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A hybrid surface integral equation partial differential equation (SIE-PDE) formulation without the boundary condition requirement is proposed to solve the transverse magnetic (TM) electromagnetic problems. In the proposed formulation, the computational domain is decomposed into two 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 by replacing them with 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 nonisotropic media are included. Through combining the surface equivalent electric current density and the inhomogeneous Helmholtz equation, a hybrid SIE-PDE formulation is derived. It requires no boundary conditions and 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 made compatible with the SIE and PDE domain interface. The accuracy and efficiency are validated through two numerical examples. Results show that the proposed SIE-PDE formulation can obtain accurate results, and significant performance improvements in terms of CPU time and memory consumption compared with the FEM are achieved.

A Hybrid SIE-PDE Formulation Without Boundary Condition Requirement for Transverse Magnetic Electromagnetic Analysis