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.
- 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 u^{i}=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)