A numerical method for time dependent acoustic scattering problems involving smart obstacles and incoming waves of small wavelengths

Abstract

In this paper we propose a highly parallelizable numerical method for time dependent acoustic scattering problems involving realistic smart obstacles hit by incoming waves having wavelengths small compared with the characteristic dimension of the obstacles. A smart obstacle is an obstacle that when hit by an incoming wave tries to pursue a goal circulating on its boundary a pressure current. In particular we consider obstacles whose goal is to be undetectable and we refer to them as furtive obstacles. These scattering problems are modelled as optimal control problems for the wave equation. We validate the method proposed to solve the optimal control problem considered on some test problems where a "smart" simplified version of the NASA space shuttle is hit by incoming waves with small wavelengths compared to its characteristic dimension. That is we consider test problems with ratio between the characteristic dimension of the obstacle and wavelength of the time harmonic component of the incoming wave up to approximately one hundred. This website contains animations and virtual reality applications showing some numerical experiments relative to the problems studied in the paper.

References

[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.

 

Warning: To see the virtual reality applications we recommend to use at least the following resources:

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

We suggest the following VRML clients:

	Windows	Mac	UNIX/Linux
Cortona	X	X
Cosmo Player	X	X
OpenVRML (Under Developement)			X
FreeWRL (Under Developement)			X

 

 

Note that in a Unix/Linux environment the clients OpenVRML and FreeWRL allow the visualization of virtual objects but not the visualization of virtual scenes.

 

 

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

 

NASA shuttle: VRML visualization

Simplified Version of the NASA space shuttle: VRML visualization

Remark. In the following animations the time values shown in eighteen frames are t=4(-j(13/21)+6.5)/c, j=1,2,...,18,

c=331.45 meters/seconds

 

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

Data: incident wave uⁱ=exp(-4(z+ct)²), c=331.45 meters/seconds (sound speed in the air), length of the simplified version of the NASA space shuttle 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 28. Remind that the incident wave is approximated with a finite linear combination of time harmonic plane wave.

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

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