The Multilevel Fast Multipole Algorithm (MLFMA) for Solving Large-Scale Computational Electromagnetics Problems

Chapter 2Solutions of Electromagnetics Problems with Surface Integral Equations

This chapter presents formulations of electromagnetics problems with surface integral equations. Surface formulations are derived for three-dimensional metallic, dielectric, and composite objects with arbitrary geometries. Various stabilization procedures are discussed for accurate solutions of low-contrast dielectric objects. Accuracy of implementations is extensively investigated for metallic objects and various discretization schemes.

2.1 Homogeneous Dielectric Objects

For dielectric objects with homogeneous material properties, a general procedure for surface formulations can be summarized as follows:

• Apply the surface operators on equivalent surface currents to obtain expressions for secondary fields.
• Enforce the boundary conditions for tangential electric and magnetic fields on the surface of the object either directly or rotationally and derive basic integral equations for the inner and outer media.
• Combine inner and outer integral equations appropriately to obtain a set of equations to solve for equivalent currents.
• Calculate radiated electric and magnetic fields from equivalent currents.

Based on the items above, various surface formulations can be derived by using different combinations of boundary conditions, testing schemes, and scaling operations. Several of them are stable, free of the internal-resonance problem, and commonly used in the literature [51–83].

2.1.1 Surface Integral ...