On the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization

• Surface and field integral-based continuous adjoint sensitivities revisited.
• The effect of grid/mesh sensitivities on accuracy, in continuous adjoint.
• Physical interpretation of the so-called Leibniz term.
• Continuous adjoint including the adjoint to hypothetical grid displacement PDE.
• New, enhanced adjoint formulation, both cheap and accurate.

The continuous adjoint method for shape optimization problems, in flows governed by the Navier–Stokes equations, can be formulated in two different ways, each of which leads to a different expression for the sensitivity derivatives of the objective function with respect to the control variables. The first formulation leads to an expression including only boundary integrals; it, thus, has low computational cost but, when used with coarse grids, its accuracy becomes questionable. The second formulation comprises a sum of boundary and field integrals; due to the field integrals, it has noticeably higher computational cost, obtaining though higher accuracy. In this paper, the equivalence of the two formulations is revisited from the mathematical and, particularly, the numerical point of view. Internal and external aerodynamics cases, in which the objective function is either the total pressure losses or the force exerted on a solid body, are examined and differences in the computed gradients are discussed. After identifying the reason behind these discrepancies, the adjoint formulation is enhanced by the adjoint to a (hypothetical) grid displacement model and the new approach is proved to reproduce the accuracy of the second adjoint formulation while maintaining the low cost of the first one.