Comparative study of inner-outer Krylov solvers for linear systems in structured and high-order unstructured CFD problems

TL;DR

The paper tackles the challenge of solving large, ill-conditioned linear systems arising from discrete adjoint formulations in turbulent aerodynamic optimization. It advocates a flexible inner-outer GMRES framework with domain-decomposition preconditioning and spectral deflation, instantiated with FV and high-order DG discretizations to assess robustness and scalability. A combination of RAS-LU-SGS and improved overlap BILU(0) preconditioners, together with deflation via GMRES-DR/FGMRES-DR, yields substantial improvements in convergence, memory efficiency, and parallel performance for transonic adjoint problems on ONERA OAT15A and M6 wing test cases. The results demonstrate strong scalability and practical numerical practices, supporting the approach as a viable, high-performance option for adjoint-based aerodynamic sensitivity analyses in complex CFD workflows.

Abstract

Advanced Krylov subspace methods are investigated for the solution of large sparse linear systems arising from stiff adjoint-based aerodynamic shape optimization problems. A special attention is paid to the flexible inner-outer GMRES strategy combined with most relevant preconditioning and deflation techniques. The choice of this specific class of Krylov solvers for challenging problems is based on its outstanding convergence properties. Typically in our implementation the efficiency of the preconditioner is enhanced with a domain decomposition method with overlapping. However, maintaining the performance of the preconditioner may be challenging since scalability and efficiency of a preconditioning technique are properties often antagonistic to each other. In this paper we demonstrate how flexible inner-outer Krylov methods are able to overcome this critical issue. A numerical study is performed considering either a Finite Volume (FV), or a high-order Discontinuous Galerkin (DG) discretization which affect the arithmetic intensity and memory-bandwith of the algebraic operations. We consider test cases of transonic turbulent flows with RANS modelling over the two-dimensional supercritical OAT15A airfoil and the three-dimensional ONERA M6 wing. Benefits in terms of robustness and convergence compared to standard GMRES solvers are obtained. Strong scalability analysis shows satisfactory results. Based on these representative problems a discussion of the recommended numerical practices is proposed.

Figures (10)

• Figure 1: 2D transonic turbulent flow over the OAT15A airfoil: (a) and (b) illustrate the structured and unstructured meshes partitioned into 16 subdomains. The corresponding steady-state density field is illustrated in (c) and (d) for the FV and DG schemes respectively.
• Figure 2: Impact of deflation for various preconditioners. The relative residual norm convergence history is plotted with respect to iterations. The impact of deflation is dramatically effective both on the LUSGS and BILU preconditioners applied to the first-order approximate Jacobian matrix. We recall the numerical parameters of the FGMRES-DR solver as follows: $m=60, m_{i}=20$ and $k=20$.
• Figure 3: Strong scalability analysis for FGMRES-DR(60,20,20) solvers on the OAT15A adjoint system.
• Figure 4: 3D transonic turbulent flow over the ONERA M6 Wing: (a) and (b) represent respectively the wing geometry for structured and unstructured meshes while the steady state pressure field is illustrated in (c) and (d) for both FV and DG computations.
• Figure 5: True and least-squares residual norm convergence histories by applying two steps of Gram-Schmidt orthogonalization procedure: least-sq. res. stands for least-squares residual while true res. stands for true residual. We recall the numerical parameters of the GMRES-DR solver as follows: $m=140$ and $k=42$