An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids

An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier-Stokes equations is presented. The method is constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system. The new approach is implemented within a three-dimensional unstructured-grid framework and results are presented for inviscid, laminar, and turbulent flows. Improvements to the baseline solution algorithm, such as line-implicit relaxation and a tight coupling of the turbulence model, are also presented. By storing nearest-neighbor terms in the residual computation, the dual scheme is computationally efficient, while requiring twice the memory of the flow solution. The current implementation allows for multiple right-hand side vectors, enabling simultaneous adjoint solutions for several cost functions or constraints with minimal additional storage requirements, while reducing the solution time compared to serial applications of the adjoint solver. The scheme is expected to have a broad impact on computational problems related to design optimization as well as error estimation and grid adaptation efforts.