A discrete adjoint method for pressure-based algorithms

Abstract

A discrete adjoint method implemented in a coupled pressure-based RANS solver is presented in this paper. The adjoint equations are solved using an adjoint fixed point iteration that inherits the convergence properties of the primal solver. Automatic differentiation is used extensively for the construction of the adjoint fixed point iteration. The concept of Krylov subspace methods was adopted to stabilize the solution procedure. A common linearization technique in collocated pressure-based algorithms is the introduction of a mass flux variable on the cell faces which is kept constant during the inner iterations. This variable is treated as an independent adjoint variable in related publications. We propose a new method that allows to treat the mass fluxes implicitly in order to take full advantage of the preconditioner of the primal solver. The adjoint solver is general and is not restricted by the commonly used frozen turbulence approximation. It can deal with any turbulence model that is supported by the flow solver as well as any boundary condition. This includes mixing planes and mesh interfaces needed for multi stage turbo machinery simulations. Furthermore, there is no restriction on the choice of objective function. The sensitivities of the adjoint solver have been validated with sensitivities obtained with finite differences.An entirely surface based interpolation method based on radial basis functions (RBF) was developed to deform the surface mesh. We propose the use of discrete geodesics instead of the classical Euclidean distance as the distance measure for the RBF interpolation. As an alternative, a direct deformation method with adjoint consistent smoothing is also described and used in the presented optimization cases.The developed adjoint solver and deformation routines were used to optimize a turbulent bend with different Reynolds numbers as well as the the rotor blade of an axial turbine.