JAMP  Vol.3 No.12 , December 2015
Shape Identification for Stokes-Oseen Problem Based on Domain Derivative Method
ABSTRACT
In this paper, we consider the shape identification problem of a body immersed in the incompressible fluid governed by Stokes-Oseen equations. Based on the domain derivative method, we derive the explicit representation of the derivative of solution with respect to the boundary. Then, according to the boundary parametrization technique, we propose a regularized Gauss-Newton algorithm for the shape inverse problem. Finally, numerical examples indicate that the iterative algorithm is feasible and effective for the practical purpose.

Cite this paper
Yan, W. and Hou, J. (2015) Shape Identification for Stokes-Oseen Problem Based on Domain Derivative Method. Journal of Applied Mathematics and Physics, 3, 1662-1670. doi: 10.4236/jamp.2015.312191.
References
[1]   Kress, R. and Rundellf, W. (1994) A Quasi-Newton Method in Inverse Obstacle Scattering. Inverse Problems, 10, 1145-1157.
http://dx.doi.org/10.1088/0266-5611/10/5/011

[2]   Hettlich, F. (1995) Frechet Derivatives in Inverse Obstacle Scattering. Inverse Problems, 11, 371-382.
http://dx.doi.org/10.1088/0266-5611/11/2/007

[3]   Hettlich, F. (1998) The Landweber Iteration Applied to Inverse Conductive Scattering Problems. Inverse Problems, 14, 931-947.
http://dx.doi.org/10.1088/0266-5611/14/4/011

[4]   Chapko, R., Kress, R. and Yoon, J.R. (1998) On the Numerical Solution of an Inverse Boundary Value Problem for the Heat Equation. Inverse Problems, 14, 853-867.
http://dx.doi.org/10.1088/0266-5611/14/4/006

[5]   Chapko, R., Kress, R. and Yoon, J.R. (1999) An Inverse Boundary Value Problem for the Heat Equation: The Neumann Condition. Inverse Problems, 15, 1033-1046.
http://dx.doi.org/10.1088/0266-5611/15/4/313

[6]   Serranho, P. (2006) A Hybrid Method for Inverse Scattering for Shape and Impedance. Inverse Problems, 22, 663-680.
http://dx.doi.org/10.1088/0266-5611/22/2/017

[7]   Harbrecht, H. and Tausch, J. (2013) On the Numerical Solution of a Shape Optimization Problem for the Heat Equation. SIAM Journal on Scientific Computing, 35, 104-121.
http://dx.doi.org/10.1137/110855703

[8]   Yan, W.J. and Ma, Y.C. (2006) The Application of Domain Derivative for Heat Conduction with Mixed Condition in Shape Reconstruction. Applied Mathematics and Computation, 181, 894-902.
http://dx.doi.org/10.1016/j.amc.2006.02.011

[9]   Yan, W.J. and Gao, Z.M. (2014) Shape Optimization in the Navier-Stokes Flow with Thermal Effects. Numerical Methods for Partial Differential Equations, 30, 1700-175.
http://dx.doi.org/10.1002/num.21818

[10]   Quarteroni, A. and Valli, A. (1994) Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin.

[11]   Pironneau, O. (1984) Optimal Shape Design for Elliptic Systems. Springer, Berlin.
http://dx.doi.org/10.1007/978-3-642-87722-3

[12]   Delfour, M.C. and Zolésio, J.P. (2002) Shapes and Geometries: Analysis, Differential Calculus and Optimization, Advance in Design and Control. Springer, Berlin.

[13]   Temam, R. (2001) Navier Stokes Equations, Theory and Numerical Analysis. AMS Chelsea Edition, American Mathematical Society, Rhode Island.

[14]   Gilbarg, D. and Trudinger, N.S. (1983) Elliptic Partial Differential Equations of Second Order. Springer, Berlin.
http://dx.doi.org/10.1007/978-3-642-61798-0

 
 
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