# The Singularity Expansion Method applied to the transient motions of a floating elastic plate

Christophe Hazard; François Loret

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 5, page 925-943
- ISSN: 0764-583X

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topHazard, Christophe, and Loret, François. "The Singularity Expansion Method applied to the transient motions of a floating elastic plate." ESAIM: Mathematical Modelling and Numerical Analysis 41.5 (2007): 925-943. <http://eudml.org/doc/250021>.

@article{Hazard2007,

abstract = {
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.
},

author = {Hazard, Christophe, Loret, François},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Laplace transform; resonance; meromorphic family of operators; integral representation.; analytic continuation; sea-keeping problem},

language = {eng},

month = {10},

number = {5},

pages = {925-943},

publisher = {EDP Sciences},

title = {The Singularity Expansion Method applied to the transient motions of a floating elastic plate},

url = {http://eudml.org/doc/250021},

volume = {41},

year = {2007},

}

TY - JOUR

AU - Hazard, Christophe

AU - Loret, François

TI - The Singularity Expansion Method applied to the transient motions of a floating elastic plate

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2007/10//

PB - EDP Sciences

VL - 41

IS - 5

SP - 925

EP - 943

AB -
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.

LA - eng

KW - Laplace transform; resonance; meromorphic family of operators; integral representation.; analytic continuation; sea-keeping problem

UR - http://eudml.org/doc/250021

ER -

## References

top- M. Abramowitz and I.A. Stegun, Handbook of mathematical functions. Dover Publications, New York, 9th edn. (1970). Zbl0171.38503
- J. Aguilar and J.M. Combes, A class of analytic perturbations for one-body Schrödinger hamiltonians. Comm. Math. Phys.22 (1971) 269–279. Zbl0219.47011
- C. Alves and T. Ha Duong, 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). Zbl0874.73077
- C. Amrouche, The Neumann problem in the half-space. C. R. Acad. Sci. Paris Ser. I335 (2002) 151–156. Zbl1019.35026
- A. Bachelot and A. Motet-Bachelot, Les résonances d'un trou noir de Schwarzschild. Ann. Henri. Poincarré59 (1993) 280–294. Zbl0774.53036
- E. Balslev and J.M. Combes, Spectral properties of many body Schrödinger operators with dilation analytic interactions. Comm. Math. Phys.22 (1971) 280–294. Zbl0219.47005
- C.E. Baum, The Singularity Expansion Method, in Transient Electromagnetic Fields, L.B. Felsen Ed., Springer-Verlag, New York (1976).
- H. Brezis, Analyse fonctionnelle, Théorie et application. Masson, Paris (1983). Zbl0511.46001
- N. Burq and M. Zworski, Resonance expansions in semi-classical propagation. Comm. Math. Phys.232 (2001) 1–12. Zbl1042.81582
- M.P. Carpentier and A.F. Dos Santos, Solution of equations involving analytic functions. J. Comput. Phys.45 (1982) 210–220. Zbl0484.65026
- P.G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model. J. Mécanique18 (1979) 315–344. Zbl0415.73072
- R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 1, Dunod, Paris (1984). Zbl0642.35001
- R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 3, Dunod, Paris (1985). Zbl0642.35001
- L.B. Felsen and E. Heyman, Hybrid ray mode analysis of transient scattering, in Low and Frequency Asymptotics, V.K. Varadan and V.V. Varadan Eds. (1986). Zbl0598.73022
- D. Habault and P.J.T. Filippi, Light fluid approximation for sound radiation and diffraction by thin elastic plates. J. Sound Vibration213 (1998) 333–374.
- C. Hazard, The Singularity Expansion Method, in Fifth International Conference on Mathematical and Numerical Aspects of Wave propagation, SIAM (2000) 494–498. Zbl0960.76072
- C. Hazard and M. Lenoir, Determination of scattering frequencies for an elastic floating body. SIAM J. Math. Anal.24 (1993) 1458–1514. Zbl0784.76013
- C. Hazard and F. Loret, Generalized eigenfunction expansions for conservative scattering problems with an application to water waves. Proceedings of the Royal Society of Edinburgh (2007) Accepted. Zbl1171.76007
- T. Kato, Perturbation theory for linear operators. Springer-Verlag, New York (1984). Zbl0531.47014
- F. Klopp and M. Zworski, Generic simplicity of resonances. Helv. Phys. Acta8 (1995) 531–538. Zbl0844.47040
- K. Knopp, Theory of functions, Part II. Dover, New York (1947).
- P. Kravanja and M. Van Barel, Computing the zeros of analytic functions. Lect. Notes Math.1727, Springer (2000). Zbl0945.65018
- N. Kuznetsov, V. Maz'ya and B. Vainberg, Linear Water Waves, a Mathematical Approach. Cambridge (2002).
- C. Labreuche, Problèmes inverses en diffraction d'ondes basés sur la notion de résonances. Ph.D. thesis, University of Paris IX, France (1997).
- P.D. Lax and R.S. Phillips, Decaying modes for the wave equation in the exterior of an obstacle. Comm. Pure Appl. Math.22 (1969) 737–787. Zbl0181.38201
- M. Lenoir, M. Vullierme-Ledard and C. Hazard, Variational formulations for the determination of resonant states in scattering problems. SIAM J. Math. Anal.23 (1992) 579–608. Zbl0801.35098
- T.-Y. Li, On locating all zeros of an analytic function within a bounded domain by a revised Delvess/Lyness method. SIAM J. Numer. Anal.20 (1983).
- F. Loret, 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).
- G. Majda, W. Strauss and M. Wei, Numerical computation of the scattering frequencies for acoustic wave equations. Comput. Phys.75 (1988) 345–358. Zbl0643.65080
- S.J. Maskell and F. Ursell, The transient motion of a floating body. J. Fluid Mech44 (1970) 303–313. Zbl0215.29003
- C. Maury and P.J.T. Filippi, 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 Vibration241 (2001) 459–483.
- M.H. Meylan, Spectral solution of time dependent shallow water hydroelasticity. J. Fluid Mech.454 (2002) 387–402. Zbl1044.74012
- L.W. Pearson, D.R. Wilton and R. Mittra, Some implications of the Laplace transform inversion on SEM coupling coefficients in the time domain, in Electromagnetics, Hemisphere Publisher, Washington DC2 (1982) 181–200.
- O. Poisson, 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.
- R.J. Prony, L'École Polytechnique (Paris), 1, cahier 2, 24 (1795).
- T.K. Sarkar, S. Park, J. Koh and S. Rao, 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 Propagation48 (2000) 612–618.
- R.H. Schafer and R.G. Kouyoumjian, Transient currents on a cylinder illuminated by an impulsive plane wave. IEEE Trans. Antennas Propagationap-23 (1975) 627–638.
- S. Steinberg, Meromorphic families of compact operators. Arch. Rational Mech. Anal.31 (1968) 372–380. Zbl0167.43002
- S.H. Tang and M. Zworski, Resonance expansions of scattered waves. Comm. Pure Appl. Math.53 (2000) 1305–1334. Zbl1032.35148
- A.G. Tijhuis and R.M. van der Weiden, SEM approach to transient scattering by a lossy, radially inhomogeneous dielectric circular cylinder. Wave Motion8 (1986) 43–63. Zbl0572.73114
- H. Überall and G.C. Gaunard, The physical content of the singularity expansion method. Appl. Plys. Lett.39 (1981) 362–364.
- B.R. Vainberg, Asymptotic methods in equations of mathematical physics. Gordon and Breach Science Publishers (1989). Zbl0743.35001
- J.V. Wehausen and E.V. Laitone, Surface waves, in Hanbuch der PhysikIX, Springer-Verlag, Berlin (1960).

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