Multilevel Fast Multipole Method (MLFMM)
MLFMM is applied to the same models treated by MoM, with no changes to the mesh. Maximally orthogonalized HOBFs, developed by WIPL-D, and system preconditioning enhance the convergence of the iterative solution algorithm, making it applicable to scatterers, antennas, metallic-dielectric structures, etc. The method is based on the acceleration of matrix-vector multiply operations, performed in each iteration of the iterative solver, by calculating interactions of the near-neighbor and distant basis functions within the model differently.
MLFMM is much more scalable then the MoM, i.e. the memory requirements and length of simulation rise much slower with electrical size. The method enables WIPL-D Pro to remain the number one choice for accurate simulation of electrically very large structures.
Basis functions are grouped in a multilevel tree-like grid. The grouping is done in two levels. At the coarse level, the entire model is divided into groups of patches according to a specified group size in wavelengths. Only non-empty groups are kept in memory, others are discarded. During simulation, distances between pairs of groups are evaluated. If the two groups are further apart than the specified threshold, interactions between the belonging basis functions are replaced by interactions between the groups. If not, the algorithm resolves the remaining basis functions interactions at the fine level of grouping, where all basis functions defined over the same quad patch are considered to belong to the same group. The algorithm is repeated at the fine level. Distance between pairs of patches are evaluated and if the two patches are further apart than the threshold, interactions between the belonging basis functions are replaced by interactions between the patches. Otherwise, their interactions are calculated in a standard MoM way. Hence, the accuracy-speed-memory trade-off is controlled through two parameters: the coarse level group size and the relative distance threshold.
|Number of unknowns (High order MoM)
||Memory [GB] (High order MoM)
||Sim. time [minutes] (MLFMM)
The benefit of the method is a dramatic reduction of memory and time resources needed for simulation of various classes of electrically large structures.
Case Study – RCS of a Fighter Airplane
The fighter airplane is excited with a plane wave, coming in from 30° under the horizon. Fuselage is 12 m long, wing span is 7 m. The airplane is simulated at 3 GHz (120 λ long) and 4 GHz (160 λ long).
Higher order MoM formulation results in:
- 152601 unknowns and 173.5 GB of memory at 3 GHz - equivalent to 1.5 million RWG unknowns,
- 307118 unknowns and 702.8 GB of memory at 4 GHz - equivalent to 3 million RWG unknowns
By applying the MLFMM, memory requirements are reduced to 3.2 GB and 7.2 GB respectively.
In both cases the simulation was done on a PC with Intel Core(TM) i7 CPU 950 @3.07 GHz, 24 GB RAM.
Radiation pattern in the incident plane at 3 GHz
Radiation pattern in the incident plane at 4 GHz