Kernel-based Fast Factorization Techniques

This chapter has focused on MLFMA as a representative kernel-based fast factorization technique. To construct a basis for further discussion, we first considered the conventional MLFMA, which is based on the plane-wave expansion of electromagnetic waves, at a formulation level. To solve multi-scale problems involving dense (uniform or non-uniform) discretizations of electrically large objects, alternative MLFMA versions are needed since the conventional MLFMA suffers from a low-frequency breakdown. We listed a variety of ways to implement low-frequency-stable MLFMAs, such as based on multipoles, inhomogeneous plane waves, coordinate shifts, and approximation techniques. We showed how MLFMA implementations can be used to solve extremely large problems via parallelization, while they can be applied to complex structures with different material properties, including plasmonic and NZI objects. Examples were given for solutions of densely discretized objects to demonstrate how MLFMA can handle such complicated problems that possess modeling challenges. Finally, problems with non-uniform discretizations that naturally arise in multi-scale simulations were considered. A rigorous implementation for stable, accurate, and efficient solutions of these problems requires a well-designed combination of a suitable formulation/discretization, an effective solution algorithm (MLFMA version), and a carefully designed clustering mechanism.