Optimal Shape Design of a Body Located in Stokes Flow


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Formulations and numerical results for optimal shape design of a body located in the incompressible Stokes fluid flow are presented. The study is based on an optimal control theory. The optimal state is defined by the reduction of drag forces subjected to the body. The cost functional should be minimized satisfying the Stokes equations. The Shape sensitivity analysis of the cost functional was derived based on the adjoint method. For the numerical study, the optimal shape of the body which has a circular shape as an initial state can be finally obtained as the streamlined shape.



Edited by:

Jun Hu and Qi Luo




X. B. Duan et al., "Optimal Shape Design of a Body Located in Stokes Flow", Advanced Materials Research, Vol. 320, pp. 303-308, 2011

Online since:

August 2011




[1] O. Pironneau: On optimum profiles in Stokes flow, Journal of Fluid Mechanics, Vol. 59, (1973), 117-128.

[2] O. Pironneau: On optimum shape design in fluid mechanics, Journal of Fluid Mechanics. Vol. 64(1) (1974), 97-110.

[3] X.B. Duan, Y.C. Ma and R. Zhang: Optimal Shape Control of Fluid Flow Using Variational Level Set Method, Physics Letters A, Vol. 9, (2008), 1374-1379.

DOI: https://doi.org/10.1016/j.physleta.2007.09.070

[4] A. Maruoka, M. Kawahara: Optimal control in Navier–Stokes equation, Int. J. Comput. Fluid Dyn. Vol. 9 (1998), 313–322.

[5] J. Matsumoto, T. Umetsu, M. Kawahara: Incompressible viscous flow analysis and adaptive finite element method using linear bubble function, J. Appl. Mech. Vol. 2 (1999), 223–232.

[6] J. Matsumoto, M. Kawahara: Stable shape identification for fluid structure interaction problem using MINI element, J. Appl. Mech. Vol. 3 (2000), 263–274.

[7] B. Mohamadi, O. Pironneau: Applied Shape Optimization for Fluids, Oxford University Press, (2001).

[8] B. A. Ton: Optimal shape control problem for the Navier-Stokes equations, SIAM Journal on control and optimization, Vol. 41(6) (2003), 1733-1747.

DOI: https://doi.org/10.1137/s0363012901391287

[9] H. Yagi, M. Kawahara: Shape optimization of a body located in low Reynolds number flow, Int. J. Numer. Methods Fluids, Vol. 48 (2005), 819–833.

DOI: https://doi.org/10.1002/fld.957

[10] E. Katamine, H. Azegami, T. Tsubata and S. Itoh: Solution to shape optimization problems of viscous flow fields, Int. J. Comput. Fluid Dyn. Vol. 19 (1) (2005), 45–51.

DOI: https://doi.org/10.1016/b978-008044268-6/50046-6