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

Chapter 1Basics

This chapter presents some of the preliminaries to the multilevel fast multipole algorithm (MLFMA). Using Maxwell's equations and boundary conditions, surface integral equations are derived to formulate electromagnetics problems involving metallic and dielectric objects. Discretizations of the surface integral equations with basis and testing functions on triangular domains lead to dense matrix equations, which can be solved iteratively via MLFMA. Numerical integrations on triangular domains, different types of excitations, iterative algorithms, and preconditioning are also discussed.

1.1 Introduction

Solving electromagnetics problems is extremely important to analyze electromagnetic interactions of electronic devices with each other and with their environments including living and nonliving objects [1]. A plethora of applications in the areas of antennas [2]–[13], radars [14], optics [15], medical imaging [16], wireless communications [17], nanotechnology [18], metamaterials [19]–[25], photonic crystals [26]–[33], remote sensing, and electronic packaging involve scattering and/or radiation of electromagnetic waves. The following are some examples of popular electromagnetics problems:

• Radiation from antennas, e.g., designing novel antennas and estimating their adverse effects on the human health.
• Indoor and outdoor propagation, e.g., optimizations of wireless electromagnetic sources to cover required areas with minimum power.
• Visualization via optical imaging ...