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

A.5 Electric-Field Volume Integral Equation

Volume integral equations can be used to formulate electromagnetics problems involving dielectric objects. Major advantages and disadvantages of volume formulations compared to surface formulations can be listed as follows:

• For thin dielectric objects, volume integral equations produce better-conditioned matrix equations than surface integral equations.
• Volume integral equations do not breakdown for low-contrast objects, as opposed to conventional surface integral equations.
• Heterogeneous dielectric objects can easily be modeled with volume integral equations, which allow different electrical parameters for different discretization elements.
• Volume integral equations usually require more numbers of unknowns than surface integral equations for a given problem.

This section presents a volume integral equation, namely, EFVIE, for nonmagnetic dielectric objects.

Consider a dielectric object with a relative permittivity of located in free space. An equivalent electric current can be defined as

A.110

where is the permittivity of free space and

A.111

is the contrast at the observation point . When is located outside the object, , hence