Multigrid Solution of the Discrete Adjoint for Optimization Problems on Unstructured Meshes

Abstract

Techniques for the efficient formulation and solution of the discrete adjoint problem for the Reynolds-averaged Navier-Stokes equations and for the mesh deformation equations are developed for use on unstructured grids in two dimensions. An explicit two-pass approach is used to construct the second-order-accurate flow adjoint equations, and a defect-correction scheme is used to solve the flow adjoint equations, employing a line-implicit agglomeration multigrid strategy to drive the defect-correction scheme. A similar line-implicit multigrid approach is used to solve the mesh deformation equations, as well as the adjoint system of these equations. For the flow sensitivity and mesh motion equations, the multigrid solver is constructed to be duality preserving, delivering similar convergence rates for the primal and dual (adjoint) problems in both cases. These techniques are demonstrated for a viscous turbulent airfoil shape optimization problem.