plications. The I-solver is seamlessly integratedin the CST DESIGN ENVIRONMENT™ (CSTDE). Engineers can take advantage of this easyto use interface and the sophisticated importsfrom a large variety of CAD formats to set upthe models. The selection of the right solver isthen just a mouse-click away.
The I-solver features a Method of Mo-ments (MoM) discretization using a surfaceintegral formulation of the electric and mag-netic field integral equations. Due to the sur-face integral formulation the new solver usesfar fewer elements than common volumemethods for the described problem classes.Nevertheless, the numerical complexity ofMoM is high and is not applicable to electri-cally large structures.
This problem is solved by applying the mul-ti-level fast multipole method (MLFMM), as de-scribed in Figure 1. This figure shows thatMoM considers the direct coupling between allelements of a mesh. In MLFMM the domain isfirst subdivided into separate blocks. Insideeach block only the coupling to one point isconsidered, representing the coupling effect ofthe elements grouped together (aggregation).On the next level the coupling between blocksis considered. In this way a multi-level hierar-chy is built-up. The coupling information is fedback through the hierarchy levels to the individ-ual elements (disaggregation).
Now the I-solver shows numerically an effi-cient complexity in operations and memory for
Research and development engineersworking in aerospace and defensecontinually strive to extend the bound-
aries of what is technically possible and thisextends to the specialized field of electromag-netic simulation technology. One branch inthis community deals with the optimization ofradar cross sections (RCS), while another con-centrates on the influence of the surround-
ings (an airplane body on theperformance of communicationor radar antennas, for exam-ple). What both of these appli-cation areas have in commonis the size of the electricalproblem, which can typicallyrun to many hundreds of wave-lengths, and that the relevantstructures are mainly surfacesand free space.
Tackling these problems isquite simply not feasible usingstandard volume discretizationmethods. CST has addressedthese problem classes with theintroduction of the IntegralEquation solver (I-solver) for itsCST MICROWAVE STUDIO
(CST MWS). It enables the userto perform accurate 3D full-wave analysis of electrically largestructures, which is particularlyuseful in military microwave ap-
Military Microwaves
SURFACE MESH
MLFMMSURFACE MESH
HIERARCHICAL COUPLING
METHOD OF MOMENTS/SURFACE DISCRETIZATION
DIRECTCOUPLING
AGGREGATION
AGGREGATION
DISAGGREGATION
DISAGGREGATION
Fig. 1 From MoM toMLFMM: The improvementof numerical complexity. ▼
electrically large structures. One of itskey strengths is the combination ofhigher order discretization elementswith an iterative or direct solver. Thehigher order discretization increasesaccuracy compared to standard firstorder methods.
To reduce the numerical complexity,the built-in discretization is also capableof applying so-called mixed order dis-cretization, where the polynomial orderof the discretization function is chosenadaptively, according to the size of themesh elements, which the automaticmesh generator chooses with respectto the structure’s features.
This means that in regions wheremany elements are required to re-solve small detai ls (an antennamount, for example), a lower orderdiscretization is used, whereas on thesmooth surface of the platform oraircraft body a higher order discretiza-tion reduces the number of neces-sary surface elements and is ideal formilitary applications.
An additional feature of the I-solver is the flexible MLFMM accura-cy control. It adjusts the MLFMM pa-rameters to the individual modeland, together with the powerful pre-conditioner, optimizes simulationtime and memory requirements.However, the I-solver is not restrictedto perfect electric conductor (PEC)surfaces. It can also take into accountlossy metals and (lossy) dielectricmaterials.
The open space boundary condi-tion best meets the demands ofmost antenna and RCS calculations.In addition, it features electric bound-ary conditions for simulating conduct-ing grounds. The available f ieldsources are discrete port and planewave excitations, and waveguideports and imported far fields will
soon be available to drive simula-tions. The latter provides an efficientmeans to use the results of a highdetail transient or frequency domainsimulation on a large structure.
THEORETICAL BACKGROUNDElectromagnetic field scattering by
three-dimensional objects can becomputed numerically using theelectromagnetic field integral equa-tions where the unknown function isthe induced current distribution J(r).The integral equations can be dis-cretized into a matrix equation sys-tem by the MoM discretization.2,3
The resulting discrete equationsystem is then solved by an iterativemethod, which usually takes N2 op-erations per iteration (for N un-knowns). However, using a multi-lev-el fast multipole method the numeri-cal complexity can be reduced toNlog(N), so that many large scaleproblems can be solved efficiently.2
For conducting objects, the electricfield integral equation (EFIE) in threespace dimensions is given by
for r on the surface S, where t is anyunit tangent vector on S and
For closed conducting objects, themagnetic field integral equation(MFIE) is given by
for r approaches to S from outside,where n is an outwardly directed nor-mal.1,2 In general, either EFIE or MFIEcan be used for closed conductingobjects but the formulation can breakdown due to interior cavity reso-nance problems. A solution is to cre-ate the combined field integral equa-tion (CFIE), which has numericallybeen proven stable for any kind ofresonance effects. The CFIE for 3D-conducting objects is a convex linear
2
4
π
π
J r n dS J r
n H
s
ik r r
r r( ) × ∇ × ( ) =
×
∫– ' '
exp( – ' )
– '
� ii r� ( )( ) 3
G r rk
ik r r
r r, ' – '
exp – '
– '
( )
( ) = ∇∇
( )1
1
2
2
t G r r J r dSi
ktE r
s
i, ' ' ' ( )( ) ( ) = ( )∫ 41
πη
combination of the EFIE and theMFIE according to
The parameter α varies from 0 to 1and can be any value within thisrange. In the literature it is consid-ered that α = 0.2 is an appropriatechoice.
The MLFMM is used for solvingthe MoM discretization of the CFIEon a surface mesh (see Figure 1).Therefore, unlike standard MoMtechniques, the I-solver reduces fullcoupling to one within multiple smallcubic volumes of the model. Theseare then coupled in a similar way toa larger volume and so on recursive-ly, until one volume remains, andscaling for numbers of cells is vastlyimproved at Nlog(N).
To apply the MoM to the CFIE, theunknown current J(r) is expanded us-ing an appropriate set of N basisfunctions into a series according to astandard Galerkin discretization.2,3 Asa result, a linear algebraic equationsystem is obtained, which reads
The matrix entries of A are simplygiven by the inner product of the se-lected basis functions and the integraloperator of the CFIE.1 Consequently,the integral equations are approximat-ed by matrix equations using theMoM. These linear equations can beevaluated efficiently by the fast multi-pole method (FMM), applying a split-ting in terms representing the interac-tions from nearby regions.
The detailed formula is shown inReference 1. The multilevel fast multi-pole method is the recursive exten-sion of the FMM, where the matrixvector multiplication is implementedin a hierarchical or multi-level multi-stage fashion, which can be written as
where Vi, Ti and Ui represent the ma-trix of aggregation, translation anddisaggregation, respectively, at the ithlevel, and NL is the total number of
Aa A a U T V a
U TVa
near NLt
NL NL
it
i ii
NL
= + +
=∑
2
1
6–
( )
A a b j Nji i ji
N
� � � , , , ( )= = …=
∑ 1 2 51
CFIE EFIEik
MFIE= + ( )α α1 4– ( )
Military MicrowavesR
CS
(dB
sm)
THETA
−20
−30
−40
−50
−600 50 100 150 180
▲ Fig. 2 Mono-static RCS for the NASAalmond for the vertical polarization at 1.19GHz.
levels. These matrices as well as Anear are sparse. In the MLFMM, Vi
and Ui (i < NL) are computed by in-terpolation and anterpolation tech-niques. For N unknowns, the compu-tational complexity in memory andsimulation time is the order Nlog(N).
APPLICATIONSMoving on to the applications of
the I-solver, the NASA almond is awell known benchmark for RCS calcu-lations due to its stealth properties.4The almond is a doubly curved sur-face with a pointed tip (see Figure2). It has a low RCS when viewed inthe tip angular sector. Elsewhere, asurface normal is always pointing backtoward the radar, creating a bright highlevel specular RCS return.
In addition to specular mecha-nisms, surface traveling waves andcreeping waves contribute to thescattering. The almond is 9.936 by3.84 by 1.26 inches in length, widthand thickness, respectively.4 Experi-mental data are available for a set offrequencies up to 9.92 GHz withhorizontal and vertical polarizationsof the plane wave excitation.
Figure 2 shows the NASA almondand displays the results of the mono-static RCS analysis for the vertical po-larization at 1.19 GHz. Figure 3shows the plot for the mono-staticRCS at 7 and 9.92 GHz for horizontaland vertical polarization, respectively.The numerical results of the IntegralEquation solver agree very well withthe experimental data. The simula-tion of a complete RCS study for asingle frequency and 180° anglesweep takes less than one hour and160 MB of memory on a PC with anIntel Xeon™ (3 GHz) CPU.
LARGE SHIP SIMULATIONThe I-solver can be used to realisti-
cally simulate an antenna placementon, or radar illumination of a largetransport ship, including the effects ofsea water, which is modeled as a di-electric layer (see Figures 4 and 5).The watercraft is modeled using PEC,and the dimensions are 132.8 by 20by 20 m. The simulated monopole an-tenna is mounted on top of the stern.
The sea water, where the ship isembedded, is modeled as a loss freedielectric with a relative permittivity ε= 80. The dimensions of the waterblock are 143 by 30 by 3.5 m. In to-tal, the surface of the geometry in-cluding the ship and water covers
about 77 by 77 square wavelengthsat 25 MHz. Figures 4 and 5 show thenumerical results for the simulationof the full geometry at 25 MHz usinga discrete port excitation for themonopole antenna. The simulationof the antenna on the watercraft in-cluding the sea water takes less than45 minutes and 5.2 GB memory.The simulation without the sea watertakes less than one hour and 1.7 GBmemory at a frequency of 150 MHz.
RCS ANALYSIS OF AN AERIALVEHICLE
The mono-static RCS for an aerialglider is calculated in the horizontalplane at 0.5 GHz with a 1° step size.The length of the glider and thewingspan is 14 m. Figure 6 displaysthe polar plot of the resulting ab-solute RCS value as a function of theincident plane wave direction. The to-tal simulation time is less than 14hours and 420 MB memory is con-sumed. I-solver features a built-infunction to perform such a mono-static RCS analysis fully automatically.
CONCLUSIONThe Integral Equation solver is a
specialist full-wave solver, which appliesMLFMM using a MoM discretization. Itsuse of higher order discretizations anda flexible accuracy control, combinedwith a powerful preconditioner, result inoutstanding performance and efficien-cy. The I-solver is dedicated to the ac-curate 3D analysis of the electromag-netic fields in electrically large struc-tures, and is therefore ideally suited forsome problem classes of particular in-terest to engineers in the military mi-crowave field.
References1. B.M. Kolundzija and A.R. Djordjevic, Electromag-
netic Modeling of Composite Metallic and Di-electric Structures, Artech House Inc., Norwood,MA, 2002.
