Computational methods for sensitivity analysis have proven to be incredibly useful to a wide range of engineers. Aerospace engineers have used these methods to optimize aerodynamic shapes and aircraft configurations, automatically adapt the computational mesh to reduce errors in Computational Fluid Dynamics (CFD) simulations, and to quantify uncertainties in these simulations. However, traditional sensitivity analysis methods, including the widely used adjoint method, break down when applied to long time averages of chaotic systems. This is problematic, because many problems of interest to aerospace engineers exhibit chaotic dynamics, most notably turbulent fluid flows when simulated with a high enough fidelity. Also, engineers are often interested in long time averaged quantities, such as long time averaged lift of a flight vehicle. To efficiently apply design optimization, mesh adaptation, and uncertainty quantification to chaotic systems and high fidelity fluid flow simulations, a new approach to sensitivity analysis is needed.
A recently proposed method, Least Squares Shadowing (LSS) presents a promising alternative that avoids the break down encountered by traditional sensitivity analysis. However, LSS has some issues, most notably a lack of robustness to certain errors and large computational costs.
This talk will discuss some methods proposed to increase the robustness and efficiency of LSS implementations. Additionally, some very preliminary results of LSS applied to a CFD simulation will be presented.