finish paper

paper/TeX/main.tex

@@ -297,7 +297,7 @@
     Replacing the implicit mapping of Figure \ref{fig:compressibility_corrections} with the surrogate model of Equation \ref{eq:laitone_surrogate} introduces negligible error, with a corresponding RMS error in $\Mcr$ of $0.0014$ over all $\Cpom$ values. This relation allows NeuralFoil to estimate the critical Mach number $\Mcr$ using only incompressible quantities with surprising accuracy, often to within $\pm 0.01$ of results computed using the full-potential code MSES \cite{mses}. To determine $\Cpom$, which is needed for Equation \ref{eq:laitone_surrogate}, the definition of $\Cpom$ is used:
 
     \begin{equation}
-        \Cpom = 1 - \left( \frac{u_{\rm max}}{u_\infty} \right)^2
+        \Cpom = 1 - \left( \frac{u_{\rm max}}{u_\infty} \right)^2 \qquad \text{at }\ \Mi = 0
     \end{equation}
 
     Following a derivation from Mason \cite{mason_transonic_2006}, an empirical relation for the shape of the drag rise beyond $\Mcr$ is included. Drag in this regime, especially beyond the drag-divergent Mach number $M_{\rm dd}$ is relatively simple and errs on the side of over-estimating wave drag. However, given that a primary goal of the NeuralFoil tool is to drive design optimization, this empirical model serves its purpose of steering the optimizer away from thick transonic airfoils with strong shocks.
@@ -408,7 +408,7 @@
     \begin{figure}[h]
         \centering
         \includesvg[width=\textwidth]{figures/daedalus-iso.svg}
-        \caption{}
+        \caption{Drawing of the \emph{MIT Daedalus} human-powered aircraft, along with its centerline airfoil, the DAE-11. This airfoil is the subject of the airfoil design optimization problem posed in this section.}
         \label{fig:daedalus_iso}
     \end{figure}
 
@@ -442,7 +442,7 @@
 
     $$w(\mathrm{airfoil}) \approx \sum_{i=2}^{N-1} \left(c_{\mathrm{lower},i+1} - 2 \cdot c_{\mathrm{lower},i} + c_{\mathrm{lower},i-1} \right)^2 + \left(c_{\mathrm{upper},i+1} - 2 \cdot c_{\mathrm{upper},i} + c_{\mathrm{upper},i-1} \right)^2$$
 
-    \noindent where $c_{\mathrm{lower},i}$ and $c_{\mathrm{upper},i}$ are the $i$th Kulfan (CST) coefficients of the lower and upper surfaces, respectively. A reasonable heuristic for this regularization constraint is to restrict this metric to no more than four times that of the original initial guess airfoil (which is typically some generic airfoil, such as a NACA0010).
+    \noindent where $c_{\mathrm{lower},i}$ and $c_{\mathrm{upper},i}$ are the $i$th Kulfan (CST) coefficients of the lower and upper surfaces, respectively. A reasonable heuristic for this regularization constraint is to restrict this metric to no more than four times that of the original initial guess airfoil (which is typically some generic airfoil, such as a NACA0012).
 
     Using the problem formulation described above, we can solve this airfoil design optimization problem by coupling NeuralFoil with the AeroSandbox aircraft design optimization framework \cite{sharpe_aerosandbox_2021}. This adds automatic differentiation capabilities to NeuralFoil, allowing efficient optimization using gradient-based methods. This optimization problem is solved in approximately 7 seconds on a standard laptop; the speed of this solution provides rapid feedback to the designer on how to improve the optimization problem formulation to capture design intent.
 
@@ -454,18 +454,18 @@
         \item \textbf{Expert-designed}, which is the original DAE-11 airfoil designed by Drela \cite{drela_lowreynoldsnumber_1988} and used on the actual \emph{MIT Daedalus} aircraft
     \end{itemize}
 
-    For comparison, Figure \ref{fig:daedalus_optimized} also includes the initial guess airfoil that was provided to the optimization algorithms (a simple NACA0012).
+    For comparison, Figure \ref{fig:daedalus_optimized} also includes the initial guess airfoil that was provided to the optimization algorithms (a simple NACA0012). This showcases that the optimization process is robust to initial guesses that are relatively far from the optimal solution.
 
     \begin{figure}[h]
         \centering
         \includesvg[width=\textwidth]{figures/neuralfoil_daedalus.svg}
-        \caption{NeuralFoil-optimized airfoils yield performance that is competitive with expert-designed airfoils and have qualitatively-similar shapes. Adversarial optimization is not observed, as the NeuralFoil-optimized and XFoil-optimized airfoils yield similar performance when analyzed using XFoil.}
+        \caption{NeuralFoil-optimized airfoils yield performance that is competitive with expert-designed airfoils and have remarkably similar shapes. Adversarial optimization is not observed, as the NeuralFoil-optimized and XFoil-optimized airfoils yield similar performance when analyzed using XFoil.}
         \label{fig:daedalus_optimized}
     \end{figure}
 
     The airfoil produced using NeuralFoil optimization is quite similar in shape to the those produced using XFoil optimization and expert-designed airfoils, when given the same design objectives and constraints. Likewise, aerodynamic performance is quite similar across all three airfoils. Notably, the aerodynamic polars depicted in Figure \ref{fig:daedalus_optimized} were produced by post-optimality analysis using XFoil. Given the minimal discrepancy between the NeuralFoil-optimized and XFoil-optimized airfoils, this suggests that NeuralFoil is not prone to model error exploitation.
 
-    Interestingly, the differences between the expert-designed and optimized airfoils in Figure \ref{fig:daedalus_optimized} are attributable to design goals that were not factored into the quantitative problem formulation. For example, the optimized airfoils both exhibit a small amount of lower-surface concavity in the vicinity of $x/c \approx 0.15$, while the expert-designed DAE-11 has a flatter lower surface. Drela discusses reasons for this discrepancy in \cite{drela_pros_1998}, noting that concavity causes the wing covering (in the case of \emph{MIT Daedalus}, a thin covering of Mylar) to lift off of the surface of the wing (creating a bubble). This undesirable effect is not captured in the quantitative problem formulation, providing a cautionary tale that an optimized airfoil is only as good as the problem formulation that led to it.
+    Interestingly, the geometric differences between the expert-designed and optimized airfoils (both from XFoil and NeuralFoil) in Figure \ref{fig:daedalus_optimized} are attributable to design goals that were not factored into the quantitative problem formulation. For example, the optimized airfoils both exhibit a small amount of lower-surface concavity in the vicinity of $x/c \approx 0.15$, while the expert-designed DAE-11 has a flatter lower surface. Drela discusses reasons for this discrepancy in \cite{drela_pros_1998}, noting that the concavity that the optimizer prefers would cause the wing covering (in the case of \emph{MIT Daedalus}, a thin shrunk covering of Mylar) to lift off of the surface of the rib, creating a bubble. This undesirable effect is not captured in the quantitative problem formulation, providing a cautionary tale that an optimized airfoil is only as good as the problem formulation that led to it.
 
     Nevertheless, the strength of NeuralFoil-powered optimization is that it allows one to generate optimized airfoils that are remarkably similar to expert-designed airfoils in mere seconds. This optimization capability is more-than-adequate for conceptual aircraft design, and it provides an excellent starting point for expert-guided airfoil design refinement.