An inverse problem for fluid-solid interaction
Pages: 83 - 120,
Volume 2, Issue 1,
Full Text (1287.3K)
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)
Any acoustic plane wave
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
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
depending on a non-negative regularization
parameter such that, for
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
Hence, numerical approximations can
be found applying finite element discretization.
Note that, though
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.