A dynamically deflated GMRES adjoint solver for aerodynamic shape optimization

Abstract

An essential step in adjoint-based aerodynamic shape optimization is to obtain the Lagrange multiplier by solving a sparse linear adjoint system of equations based on the Jacobian matrices from the converged flow states. Such an approach has been applied widely within the aerospace community for the design of aircraft and other optimization problems for aerospace applications. However, the need to resolve the flow over complex geometries often requires highly stretched grids and gives rise to anisotropic flow fields which increase the stiffness of the discrete Jacobian needed for the solution of the adjoint system. When a GMRES algorithm preconditioned by an Incomplete LU factorization is used, this stiff linear system requires the use of a large number of Krylov subspace vectors and a high level of fill-in; both require an increase in the amount of memory. Deflated restarting, which distributes spectral eigenpairs, has proven to be an effective method to enhance the convergence rates when solving an ill-conditioned linear system of equations. In this paper, we propose an algorithm to dynamically determine the number of deflated vectors and to deactivate deflation to reduce memory utilization. We demonstrate its efficiency for a two-dimensional NACA 0012 airfoil, the three-dimensional Common Research Model (CRM) wing, and two three-dimensional complete aircraft configurations in viscous flow.