Mapping the 2 Dimensional Problem to 1 Dimension

Since we aim to construct grid in $\Omega \subset \R^2$, we need a method to convert the 2D problem into a 1D problem. While we have the 1D equation

\[\begin{align*} -8 \sigma^4 M^2 x_s^2 x_{ss} - 4 \sigma^4 M M_x x_s^4 - 4 \sigma^2 m M x_{ss} - 2 \sigma^2 m M_x x_s^2 = 0, \end{align*}\]

we need to reduce the dimension of $x \in \R^2$ and $M \in \R^{2 \times 2}$. This will consistent of four steps:

Results

Putting everything together and using the analytical solver, we get the following results:

Uniform.

uniform

Clustering at $x=0.0$

x=0

Example with sparse sampling of the airfoil:

x=0

Clustering at $x=1$

x=1

Zooming in on the leading edge of the airfoil, we see a potential issue: the solution boundary is no longer aligned with the real boundary:

x=1

Real Metric Data

Here we have real metric data:

real-metric

Since the spacing of the points is hard to see, the difference of the 1D points is plotted in red.

Here we show real metric data but scaled down by $0.001$:

real-metric