Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes

Anna Kaźmierczak; Jan Sokolowski; Antoni Zochowski

doi:10.3934/mcrf.2018004

Article Contents

2018, Volume 8, Issue 1: 89-115. Doi: 10.3934/mcrf.2018004

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Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes

1.
Faculty of Mathematics and Computer Science, Banacha 22, 90-238 Lodz, University of Lodz, Poland
2.
Institut Elie Cartan de Nancy, UMR 7502, 54506 Vandoeuvre lès Nancy Cedex, France
3.
Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-338 Warsaw, Poland

Received: April 2017

Revised: September 2017

Published: January 2018

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Abstract

In the paper the shape optimization problem for the static, compressible Navier-Stokes equations is analyzed. The drag minimizing of an obstacle immersed in the gas stream is considered. The continuous gradient of the drag is obtained by application of the sensitivity formulas derived in the works of one of the co-authors. The numerical approximation scheme uses mixed Finite Volume - Finite Element formulation. The novelty of our numerical method is a particular choice of the regularizing term for the non-homogeneous Stokes boundary value problem, which may be tuned to obtain the best accuracy. The convergence of the discrete solutions for the model under considerations is proved. The non-linearity of the model is treated by means of the fixed point procedure. The numerical example of an optimal shape is given.

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Mathematics Subject Classification: Primary: 49Q10, 35Q30; Secondary: 74S10.

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Figure 1. Computational domain $\Omega = B\setminus S$

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Figure 2. On the left: the plot of $J$ versus number of steps; on the right: the final shape of the obstacle $\Gamma$ after minimization of the drag

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