A novel boundary element formulation for solving problems involving eddy currents in the thin skin depth approximation is developed. It is assumed that the time-harmonic magnetic field outside the scatterers can be described using the quasistatic approximation. A two-term asymptotic expansion with respect to a small parameter characterizing the skin depth is derived for the magnetic and electric fields outside and inside the scatterer, which can be extended to higher order terms if needed. The introduction of a special surface operator (the inverse surface gradient) allows the reduction of the computational complexity of the solution. A method to compute this operator is developed. The obtained formulation operates only with scalar quantities and requires the computation of surface integral operators that are customary in boundary element (method of moments) solutions to the Laplace equation. The formulation can be accelerated using the fast multipole method. The resulting method is much faster than solving the vector Maxwell equations. The obtained solutions are compared with the Mie solution for scattering from a sphere, and the error of the solution is studied. Computations for much more complex shapes of different topologies, including for magnetic and electric field cages used in testing, are also performed and discussed.

中文翻译：

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低频薄皮深下非完美导体电磁场的边界元解

开发了一种新的边界元公式，用于解决涉及薄趋肤深度近似中的涡流问题。假设散射体外部的时谐磁场可以使用准静态近似来描述。对于散射体外部和内部的磁场和电场，推导出关于表征趋肤深度的小参数的两项渐近展开式，如果需要，可以将其扩展到更高阶项。特殊曲面算子（逆曲面梯度）的引入可以降低解的计算复杂度。开发了一种计算该运算符的方法。所获得的公式仅适用于标量，并且需要计算表面积分算子，这些算子在拉普拉斯方程的边界元（矩法）解中很常见。可以使用快速多极方法加速该公式。结果方法比求解向量麦克斯韦方程组快得多。将得到的解与球体散射的 Mie 解进行比较，研究解的误差。还执行和讨论了不同拓扑结构的更复杂形状的计算，包括测试中使用的磁场和电场笼。将得到的解与球体散射的 Mie 解进行比较，研究解的误差。还执行和讨论了不同拓扑结构的更复杂形状的计算，包括测试中使用的磁场和电场笼。将得到的解与球体散射的 Mie 解进行比较，研究解的误差。还执行和讨论了对不同拓扑的更复杂形状的计算，包括用于测试的磁场和电场笼。