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     In the direct manual method (or ``geometric design'' method, using parlance from XFoil), an engineer formulates an airfoil shape directly and modifies this iteratively by hand. Aerodynamic analysis is performed by any code that will perform the forward problem (geometry $\rightarrow$ aerodynamics), such as XFoil, MSES \cite{mses}, or any RANS-based CFD code \cite{adler_cfd_2022}. This allows for easier satisfaction of non-aerodynamic constraints, but it is difficult to directly target aerodynamic quantities. Significant user expertise is also required to identify the most relevant geometric parameters and to make effective changes to the airfoil shape.
 
-    In the optimization approach, a parameterized airfoil shape is optimized to minimize a cost function and satisfy specified constraints. At first glance, this appears to be an automation of the direct manual method. However, Drela and others note the surprising difficulty of posing the correct optimization problem \cite{drela_pros_1998, kroo_multidisciplinary_1997}, so this approach requires just as much (if not more) human expertise than the direct manual method. As stated by Drela \cite{drela_pros_1998}, optimization-based airfoil design ``is still an iterative cut-and-try undertaking. But compared to [direct] techniques, the cutting-and-trying is not on the geometry, but rather on the precise formulation of the optimization problem.'' To support this, Drela gives compelling case studies of how airfoil design optimization can go awry in the absence of user review and care. Some codes, like the LINDOP \cite{mses} optimization routine coupled with XFoil, alleviate this somewhat by using a hybrid of the direct and optimization approaches: update directions are computed by an optimizer, but the actual changes are reviewed and implemented by a human between iterations. Despite the potential challenges of this optimization-based airfoil design approach, it offers compelling benefits: resulting airfoil performance is competitive with expert-designed airfoils \cite{drela_pros_1998}, and optimization can provide a systematic and robust approach that solves challenging design problems from poor initial guesses \cite{he2019robust}.
+    In the optimization approach, a parameterized airfoil shape is optimized to minimize a cost function and satisfy specified constraints. At first glance, this appears to be an automation of the direct manual method. However, Drela and others note the surprising difficulty of posing the correct optimization problem \cite{drela_pros_1998, kroo_multidisciplinary_1997}, so this approach requires just as much (if not more) human expertise than the direct manual method -- though that is not to say it is without benefits. As stated by Drela \cite{drela_pros_1998}, optimization-based airfoil design ``is still an iterative cut-and-try undertaking. But compared to [direct] techniques, the cutting-and-trying is not on the geometry, but rather on the precise formulation of the optimization problem.'' To support this, Drela gives compelling case studies of how airfoil design optimization can go awry in the absence of user review and care. Some codes, like the LINDOP \cite{mses} optimization routine coupled with XFoil, alleviate this somewhat by using a hybrid of the direct and optimization approaches: update directions are computed by an optimizer, but the actual changes are reviewed and implemented by a human between iterations. Despite the potential challenges of this optimization-based airfoil design approach, it offers compelling benefits: resulting airfoil performance is equal or exceed that of expert-designed airfoils \cite{drela_pros_1998}, particularly on problems that with unique or otherwise non-intuitive constraints. This optimization process also has the potential to require orders of magnitude less engineering time, and it provides a systematic and disciplined approach that is especially suited to the most challenging design problems \cite{he2019robust}.
 
-    In all of these methods, some form of a computational tool for airfoil aerodynamics analysis is required. For subsonic airfoils, the gold standard of such tools is XFoil \cite{drela_xfoil_1989}. XFoil is more accurate than RANS-CFD-based tools \cite{morgado2016xfoil}, yet it has a computational cost that is roughly 1,000x lower -- a testament to the power of its modeling approach, which couples integral boundary layer and potential flow methods. A complete description of this modeling approach is available in Drela's \textit{Aerodynamics of Viscous Fluids} \cite{drela_aerodynamics_2019}, and in recent state-of-the-art work by Zhang \cite{zhang_threedimensional_2022, zhang_nonparametric_2017}. However, XFoil has several attributes that make it less-than-ideal for directly driving numerical optimization studies:
+    In all of these methods, some form of a computational tool for airfoil aerodynamics analysis is required. For subsonic airfoils, the gold standard of such tools is XFoil \cite{drela_xfoil_1989}. XFoil is more accurate than RANS-CFD-based tools \cite{morgado2016xfoil}, yet it has a computational cost that is roughly 1,000x lower -- a testament to the power of its modeling approach, which couples integral boundary layer and potential flow methods. A complete description of this modeling approach is available in Drela's \textit{Aerodynamics of Viscous Fluids} \cite{drela_aerodynamics_2019}, and in recent state-of-the-art work by Zhang \cite{zhang_threedimensional_2022, zhang_nonparametric_2017}. However, XFoil has several attributes that make it less-than-ideal for directly driving numerical optimization studies. Among these:
 
     \begin{itemize}
-        \item XFoil is not guaranteed to produce a solution. When an ``ambitious'' calculations are attempted, XFoil often fails to provide a converged solution; the unconverged result often has wildly-diverging values and is effectively unusable. In some cases, calculations can lead to infinite loops or process crashes due to unhandled exceptions. While this is acceptable in certain applications (e.g., manual direct analysis), it is generally unacceptable for use in numerical optimization. Instead, optimization strongly benefits from a robust analysis tool that always produces a result, even if that result has slightly degraded accuracy. This allows the analysis to steer the optimizer back towards the design space of reasonable airfoils \cite{he2019robust}. A particularly useful case is when the model is deliberately made to be slightly pessimistic (e.g., overestimate drag) in regions of the design space with high uncertainty, adding further optimization pressure towards reasonable designs.
+        \item XFoil is not guaranteed to produce a solution. When an ``ambitious'' calculations are attempted, XFoil often fails to provide a converged solution; the unconverged result often has wildly-diverging values and is effectively unusable. In some cases, calculations can lead to infinite loops or process crashes due to unhandled exceptions. While this is acceptable in certain applications (e.g., manual direct analysis), it is generally unacceptable for use in numerical optimization. Instead, optimization strongly benefits from a robust analysis tool that always produces a result, even if that result has slightly degraded accuracy; this allows the analysis to steer the optimizer back towards the design space of reasonable airfoils \cite{he2019robust}. A particularly useful case is when the model is deliberately made to be slightly pessimistic (e.g., overestimate drag) in regions of the design space with high uncertainty, adding further optimization pressure towards reasonable designs with low performance uncertainty.
         \begin{itemize}
             \item More generally, design optimization is not the only application that strongly benefits from an aerodynamics analysis tool that always produces an answer. Other examples where a non-answer, infinite loop, or crashed process are unacceptable include real-time control (e.g., as an aerodynamic model for a model-predictive controller onboard an aircraft) and flight simulation.
         \end{itemize}
         \item XFoil solutions are not necessarily unique, and slightly different solutions can be obtained for the same analysis problem (airfoil shape, angle of attack, and Reynolds and Mach numbers). In practice, this manifests as an effective hysteresis depending on whether angle of attack is swept up or down. This flow non-uniqueness is in fact a real physical\footnote{For example, flow over an airfoil may separate as its angle of attack increases past $12\degree$, but it may not fully reattach until the angle of attack descends back to below $11\degree$} effect \cite{jameson_airfoils_1991, kuzmin2012non, he2019robust}. However, this non-uniqueness can be exceptionally problematic for numerical optimization, as there is no limit to how sensitive performance can be to input parameters (precluding approaches such as finite-differencing for gradient-based optimization).
         \item XFoil solutions are non-smooth\footnote{precisely, they are $C^0$ continuous but not $C^1$ continuous} with respect to input parameters, which makes it fundamentally incompatible with gradient-based optimization. This is a consequence of how laminar-turbulent transition is handled by XFoil's integral boundary layer solver. This solve requires the use of laminar and turbulent boundary layer \emph{closure models}, which are curve-fitted functions that yield various necessary quantities ($H^*$, $c_f$, $c_\mathcal{D}$, etc.) of the von Karman integral momentum and kinetic energy equations as a function of the two values that parameterize the boundary layer ($H$, $\Rey_\theta$). The laminar and turbulent versions of these functions differ. XFoil implements a cut-cell approach on the transitioning interval, which restores $C^0$-continuity (i.e., transition won't truly ``jump'' from one node to another discretely); however, a sharp change in gradient occurs whenever an individual node switches its equation from laminar to turbulent. Adler et al. provide a graphical depiction of this phenomenon \cite{adler_cfd_2022}.
-        \item Most interfaces between an optimizer and XFoil communicate through a series of text files (i.e., hard disk), rather than by sharing data in memory (i.e., RAM). Given the quick speed of an individual XFoil run, this input-output overhead can impose a significant performance penalty.
+        \item Most interfaces between an optimizer and XFoil communicate through a series of text files (i.e., hard disk), rather than by sharing data in memory (i.e., RAM). Given the quick speed of an individual XFoil run, this input-output overhead can impose a significant performance penalty. While this programming-language-agnostic interface is arguably one of the reasons for XFoil's long-enduring popularity, it comes at the cost of runtime performance if the tool is to be used in a high-throughput setting (e.g., for design optimization).
     \end{itemize}
 
-    This motivates the development of a new airfoil aerodynamics analysis tool that captures the advantages of XFoil (accuracy, speed) while mitigating these drawbacks. In recent years, many fields have benefited from a hybrid data-and-theory approach, where data-driven models are used to augment traditional physics-based models with learned closures. This work presents a similar physics-informed approach as applied to analyzing airfoil aerodynamics.
+    This motivates the development of a new airfoil aerodynamics analysis tool that captures the advantages of XFoil (accuracy, speed) while mitigating these drawbacks (i.e., incompatibility with optimization). In recent years, many fields have benefited from a hybrid data-and-theory approach, where data-driven models are used to augment traditional physics-based models with learned closures. This work presents a similar physics-informed approach as applied to analyzing airfoil aerodynamics.
 
     % TODO write about other physics-informed ML approaches, webfoil?