An exact discrete adjoint of an unstructured finite-volume solver for the RANS equations has been developed. The adjoint is exact in the sense of being based on the full linearization of all terms in the solver, including all turbulence model contributions. From this starting point various approximations to the adjoint are derived with the intention of simplifying the development and memory requirements of the method; considered are many approximations already seen in the literature. The effect of these approximations on the accuracy of the resulting design gradients, and the convergence and final solution of optimizations is studied, as it applies to a two-dimensional high-lift configuration.
This paper describes the work performed by ONERA and Airbus to solve several aerodynamic optimization problems proposed in 2013 by the AIAA Optimization Discussion Group (ADODG). Three of the four test cases defined by this group have been addressed, respectively a 2D invicid, non-lifting, transonic airfoil optimization problem, a 2D RANS transonic airfoil optimization problem and a 3D RANS transonic wing optimization problem. All three problems have been investigated using local, gradient-based, optimization techniques and the elsA[1][2] CFD software and its adjoint capability. Through these three optimization exercises, several generic issues introduced by aerodynamic gradient-based optimization have been investigated. Among the investigated aspects are the impact of the geometry parameterization (nature and dimension), of the accuracy of the gradient calculation method, optimization algorithm and presence of constraints in the optimization problem.
NomenclatureC p = pressure coefficient CD = total drag coefficient CDp = pressure drag coefficient CDf = friction drag coefficient CDw = wave drag coefficient CDvp = viscous pressure drag coefficient CL = lift coefficient CM = pitching moment coefficient c ref = chord reference d.c. = drag counts (0.0001) Ma = Mach number Re = Reynolds number AoA = Angle of attack f = objective function g = inequality constraint 1
The complete discrete adjoint equations for an unstructured finite volume compressible Navier-Stokes solver are discussed with respect to the memory and time efficient evaluation of their residuals, and their solution. It is seen that application of existing iteration methods for the non-linear equation -suitably adjointed -have a property of guaranteed convergence provided that the non-linear iteration is well behaved. For situations where this is not the case, in particular for strongly separated flows, a stabilization technique based on the Recursive Projection Method is developed. This method additionally provides the dominant eigenmodes of the problem, allowing identification of flow regions that are unstable under the basic iteration. These are found to be regions of separated flow. Finally an adjoint based optimization with 96 design variables is performed on a wing-body configuration. The initial flow has large regions of separation, which are significantly diminished in the optimized configuration.
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