Equations

Virtual Casing Principle

Let Gamma be a closed toroidal surface with outward unit normal n. Let B be the total magnetic field on Gamma, decomposed into interior and exterior contributions:

\[\mathbf{B} = \mathbf{B}_{\mathrm{int}} + \mathbf{B}_{\mathrm{ext}}.\]

The virtual casing principle provides an explicit expression for B_ext on the surface in terms of singular layer potentials:

\[\mathbf{B}_{\mathrm{ext}}(\mathbf{r}) = \frac{1}{2}\mathbf{B}(\mathbf{r}) + \nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r}) + \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}),\]

where G denotes the Laplace single-layer potential:

\[G[\sigma](\mathbf{r}) = \frac{1}{4\pi}\int_{\Gamma} \frac{\sigma(\mathbf{r}')}{\lVert \mathbf{r} - \mathbf{r}' \rVert} \, d a(\mathbf{r}').\]

The interior field is obtained by reversing the signs:

\[\mathbf{B}_{\mathrm{int}}(\mathbf{r}) = \frac{1}{2}\mathbf{B}(\mathbf{r}) - \nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r}) - \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}).\]

Internal vs External Off-Surface

For off-surface targets, the jump term is absent. The external and internal fields are:

\[\mathbf{B}_{\mathrm{ext}}(\mathbf{r}) = \nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r}) - \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}),\]

\[\mathbf{B}_{\mathrm{int}}(\mathbf{r}) = -\nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r}) + \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}).\]

The off-surface gradients are obtained by differentiating these expressions and using the second-derivative Laplace kernel.

The two vector layer potentials used are:

• A Laplace single-layer gradient (FxdU in the reference code)

• A Biot-Savart single-layer (FxU)

The kernels are (up to scaling) derivatives of 1 / |r|.

Kernel Formulas

Let r = x - x' be the displacement from a source to a target point. The kernels implemented follow the BIEST convention with a 1/(4*pi) factor.

Laplace single-layer:

\[\begin{split}G(r) = \\frac{1}{4\\pi \\lVert r \\rVert}\end{split}\]

Gradient of single-layer:

\[\begin{split}\\nabla G(r) = -\\frac{r}{4\\pi \\lVert r \\rVert^3}\end{split}\]

Second derivatives:

\[\begin{split}\\partial_i\\partial_j G(r) = \\frac{1}{4\\pi}\\left(-\\delta_{ij}\\lVert r \\rVert^{-3} + 3 r_i r_j \\lVert r \\rVert^{-5}\\right)\end{split}\]

Biot-Savart (single-layer):

\[\begin{split}\\mathbf{K}(r) = \\frac{1}{4\\pi} \\frac{\\mathbf{f} \\times r}{\\lVert r \\rVert^3}\end{split}\]

The derivative FxdU for Biot-Savart is implemented explicitly to match the reference code in biest/kernel.hpp.

Off-Surface Evaluation

For off-surface targets r:

\[\mathbf{B}_{\mathrm{ext}}(\mathbf{r}) = \nabla G[\mathbf{B}\cdot\mathbf{n}](\mathbf{r}) - \nabla \times G[\mathbf{n}\times\mathbf{B}](\mathbf{r}).\]

The singular +1/2 jump term is absent off the surface.

Field-Period Symmetry

The surface and field may be defined on a half field period using stellarator symmetry. The toroidal grid is shifted by half a grid point for the symmetric case.

A full description of the grid conventions is in the reference code and reproduced in the Numerics section.