Optimization Using Arbitrary Lagrangian–Eulerian Formulation of the Navier–Stokes Equations

In this paper, we present a new shape optimization method by using sensitivities obtained from the Arbitrary Lagrangian–Eulerian (ALE) form of the Navier–Stokes equations. In the ALE description, the nodes of the computational domain may be moved with the fluid as in the Lagrangian description, held fixed in space as in the Eulerian description, or moved in some arbitrary way in between. Applying the adjoint method with respect to mesh motion allows the whole sensitivity field for the shape changes to be calculated using only two solver calls, a primal solver call and an adjoint solver call. We show that the sensitivities with respect to the mesh motion can be calculated in a postprocessing step to the primal and adjoint flow simulations. The resulting ALE sensitivities are compared to sensitivities obtained using a finite difference approach. Finally, the sensitivities are coupled to a mesh motion smoothing algorithm, and a duct is optimized with respect to the total pressure drop using the proposed method.