Abstract
We present a penalization parameter method for obstacle identification in an incompressible fluid flow for a modified version of the Oseen equations. The proposed method consists in adding a high resistance potential to the system such that some subset of its boundary support represents the obstacle. This allows to work in a fixed domain and highly simplify the solution of the inverse problem via some suitable cost functional. Existence of minimizers and first and second order optimality conditions are derived through the differentiability of the solutions of the Oseen equation with respect to the potential. Finally, several numerical experiments using Navier–Stokes flow illustrate the applicability of the method, for the localization of a bi-dimensional cardiac valve from MRI and ultrasound flow type imaging data.
Highlights
The pathway of blood flow through the heart is regulated by four membrane structures or valves, opening and closing by differential blood pressures
When the valves are open, these parabolas are rotated with respect to a reference system whose origin is at the point where the valve coincides with the walls of the structure given by ΓW
We have presented a new parameter identification problem for the Oseen equation, contributing to the detection of obstacles in fluid dynamic studies
Summary
The pathway of blood flow through the heart is regulated by four membrane structures or valves, opening and closing by differential blood pressures. Visual inspection of 3D Flow MRI Imaging (4D Flow, see [17]) data sets clearly shows that the valvular shape could be retrieved from the flow pattern in the valve surroundings This fact motivates to formulate, analyze and numerically assess the inverse problem of inferring rigid obstacles from interior velocity measurements, with the ultimate goal of recovering the shape of cardiac valves at the opening position from velocity images. The distributed resistance term that we propose here allows us to work in a fixed domain to solve the valve shape identification problem This distributed resistance method can be useful to estimate porosity in porous media flows following Brinkman’s law [12, 6].
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