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.
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