On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives

Fabien  Caubet; Carlos Conca; Matías  Godoy

doi:10.3934/ipi.2016003

Article Contents

2016, Volume 10, Issue 2: 327-367. Doi: 10.3934/ipi.2016003

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On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives

1.
Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse Cedex 9
2.
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago

Received: January 31, 2015

Revised: April 30, 2015

Published: April 2016

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Abstract

We consider the inverse problem of detecting the location and the shape of several obstacles immersed in a fluid flowing in a larger bounded domain $\Omega$ from partial boundary measurements in the two dimensional case. The fluid flow is governed by the steady-state Stokes equations. We use a topological sensitivity analysis for the Kohn-Vogelius functional in order to find the number and the qualitative location of the objects. Then we explore the numerical possibilities of this approach and also present a numerical method which combines the topological gradient algorithm with the classical geometric shape gradient algorithm; this blending method allows to find the number of objects, their relative location and their approximate shape.

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Mathematics Subject Classification: Primary: 49Q10, 35R30, 49Q12; Secondary: 76D55.

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