A masking problem in time dependent acoustic obstacle scattering

Lorella Fatone
Dipartimento di Matematica Pura ed Applicata, Università di Modena e Reggio Emilia, 
Via Campi 213/b, 41100 Modena, Italy
Maria Cristina Recchioni
Istituto di Teoria delle Decisioni e Finanza Innovativa (DE.F.IN.), Università di Ancona, 
Piazza Martelli 8, 60121 Ancona, Italy
Francesco Zirilli
Dipartimento di Matematica ``G. Castelnuovo'', Università di Roma ``La Sapienza'',
Piazzale Aldo Moro 2, 00185 Roma, Italy 


In this paper we consider a masking problem in time dependent three dimensional acoustic obstacle scattering. The masking problem consists in making ``masked" an object W characterized by an acoustic boundary impedance c ³ 0 and immersed in a homogeneous isotropic medium when, hit by an acoustic incident wave, generates a scattered acoustic field. To realize this masking effect we try to make the acoustic field scattered by the obstacle (W;c) "similar" to the field scattered, when hit by the same acoustic incident wave, by a given obstacle D with impedance c¢. We refer to (D;c¢) as the ``mask" and we require that the closure of D is contained in W . We formulate a mathematical model for the masking problem consisting in an optimal control problem for the wave equation. That is we try to make the field scattered by (W;c) similar to the field scattered by the mask (D ;c¢) choosing a control function, that is a pressure current, defined on the boundary of the obstacle for all times, in such a way that a suitable cost functional is minimized. The cost functional depends on the control function, and on the scattered acoustic fields generated by the obstacle (W; c) and by the mask (D;c¢). Using the Pontryagin maximum principle we show that the first order optimality condition for the optimal control problem considered can be formulated as an exterior problem defined outside the obstacle for a system of two coupled wave equations. To solve numerically this exterior problem we develop a highly parallelizable method that is a slight modification of the operator expansion method proposed in [1]. We validate the mathematical model and the numerical method proposed on some test problems and we discuss the results obtained with a parallel implementation of the numerical method from the numerical and the physical point of view.


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).


Warning: To see the virtual reality animations we reccomend 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)



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



We propose two animations involving the following two obstacles:


Double Cone Double Cone (virtual object): VRML visualization

Animation 1: Double Cone (gif file)



Excavated Sphere Excavated Sphere (virtual object): VRML visualization

Animation 2: Excavated Sphere (Virtual scene: VRML file)


Animation 3: Excavated Sphere (Virtual scene: VRML file)

Acknowledgment: It is a pleasure to thank Mr. Adriano Tittarelli of Università di Ancona for the really helpful assistance in the realization of the virtual reality applications contained in this website.




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