2010, Elschner, Johannes, Hsiao, George C., Rathsfeld, Andreas

We consider the inverse problem for an elastic body emerged in a fluid due to an acoustic wave. The shape of this obstacle is to be reconstructed from the far-field pattern of the scattered wave. For the numerical solution in the two-dimensional case, we compare a simple Newton type iteration method with the Kirsch-Kress algorithm. Our computational tests reveal that the Kirsch-Kress method converges faster for obstacles with very smooth boundaries. The simple Newton method, however, is more stable in the case of not so smooth domains and more robust with respect to measurement errors.

2008, Elschner, Johannes, Hsiao, George C., Rathsfeld, Andreas

We consider the interaction between an elastic body and a compressible inviscid fluid, which occupies the unbounded exterior domain. The inverse problem of determining the shape of such an elastic scatterer from the measured far field pattern of the scattered fluid pressure field is of central importance in detecting and identifying submerged objects. Following a method proposed by Kirsch and Kress, we approximate the acoustic and elastodynamic wave by potentials over auxiliary surfaces, and we reformulate the inverse problem as an optimisation problem. The objective function to be minimised is the sum of three terms. The first is the deviation of the approximate far field pattern from the measured one, the second is a regularisation term, and the last a control term for the transmission condition. We prove that the optimisation problem has a solution and that, for the regularisation parameter tending to zero, the minimisers tend to a solution of the inverse problem. In contrast to a numerical method from a previous paper, the presented method does require neither a direct solution method nor an additional treatment of possible Jones modes.

2007, Elschner, Johannes, Hsiao, George C., Rathsfeld, Andreas

Any acoustic plane wave incident to an elastic obstacle results in a scattered field with a corresponding far field pattern. Mathematically, the scattered field is the solution of a transmission problem coupling the reduced elastodynamic equations over the domain occupied by the obstacle with the Helmholtz equation in the exterior. The far field pattern is obtained applying an integral operator to the scattered field function restricted to a simple smooth surface surrounding the obstacle. The subject of our paper is the inverse problem, where the shape of the elastic body represented by a parametrization of its boundary is to be reconstructed from a finite number of measured far field patterns. We define a family of objective functionals depending on a non-negative regularization parameter such that, for regularization parameter zero, the shape of the sought elastic obstacle is a minimizer of the functional. For any positive regularization parameter, there exists a regularized solution minimizing the functional. Moreover, for the regularization parameter tending to zero, these regularized solutions converge to the solution of the inverse problem provided the latter is uniquely determined by the given far field patterns. The whole approach is based on the variational form of the partial differential operators involved. Hence, numerical approximations can be found applying finite element discretization. Note that, though the transmission problem in its weak formulation may have non-unique solutions for domains with so-called Jones frequencies, the scattered field and its far field pattern is unique and depend continuously on the shape of the obstacle.