A Consistent Fixed-point Discrete Adjoint Method for Segregated Navier--Stokes Solvers

Abstract

The adjoint method is an efficient way to compute derivatives for gradient-based optimization problems with a large number of design variables. In a previous study, we developed a Jacobian-free Krylov adjoint method for steady-state Navier--Stokes (NS) solvers in OpenFOAM. Despite its advantage in speed, the fully-coupled Krylov adjoint convergence is inconsistent with the OpenFOAM's primal solvers, which commonly use a segregated solution strategy. Furthermore, the Krylov adjoint uses a relatively large amount of memory because it needs to store the full Jacobian matrix for computing the precondition. Fixed-point adjoint is an alternative approach that generally converges slower but uses much less memory than the Krylov adjoint. However, existing fixed-point adjoint algorithms cannot be directly applied to OpenFOAM's primal solvers because they typically assume a coupled primal solution approach. In this paper, we propose a fully-segregated, fixed-point discrete adjoint approach to ensure consistency with the segregated NS primal solvers. The central recipe is that we rewrite the segregated primal solution process into a fully coupled left-preconditioned Richardson format. With this treatment, we can then transpose the preconditioner matrix to easily derive a consistent fixed-point adjoint formulation. The transposed matrix-vector products in the preconditioner are computed using the reverse-mode automatic differentiation (AD) without saving the matrix in memory. We implement the proposed algorithm to an incompressible segregated solver (simpleFoam) and evaluate the adjoint performance using a U-bend channel and a low-speed rectangular wing. The proposed adjoint approach exhibits a consistent convergence rate with the primal solution for both cases. The proposed fixed-point adjoint approach has the potential to make the adjoint solution more robust and memory efficient for any segregated NS solvers.