GVEC Tokamak Interface Test

Note

For general information about finite element discretization, basis functions, mesh parameters, polynomial degrees, boundary conditions, and matrix/operator dimensions, see Overview.

This script tests the interface with GVEC data for tokamak configurations. The script is located at scripts/interactive/test_gvec_tokamak.py.

Problem Statement

This script tests the interface between MRX and GVEC (Generalized Variational Equilibrium Code) data for tokamak geometries. The script does not solve a PDE directly, but rather:

  1. Loads GVEC equilibrium data from HDF5 files

  2. Interpolates the GVEC coordinate mapping into MRX spline space

  3. Tests projection accuracy for functions defined on the GVEC geometry

The test function is:

\[f(r,\theta,\zeta) = \sin(2\pi\theta) \sin(\pi r)\]

which is projected onto the 0-form finite element space and the projection error is computed.

GVEC Data Structure

GVEC provides equilibrium data in the form: - \(\rho \in [0,1]\): Normalized radial coordinate - \(\theta \in [0,2\pi]\): Poloidal angle - \(\theta^*(\rho,\theta)\): Modified poloidal angle mapping - \(X_1(\rho,\theta^*), X_2(\rho,\theta^*)\): Physical coordinates \((R, Z)\) in cylindrical coordinates

The mapping from logical coordinates \((\rho, \theta^*, \zeta)\) to physical coordinates is:

\[\begin{split}F(\rho, \theta^*, \zeta) = \begin{bmatrix} X_1(\rho, \theta^*) \cos(2\pi\zeta) \\ X_1(\rho, \theta^*) \sin(2\pi\zeta) \\ X_2(\rho, \theta^*) \end{bmatrix}\end{split}\]

Least-Squares Interpolation

The functions \(X_1(\rho,\theta^*)\) and \(X_2(\rho,\theta^*)\) are interpolated into MRX spline space using least-squares fitting:

\[\min_{\mathbf{c}_1, \mathbf{c}_2} \sum_{j=1}^{m_\rho m_\theta} \left\| \sum_{i=1}^{N_0} c_{1,i} \Lambda_0^i(\rho_j, \theta^*_j, 0) - X_1(\rho_j, \theta_j) \right\|^2 + \left\| \sum_{i=1}^{N_0} c_{2,i} \Lambda_0^i(\rho_j, \theta^*_j, 0) - X_2(\rho_j, \theta_j) \right\|^2\]

This leads to the linear system:

\[M \mathbf{c} = \mathbf{y}\]

where \(M \in \mathbb{R}^{(m_\rho m_\theta) \times N_0}\) is the design matrix and \(\mathbf{c} \in \mathbb{R}^{N_0 \times 2}\) contains coefficients for both \(X_1\) and \(X_2\).

Projection Error Test

The projection error is computed as:

\[\text{error} = \frac{\|f - f_h\|_{L^2(\Omega)}}{\|f\|_{L^2(\Omega)}}\]

where \(f_h = \sum_{i=1}^{N_0} \hat{f}_i \Lambda_0^i\) is the projected function and \(\hat{f} = P_0(f)\) is the projection of \(f\) onto the 0-form space.

Usage:

python scripts/interactive/test_gvec_tokamak.py

The script generates plots showing the tokamak configuration and projection errors.

Code Walkthrough

Block 1: Imports and Data Loading (lines 1-27)

Imports xarray for reading HDF5/NetCDF data and MRX modules. Loads GVEC tokamak equilibrium data from ``data/gvec_tokamak.h5`:

  • \(\theta^*\): Modified poloidal angle mapping \(\theta^*(\rho,\theta)\)

  • \(\rho, \theta\): Radial and poloidal coordinate grids

  • \(X_1, X_2\): Physical coordinates \((R, Z)\) as functions of \((\rho, \theta^*)\)

Block 2: Mapping Interpolation (lines 29-59)

Interpolates GVEC coordinate mapping into MRX spline space:

  • Creates DeRham sequence mapSeq for approximating \(X_1(\rho,\theta^*)\) and \(X_2(\rho,\theta^*)\)

  • Builds design matrix \(M\) by evaluating 0-form basis functions at GVEC grid points

  • Solves least-squares problem:

\[M \mathbf{c} = [X_1, X_2]\]

to find spline coefficients \(\mathbf{c}\). - Constructs gvec_stellarator_map from interpolated coordinates (note: uses

stellarator map function but with nfp=1 for axisymmetric tokamak)

Block 3: Projection Accuracy Test (lines 61-140)

Tests projection accuracy for various mesh sizes:

  • Defines test function:

\[f(r,\theta,\zeta) = \sin(2\pi\theta) \sin(\pi r)\]

  • Creates DeRham sequence using GVEC mapping

  • Projects function into finite element space: \(\hat{f} = P_0(f)\)

  • Reconstructs function: \(f_h = \text{DiscreteFunction}(\hat{f}, \Lambda_0, E_0)\)

  • Computes L2 projection error using jax.lax.scan to avoid memory issues

  • Tests convergence as mesh is refined (\(n \in \{4,6,8,\ldots,18\}\))

Block 4: Visualization (lines 142-241)

Generates plots:

  • GVEC coordinate mapping visualization

  • Projection error convergence plot

  • Comparison of exact vs. projected function

This script validates that MRX can accurately represent geometries from external equilibrium codes like GVEC, enabling the use of MRX solvers on experimentally relevant configurations.