Radar Cross-Section Calculation of Aircraft
Application of higher order basis functions (HOBFs)
on a quad mesh is the foundation of WIPL-D Pro EM
solver efficiency. As a result, very large structures aresimulated on inexpensive computers.
WIPL-D Pro models a large structure with about ten
times less unknowns than MoM codes using triangularmesh and Rao-Wilton-Glisson (RWG) basis functions[1]. However, demands for simulation of ever larger
structures (e.g. in car, aircraft or ship industries), aregoing beyond the capabilities of higher order MoM.
Hence, Multilevel Fast Multipole Method (MLFMM)
has been applied to the higher order MoM setup toextend these limits. MLFMM is applied to the samemodels previously treated by MoM, with no changes
to the mesh. Maximally orthogonalized HOBFs [2]and system preconditioning enhance the convergenceof the iterative solution algorithm, making it
applicable to scatterers, antennas, metallic-dielectric
structures, etc.
In MoM, interactions between all basis functions in themodel are calculated independently. The MLFMM
groups basis functions. In case when groups arefar-apart in the model, it calculates interactions
between groups, rather than between individual basis
functions. This way, the MLFMM algorithm provides
dramatic memory savings compared to the MoM.

Figure 1. 160 λ long fighter airplane model
Table 1. Simulation reports
Higher Order MoM

MLFMM

No. ofunknowns

RAM[GB]

Iterations

RAM[GB]

Simulation
time

153646

189

65

3.2

0.8 hours

307170

754

83

7.2

3.5 hours

The fighter airplane (Fig 1) is excited with a planewave, coming in from 30° under the horizon. Fuselage
is 12 m long, wing span is about 7 m. The airplane issimulated at 3 GHz (it is 120 λ long) and 4 GHz (it is160 λ long). The simulation report is given in Table 1.
Higher order MoM models requiring 153646 and307170 unknowns are equivalent to around 1.5 million
and 3 million RWG unknowns, respectively. By
applying the MLFMM, memory requirements arereduced to just 3.2 GB and 7.2 GB respectively. The
calculated RCS in the incident plane, at 4 GHz, isshown in Fig 2.
At 3 GHz the simulation was done on a Intel Core2
Quad CPU at 2.66 GHz clock and 4 GB of RAM. At 4GHz the simulation was done on a Dell PowerEdge
2900 workstation comprising 2.33 GHz Quad Xeon
and 24 GB RAM.

Figure 2. Radiation pattern, incident plane, at 4 GHz

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accurate and reliable results are acquired in reasonabletime.

Conclusion
WIPL-D Pro offers several sophisticated techniques
forsolving
challenging
radar
cross-section
computation problems, such as MLFMM. In addition,
the out-of-core solver can be used in case when there
is not enough RAM to perform simulation, at a cost ofslight decrease in simulation speed.
It is often not necessary to turn to asymptotic methods
to solve RCS of electrically large aircraft. Using theadvanced algorithms of WIPL-D Pro instead, bothReferences[1] B. M. Kolundzija and A. M. Djordjevic, Electromagnetic
modeling of composite metallic and dielectric structures, Artech
House, Norwood, MA, USA, 2002.
[2] D. S. Sumicand B. M. Kolundzija: “Efficient Iterative
Solution of Surface Integral Equations Based on Maximally
Orthogonalized Higher Order Basis Functions”, Proceedings of the2005 international IEEE AP-S Symposium, session #116, paper #5,
July 2005.

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