Shape determination of wind-resistant wings attached to an oscillating bridge using adjoint equation method

Abstract

The purpose of this study is to determine the angle of a wing that is attached to an oscillating bridge located in transient incompressible viscous flows, using the arbitrary Lagrangian–Eulerian (ALE) finite element method and optimal control theory, in which a performance function is expressed by the displacement of the bridge. Currently, some bridges have wings attached to them to prevent oscillation caused by wind flows. When the angle of the wing changes, the state of oscillation also changes. Therefore, the angle of the wing is a very important parameter to consider the minimization of the oscillation of the bridge. In this research, the angle of the wing is determined based on optimal control theory. To minimize the oscillation of a bridge, the performance function is introduced as the minimization index. The performance function is defined by the square sum of the displacements of a bridge. This problem can be transformed into a unconstrained minimization problem by the Lagrange multiplier method. The adjoint equations can be obtained by using the stationary condition of the extended performance function. The gradient used for updating the angle of the wing can be derived by solving the adjoint and state equations. The weighted gradient method is applied as a minimization technique. In this study, the determination of the angle at which the oscillation of the bridge is minimized is presented using this theory. To express the motion of fluids around a bridge, the Navier–Stokes equations described in the ALE form are employed as the state equations. The motion of the bridge is expressed by the motion equations by using the displacements and rotational angle of the body supported by springs. As a numerical study, the optimal control of the angle of the wing is demonstrated at low Reynolds number flows. Thus, the angle of the wing at which the oscillation of the bridge becomes minimum can be determined. Numerical results obtained correspond to the angle of actual bridge wing.