Panel Definition

In order to compute and/or solve the boundary integral equation, we discretize the numerical boundaries of our domain through numerical elements, forming what is called a boundary element method. or BEM. The geometry of our problem is discretized into panels. To each panel then we associate one or multiple types of elements described in this section.

Given a planar polygonal panel, we define the following unitary vectors following the right-hand rule as follows

\[\begin{align*} \text{Tangent vector}\qquad & \hat{\mathbf{t}} = \frac{\mathbf{p}_2 - \mathbf{p}_1}{\Vert \mathbf{p}_2 - \mathbf{p}_1 \Vert} \\ \text{Normal vector}\qquad & \hat{\mathbf{n}} = \frac{ \left( \mathbf{p}_2 - \mathbf{p}_1 \right) \times \left( \mathbf{p}_3 - \mathbf{p}_1 \right) }{\left\Vert \left( \mathbf{p}_2 - \mathbf{p}_1 \right) \times \left( \mathbf{p}_3 - \mathbf{p}_1 \right) \right\Vert} \\ \text{Oblique vector}\qquad & \hat{\mathbf{o}} = \hat{\mathbf{n}} \times \hat{\mathbf{t}} \end{align*}\]

This defines the orthonormal basis $\left( \hat{\mathbf{t}},\, \hat{\mathbf{o}},\, \hat{\mathbf{n}} \right)$ which, along with the panel's centroid, define the panel's local coordinate system. This basis follows $\hat{\mathbf{t}} \times \hat{\mathbf{o}} = \hat{\mathbf{n}}$, $\hat{\mathbf{o}} \times \hat{\mathbf{n}} = \hat{\mathbf{t}}$, and $\hat{\mathbf{n}} \times \hat{\mathbf{t}} = \hat{\mathbf{o}}$.