Compute stiffness matrix for polygonal domains

[carlos-adir]

According to the documentation, the matrix M must be computed for every section.

$$ M_{ij} = \int_{t_{min}}^{t_{max}} \varphi_{j} \cdot \dfrac{\mathbf{r}_i \times \mathbf{p'}}{\langle \mathbf{r}_i, \ \mathbf{r}_i \rangle} \ dt $$

• Diagonal matrix, which is when the integration happens on the same curve as
  • Outside integration, when the integration happens when the radius is not zero
  • Inside integration, when at some point, the radius goes to zero.
• Off diagonal matrix, the source points are not in the same curve which we integrate

[carlos-adir]

Note that, when the source point lies on a straight segment (for polygonal edges for example), the inside integration may be skipped. If the source point lies in a corner of a polygon, this integration must not be skipped

$$\mathbf{r}(t) = \left(\tau - \tau_{i}\right) \cdot \left( \mathbf{P}_{k+1} - \mathbf{P}_k \right)$$

$$\mathbf{r}(t) \times \mathbf{p}'(t) = 0$$

With $t_i$ being the position of the source point

$$\tau = \dfrac{t-t_{k}}{t_{k+1}-t_{k}}$$