Keywords: arbitrary Lagrangian Eulerian finite element method, fluid-structure interaction, optimal control theory, performance function, first order adjoint equation, weighted gradient method.

The purpose of this paper is to determine the angle of a wing which is attached to a oscillating bridge 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 using the velocity of the bridge. At present, there are some bridges with wings to prevent oscillation by the wind. The angles of these wings are determined by the wind tunnel experiment, but the experiment is expensive and it takes a lot of time to prepare the models. Therefore, it is necessary to determine the angle by numerical analysis.

The body is assumed to be a bridge with a wing supported by the elastic spring. The Navier-Stokes equation described in the ALE form is employed to express the motion of fluid around the body. The performance function is given in the optimal control theory. The vertical velocity of the body is computed so as to minimize the performance function under the constraint conditions. The performance function is defined by the square sum of the velocity on the surface of the body. The oscillation of the body can be minimized by the control variable. The extended performance function is defined by using the Lagrange multiplier method. The first order adjoint equations can be obtained using the stationary condition of the extended performance function. We can derive the gradient by solving the adjoint equations and the state equations. It is necessary to change coordinate system of the gradient into the polar coordinate system. The weighted gradient method is applied as the minimization technique. The performance function is minimized by calculating the angle in the weighted gradient method.

In the numerical study, an optimal control of an oscillating bridge with a wing is carried out. The Reynolds numbers are assumed to be 250 in the study, the computational domain is modelled as a two-dimensional surface. A uniform stream is given on the inflow boundary for the horizontal direction. The angle of the wing which the oscillation of the body becomes minimum is shown in the numerical results.

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