J. Fluid Mech. 454 (2002) 387-402. | Zbl 1044.74012
[33] , and, Some implications of the Laplace transform inversion on SEM coupling coefficients in the time domain, in Electromagnetics, Hemisphere Publisher, Washington DC 2 (1982) 181-200.
[34] , Étude numérique des pôles de résonance associés à la diffraction d'ondes acoustiques et élastiques par un obstacle en dimension 2. RAIRO Modèle. Anal. Numér. 29 (1995) 819-855. | Numdam | Zbl 0851.65072
[35] , L'École Polytechnique (Paris), 1, cahier 2, 24 (1795).
[36] ,, and, Application of the matrix pencil method for estimating the SEM (Singularity Expansion Method) poles of source-free transient responses from multiple look directions. IEEE Trans. Antennas Propagation 48 (2000) 612-618.
[37] and, Transient currents on a cylinder illuminated by an impulsive plane wave. IEEE Trans. Antennas Propagation ap-23 (1975) 627-638.
[38] , Meromorphic families of compact operators. Arch. Rational Mech. Anal. 31 (1968) 372-380. | Zbl 0167.43002
[39] and, Resonance expansions of scattered waves. Comm. Pure Appl. Math. 53 (2000) 1305-1334. | Zbl 1032.35148
[40] and, SEM approach to transient scattering by a lossy, radially inhomogeneous dielectric circular cylinder. Wave Motion 8 (1986) 43-63. | Zbl 0572.73114
[41] and, The physical content of the singularity expansion method. Appl. Plys. Lett. 39 (1981) 362-364.
[42] , Asymptotic methods in equations of mathematical physics. Gordon and Breach Science Publishers (1989). | MR 1054376 | Zbl 0743.35001
[43] and, Surface waves, in Hanbuch der Physik IX, Springer-Verlag, Berlin (1960). | MR 119656