Keywords: shape optimization, optimal control theory, domain variation.

What is the shape of the body which has minimum drag when moved at constant speed in a viscous flow field? Not much is known theoritically about this problem. Therefore it is very difficult to design a shape of such bodies. The extreme shape of such a body would be a dot of which volume is zero. If it is tried to design a shape with a given volume, the shape would be a flat plate which length is as long as possible. Then, what would be the shape which has minimum drag with a given volume and a given length? To suggest such a body is not easy. The purpose of this study is to apply a formulation of shape optimization to a numerical simulation of the body located in an incompressible viscous flow field and to find out the solution of this problem.

The shape optimization is based on the optimal control theory. In an optimal control theory, a control value which makes phenomenon an optimal state can be obtained. In this theory, a performance function should be introduced. When the performance function is minimized, it is assumed that the state is optimized, and then the control value is obtained. The shape optimization is to find out the surface coordinates of the body so as to minimize the performance function. The performance function is defined by the fruid forces subjected to the body. The fluid forces can be derived by integrating the traction on the boundary of the body.

The performance function should be minimized satisfying the state equation. This problem can be transformed into the minimization problem without constraint condition by Lagrange multiplier method. The optimal control problem with the constraint condition of the state equations results in solving a stationary condition of the extended performance function instead of minimizing the original performance function. The necessary condition to minimize the performance function is stationary condition of the extended performance function. Setting each term equal to zero to satisfy the optimal condition, adjoint equations, adjoint boundary condition can be obtained. To minimize the performance function , the gradient of the extended performance function with respect to geometrical surface coordinates should be obtained. The gradient can be computed by using both of the state and adjoint values.

For the minimization of the performance function, Sakawa-Shindo method is employed. In this method a modified performance function which can be obtained by adding a penalty term to the performance function is introduced. When the modified performance function converges to minimum, the penalty term will be zero. To minimize the modified performance function is equal to minimize the extended performance function. The state equation for this problem is the Stokes equation.

The Pressure Stabilization / Petrov-Galerkin (PSPG) formulation is employed for discretization both of state and adjoint equations. In this formulation a stability parameter that is defined in each element. The stability parameter gives the optimal viscaucity to each elemant to solve stable the state equation.

The shape optimization is to find out the surface coordinates of the body. That is to determine the domain that minimizes the extended performance function. When the domain variation is treated, the material derivertive concept should be considered.

1

A. Maruoka and M. Kawahara, "Optimal control in Navier-Stokes equtions", IJCFD, 9, 1998, 313-322.

2

Zolesio, J.P., "The material derivertive (or speed) method for shape optimization", Optimization of Distributed Parameter Structures, edid by Haung,E.J. and Cea,J.,Vol.2, Sijthoff - Noordhoff, Alphen aan den Rijn, 1981, 1089-1151.

3

Zolesio, J.P., "Domain variational formulation for free boundary problems", Optimization of Distributed Parameter Structures, edid by Haung,E.J. and Cea,J.,Vol.2, Sijthoff and Noordhoff, Alphen aan den Rijn, 1981, 1152-1194.

4

Jean Cea, "Problems of Shape Optimal Design", Optimization of Distributed Parameter Structures, edid by Haung,E.J. and Cea,J.,Vol.2, Sijthoff and Noordhoff, Alphen aan den Rijn, 1981, 1005-1048.

5

T.E. Tezduyar, S. Mittal, S.E. Ray and R. Shih, "Incompressible Flow Computations with Stabilized Bilinear and Linear Equal-Order-Interpolation Velocity-Pressure Element", Comput Methods Appl.Mech.Engng.95, 1992, 221-242. doi:10.1016/0045-7825(92)90141-6

6

Max D. Gunzburger, "Flow Control", The IMA volumes in mathmatics and its applications vol. 68.

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