AMPERe: an exactly divergence-free quadrature-by-expansion evaluator for near-surface Biot--Savart and virtual-casing fields in fusion geometries
Abstract
Accurate magnetic-field evaluation on and near current-carrying surfaces is a recurring computational bottleneck in magnetic-confinement fusion. As targets approach the source surface, Biot--Savart and virtual-casing kernels become singular, causing standard Gauss--Legendre and upsampled trapezoidal quadratures to lose accuracy. We introduce AMPERe, a quadrature-by-expansion (QBX) evaluator for vector magnetostatic kernels on general three-dimensional, non-axisymmetric tokamak and stellarator geometries. The method develops a spherical-harmonic QBX formulation for the vector Biot--Savart and virtual-casing (Stratton--Chu) fields and expresses the resulting magnetic field as the analytic curl of a regular-solid-harmonic vector potential. This construction yields a divergence-free discrete field at every expansion order, provides analytic derivatives, and enables an a priori separation of truncation, quadrature, and geometric errors. On the rotating-ellipse benchmark of Malhotra et al. [1], QBX maintains a relative error of 2--5 x 10^{-6} throughout the near-surface regime at distance-independent cost, whereas upsampled trapezoidal quadrature exhibits O(1) errors and requires node counts scaling as (h/d)^2. AMPERe is implemented as a header-only, performance-portable Kokkos library supporting CPU and GPU backends, with target-specific and local-correction accelerations validated against the baseline. The method is further demonstrated on realistic Wendelstein 7-X configurations, including sheet-current and virtual-casing field evaluations.
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BibTeX
@misc{betar2026ampere,
title = {{AMPERe: an exactly divergence-free quadrature-by-expansion evaluator for near-surface Biot--Savart and virtual-casing fields in fusion geometries}},
author = {H. Betar and Y. Sarazin and V. Grandgirard and D. Zarzoso},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41544},
eprint = {2610.41544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41544},
}
RIS
TY - UNPB
TI - AMPERe: an exactly divergence-free quadrature-by-expansion evaluator for near-surface Biot--Savart and virtual-casing fields in fusion geometries
AU - H. Betar
AU - Y. Sarazin
AU - V. Grandgirard
AU - D. Zarzoso
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41544
UR - https://arxiv.org/abs/2610.41544
AB - Accurate magnetic-field evaluation on and near current-carrying surfaces is a recurring computational bottleneck in magnetic-confinement fusion. As targets approach the source surface, Biot--Savart and virtual-casing kernels become singular, causing standard Gauss--Legendre and upsampled trapezoidal quadratures to lose accuracy. We introduce AMPERe, a quadrature-by-expansion (QBX) evaluator for vector magnetostatic kernels on general three-dimensional, non-axisymmetric tokamak and stellarator geometries. The method develops a spherical-harmonic QBX formulation for the vector Biot--Savart and virtual-casing (Stratton--Chu) fields and expresses the resulting magnetic field as the analytic curl of a regular-solid-harmonic vector potential. This construction yields a divergence-free discrete field at every expansion order, provides analytic derivatives, and enables an a priori separation of truncation, quadrature, and geometric errors. On the rotating-ellipse benchmark of Malhotra et al. [1], QBX maintains a relative error of 2--5 x 10^{-6} throughout the near-surface regime at distance-independent cost, whereas upsampled trapezoidal quadrature exhibits O(1) errors and requires node counts scaling as (h/d)^2. AMPERe is implemented as a header-only, performance-portable Kokkos library supporting CPU and GPU backends, with target-specific and local-correction accelerations validated against the baseline. The method is further demonstrated on realistic Wendelstein 7-X configurations, including sheet-current and virtual-casing field evaluations.
ER -