a code-coupling approach to the implementation of discrete adjoint solvers based on automatic differentiation

Abstract

We propose a method for selectively applying automatic differentiation (AD) by operator overloading to develop the discrete adjoint of a turbomachinery flow solver. A fully differentiated version of the solver is generated by operator overloading using the tapeless tangent mode of ADOL-C. The differentiated solver is coupled to an undifferentiated version of the same code using message passing. The automatic differentiation is used to calculate derivatives of the flux calculation routines. The flux derivatives depending on inner cell states are sparse, and this sparsity is exploited using analytical differentiation of the spatial discretization scheme. Subsequently the sparse matrix is communicated to the undifferentiated code for solution. Turbomachinery boundary conditions may have dense Jacobians and are therefore only evaluated during the solution process. The solution of the adjoint system of equations is achieved through a preconditioned GMRES, implemented inside the undifferentiated code. A modern three dimensional contra-rotating fan stage with engineering parameterization serves as application example in order to demonstrate the technique and to perform numerical validations. The validation of gradient results is performed by comparing against results from finite differences, and the tangent forward mode.