The volume integral equation method based on the current vector potential approximated by means edge element basis function is a well-established approach for 3-D eddy currents computation. The application of the method is straightforward when simply connected geometries and no connection with external circuits are involved. In this case, in fact, the solving system is easily obtained based on tree–cotree decomposition of the primal graph. However, when multiply connected geometries or external generators are considered, “additional” degrees of freedom, not related to interior cotree edges of the primal graph, need to be identified and involved to assure the consistency of the numerical solution. In this article, the link between the volume integral equation method and the circuit is investigated in detail, and the circuit view is used as a guide for systematically finding the additional degrees of freedom arising in case of multiply connected geometries and/or external circuits. In particular, the dual graph is introduced as the support of the circuit and it is shown that this is the natural frame for taking the topology into account. For multiply connected geometries, the additional degrees of freedom are related to loop currents crossing one only time (or an odd number of times) any cutting surface making the domain simply connected, and are found by applying a minimum path algorithm on the dual graph forming the circuit. In case of conducting domain connected to an external generator, an extended dual graph is introduced for finding the further additional degrees of freedom. This article also researches into the possibility to replace the usual current density of the elements, obtained via facet-element shape functions and exactly matching the current of the faces, with a uniform current density obtained by means of a minimum error procedure and approximately matching the current of the faces. The use of this uniform current density, besides improving the calculation time and the accuracy of the coupling coefficients, also allows the extension of the volume integral equation method to discretizations of the problem domain made of arbitrarily shaped polyhedral elements.

中文翻译：

---

具有体积积分方程的拓扑非平凡域中 3D 涡流计算的循环和网格公式

基于通过边缘元基函数近似的电流矢量势的体积积分方程方法是一种行之有效的 3-D 涡流计算方法。当简单连接的几何结构且不涉及与外部电路的连接时，该方法的应用是直接的。在这种情况下，实际上，基于原始图的树-协树分解很容易获得求解系统。然而，当考虑多重连接几何或外部生成器时，需要识别和涉及与原始图的内部 cotree 边无关的“附加”自由度，以确保数值解的一致性。在本文中，详细研究了体积积分方程方法与电路之间的联系，并且电路视图被用作系统地寻找在多重连接的几何形状和/或外部电路的情况下产生的额外自由度的指南。特别是，引入了对偶图作为电路的支持，并表明这是考虑拓扑的自然框架。对于多连通几何，附加自由度与回路电流仅通过一次（或奇数次）任何使域简单连接的切割表面有关，并且通过在对偶图形成上应用最小路径算法来找到电路。在传导域连接到外部发生器的情况下，引入了扩展对偶图以找到进一步的附加自由度。本文还研究了用通过最小误差程序获得的均匀电流密度并近似匹配面元形状函数和精确匹配面电流获得的元件通常电流密度的可能性。面部的电流。使用这种均匀的电流密度，除了提高计算时间和耦合系数的精度外，还允许将体积积分方程方法扩展到由任意形状的多面体单元组成的问题域的离散化。