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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.
Mathematics Subject Classification: Primary: 35R30, 76Q05; Secondary: 35J05, 35J20, 70G75.
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