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.
This paper presents a new algorithm to construct a smooth and invertible mapping between arbitrary toroidal domains and boundary-conforming coordinates. This is useful for plasma physics simulations, where the magnetic field geometry needs to be accurately represented, even in complex shapes like stellarators.