Detecting immersed obstacle in Stokes fluid flow using the coupled complex boundary method

Abstract

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.