Optimal Control of an Oscillating Body using the Adjoint Equation and Arbitrary Lagrangian Eulerian Finite Element Methods

The purpose of this study is to determine the angle of a wing which is attached to a oscillating body located in a transient incompressible viscous flow using the Arbitrary Lagrangian Eulerian (ALE) finite element method and the optimal control theory in which a performance function is expressed by the velocity of the body. In order to minimize the oscillation of body, the performance function is introduced. The performance function is defined by the square sum of the velocity on surface of body. This problem can be transformed into the minimization problem by the Lagrange multiplier method. The adjoint equations can be obtained by the stationary condition of the extended performance function. We can derive the gradient to update the angle of the wing from solving state and adjoint equations. As a minimization technique, the weighted gradient method is applied. In this study, the angle which the oscillation of the body become to minimize is presented by these theory. To express the motion of fluid around a body, the Navier-Stokes equations described in the ALE form is employed as the state equation. The motion of the body is expressed by the motion equations. As a numerical study, the optimal control of the angle of the wing is shown at low Reynolds number 250.0. As the numerical result, the angle of the wing which the oscillation of the body becomes to minimize is shown.