A parallel numerical method to solve high frequency ghost obstacle acoustic scattering problems^*

Abstract

A highly parallelizable numerical method for time dependent high frequency acoustic scattering problems involving realistic smart obstacles is proposed. A scatterer becomes smart when hit by an incoming wave reacts circulating on its boundary a pressure current to pursue a given goal. A pressure current is a quantity whose physical dimension is pressure divided by time. In particular we consider obstacles that when hit by an incoming acoustic wave try to generate a virtual image of themselves in a location in space different from their actual location. The virtual image of the obstacle (i.e.: the ghost obstacle) is seen outside a given set containing the obstacle and its virtual image in the apparent location. We call this problem ghost obstacle scattering problem. We model this acoustic scattering problem and several other acoustic scattering problems concerning other types of smart obstacles as optimal control problems for the wave equation. Using the Pontryagin maximum principle the first order optimality conditions associated to these control problems are formulated. The numerical method proposed to solve these optimality conditions is a variation of the operator expansion method and reduces the solution of the optimal control problem to the solution of a sequence of systems of integral equations. These systems of integral equations are solved using suitable wavelet bases to represent the unknowns, the data and the integral kernels. These wavelet bases are made of piecewise polynomial functions and have the property that the matrices that represent the integral operators on these wavelet bases can be approximated satisfactorily with very sparse matrices. This property of the wavelet bases makes possible to approximate the optimal control problems considered with linear systems of equations with hundreds of thousands or millions of unknowns and equations that can be stored and solved with affordable computing resources, that is it makes possible to solve satisfactorily problems with realistic obstacles hit by waves of small wavelength. We validate the method proposed solving some test problems, these problems are optimal control problems involving a ``smart" simplified version of the NASA space shuttle hit by incoming waves with small wavelengths compared to its characteristic dimension. We consider test problems with ratio between the characteristic dimension of the obstacle and the wavelength of the time harmonic component of the incoming wave up to approximately sixty. The numerical results obtained are very satisfactory. This website contains stereoscopic and virtual reality applications showing some numerical experiments relative to the problems studied in the paper associated to the website. A more general reference to the work in acoustic and electromagnetic scattering of the authors and of their coauthors is the website: http://www.econ.univpm.it/recchioni/scattering.

References

[1]

L.Fatone, G. Pacelli, M.C. Recchioni, F.Zirilli, "Optimal control methods for two new classes of smart obstacles in time dependent acoustic scattering", Journal of Engineering Mathematics, 56, 385--413 (2006).

[2]

F. Mariani, M.C. Recchioni and F. Zirilli, "The use of the Pontryagin maximum principle in a furtivity problem in time-dependent acoustic obstacle scattering", Waves in Random Media 11, 549-575 (2001).

[3]

L. Fatone, M. C. Recchioni and F. Zirilli, "A masking problem in time dependent acoustic obstacle scattering", ARLO - {Acoustics Research Letters Online}, 5, Issue 2, 25-30 (2004).

[4]

P. Corna, L. Fatone, M. C. Recchioni and F. Zirilli, "Some mathematical models of furtivity and masking problems in time dependent acoustic obstacle scattering", in Mathematical Modelling of Wave Phenomena 2002, Mathematical Modelling in Physics, Engineering and Cognitive Sciences, edited by B. Nilsson, L. Fishman, Vaxjo University Press, Vaxjo, Sweden, 7, 79-89, (2003).

[5]

L. Fatone, M. C. Recchioni and F. Zirilli, "Furtivity and masking problems in time dependent electromagnetic obstacle scattering", Journal of Optimization Theory and Applications, 121, 223-257 (2004).

[6]

L. Fatone, M. C. Recchioni and F. Zirilli, "Mathematical models of ``active" obstacles in acoustic scattering", in Control and Boundary Analysis, edited by J. Cagnol and J. P. Zolesio, Marcel Dekker/CRC Press, Boca Raton, Fl. USA, Lecture Notes in Pure and Applied Mathematics 240, 2005, pp. 119--129.

[7]

L. Fatone, M. C. Recchioni and F. Zirilli, "A parallel numerical method to solve high frequency ghost obstacle acoustic scattering problems", submitted to ACES journal

The numerical experience reported in this paper has been obtained using the computing resources of CASPUR (Roma, Italy) under grant: "Algoritmi di alte prestazioni per problemi di scattering acustico". The support and sponsorship of CASPUR are gratefully acknowledged.

It is a pleasure to thank Mr. Fabio Bonaccorso, Mr. Piero Lanucara and Mrs. Claudia Truini of CASPUR (Roma, Italy) for the helpful assistance in the realization of the stereoscopic applications contained in this website.

- screen resolution: 1024 x 768
- video adapter: 8Mb
- processor: Pentium II or equivalent

 

To see the stereoscopic applications click here and follow the instructions

 

 

Time dependent scattering from a smart simplified version of the NASA space shuttle

 

NASA space shuttle: VRML visualization

Simplified Version of the NASA space shuttle: VRML visualization

 

 

Setting of the ghost obstacle experiment

	W= smart obstacle, WG= ghost obstacle, We= auxiliary set (ellipsoid).
	Virtual scene: VRML visualization

 

 

Time dependent scattering from the smart simplified version of the Nasa space shuttle

Data: incident wave uⁱ=exp(-p²(a₁x+a₂y+a₁z-ct)²), where a₁=sin(p/4)cos(p/4), a₂=sin(p/4)sin(p/4), a₃=cos(p/4), c=331.45 meters/seconds (sound speed in the air), the length of the simplified version of the NASA space shuttle in the direction of its main body is L=56.14   meters. The z axis is the symmetry axis of the main body of the obstacle. Let R_T be the ratio between the characteristic dimension of the obstacle and the wavelength of the time harmonic components of the incident wave considered. The ratio R_T in this experiment ranges approximately between 3 to 60. Remind that the incident wave is approximated with a finite linear combination of time harmonic plane waves.

Animation 1: Click here to see the scattering phenomenon (Virtual scene: VRML file)

Remark. In the animation the time values shown in twenty frames are t=(-(j-1)(15/21)+7.5)/c, j=1,2,...,20,

c=331.45 meters/seconds

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