Select source points automatically

[carlos-adir]

The warping function $\omega(x, \ y)$ is needed to compute torsion. Although we can let the user define the place of the source points, a method to decide the best points is desirable.

Description

The approximative solution $\overline{\omega}$ should be as near as possible of the exact (and unknown) solution $\omega$.

At first sight, there are three ways to mesure how good a solution $\overline{\omega}$ is

1. Torsion constant $J$

$$J = I_{xx} + I_{yy} - \int_{t_{min}}^{t_{max}} \omega \cdot \langle \mathbf{p}, \ \mathbf{p}'\rangle \ dt$$

2. Residual square on the domain

$$I_{\omega \omega} = \int_{\Omega} \omega^2 \ dx \ dy$$

3. Residual square on the boundary

$$I = \int_{t_{min}}^{t_{max}} \omega^2 \ \langle \mathbf{p}, \ \mathbf{p}'\rangle dt$$