Shape optimisation of an oscillating body in fluid flow by adjoint equation and ALE finite element methods

The purpose of this paper is to determine the shape of an oscillating body by minimising drag and lift forces, located in a transient incompressible viscous fluid flow by means of the Arbitrary Lagrangian Eulerian finite element method and an optimal control theory. A performance function is expressed by the drag and lift forces. The performance function should be minimised satisfying the state equation and the constant volume condition. Therefore, this problem can be transformed into a minimisation problem without constraint by the Lagrange multiplier method. The adjoint equation and the gradient of the performance function are used to update the shape of the body. In this study, as a minimisation technique, the weighted gradient method is applied. The final shape is obtained of which drag and lift forces are reduced by 66.2% and 92.8%, respectively. The final shape obtained by this study is compared with the final shape of the non-oscillating body. The obtained final shape of the oscillating body is significantly different from the non-oscillating body.