电磁线圈自磁场、自力和自感的高效计算
Efficient calculation of the self magnetic field, self-force, and self-inductance for electromagnetic coils
AI总结：
本文提出基于非奇异降维积分公式的高效方法，用于计算电磁线圈的自磁场、自力和自感，并在HSX仿星器线圈上验证了其准确性与效率。
摘要：
电磁线圈的设计可能需要评估若干在数值计算上具有挑战性的量。这些量包括洛伦兹力，其可能因应力而成为限制因素；内部磁场，其对于确定应力以及超导线圈接近其失超极限的程度至关重要；以及电感，其决定存储的磁能和动力学。当计算一个线圈因另一线圈中的电流而产生的影响时，这些量通常可以通过将线圈视为无限细而快速近似。当计算一个线圈因自身电流而产生的影响（例如，自力或自感）时，由于奇异性的存在，评估变得困难；线圈不能被视为无限细，因为每个量在导体宽度为零时发散。在此，我们提出了新颖且良性的方法，利用降维的非奇异积分公式来评估这些量。这些公式通过将磁矢势的积分域划分为两个区域，在每个区域中利用适当的近似，并针对高纵横比进行展开，从而严格确定。我们的公式即使在低纵横比下也与全有限厚度计算显示出良好的一致性，无论是解析地针对环面，还是数值地针对仿星器聚变装置——螺旋对称实验（HSX）——的非平面线圈。由于这些公式的被积函数随着小半径变得无限细而发展出精细结构，我们还开发了一种通过解析积分这一尖锐特征来更高效地评估自力和自感的方法。我们证明，使用该方法，HSX线圈的自力可以仅用12个网格点就精确计算。
英文摘要：
The design of electromagnetic coils may require evaluation of several quantities that are challenging to compute numerically. These quantities include Lorentz forces, which may be a limiting factor due to stresses; the internal magnetic field, which is relevant for determining stress as well as a superconducting coil's proximity to its quench limit; and the inductance, which determines stored magnetic energy and dynamics. When computing the effect on one coil due to the current in another, these quantities can often be approximated quickly by treating the coils as infinitesimally thin. When computing the effect on a coil due to its own current (e.g., self-force or self-inductance), evaluation is difficult due to the presence of a singularity; coils cannot be treated as infinitesimally thin as each quantity diverges at zero conductor width. Here, we present novel and well-behaved methods for evaluating these quantities using non-singular integral formulae of reduced dimensions. These formulae are determined rigorously by dividing the domain of integration of the magnetic vector potential into two regions, exploiting appropriate approximations in each region, and expanding in high aspect ratio. Our formulae show good agreement to full finite-thickness calculations even at low aspect ratio, both analytically for a torus and numerically for a non-planar coil of a stellarator fusion device, the Helically Symmetric eXperiment (HSX). Because the integrands of these formulae develop fine structure as the minor radius becomes infinitely thin, we also develop a method of evaluating the self-force and self-inductance with even greater efficiency by integrating this sharp feature analytically. We demonstrate with this method that the self-force can be accurately computed for the HSX coil with as few as 12 grid points.
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