  A non-conventional shape optimization approach is introduced to address the
identification of an obstacle immersed in a fluid described by the Stokes
equation within a larger bounded domain, relying on boundary measurements on
the accessible surface. The approach employs tools from shape optimization,
utilizing the coupled complex boundary method to transform the over-specified
problem into a complex boundary value problem by incorporating a complex Robin
boundary condition. This condition is derived by coupling the Dirichlet and
Neumann boundary conditions along the accessible boundary. The identification
of the obstacle involves optimizing a cost function constructed based on the
imaginary part of the solution across the entire domain. The subsequent
calculation of the shape gradient of this cost function, rigorously performed
via the rearrangement method, enables the iterative solution of the
optimization problem using a Sobolev gradient descent algorithm. The
feasibility of the method is illustrated through numerical experiments in both
two and three spatial dimensions, demonstrating its effectiveness in
reconstructing obstacles with pronounced concavities under high-level
noise-contaminated data, all without perimeter or volume functional
penalization.
