Source: real_space2bezier.f90 (STARWALL)

Note: To be adapted to include the scale factors!

Overview

As JOREK works with Bézier finite elements and STARWALL with a set of discrete points, a conversion is needed from points representation to the Bézier representation (and vice versa).

Bézier Representation

The Bézier representation is defined as:

\[f(s) = \sum_{i=1}^{2} \sum_{j=1}^{N} \alpha_{ij} H_{ij}(s)\]

where:

• $i$ numbers the basis functions (2 basis functions per element)
• $j$ numbers the nodes
• $\alpha_{ij}$ are the Bézier coefficients to be determined
• $H_{ij}(s)$ are the Bézier basis functions

The target function $g(s)$ is given on $N_g$ discrete points.

Least Squares Fit

The Bézier coefficients $\alpha_{ij}$ are calculated using a least squares fit by minimizing:

\[W = \int \left( f(s) - g(s) \right)^2 ds\]

Setting $\frac{\partial W}{\partial \alpha_{ij}} = 0$, we obtain:

\[\int H_{kl}(s) \sum_{i=1}^{2} \sum_{j=1}^{N} \alpha_{ij} H_{ij}(s) \, ds = \int H_{kl}(s) g(s) \, ds\]

This yields a system of linear equations with coefficients:

\[A_{kl,ij} = \int H_{kl}(s) H_{ij}(s) \, ds\]

Elemental Matrix Contributions

For a single element, integrating from $s=0$ to $1$:

\[A_{elm} = \frac{1}{140} \begin{pmatrix} 52 & 22 & 18 & -13 \\ 22 & 12 & 13 & -9 \\ 18 & 13 & 52 & -22 \\ -13 & -9 & -22 & 12 \end{pmatrix}\]

Regular Periodic Boundaries

For a regular periodic boundary without corners, elemental matrices are assembled to form repeated identical blocks:

\[\frac{1}{140} \begin{pmatrix} 18 & 13 & 104 & 0 & 18 & -13 \\ -13 & -9 & 0 & 24 & 13 & -9 \end{pmatrix}\]

Boundaries with Corners

Boundaries with corners require special treatment. At a corner node:

• The node has 3 degrees of freedom
• Elements sharing a corner have the same coefficient for the first Bézier function
• The second function (representing the derivative) is independent on the left and right
• This creates a discontinuity in the derivative along the boundary

Discussion Points

• Integration consistency: Why integrate the Bézier basis functions exactly when the integral over $g(s)$ is computed discretely? Should both sides of the equation use the same integration accuracy?
• Quadrature rule improvement: STARWALL currently uses equidistantly spaced points (typically 10) inside each boundary element. For better least-square approximation accuracy, consider using 4 Gaussian points per element with 4th-order Gaussian integration instead of 2nd-order trapezoidal integration.

To be continued…