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.
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
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 |
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, W_{G}= ghost obstacle, W_{e}= auxiliary set (ellipsoid). | |
Virtual scene: VRML visualization |
Time dependent scattering from the smart simplified version of the Nasa space shuttle
Data: incident wave u^{i}=exp(-p^{2}(a_{1}x+a_{2}y+a_{1}z-ct)^{2}), where a_{1}=sin(p/4)cos(p/4), a_{2}=sin(p/4)sin(p/4), a_{3}=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