Application of the discrete adjoint method to coupled multidisciplinary unsteady flow problems for error estimation and optimization

Adjoint methods have found applications in several key areas of computational fluid dynamics (CFD), namely, shape optimization and goal based adaptive solutions. CFD has become an essential tool in the design process by enabling the rapid testing of multiple designs, and currently it is normal practice to use CFD in conjunction with optimization algorithms for design improvement. In the context of shape optimization problems based on CFD, adjoint methods offer the significant advantage of computing sensitivity derivatives of the optimization cost function with respect to the set of design parameters, at a cost that is essentially independent of the number of design parameters. Adjoint methods reduce the cost of obtaining the complete gradient vector at any point in the design space equivalent to that of a single flow solution at the same point in the design space. This immediately enables the use of all gradient based optimization algorithms and lifts any restrictions on the number of design parameters required for the adequate definition of the optimization problem.;Adaptive techniques in CFD constitute the other aspect where adjoint methods have have made great inroads. Typical adaptive solutions of the governing flow equations rely on estimating the local error in an evolving solution to target regions of the computational mesh for increased discrete resolution. The main goal of any adaptive solution method is the overall increase in solution accuracy with minimal increase in computational cost. However, targeting local error in the solution does not translate into efficient use of computational resources, since ultimately it is the accurate estimation of boundary integrated functional quantities such as load coefficients that are of importance to the user. Contrary to local error-based methods, adjoint methods allow the adaptation of the computational mesh specifically for the improvement of functionals such as load coefficients. This is achieved by mathematically establishing a clear relationship between the functional of interest and the regions of the computational mesh that are most relevant to it.;The current work extends the use of adjoint methods to multiple governing disciplines that are tightly coupled, and more importantly unsteady in nature. The adjoint method is derived in a very general form for the purpose of computing the gradient vector for use in shape optimization in the context of coupled multidisciplinary unsteady equations. It is shown that computing the gradient vector in unsteady problems involves solving the analysis problem forward in time and then solving the adjoint problem backward in time. While adjoint methods have been used successively to drive spatial mesh adaptation, the current work extends the use of the computed unsteady adjoint variables for estimating temporal discretization error, which is then applied to temporal mesh adaptation. Additionally, the computed adjoint variables are also used for the estimation of algebraic error in the solution arising due to intentional or nonintentional partial convergence of the governing equations. Results indicate that the adaptation of the temporal resolution and convergence tolerance limits using adjoint-based error estimates is able to outperform traditional adaptation methods such as uniform refinement and those based on local error estimates. All of the development is carried out in a fully unstructured mesh framework with dynamic deformation of the computational spatial mesh.