We present quantitative means for assessing the numerical accuracy of static magnetic field calculations in finite-element models. Our calculations use the three-dimensional Opera simulation software suite of Dassault Systèmes. Our need to assess the effects of fringe fields requires such a 3D algorithm. While we do discuss and compare our approach to a method of accuracy estimation used in the Opera post-processor, our methods are generally applicable to any model using relaxation techniques in finite-element systems. For purposes of illustration, we present modeling and analysis of two types of quadrupole electromagnets presently in operation in the south arc of the Cornell Electron Storage Ring (CESR). Calculations of field multipole expansion coefficients and numerical deviations from Maxwell’s equations in source-free regions are discussed, with emphasis on the dependence of their accuracy on changes to the finite-element model. Successive refinement steps in the finite-element model for the non-extraction type of CESR south arc quadrupole achieve a reduction in the RMS value of the longitudinal component of the curl vector on the magnet axis by a factor of nearly 70 from $4.38\times 10^{-2}$ T/m to $6.36\times 10^{-4}$ T/m, which is 0.0021% of the field gradient. An accuracy of 2.9% is achieved for a dodecapole coefficient of $2.4\times 10^{-4}$ of that of the quadrupole in the longitudinal integral of the vertical field gradient at a radius of 1 cm.

我们提出了定量方法，用于评估有限元模型中静磁场计算的数值精度。我们的计算使用DassaultSystèmes的三维Opera仿真软件套件。我们需要评估边缘场的影响，因此需要这样的3D算法。虽然我们确实讨论和比较了我们的方法与Opera后处理器中使用的准确性估计方法，但我们的方法通常适用于有限元系统中使用松弛技术的任何模型。为了便于说明，我们介绍了两种类型的四极电磁体的建模和分析。目前在康奈尔电子存储环（CESR）的南弧中运行。讨论了在无源区域中场多极膨胀系数的计算和麦克斯韦方程组的数值偏差，重点是其精度对有限元模型变化的依赖。对于非提取类型的CESR南弧四极杆，有限元模型中的连续细化步骤使卷曲矢量在磁体轴上的纵向分量的RMS值降低了近70倍。$4。38\times \mathrm{1个}{0}^{-\mathrm{2个}}$ T / m至 $6。36\times \mathrm{1个}{0}^{-4}$ T / m，为场梯度的0.0021％。十二极系数为时，精度为2.9％$\mathrm{2个}。4\times \mathrm{1个}{0}^{-4}$ 在半径为1 cm的垂直场梯度的纵向积分中四极杆的半径的四分之一。