A time-dependent three-dimensional acoustic scattering problem is considered. An incoming wave packet is scattered by a bounded, simply connected obstacle with locally Lipschitz boundary. The obstacle is assumed to have a constant boundary acoustic impedance. The limit cases of acoustically soft and acoustically hard obstacles are considered. The scattered acoustic field is the solution of an exterior problem for the wave equation. A new numerical method to compute the scattered acoustic field is proposed. This numerical method obtains the time-dependent scattered field as a superposition of time-harmonic acoustic waves and computes the time-harmonic acoustic waves by a new "operator expansion method." That is, the time-harmonic acoustic waves are solutions of an exterior boundary value problem for the Helmholtz equation. The method used to compute the time-harmonic waves improves on the method proposed by Misici, Pacelli, and Zirilli [J. Acoust. Soc. Am. 103, 106-113 (1998)] and is based on a "perturbative series" of the type of the one proposed in the operator expansion method by Milder [J. Acoust. Soc. Am. 89, 529-541 (1991)]. Computationally, the method is highly parallelizable with respect to time and space variables. Some numerical experiments on test problems obtained with a parallel implementation of the numerical method proposed are shown and discussed from the numerical and the physical point of view. The website: http://www.econ.unian.it/recchioni/w1 shows four animations relative to the numerical experiments.
A numerical method for a three-dimensional inverse acoustic scattering problem is considered. From the knowledge of several far fields patterns of the Helmholtz equation a closed surface 0 D representing the boundary of an unknown obstacle D is reconstructed. The obstacle D is supposed to be acoustically soft or acoustically hard or characterized by a given acoustic impedance.
obtained at the same degree of convergence without any regularization, with a Tikhonov (TK) regularization, and with our EP regularization scheme. The reconstruction without regularization shows a blurred profile, with a coarse shape description. The use of a TK regularization smoothes the profile, and the edges are not preserved. The new regularization scheme improves the perfonnance of the CG algorithm: the edges are clearly preserved, while the homogeneous areas are smoothed.We proposed, then, two other results, obtained from two mystery objects, named IPS005 and IPS007. The scattered fields were collected from 36 viewing angles, 9, ~[0",350"], with a sample spacing of lo", over the observation sector 8, SOs <8,+170",withasamplespacingof AOs =lo". Theonly one piece of information given with these data sets was the radius of the minimum circumscribing circle. We show, in Figure 2 and Figure 3, the results obtained by using the CG algorithm, without any regularization (no a priori information used), and also the corresponding original profiles revealed at the 1996 Symposium. The domain, L, is discretized into 29 x 29 square cells of 1 mm', for target IPS005, and of 0.5 mm2, for target IPS007. The main problem was to find a satisfying calibration factor for each mystery target. After several blind tests, we were able to propose two suitable reconstructions, using valid but non-optimum calibration parameters. These reconstructions gave quite good spatial resolution of the t w~ mystery objects. ConclusionThis paper presents some evidence of the effectiveness of adding the EP regularization to the CG method, in reconstructing the shape and permittivity profile of dielectric objects. Without any a priori information, our CG algorithm is also still efficient, and succeeded in reconstructing two mystery targets. As the two targets are now known, we hope we can greatly enhance the reconstruction quality by choosing better values for the calibration factors, and by applying our EP regularization to these data. References1. P. Lobel, R. Kleinman, Ch. Pichot, L.
Let ⍀ʚR 3 be an obstacle that is a simply connected bounded domain. The exterior Dirichlet problem for the Helmholtz equation in R 3 گ⍀ with the Sommerfeld radiation condition at infinity is considered. Based on an integral representation formula, a new method to compute the solution of the exterior boundary value problem mentioned above is proposed. This method generalizes the formalism introduced for an unbounded obstacle by Milder ͓J. Acoust. Soc. Am. 89, 529-541 ͑1991͔͒ and consists in computing a perturbation series whose coefficients are integrals. These integrals are independent one from the other so that the computation of the series is fully parallelizable. Finally, some numerical results obtained on test problems are shown. In particular, numerical experiments for obstacles with nonsmooth boundaries such as polyhedra and obstacles with multiscale corrugations are shown.
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