Shape Optimization for the Navier–stokes Equations Based on Optimal Control Theory

This paper presents a new approach to a shape optimization problem of a body located in the unsteady incompressible viscous flow field based on an optimal control theory. The optimal state is defined by the reduction of drag and lift forces subjected to the body. The state equation used is the transient incompressible Navier–Stokes equations. The shape optimization problem can be formulated to find out geometrical coordinates of the body to minimize the performance function that is defined to evaluate forces subjected to the body. The mixed finite element method by the MINI element is used for the spatial discretization, while the fractional step method with implicit temporal integration is used for the temporal discretization. For the numerical study, the optimal shape of the body which has circular shape as the initial state can be finally obtained as the streamlined shape.