The identification of obstacles immersed in a steady incompressible viscous fluid

Abstract

In this paper, the identification of immersed obstacles in a steady incompressible Navier-Stokes viscous fluid flow from fluid traction measurements is investigated. The solution of the direct problem is computed using the finite element method (FEM) implemented in the Freefem++ commercial software package. The solution of the inverse geometric obstacle problem (parameterized by a small set of unknown constants) is accomplished iteratively by minimizing the nonlinear least-squares functional using an adaptive moment estimation algorithm. The numerical results for the identification of an obstacle in a viscous fluid flowing in a channel with open ends, show that when the fluid traction is measured on the top, bottom and inlet boundaries, then the algorithm provides accurate and robust reconstructions of an obstacle parameterized by a small number of parameters in a Fourier trigonometric finite expansion. Stable reconstructions with respect to noise in the measured fluid traction data are also achieved, although for complicated shapes parameterized by larger degrees of freedom nonlinear Tikhonov regularization of the least-squares functional may need to be employed. Multiple-component obstacles may also be identified provided that a good initial guess is provided. In case of limited data being available only at the inlet boundary the pressure gradient provides more information for inversion than the fluid traction.