Multi-point aerodynamic shape optimization of cars based on continuous adjoint

This article presents a continuous adjoint-enabled, gradient-based optimization tool for multi-point, multi-objective industrial optimization problems and its application to the shape optimization of a concept car. Apart from the adjoint to the incompressible Reynolds-averaged Navier–Stokes equations, the adjoint to the Spalart–Allmaras turbulence model equation is also solved, in order to support the optimization with accurate gradients. Part of the mathematical development related to the sensitivity derivative terms resulting from the differentiation of the Reynolds-averaged Navier–Stokes (RANS) variant of the Spalart–Allmaras model when using an adjoint formulation consisting of field integrals is presented for the first time in the literature. In the industrial application, two operating points are considered, corresponding to two flow velocity angles with respect to the car symmetry plane, with a different objective (drag and yaw moment coefficients) for each of them. With the aforesaid targets, the Pareto front of optimal solutions is computed and discussed. Each point on this front is computed by minimizing a single objective function, resulting from the linear combination of the objective functions defined on the different operating points, using appropriate weights. Finally, some of the Pareto front members are re-evaluated using delayed detached eddy simulation (DDEs). The overall optimization tool is developed in the open-source CFD toolbox OpenFOAM.