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. The website: http://www.econ.univpm.it/recchioni/w14 contains animations and virtual reality applications showing some numerical experiments relative to the problems studied in this paper.

Wavelet Bases Made of Piecewise Polynomial Functions: Theory and Applications

“…Property (15) follows from definition (4) and Equation (6). The proof of (16) 16is obvious, when the supports are contained one into the other condition (16) follows from (15). Finally when = , m m = ,    = j j condition (16) follows from Equation 7.…”

“…The previous construction based on the tensor product can be easily extended to the case when A is a parallelepiped in dimension 3 s  . Note that with straightforward generalizations of the material presented here, it is easy to construct wavelet bases for   2 L A when A is a sufficiently simple subset of a real Euclidean space (see [12,16] to find several choices of A useful in some scattering problems). The analysis that follows of the wavelet bases when   = 0,1 A can be extended with no substantial changes to the other choices of A considered here.…”

“…This is a relevant task, see for example [10,15], since it makes possible, among other things, the approximation of some integral operators with sparse matrices allowing the approximation and the solution of the corresponding integral equations in very high dimensional subspaces at an affordable computational cost. In [11,12,16], for example, we exploit this property to solve some time dependent acoustic obstacle scattering problems involving realistic objects hit by waves of small wavelength when compared to the dimension of the objects. Let us note that these scattering problems are translated mathematically in problems for the wave equation and that they are numerically challenging, moreover thanks to the use of the wavelet bases, when boundary integral methods or some of their variations are used, they can be approximated by sparse systems of linear equations in very high dimensional spaces (i.e.…”