An inverse problem for fluid-solid interaction

Pages: 83 - 120, Volume 2, Issue 1, February 2008

doi:10.3934/ipi.2008.2.83       Abstract        Full Text (1287.3K)       Related Articles

Johannes Elschner - Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany (email)
George C. Hsiao - Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, United States (email)
Andreas Rathsfeld - Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany (email)

Abstract: 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 obstacle with the Helmholtz equation in the exterior. The inverse problem is to reconstruct the elastic body represented by a parametrization of its boundary.
    We define an objective functional depending on a non-negative regularization parameter such that, 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 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.

Keywords:  acoustic and elastic waves, inverse scattering, gradients, Gauß-Newton method.
Mathematics Subject Classification:  Primary: 35R30, 76Q05; Secondary: 35J05, 35J20, 70G75.

Received: October 2007;      Published: January 2008.