This paper presents a number of algorithm developments for adjoint methods using the `discrete' approach in which the discretisation of the nonlinear equations is linearised and the resulting matrix is then transposed

The linear analysis of turbomachinery aeroelasticity relies on the assumption of small level of unsteadiness and requires the solution of both the nonlinear steady and the linear unsteady flow equations. The objective of the analysis is to compute a complex flow solution that represents the amplitude and phase of the unsteady flow perturbation for the frequency of unsteadiness of interest. The solution procedure of the linear harmonic Euler/Navier–Stokes solver of the HYDRA suite of codes consists of a preconditioned fixed-point iteration, which in some circumstances becomes numerically unstable. Previous work had already highlighted the physical origin of these numerical instabilities and demonstrated the code stabilization achieved by wrapping the core part of the linear code with a Generalized Minimal Residual (GMRES) solver. The implementation and the use of an alternative algorithm, namely, the Recursive Projection Method, is summarized. This solver is shown to be well suited for both stabilizing the fixed-point iteration and improving its convergence rate in the absence of numerical instabilities. In the framework of the linear analysis of turbomachinery aeroelasticity, this method can be computationally competitive with the GMRES approach. I.

This paper presents a review of developments in aerodynamics during the last 50 years. Progress in aerodynamic design, theoretical aerodynamics, wind tunnel testing, and especially computational fluid dynamics (CFD) is discussed. Where appropriate, applications to aircraft design are presented, as are new aircraft concepts. Topics of Canadian interest are presented and the paper includes several examples of research and development in aerodynamics at de Havilland. RÉSUMÉ Le document présente une revue du secteur de l’aérodynamique depuis les 50 dernières années. On y traite des progrès réalisés en conception aérodynamique, en aérodynamique théorique, dans les essais en soufflerie et, surtout, en simulation numérique en mécanique des fluides (CFD). Le cas échéant, des applications dans la conception des aéronefs sont présentées, comme de nouveaux aéronefs concepts. Des sujets d’intérêt pour le Canada sont présentés, et le document renferme plusieurs exemples de recherche et de développement en aérodynamique chez de Havilland.

A number of algorithm developments are presented for adjoint methods using the “discrete ” approach in which the discretization of the nonlinear equations is linearized and the resulting matrix is then transposed. With a new iterative procedure for solving the adjoint equations, exact numerical equivalence is maintained between the linear and adjoint discretizations. The incorporation of strong boundary conditions within the discrete approach is discussed, and dif � culties associated with the use of linear perturbation and adjoint methods for applications with strong shocks are also examined. Nomenclature B = boundary condition projection operator I.z / = imaginary part of complex variable z i = p ¡1 J = nonlinear output functional, for example, lift QJ = linear perturbation to functional L = linearized discrete operator R = nonlinear discrete residual operator U = nonlinear variables u = linear perturbation variables v = adjoint variables ® = design/sensitivity variable Subscripts? = orthogonal to boundary conditions k = parallel to boundary conditions Superscripts.m / = subiteration index n = iteration index T = transpose

Many unsteady problems equilibrate to periodic behavior. For these problems the sensitivity of periodic outputs to system parameters are often desired, and must be estimated from a finite time or frequency domain calculation. Sensitivities computed in the time domain over a finite time span can take excessive time to converge, or fail altogether to converge to the periodic value. We derive a theoretical basis for this error and demonstrate it using two examples: a van der Pol oscillator and vortex shedding from a low Reynolds number airfoil. We show that output windowing enables the accurate computation of periodic output sensitivities, and may allow for decreased simulation time to compute both time-averaged outputs and sensitivities. We classify two distinct window types: long-time, over a large, not necessarily integer number of periods; and short-time, over a small, integer number of periods. Finally, from these two classes we investigate several examples of window shape and demonstrate their convergence with window size and error in the period approximation, respectively. Keywords: sensitivity analysis, unsteady, limit cycles, periodic