When the “ exact ” discrete gradient is not the best choice in optimal shape design ?

In aerodynamics, Optimal Shape Design (OSD) aims to find the minimum of an objective function describing an aerodynamic property, by controlling the Partial Differential Equation (PDE) modeling the dynamics of the flow that surrounds an aircraft. The objective function minimization is usually achieved by means of an iterative process which requires the computation of the gradients of the cost function with respect to the design variables. Gradients can be computed in a variety of ways but, to the best of our knowledge, the most efficient method to compute the cost function gradient is the so-called “adjoint method”. At the computational level there are two approaches to the adjoint system: the continuous method and the discrete one. In the continuous approach, the adjoint equations are derived from the flow equations and then subsequently discretized (continuous adjoint methodology ), whereas in the discrete approach the adjoint equations are directly derived from the discretized governing equations (discrete adjoint, automatic differentiation, finite differences or complex step methodologies). In the aeronautical practice, the target is to minimize a discrete objective function and, consequently, the “discrete approach” seems to be more “natural”. However, as shown in this paper, there are relevant situations (strong shock waves or wrong numerical grid orientation) in which the gradient obtained using a discrete approach shows a non-physical oscillatory behavior. In those cases, the discrete objective function (at the smallest scales) fails to capture the behavior of the continuous objective function and, consequently, the discrete approach to compute the objective function gradient could be inappropriate. In this article, we will analyze the origin of the aforementioned numerical oscillations in different systems (Burgers’ equation, quasi 1D Euler’s equations and 2D Euler’s equations). Some numerical experiments will be shown and their impact on the optimization process will be explained. Finally some conclusions and further recommendation will be presented.