Robust and efficient adjoint solver for complex flow conditions

- Krylov solvers with subspace recycling are used to solve the adjoint for stiff cases.
- Significant convergence acceleration and memory reduction are demonstrated.
- The memory threshold for convergence stall is significantly reduced using recycling.
- GCRO-DR significantly outperforms the baseline solver using GMRES.
- GCROT also shows improvement but is less efficient and robust than GCRO-DR.

A key step in gradient-based aerodynamic shape optimisation using the Reynolds-averaged Navier-Stokes equations is to compute the adjoint solution. Adjoint equations inherit the linear stability and the stiffness of the nonlinear flow equations. Therefore for industrial cases with complex geometries at off-design flow conditions, solving the resulting stiff adjoint equation can be challenging. In this paper, Krylov subspace solvers enhanced by subspace recycling and preconditioned with incomplete lower-upper factorisation are used to solve the stiff adjoint equations arising from typical design and off-design conditions. Compared to the baseline matrix-forming adjoint solver based on the generalized minimal residual method, the proposed algorithm achieved memory reduction of up to a factor of two and convergence speedup of up to a factor of three, on industry-relevant cases. These test cases include the DLR-F6 and DLR-F11 configurations, a wing-body configuration in pre-shock buffet and a large civil aircraft with mesh sizes ranging from 3 to 30 million. The proposed method seems to be particularly effective for the more difficult flow conditions.