A hybrid surface integral equation partial differential equation (SIE-PDE) formulation without the boundary condition requirement is proposed to solve the electromagnetic problems. In the proposed formulation, the computational domain is decomposed into two \emph{overlapping} domains: the SIE and PDE domains. In the SIE domain, complex structures with piecewise homogeneous media, e.g., highly conductive media, are included. An equivalent model for those structures is constructed through replacing them by the background medium and introducing a surface equivalent electric current density on an enclosed boundary to represent their electromagnetic effects. The remaining computational domain and homogeneous background medium replaced domain consist of the PDE domain, in which inhomogeneous or non-isotropic media are included. Through combining the surface equivalent electric current density and the inhomogeneous Helmholtz equation, a hybrid SIE-PDE formulation is derived. Unlike other hybrid formulations, where the transmission condition is usually used, no boundary conditions are required in the proposed SIE-PDE formulation, and it is mathematically equivalent to the original physical model. Through careful construction of basis functions to expand electric fields and the equivalent current density, the discretized formulation is compatible on the interface of the SIE and PDE domain. Finally, its accuracy and efficiency are validated through two numerical examples. Results show that the proposed SIE-PDE formulation can obtain accurate results including both near and far fields, and significant performance improvements in terms of CPU time and memory consumption compared with the FEM are achieved.

中文翻译：

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一种无边界条件要求的混合 SIE-PDE 公式，用于电磁分析

提出了一种没有边界条件要求的混合表面积分方程偏微分方程 (SIE-PDE) 公式来解决电磁问题。在提出的公式中，计算域被分解为两个 \emph{overlapping} 域：SIE 和 PDE 域。在 SIE 域中，包括具有分段均匀介质的复杂结构，例如高导电介质。通过将它们替换为背景介质并在封闭边界上引入表面等效电流密度来表示它们的电磁效应，构建了这些结构的等效模型。剩余的计算域和均匀背景介质替换域由 PDE 域组成，其中包括非均匀或非各向同性介质。通过结合表面等效电流密度和非齐次亥姆霍兹方程，推导出混合 SIE-PDE 公式。与通常使用传输条件的其他混合公式不同，所提出的 SIE-PDE 公式中不需要边界条件，并且在数学上与原始物理模型等效。通过精心构建基函数以扩展电场和等效电流密度，离散化公式在 SIE 和 PDE 域的界面上兼容。最后，通过两个数值例子验证了其准确性和效率。结果表明，所提出的 SIE-PDE 公式可以获得准确的结果，包括近场和远场，