General Applicability of the Technique

MLFMM boxes at the 3rd finest level.

Aggregation, translation and disaggregation.

The MLFMM is an alternative formulation of the technology behind the MoM and is applicable to much larger structures than the MoM, making full-wave current-based solutions of electrically large structures a possibility. This fact implies that it can be applied to most large models that were previously treated with the MoM without having to change the mesh.

Technical Foundation

The agreement between the MoM and MLFMM is that basis functions model the interaction between all triangles. The MLFMM differs from the MoM in that it groups basis functions and computes the interaction between groups of basis functions, rather than between individual basis functions. FEKO employs a boxing algorithm that encloses the entire computational space in a single box at the highest level, dividing this box in 3 dimensions into a maximum of 8 child boxes and repeating the process iteratively until the side length of each child box is approximately a quarter wavelength at the lowest level. Only populated boxes are stored at each level, forming an efficient tree-like data structure. In the MoM framework the MLFMM is implemented though a process of aggregation, translation and disaggregation of the different levels.

The MoM treats each of the N basis functions in isolation, thus resulting in an N² scaling of memory requirements (to store the impedance matrix) and N³ in CPU-time (to solve the linear set of equations). It is thus clear that processing requirements for MoM solutions scale rapidly with increasing problem size. The MLFMM formulation's more efficient treatment of the same problem results in N.log(N) scaling in memory and N.log(N).log(N) in CPU time. In real applications this reduction in solution requirements can range to orders of magnitude.

Significant effort has also been invested in improving the parallel MLFMM formulation to achieve exceptionally high efficiency when distributing a simulation over multiple processors.

Asymptotic Memory Estimates for Different Applications
Application and Frequency	Number of Unknowns	Memory allocated
		MoM	MLFMM
Military aircraft at 690 MHz, Ship (115m x 14 m) at 107 MHz, Reflector antenna with aperture size 19 lambda	100 000	150 GByte	1 GByte
Military aircraft at 1.37 GHz, Ship (115m x 14 m) at 214 MHz, Reflector antenna with aperture size 38 lambda	400 000	2.4 TByte	4.5 GByte
Military aircraft at 2.65 GHz, Ship (115m x 14 m) at 414 MHz, Reflector antenna with aperture size 73 lambda	1 500 000	33.5 TByte	18 GByte

Total run-time efficiency (all solution phases) for parallel MLFMM solution of a problem with 3.18 million unknowns.

Typical Application of the MLFMM

Possible applications of the MLFMM span a wide range of problems. It is best suited to problems that include electrically large structures, e.g. antenna coupling on ships, antenna placement on aircraft and radiation hazard analysis behind a reflector antenna.

MLFMM analysis of a ship.	Modelling antenna placement on a military aircraft with the MLFMM.