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Mathematical Problems in Engineering
Volume 2013, Article ID 728534, 11 pages
Research Article

The Application of Shape Gradient for the Incompressible Fluid in Shape Optimization

1School of Mathematics and Statistics, Xi'an Jiaotong University, Shaanxi 710049, China
2Center for Computational Geosciences, Xi'an Jiaotong University, Shaanxi 710049, China
3School of Science, Chang'an University, Shaanxi 710064, China

Received 20 November 2012; Revised 4 March 2013; Accepted 7 March 2013

Academic Editor: Jun Jiang

Copyright © 2013 Wenjing Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is concerned with the numerical simulation for shape optimization of the Stokes flow around a solid body. The shape gradient for the shape optimization problem in a viscous incompressible flow is evaluated by the velocity method. The flow is governed by the steady-state Stokes equations coupled with a thermal model. The structure of continuous shape gradient of the cost functional is derived by employing the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique. A gradient-type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose, and the proposed algorithm is feasible and effective.