Toroid Cavity

Note

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

This script solves eigenvalue problems for a toroidal cavity. The script is located at scripts/interactive/toroid_cavity.py.

Problem Statement

This script computes electromagnetic eigenmodes for a toroidal cavity (2D cross-section with periodic boundary in toroidal direction). The eigenvalue problem is:

\[\nabla \times \nabla \times \mathbf{E} = k^2 \mathbf{E} \quad \text{in } \Omega\]

with the constraint:

\[\nabla \cdot \mathbf{E} = 0 \quad \text{in } \Omega\]

and boundary conditions:

\[\mathbf{E} \times \mathbf{n} = 0 \quad \text{on } \partial\Omega\]

where: - \(\mathbf{E}: \Omega \to \mathbb{R}^3\) is the electric field (1-form) - \(k^2\) is the eigenvalue (square of the wavenumber) - \(\Omega\) is a toroidal domain - \(\mathbf{n}\) is the outward unit normal vector on the boundary

Toroidal Geometry

The toroidal domain is parameterized by: - Minor radius: \(a=1\) - Major radius: \(R_0=2.1\) - Aspect ratio: \(\epsilon = a/R_0 \approx 0.476\)

The mapping from logical coordinates \((r, \chi, \zeta)\) to physical coordinates is:

\[\begin{split}F(r, \chi, \zeta) = \begin{bmatrix} (R_0 + \epsilon r \cos(2\pi\chi)) \cos(2\pi\zeta) \\ (R_0 + \epsilon r \cos(2\pi\chi)) \sin(2\pi\zeta) \\ \epsilon r \sin(2\pi\chi) \end{bmatrix}\end{split}\]

Boundary Conditions

  • Radial direction: Clamped (perfect conductor boundary)

  • Poloidal direction: Periodic (rotational symmetry)

  • Toroidal direction: Constant (2D cross-section)

The script demonstrates:

  • Computing eigenvalues and eigenmodes for toroidal cavities

  • Visualizing eigenmodes

  • Analyzing cavity resonances

Usage:

python scripts/interactive/toroid_cavity.py

The script generates plots showing eigenvalues and eigenmode visualizations.

Finite Element Discretization

The electric field is represented as a 1-form:

\[V_1 = \text{span}\{\Lambda_1^i\}_{i=1}^{N_1}\]

where \(N_1\) is the number of 1-form DOFs.

Matrix and Operator Dimensions

The 1-form mass matrix \(M_1 \in \mathbb{R}^{N_1 \times N_1}\) is used.

The double curl operator is constructed as:

\[C = M_1 (\Delta_1 + \nabla_h \circ (\nabla \cdot)_h) = (\nabla \times)_h^T M_2 (\nabla \times)_h\]

This represents the curl-curl operator \(\nabla \times (\nabla \times)\) for electromagnetic modes.

Eigenvalue Problem

The eigenvalue problem is:

\[C \mathbf{v} = \lambda M_1 \mathbf{v}\]

where \(C \in \mathbb{R}^{N_1 \times N_1}\) is the double curl matrix, \(M_1 \in \mathbb{R}^{N_1 \times N_1}\) is the mass matrix, \(\mathbf{v} \in \mathbb{R}^{N_1}\) is the eigenvector, and \(\lambda = k^2\) is the eigenvalue.

Toroidal Curvature Effects

The toroidal geometry introduces curvature through: - Jacobian determinant: \(J(x) = \det(DF(x)) = \epsilon (R_0 + \epsilon r \cos(2\pi\chi))\) - Metric tensor: \(G(x) = DF(x)^T DF(x)\) (accounts for non-orthogonal coordinates) - Inverse metric: \(G^{-1}(x)\) (used in mass matrix computation)

These geometric factors modify the eigenmode structure compared to cylindrical cavities.

Code Walkthrough

Block 1: Imports and Configuration (lines 1-21)

Imports modules and sets up output directory. Configures discretization parameters: \(n=8\) elements, \(p=3\) polynomial degree, creating a 2D toroidal cross-section.

Block 2: Domain Setup (lines 23-38)

Defines toroidal geometry:

  • Minor radius: \(a=1\)

  • Major radius: \(R_0=2.1\)

  • Aspect ratio: \(\epsilon = a/R_0 \approx 0.476\)

  • Uses toroid_map for coordinate transformation

Block 3: DeRham Sequence and Operators (lines 40-56)

Sets up finite element spaces:

  • Boundary conditions: clamped in radial, periodic in poloidal, constant in toroidal

  • Assembles mass matrices: \(M_0, M_1, M_2, M_3\)

  • Constructs gradient operator: \(D_0 = \nabla_h\) (strong gradient)

  • Builds double curl matrix:

\[C = M_1 (\Delta_1 + \nabla_h \circ (\nabla \cdot)_h) = (\nabla \times)_h^T M_2 (\nabla \times)_h\]

This represents the curl-curl operator \(\nabla \times (\nabla \times)\) for electromagnetic modes.

Block 4: Eigenvalue Computation (lines 58-68)

Solves generalized eigenvalue problem:

\[C \mathbf{v} = \lambda M_1 \mathbf{v}\]

  • Extracts real eigenvalues and eigenvectors

  • Filters finite eigenvalues

  • Sorts in ascending order

Block 5: Visualization (lines 70-215)

Generates plots:

  • Eigenvalue spectrum: First 40 eigenvalues normalized by \(\pi^2\)

  • Eigenmode visualization: Plots pushforward of eigenvectors on 2D cross-section (\(\zeta=0.5\)) showing field magnitude as contour plots

  • Creates grid of eigenmode plots (first 25 modes) using plot_eigenvectors_grid()

The toroidal geometry introduces curvature effects that modify the eigenmode structure compared to cylindrical cavities, making this a more complex test case for the electromagnetic solver.