A coupled fully implicit method for solving drag-adjoint equations of steady incompressible flows

Numerical instability is one of the main concerns when applying adjoint-based gradient computation in aerodynamic design optimization for a variety of engineering problems. An adjoint pressure-adjoint velocity-coupled fully implicit method for solving the adjoint equations of steady incompressible flows is developed in the present work, taking the aerodynamic drag as the optimization target. The finite volume method for cell-centered, collocated variables on unstructured grids is used for spatial discretization. A stabilization term based on pressure-weighted interpolation (PWI) of the adjoint velocity is added to the adjoint continuity equation to avoid spurious oscillations of adjoint pressure. All the terms of the continuous adjoint equations are discretized implicitly, including the adjoint transpose convection (ATC) term, which is regarded as one of the major sources of numerical instability. The resulting discretized equations are integrated into one linear system, and the adjoint fields can be obtained simultaneously. To prevent the diagonal element of the coefficient matrix of the linear system from getting too small, a singular value decomposition is applied to the diagonal block of the coefficient matrix, and a limiter is applied to the diagonal elements. The condition number of the coefficient matrix may be quite large, so a parallel sparse linear solver that effectively combines direct and iterative methods through a Schur complement approach is applied to solve the linear system. Compared to the staggered iterative solver, no damping or masking is needed for the ATC term in the proposed fully implicit method. There is no need to sacrifice numerical accuracy for numerical stability. This method has been tested in several benchmark cases, and the numerical results show good numerical stability and accuracy. The present method provides a sound basis for industrial applications of adjoint-based aerodynamic optimization methods.