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:
- Convert the 2D metric to 1D
- Convert the 2D curve to 1D
- Numerically solve the Spacing ODE
- Convert 1D distribution of points back to curve
Results
Putting everything together and using the analytical solver, we get the following results:
Uniform.
Clustering at $x=0.0$
Example with sparse sampling of the airfoil:
Clustering at $x=1$
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:
Real Metric Data
Here we have real metric data:
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 $0.001$: