Construction of an invertible mapping to boundary conforming coordinates for arbitrarily shaped toroidal domains

TL;DR

This paper solves the initialization challenge for boundary-conforming coordinates in arbitrarily shaped toroidal domains by constructing a harmonic map $g:\overline{\Omega}\to\overline{D}$ through two Dirichlet-Laplace problems solved via boundary integrals. The inverse $f=g^{-1}$ provides a diffeomorphic mapping from the unit disk to the cross-section, yielding robust $(\rho,\theta)$-coordinates with $\rho\in[0,1]$ and $\theta\in[0,2\pi)$. The authors prove existence and regularity of the harmonic map, present a discretization via a Zernike basis that preserves invertibility, and demonstrate applications to strongly shaped boundaries, including optimized stellarator geometries, with numerical experiments showing positive Jacobians and spectral convergence. This approach enables reliable, parameter-free initialization of 3D MHD equilibrium solvers and offers a general boundary-conforming meshing strategy for toroidal and cylinder-like domains, implemented in the map2disc package.

Abstract

Boundary conforming coordinates are commonly used in plasma physics to describe the geometry of toroidal domains, for example, in three-dimensional magnetohydrodynamic equilibrium solvers. The magnetohydrodynamic equilibrium configuration can be approximated with an inverse map, defining nested surfaces of constant magnetic flux. For equilibrium solvers that solve for this inverse map iteratively, the initial guess for the inverse map must be well defined and invertible. Even if magnetic islands are to be included in the representation, boundary conforming coordinates can still be useful, for example to parametrize the interface surfaces in multi-region, relaxed magnetohydrodynamics or as a general-purpose, field-agnostic coordinate system in strongly shaped domains. Given a fixed boundary shape, finding a valid boundary conforming mapping can be challenging, especially for the non-convex boundaries from recent developments in stellarator optimization. In this work, we propose a new algorithm to construct such a mapping, by solving two Dirichlet-Laplace problems via a boundary integral method. We can prove that the generated harmonic map is always smooth and has a smooth inverse. Furthermore, we can find a discrete approximation of the mapping that preserves this property.

Figures (8)

• Figure 1: Sketch of the unit circle $S^1$, a Jordan curve $\Gamma\subset\mathbb{R}^2$ and one possible parametrization $\gamma: S^1\to\mathbb{R}^2$ with $\gamma\!\left(S^1\right)=\Gamma$. Equidistant points of $\theta$ are purple, with the point corresponding to $\theta=0$ marked with a cross.
• Figure 2: Sketch of the mapping between the (open) unit disk $D$ and a domain $\Omega\subset\mathbb{R}^2$. Lines of constant $\rho,\theta$ are purple, lines of constant $\xi,\eta$ are white and the boundary is black. A blue cross marks an exemplary point in both domains, $\xi_\ell +\rmi\eta_\ell = \rho_\ell\exp(\rmi\theta_\ell)\leftrightarrow(x_\ell,y_\ell)$.
• Figure 3: Sketch of an elliptical domain with boundary $\Gamma$ in black, semi-major axis $a$ in orange, semi-minor axis $b$ in blue and equidistant contours of $\xi$ and $\eta$ in grey.
• Figure 4: The $\phi=\pi/8$ poloidal cross-section of the optimized two-field-period quasi-helically symmetric stellarator generated from the near-axis methodlandreman_mapping_2022pyQSC_0.1.2. The color shows the normalized Jacobian determinant eq:Jac_n, with yellow depicting regions with high $\mathcal{J}_n$, orange depicting regions of low $\mathcal{J}_n$, and red depicting invalid regions with negative $\mathcal{J}_n$. Contours of $\rho$ and $\theta$ are grey, the prescribed boundary and/or axis is black. (a) is the near-axis solution. (b) is constructed using radial blending (see \ref{['app:radial_blending']}). (c) is constructed by minimizing the squared Jacobian determinanttecchiolli_constructing_2024 (with a straightness weighting coefficient of $\omega=0.001$) and (d) is constructed using the proposed method.
• Figure 5: The two components of the conformal map $f_c=f_2 \circ f_1$. $f_1$ maps the unit disk to the ellipse and $f_2$ maps the plane to a polar domain. For $f_1$ the contours of $\rho$ and $\theta$ are shown in blue and orange and for $f_2$ the contours of $\xi$ and $\eta$ are shown in green and red with the $\rho$ and $\theta$ contours of the ellipse overlaid in grey.

Theorems & Definitions (3)

• Theorem 1
• Remark
• proof