Displacement Calculation

For the displacement calculation the polygons resulting from these cuts is used

• 1D – the element cut of the plane is expanded to the enclosing box
• 2D – the element cut line is expanded to the top and bottom surfaces
• 3D – the polygon on the cut plane

The displacement and rotations are based on a displacement plane described for each component $j$ by

$u_{j} = a_{j} + b_{j}y + c_{j}z$

A least-squares fit across the $n$ points gives a set of equations (one for each of $x,y,z$ directions)

$\begin{bmatrix} n & \sum_{}^{}y_{i} & \sum_{}^{}z_{i} \\ \sum_{}^{}y_{i} & \sum_{}^{}{y_{i}}^{2} & \sum_{}^{}{y_{i}z_{i}} \\ \sum_{}^{}z_{i} & \sum_{}^{}{y_{i}z_{i}} & \sum_{}^{}{z_{i}}^{2} \\ \end{bmatrix}\begin{Bmatrix} a_{j} \\ b_{j} \\ c_{j} \\ \end{Bmatrix} = \begin{Bmatrix} \sum_{}^{}u_{i,j} \\ \sum_{}^{}u_{i,j}y_{i} \\ \sum_{}^{}u_{i,j}z_{i} \\ \end{Bmatrix}$

Then the displacement are

\begin{aligned}{u}_{a,x} = a_{x} \\ {u}_{a,y} = a_{y} \\ {u}_{a,z} = a_{z}\end{aligned}

And rotations are about $y$ and $z$ come from the curvature terms in the displacement equation in the $x$ direction

$\theta_{a,y} = c_{x}$
$\theta_{a,z} = b_{x}$

The twisting rotation is calculated from

$\theta_{a,x} = \frac{1}{n}\sum_{i = 1}^{n}\left( \frac{\left( u_{i,z} - {\widetilde{u}}_{z} \right)y_{i} - \left( u_{i,y} - {\widetilde{u}}_{y} \right)z_{i}}{\left( {y_{i}}^{2} + {z_{i}}^{2} \right)} \right)$