Keywords: shape optimization, optimal control theory, finite element method, mixed interpolation, adjoint equation method, Delaunay triangulation.

This paper demonstrates the shape optimization of a body located in an incompressible flow using the adjoint equation method. Shape optimization using the finite element method is one of the main topics in optimal control theory. Pironneau [1,2] was the first to carry out shape optimization of this type using a computer. He tried to determine the minimum drag shape in the Stokes approximation and suggested the algorithm for the shape optimization. Then shape optimization for the Navier-Stokes equation was considered using the finite element method [3,4,5,6,7,8]. In these studies, better shapes were obtained from the initial shape. But there is a problem that is the shapes obtaine ddepend on the initial shapes when using the gradient method. Therefore in this study, the final shapes are calculated from the initial shapes of three types. Then these final shapes are compared. How these initial shapes effect the final shapes is shown. In the numerical study, the Reynolds' number is set to 40.0,100.0 and 250.0. The final shapes are calculated from initial shapes of three different types, one is selected from a previous case. The second type is selected from the NACA Airfoil series. The third type is selected as a circle. Then these final shapes are compared. It is concluded that the drag and lift forces may be reduced by changing the surface coordinates of the body. In each shape obtained, the smooth surface of the body was calculated using the structured mesh around the body. The shape of three different types was applied as the initial shape in each case. Then the final shape for which the performance function was reduced was obtained from each initial shape. However, the final shapes became different shapes in each case. This result shows that the final shapes depend on initial shapes and the possibility that these final shapes are local minimum. Therefore the development of method that has a low dependency of the initial shape and how to select the initial shape near to the optimal shape are future work.

1

J. Matumoto, T. Umesu, M. kawahara, "Incompressible Viscous Flow Analysis and Adaptive Finite Element Method Using Linear Bubble Function", Journal of Applied Mechanics, 2, 223-232, 1999.

2

O. Pironneau, "Consist Approximations, Automatic Differentiation and Domain Decomposition For Optimal Shape Design", International Series, Mathematical Science and Applications, 16, Computational Methods for Control Application, 167-178, 2001.

3

H. Okumura, M. Kawahara, "Shape Optimization of the Body Located in Incompressible Viscous Flow Based on Optimal Control Theory", CMES, 1(2), 2000.

4

Y. Ogawa, M. Kawahara, "Optimization a Body located in Incompressible Viscous Flow Based on Optimal Control Theory", Int. J. Comp. Fluid Dyn, 17(4), 243-251, 2003. doi:10.1080/1061856031000135091

5

H. Yagi, M. Kawhara, "Shape Optimization of a Body Located in Incompressible Viscous Flow Using Adjoint Method", in Proc. ECCOMAS 2004.

6

H. Yagi, M. Kawahara, "Shape optimization of a body located in low Reynolds number flow", Int. J. Numer. Meth. Fluids, 48, 819-833, 2005. doi:10.1002/fld.957

7

H. Yoshida, "Shape optimization of an oscillating body in fluid flow by adjoint equation and ALE finite element methods", Int. J. Numer. Meth. Fluids, 0, 1-6, 2006.

8

S. Nakajima, M. Kawahara, "Shape Optimization of a Body in Compressible Inviscid Flows", Comp. Meth. Appl. Mech. Engrg., 197, 4521-4530, 2008. doi:10.1016/j.cma.2008.05.013

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