## 4

**4 **

**Improvement of the CBFM **

In this section two innovation of the Characteristic Basis Function Method will be presented. The first one concerns a important feature of the CBFs, i.e. the ability to represent the electromagnetic behaviour of an metallic object in a wide frequency range. The second one gives the possibility to solve electromagnetic problem related at the radiation of aperture antenna array. In both cases there is a gain in terms of time savings and employment of computing resources, compared to conventional numerical methods.

**4.1**

**4.1**

** Introduction to Ultra wide band CBFs **

**Introduction to Ultra wide band CBFs**

The response of a system over a wide band is required in many applications. Wide band electromagnetic scattering can be addressed either in time or in frequency domain; in the latter case, a frequency sampling in the band of interest has to be performed, and for each frequency sample, the harmonic solution has to be calculated anew leading to a dramatic increase of the total CPU time.

Frequency domain techniques, such as the Method of Moments (MoM), can be applied for accurate solution of such electromagnetic problems, but they place a heavy burden on the CPU time as well as memory requirements when electrically large structures are analyzed with the usual discretization step of λ/20 to λ/10. Moreover, they require the impedance matrix to be generated and solved anew for each frequency sample; hence, frequency sweep over a band can be computationally expensive, particularly for large objects for which the frequency step must be small to capture the nuances of the response.

Several techniques have been proposed to alleviate this problem. In [17], wide band data are obtained from MoM by interpolating the impedance matrix; this method leads to a CPU time reduction to the detriment of increased memory requirement. In [18], the model-based parameter estimation (MBPE) is used to obtain the wide band data from frequency and frequency-derivative data. In the asymptotic waveform evaluation (AWE) technique [19], the equivalent surface current is expanded in a Taylor’s series around a frequency in the desired frequency band. The Taylor series coefficients are then matched via the Padè approximation to

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ 68

a rational function. Then, instead of solving the equivalent current at each frequency sample, it is only necessary to solve for the coefficients of the series. However, the AWE technique needs to store both the impedance matrix and its frequency derivatives, limiting its applicability to small scale problems only. In [20], AWE technique is combined with the pre-corrected fast Fourier transform (PFFT) /adaptive integral method (AIM). Furthermore, PFFT/AIM method is employed to accelerate all matrix-vector products encountered in the iterative solution of the matrix equation for the unknown coefficients of the series. To reduce the memory requirement, all the full matrices are stored in a sparse form. In addition, a preconditioner is used to improve convergence of the matrix equation for the coefficients of the series at each expansion point. Similarly, in [21] the impedance matrix interpolation is combined with the sparse approximate inversion (SAI) preconditioning method.

However, all the previous techniques are still limited by the discretization size ranging from 10 to 20 basis functions per wavelength. Moreover, they resort to iterative solvers: thus, multiple angular excitations cannot be handled very efficiently.

In the previous chapter has been presented a technique developed that reduces the size of the MoM matrix, i.e. the Characteristic Basis Function Method (CBFM) [14-15]. In CBFM, the object is divided into a number of blocks, and high-level basis functions called CBFs are derived for these blocks, which are discretized by using the conventional triangular patch segmentation and RWG basis functions [8]. The use of CBFs has several advantages. First, the technique is more general and can be used for any geometry formed by surfaces and wires. Moreover, the CBFs assure that the solution will naturally tend to the asymptotic limit and also their use obviates the need to hybridize them with other basis function derived by asymptotic methods, e.g., the GTD or PO/PTD. The CBFs allow a decrease in the matrix size compared to that obtained by RWG. This matrix, named reduced matrix, is sparse and well-conditioned in nature, which is not the case when conventional entire-domain basis functions are used instead. Last but not least, the CBFM utilizes direct solvers rather than iterative methods; hence, it does not suffer from convergence problem and can solve multiple excitation problems efficiently.

The computation of the CBFs comprises the most time-consuming and memory demanding task in the CBFM approach. Since the CBFs depend upon the frequency, they need to be regenerated anew for each frequency.

In this section, we introduce a new version of the CBFs –referred to herein as Ultra-wide band Characteristic Basis Functions (UCBFs) [22,23] – that can be used on a wide band, without having to repeat the generation of CBFs for each frequency. The proposed approach begins by

________________________________________________________________________________ analyzing the problem at the highest frequency of interest and following the usual procedure. The CBFs calculated at the highest frequency -- termed UCBFs-- entail the electromagnetic behavior at lower frequency range; thus, it follows that they can also be employed at lower frequencies without going through the time-consuming step of generating them anew. To validate the proposed procedure, we show numerical results of a class of test targets that serve to illustrate the applicability, accuracy, and efficiency of UCBFs.

**4.2**

**4.2**

** Formulation of UCBFs **

**Formulation of UCBFs**

It is apparent that if we are interested in carrying out a frequency sweep, the procedure introduced in paragraph 3.1 must be applied anew for each frequency in the range, and this can be very time-consuming. The most intensive task is represented by the generation of the CBFs, since it requires an LU decomposition for each block, whose size can range from 1000 to 100000 unknowns.

The key feature of the approach proposed in this paper concerns the possibility of re-using the same CBFs for other frequencies. In order to understand the physical basis of the approach, let us consider the following example, where the CBFs are calculated for a 8

### λ

strip, TM case. After the SVD and threshold procedure, 24 CBFs are retained, the first four of which are*plotted in Figure 4.1. Specifically, the first CBF (k=1) refers to the even basis of first order,*

*while the second CBF (k=2) refers to the odd basis of first order, and so on up to k=24.*

**Figure 4.1 CBFs after SVD for a 8**

### λ

**strip, TM case**

Note that the above CBFs have all the desired features of wavelets, though in contrast to the wavelets, they are tailored to the geometry of the object. Next, a

### 4λ

strip is considered; in this0 2 4 6 8 0 0.01 0.02 0.03 0.04 0.05 CBFs (magnitude) x A /m m=1 m=2 m=3 m=4 0 2 4 6 8 -4 -2 0 2 4 CBFs (angle) x ra d m=1 m=2 m=3 m=4

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ 70

case, the SVD and threshold procedure leads to 15 CBFs, the firsts four of which are plotted in
*Figure 4.2. Again, the first CBF (k=1) refers to the even basis of first order, while the second *
*CBF (k=2) refers to the odd basis of first order, and so on up to k=15. By comparing the two *
previous figures, it is possible to note that the CBFs generated at the highest frequency,
embody all the spatial behaviors we would need to capture the corresponding behaviors of the
CBFs at lower frequencies.

**Figure 4.2 CBFs after SVD for a **

### 4λ

**strip, TM case**

*Thus, let us suppose that we calculate the CBFs for a given frequency, i.e. for a given space; *
through these bases it is possible to describe the induced current density. Let us suppose now to
*reduce the frequency: the same CBFs are sufficient to construct the induced current density on *
*this new space too. This feature opens the way to the use of the Universal Characteristic Basis *
*Functions (UCBFs), i.e. the CBFs generated at the highest frequency. Since the UCBFs can *
adequately represent the solution in the entire band of interest, they are used for lower
frequencies without going through the time-consuming step of generating them anew. It
follows that using UCBFs leads to a considerable reduction of CPU time with respect to
conventional CBFs.

In Figure 4.3 we show the flow diagrams of the two methodologies. Conventional CBFs require all the operation of flow diagram (a) to be repeated for all frequency samples; conversely, UCBFs implies to use the same bases–calculated at the highest frequency–for all the frequency samples, as shown in flow diagram (b).

0 1 2 3 4 0 0.02 0.04 0.06 0.08 CBFs (magnitude) x A /m m=1 m=2 m=3 m=4 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 CBFs (angle) x ra d m=1 m=2 m=3 m=4

________________________________________________________________________________

**Figure 4.3 Flow diagram for: a) conventional CBFs; b) UCBFs **

**4.3**

**4.3**

** Numerical Results **

**Numerical Results**

The proposed approach has been applied to several different test examples for calculating the
bistatic RCS, and the results have been compared with those derived by using conventional
CBFs and/or commercial MoM (FEKO). The test cases are: i) cube, ii) sphere, iii) boat, iiii)
*airplane. All the objects are made of PEC and are illuminated by a normally incident *
theta-polarized plane wave; a conventional triangular patch segmentation and RWG basis functions
is employed [8]. For the simulations, a PC with processor Intel(R) Xeon 3,4 GHz (2 cores, 2
threads) and 3 GB of RAM has been used.

**4.3.1 PEC cube **

A cube of 1 meter side is considered first. We present the result for the problem of scattering over a frequency range from 300 MHz to 600 MHz (1:2 band) [23]. The geometry is divided into 2 blocks, as shown in Figure 4.4. The discretization in triangular patches is carried out at the highest frequency, with a mean edge length of 0.1λ; this leads to a number of the unknowns of 5090 and 5170 on the two blocks.

Chapter _________________________________

________________________________________________________________________________

**Figure **

The slight discrepancy in the number

asymmetry in the discretization introduced by the mesher. The number of UCBFs obtained (after SVD) is 233 and 235, respectively. These UCBFs

for any frequency in the range (without th

RCS in dBsm (dB with respect to square meter) at the frequency of 300 MHz is shown for H plane and E-plane; the continuous line refers to the UCBFs solutions, wh

refer to conventional CBFs solutions. The simulation times are summarized in From the table, it is possible to note that a time saving of

using UCBFs; note that this time saving holds for each frequency sample. gain is calculated as:

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________

**Figure 4.4 Block splitting of PEC Cube**

The slight discrepancy in the number of the unknowns on the two blocks is related to the
asymmetry in the discretization introduced by the mesher. The number of UCBFs obtained
*(after SVD) is 233 and 235, respectively. These UCBFs–calculated at f=600 MHz*

nge (without the need of recalculation). In Figure

RCS in dBsm (dB with respect to square meter) at the frequency of 300 MHz is shown for H plane; the continuous line refers to the UCBFs solutions, while the dotted lines refer to conventional CBFs solutions. The simulation times are summarized in

to note that a time saving of 63% (percent time gain)

time saving holds for each frequency sample. The percent time

1 *reduced* 100
*reference*
*T*
*T*
− ∗
_______________________________________________
________________________________________________________________________________
of the unknowns on the two blocks is related to the
asymmetry in the discretization introduced by the mesher. The number of UCBFs obtained
=600 MHz–can be used
Figure 4.5, the bistatic
RCS in dBsm (dB with respect to square meter) at the frequency of 300 MHz is shown for

H-ile the dotted lines refer to conventional CBFs solutions. The simulation times are summarized in TABLE 2. (percent time gain) holds when The percent time

________________________________________________________________________________

a)

b)

**Figure 4.5 Bistatic RCS of PEC cube at 300 MHz: a) E plane; b) H plane **

**TABLE 2 Simulation time for PEC cube **

**UCBFs (f=300 MHZ) CBFs (f=300 MHZ) **

CPU Time 797.98 s 2113.46 s

Saving 63% -

It is worthwhile pointing out that the simulations time summarized in TABLE 2 refers to the discretization carried out at 600 MHz. If conventional CBFs is used employing a discretization carried out at 300 MHz, a reduction of the CPU time will be achieved, due to a reduction of the number of the unknowns; however, the CPU time should be increased by the time required

**-40**
**-30**
**-20**
**-10**
**0**
**10**
**20**
**0** **30** **60** **90** **120** **150** **180**
CBF
UCBF
**B**
**is**
**ta**
**ti**
**c**
** R**
**C**
**S**
** [**
**d**
**B**
**s**
**m**
**],**
** φφφφ**
** =**
** 0**
**°**
θ
θθ
θ
**-5**
**0**
**5**
**10**
**15**
**0** **30** **60** **90** **120** **150** **180**
CBF
UCBF
**B**
**is**
**ta**
**ti**
**c**
** R**
**C**
**S**
** [**
**d**
**B**
**S**
**M**
**],**
** φφφφ**
** =**
** 9**
**0**
**°**
θ
θ
θ
θ

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ 74

to generate the new mesh and upload the new discretization (operations which are time-consuming). Moreover, a dicretization carried out at a lower frequency leads to a reduction of the number of unknowns only when dealing with problems containing smooth or flat surfaces; the same reduction is not achieved when dealing with curved surfaces or geometries with fine details, i.e. when the geometry put a constraint on the discretization step.

**4.3.2 Sphere **

A sphere of 1 meter of radius is shown in Figure 4.6. The UCBFs are calculated at 600MHz and re-used for 300MHz, 30MHz and 6 MHz (100:1 band). The sphere is discretized in triangular patches at highest frequency, with a mean edge length of 0.1λ. We divide the object into 8 blocks, each having about 7500 unknowns. The number of UCBFs obtained after SVD for each block is 183.

**Figure 4.6 Block splitting of PEC sphere **

In Figure 4.7, we present the bistatic RCS calculated at a) 300MHz, b) 30MHz and c) 6 MHz (E plane). The results are compared with Mie series solutions. An excellent agreement between the methods can be observed; this permits extending the limit of applicability of UCBFs in the

________________________________________________________________________________
low frequency regime (note that the radius of the sphere is 2λ at 600MHz, λ at 300MHz, 0.1λ
at 30MHz, 0.02λ at 6MHz).
**-20**
**-10**
**0**
**10**
**20**
**30**
**-180** **-120** **-60** **0** **60** **120** **180**
UCBFs
MIE
**B**
**is**
**ta**
**ti**
**c**
** R**
**C**
**S**
** [**
**d**
**B**
**s**
**m**
**],**φ
φφ φ
**=**
** 0**
**°**
θ
θ
θ
θ
**-20**
**-15**
**-10**
**-5**
**0**
**5**
**10**
**-180** **-120** **-60** **0** **60** **120** **180**
UCBFs
MIE
**B**
**is**
**ta**
**ti**
**c**
** R**
**C**
**S**
** [**
**d**
**B**
**s**
**m**
**],**
φφφφ
** =**
** 0**
**°**
θ
θ
θ
θ
a) b)
**-70**
**-60**
**-50**
**-40**
**-30**
**-20**
**-10**
**-180** **-120** **-60** **0** **60** **120** **180**
UCBFs
MIE
**B**
**is**
**ta**
**ti**
**c**
** R**
**C**
**S**
** [**
**d**
**B**
**s**
**m**
**],**φ
φφ φ
**=**
** 0**
**°**
θ
θ
θ
θ
c)

**Figure 4.7 Bistatic RCS of PEC sphere at a) 300MHz, b) 30MHz and c) 6 MHz; E plane **

**4.3.3 Boat **

The boat considered in this example has a length of 10 m, a width of 2 m, and a height of 2 m. The frequency range starts from 20 MHz and terminates at 200MHz (1:10 band); a length of 10 m is approximately equal to 7λ at 200MHz. The boat is divided in 4 blocks, as shown in

Chapter _________________________________

________________________________________________________________________________ Figure 4.8. The discretization in triangular patches is carried out at the highest frequency, with a mean edge length of 0.1λ; this leads

(block #2), 4073 (block #3), 6429 (block #4). After SVD procedure we obtain 216, 296, 217 and 298 UCBFs, respectively. These UCBFs

frequency in the range (without the need of recalculation).

**Figure **

In Figure 4.9, the bistatic RCS calculated at 100MHz is shown for both H The continuous line refers to the UCBFs solutions while the

CBFs: an excellent match can be observed. Next, we present in calculated at 20MHz, for both

H-UCBFs solutions while the dotted lines to conventional CBFs. Some discrepancies can be observed between the two curves.

procedures are employed at 20MHz using the discretization carried out at 200 MHz (the
highest frequency): this could lead to an increase of the condition number when calculating the
impedance matrix. However, in UCBFs only the reduced matrix is calculated anew, while
CBFs requires computing again *Z*

refers to conventional MoM (FEKO) solution obtained by using the discretization calculated at 20MHz, i.e. generating a new mesh at 20MHz and uploading the correspondent di

this latter solution perfectly matches with the UCBFs. Thus, it follows that using UCBFs guarantees that the reduced matrix remains well conditioned at lower frequencies.

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ . The discretization in triangular patches is carried out at the highest frequency, with

; this leads to a number of unknowns of 4086 (block #1),
(block #2), 4073 (block #3), 6429 (block #4). After SVD procedure we obtain 216, 296, 217
*and 298 UCBFs, respectively. These UCBFs –calculated at f=200 MHz– can be used for any *
frequency in the range (without the need of recalculation).

**Figure 4.8 Block splitting of PEC boat **

, the bistatic RCS calculated at 100MHz is shown for both H-plane and E

The continuous line refers to the UCBFs solutions while the dotted lines to conventional CBFs: an excellent match can be observed. Next, we present in Figure 4.10 the bistatic RCS

-plane and E-plane. Again, the continuous line refers

ile the dotted lines to conventional CBFs. Some discrepancies can be observed between the two curves. The discrepancies can be explained recalling that both the procedures are employed at 20MHz using the discretization carried out at 200 MHz (the requency): this could lead to an increase of the condition number when calculating the impedance matrix. However, in UCBFs only the reduced matrix is calculated anew, while

*ii*

*Z* (see Figure 4.3). In the sameFigure 4.10, the dashed curve
refers to conventional MoM (FEKO) solution obtained by using the discretization calculated at
20MHz, i.e. generating a new mesh at 20MHz and uploading the correspondent di

this latter solution perfectly matches with the UCBFs. Thus, it follows that using UCBFs guarantees that the reduced matrix remains well conditioned at lower frequencies.

_______________________________________________

________________________________________________________________________________ . The discretization in triangular patches is carried out at the highest frequency, with to a number of unknowns of 4086 (block #1), 6323 (block #2), 4073 (block #3), 6429 (block #4). After SVD procedure we obtain 216, 296, 217 can be used for any

plane and E-plane. dotted lines to conventional the bistatic RCS plane. Again, the continuous line refers to the ile the dotted lines to conventional CBFs. Some discrepancies can be The discrepancies can be explained recalling that both the procedures are employed at 20MHz using the discretization carried out at 200 MHz (the requency): this could lead to an increase of the condition number when calculating the impedance matrix. However, in UCBFs only the reduced matrix is calculated anew, while , the dashed curve refers to conventional MoM (FEKO) solution obtained by using the discretization calculated at 20MHz, i.e. generating a new mesh at 20MHz and uploading the correspondent discretization; this latter solution perfectly matches with the UCBFs. Thus, it follows that using UCBFs guarantees that the reduced matrix remains well conditioned at lower frequencies.

________________________________________________________________________________

a)

b)

**Figure 4.9 Bistatic RCS for PEC boat at 100 MHz: a) H plane; b) E plane **

a)
**10**
**15**
**20**
**25**
**30**
**35**
**0** **60** **120** **180** **240** **300** **360**
CBFs
UCBFs
**B**
**is**
**ta**
**ti**
**c**
** R**
**C**
**S**
** [**
**d**
**B**
**s**
**m**
**] **
**, **
φφφφ
** =**
** 9**
**0**
**°**
θ
θθ
θ
**-20**
**-10**
**0**
**10**
**20**
**30**
**40**
**0** **60** **120** **180** **240** **300** **360**
CBFs
UCBFs
**B**
**is**
**ta**
**ti**
**c**
** R**
**C**
**S**
** [**
**d**
**B**
**s**
**m**
**],**
φφφφ
** =**
** 0**
**°**
θ
θ
θ
θ

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ 78

b)

**Figure 4.10 Bistatic RCS for PEC boat at 20 MHz: a) E plane; b) H plane **

A further check on the limit of applicability of the UCBFs can be obtained by checking the condition number of the reduced matrix. Figure 4.11 shows the condition number for some frequency samples: the figure highlights that the condition number does not grows significantly when moving from 200MHz to lower frequencies. Thus, UCBFs can be re-used down to very low frequency region; however, it should be noted that since the UCBFs have been generated at the highest frequency, their number is higher than necessary. This feature is highlighted in TABLE 3 where the number of UCBFs and CBFs of the four blocks is given for some frequency samples. Obviously, the number of UCBFs remains the same, while the number of CBFs decreases with the frequency. Again, it has to be clearly pointed out that this over-estimation on the number of basis functions will not lead to an ill-conditioned reduced matrix; the only drawback is that the reduced matrix is slightly larger than necessary. Further research will be focused on an appropriate way of cutting down the number of UCBFs.

________________________________________________________________________________

**Figure 4.11 Condition number of the reduced matrix for boat problem **

**TABLE 3 Comparison between UCBFs and CBFs **

**Frequency Number of UCBFs ** **Number of CBFs **

200 MHz Block #1 - 216 Block #2 - 298 Block #3 - 217 Block #4 - 298 Block #1 - 216 Block #2 - 298 Block #3 - 217 Block #4 - 298 100 MHz Block #1 - 216 Block #2 - 298 Block #3 - 217 Block #4 - 298 Block #1 - 92 Block #2 - 126 Block #3 - 92 Block #4 - 126

**4.3.4 Airplane **

In the final example, we consider an Unmanned Aerial Vehicle (UAV) model. The dimensions of object are: 0.8 m length to tip-to-tail, 0.8 m wing span, 0.4 m height. The frequency range of interest goes from 1 GHz to 3 GHz. This example has been chosen since a similar problem has been addressed in [24]. The airplane is divided in 2 blocks (Figure 4.12 a). The discretization in triangular patches is carried out at the highest frequency (Figure 4.12 b), with a mean edge length of 0.1λ; this leads to a number of unknowns of 8755 (block #1), 8767 (block #2). After SVD procedure, we obtain 282 and 287 UCBFs, respectively. These UCBFs –calculated at

Chapter _________________________________

________________________________________________________________________________

**Figure 4.12 a) Block splitting of PEC airplane, b) triangular mesh of PEC airplane**

Next, the bistatic RCS for E-plane is

4.13). We get a total simulation time of about conventional CBFs leads to 158 hours (see

approach requires approximately 100 hours for solving a similar Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ a)

b)

**a) Block splitting of PEC airplane, b) triangular mesh of PEC airplane**

plane is computed at 64 frequencies in the band of interest ( ). We get a total simulation time of about 38 hours with UCBFs, while the use of conventional CBFs leads to 158 hours (see TABLE 4). A conventional frequency

approach requires approximately 100 hours for solving a similar problem [24].

_______________________________________________

________________________________________________________________________________

**a) Block splitting of PEC airplane, b) triangular mesh of PEC airplane **

computed at 64 frequencies in the band of interest (Figure 38 hours with UCBFs, while the use of tional frequency-domain AIM

________________________________________________________________________________

**Figure 4.13 Airplane bistatic RCS in the frequency band 1-3 GHz, E plane **

**TABLE 4 Total simulation time for airplane problem **

**UCBFs (64 frequency **
**samples) **
**CBFs (64 frequency **
**samples) **
CPU Time 38 h 158 h
Saving 76% -

In the end, in Figure 4.14 it displayed the electric surface density current on the airplane, for the whole object (a) e for the two blocks (b).

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ 82

b)

**Figure 4.14 Electric surface density current on the: a) airplane, b) blocks **

**4.4**

**4.4**

** Problem structure with apertures (slots) **

**Problem structure with apertures (slots)**

Recent year have drawn increasing attention to the problem of efficient and accurate analysis of antennas mounted on large structure. Typical examples include slotted array antennas mounted on large platforms, or on a ship or aircraft surface. This type of antenna has the advantage of a low impact on the object RCS. Hybrid approaches are often used to address this task [25,26]. Specifically, the antenna array itself may be analyzed with a state-of-the-art numerical method such as Finite Element Method (FEM) or the Finite Difference Time Domain (FDTD) technique. However, the much larger platform on which the antenna is mounted remains beyond the capability of numerical methods without the use of very large-scale computer resources.

The CBFM can be properly extended for addressing radiation of apertures arrays mounted on
Large Conducting Structures. Specifically, we will show that the problem of radiation of
conducting object with apertures can be solved by filling the apertures with PEC and applying
CBFM to the entire structure. Then the apertures are replaced with equivalent currents
radiating in the presence of the filled structure. The equivalent currents are determined by
*solving the apertures arrays problems locally. *

Note that the proposed procedure permits to address the radiation of structures with apertures by simply using the same CBFs obtained when filling the apertures. Moreover, it exploits the localization properties of the CBFM.

________________________________________________________________________________ To validate the proposed procedure problems of radiation from apertures arrays mounted on PEC plate and PEC cylinder have been considered. The results will be compared with those obtained through commercial software. Moreover, concerning the radiation from cylinder, Mode Matching (MM)/Finite Element Method (FEM) will be used, and spectral rotation will be employed to efficiently calculate the coupling between apertures. In order for the spectral rotation approach to be applicable to the cylindrical geometry too, the conventional spectral rotation technique [31] will be properly extended [32].

**4.5**

**4.5**

** Formulation **

**Formulation**

Let us consider a PEC body, which can be either open or closed, with an aperture (slot). Our
objective is to compute the field radiated by the aperture. Let us consider, for instance, the
example shown in Figure 4.15 a), where we have complex object with a slot. The slot is excited
by an electric field *Ea*. The problem can be solved by using the CBFM. As a first step, we

divide the body into several blocks, say *P* (see Figure 4.15 b). For convenience of analysis, we
define the domain of block 1 such that the slot is contained well-within it. Next, we proceed as
follows:

i. fill the slot with PEC (see Figure 4.15 c);

ii. calculate the magnetic current density *M* =*Ea*×*n*ˆ lying in the region of the slot ( ˆ*n*

represents the unit vector normal to the surface). This current density can be calculated separately, by solving local problem;

iii. compute the excitation vector *V* as concatenation of two kind of excitations, plane
wave spectrum and electric near field *Ea*

### ( )

*M that represents the field radiated by the*

equivalent magnetic current density *M*. In compact form:

( )

*PWS* *E M*

*i* *i* *i*

*V* =*V* ⊕*V* (4.2)

where

### ⊕

is the horizontal concatenation operator and *PWS*
*PWS*
*i*
*i*
*N* *N*
*V*
×
is the matrix of
the plane wave excitations (*N _{PWS}* number of plane waves,

*N*number of unknowns in the

_{i}*th*

*i*block) and ( )

*E M*

*i*

*exc*

*i*

*blk*

*N*

*N*

*V*×

is the electric near field source matrix ( *exc*
*blk*

*N* number
of block with slot excitation, which has been assumed equal to 1 in this example). In
this way, we can better take account the near field behavior;

Chapter _________________________________

________________________________________________________________________________ iv. apply the CBFM to the modified structure (with the slot filled by PEC)

CBFs (see paragraph 3.1):

The use of the Galerkin

functions, leads to a linear system of algebraic equations that are cast in the matrix form
*as A X* =*B, where A is the known coefficient matrix of size *

known excitation vector, alternatively referred to as the right hand side (RHS); unknown solution vector of size

v. compute the excitation vector ˆ

*a*

*M* =*E* × lying on the surface of the now*n*

located (see Figure 4.15. d).

**Figure 4.15 a) Complex PEC body with a slot excited by a electric field radiated by magnetic current; b) the body is **

**divided into P blocks; c) we fill the slot with PEC and apply the CBFM to **

**absence of the slot) and calculate the coefficient matrix; d) the original problem is solved using **

The problem of determining *Ea* can be addressed by solving a local problem which can handle

the localized problem even when the region around the slot is inhomogeneous.

It is worthwhile pointing out that in the aforementioned approach we wish to address the problem of radiating by structures with apertures by using

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ apply the CBFM to the modified structure (with the slot filled by PEC)

whit 1, 2,...,

*i*

*J* *i*= *M*

procedure, coupled with the employment of above basis functions, leads to a linear system of algebraic equations that are cast in the matrix form

is the known coefficient matrix of size *M M*× ;
known excitation vector, alternatively referred to as the right hand side (RHS);
unknown solution vector of size *M*×1;

*compute the excitation vector B for considering as source a magnetic current density *
lying on the surface of the now-closed object where the slot was originally

d).

**a) Complex PEC body with a slot excited by a electric field radiated by magnetic current; b) the body is **
**divided into P blocks; c) we fill the slot with PEC and apply the CBFM to the entire structure to extract the CBFs (in the **
**absence of the slot) and calculate the coefficient matrix; d) the original problem is solved using ***M*** as further source (in the **

**absence of the slot).**

can be addressed by solving a local problem which can handle the localized problem even when the region around the slot is inhomogeneous.

It is worthwhile pointing out that in the aforementioned approach we wish to address the of radiating by structures with apertures by using the set of CBFs, already obt

_______________________________________________

________________________________________________________________________________ apply the CBFM to the modified structure (with the slot filled by PEC) and extract the

(4.3)
procedure, coupled with the employment of above basis
functions, leads to a linear system of algebraic equations that are cast in the matrix form
; *B*is the *M*×1

known excitation vector, alternatively referred to as the right hand side (RHS); *X* is the

for considering as source a magnetic current density the slot was originally

**a) Complex PEC body with a slot excited by a electric field radiated by magnetic current; b) the body is **
**the entire structure to extract the CBFs (in the **

**as further source (in the **

can be addressed by solving a local problem which can handle the localized problem even when the region around the slot is inhomogeneous.

It is worthwhile pointing out that in the aforementioned approach we wish to address the already obtained

________________________________________________________________________________
when filling the aperture, is increased by one by adding the CBFs originated when considering
as source the magnetic current density *M* lying in the region of the slot. Thus, the procedure
differs from other approaches based on addressing the entire problem in one step, by
formulating the aperture problem in terms of an integral equation for the magnetic current
which is then solved numerically.

**4.5.1 RHS calculation **

An important task of the procedure is the calculation of the right hand side matrix, that is the matrix that contains the excitations of the problems. We have already seen in paragraph 3.1 haw build the plane wave excitation vector, then in this section it will be presented how calculate the excitation vector related at the electric field on the aperture. The radiated electric field in near field configuration can be represent as [4]:

3
3
3
1 1
[( ') ( ') ] '
4
1 1
[( ') ( ') ] '
4
1 1
[( ') ( ') ] '
4
*j R*
*Fx* *y* *z*
*S*
*j R*
*Fy* *z* *x*
*S*
*j R*
*Fz* *x* *y*
*S*
*j R*
*E* *z* *z M* *y* *y M* *e* *dS*
*R*
*j R*
*E* *x* *x M* *z* *z M* *e* *dS*
*R*
*j R*
*E* *y* *y M* *x* *x M* *e* *dS*
*R*
β
β
β
β
π
β
π
β
π
−
−
−
+
= − − − −
+
= − − − −
+
= − − − −

### ∫∫

### ∫∫

### ∫∫

(4.4)where *M* is the magnetic surface current on the aperture,

*S*

is the surface of the aperture and
*R* represent the distance from aperture to an observation point. Hence, this formulation is
used to calculate the electric field on every triangular patches of surface external to the slot,
due to a magnetic current on the slot. For the patch interior to the slot, considering the
paragraph 1.9.1 , as shown in Figure 4.16 , the electric field is given by:

ɵ 2

*F*

*M* *n*

Chapter _________________________________

________________________________________________________________________________

**Figure 4.16 Contribution of magnetic current on patches interior and external to the aperture**

**4.5.2 Far Field calculation **

To calculate the electric far field we have

The electric current is calculated through the CBFM proc

given by solving the local problem. This can be summarized by the following expressions:

where:

( cos cos cos sin sin ) '

( cos cos cos sin sin ) '

( sin cos ) '
( sin cos ) '
*x* *y* *z*
*S*
*x* *y* *z*
*S*
*x* *y*
*S*
*x* *y*
*S*
*N* *J* *J* *J* *e* *dS*
*L* *M* *M* *M* *e* *dS*
*N* *J* *J* *e* *dS*
*L* *M* *M* *e* *dS*
θ
θ
φ
φ
θ φ θ φ θ
θ φ θ φ θ
φ φ
= + −
= + −
= − +
= − +

### ∫∫

### ∫∫

### ∫∫

### ∫∫

*J* is the electric surface current obtained from the solution of CBFM, while
Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________

**Contribution of magnetic current on patches interior and external to the aperture**

To calculate the electric far field we have to employ both the electric that the magnetic one. The electric current is calculated through the CBFM procedure, while the magnetic current is given by solving the local problem. This can be summarized by the following expressions:

0
4
4
*r*
*j r*
*j r*
*E*
*L*
*j* *e*
*E* *N*
*r*
*L*
*j* *e*
*E* *N*
*r*
β
φ
θ θ
β
ϑ
φ φ
ηβ
π η
ηβ
π η
−
−
− +
− _{} − _{}
≃
≃
≃
'cos
'cos
'cos
'cos

( cos cos cos sin sin ) '

( cos cos cos sin sin ) '

( sin cos ) '
( sin cos ) '
*j r*
*x* *y* *z*
*j r*
*x* *y* *z*
*j r*
*x* *y*
*j r*
*x* *y*
*N* *J* *J* *J* *e* *dS*
*L* *M* *M* *M* *e* *dS*
*N* *J* *J* *e* *dS*
*L* *M* *M* *e* *dS*
β ψ
β ψ
β ψ
β ψ
θ φ θ φ θ
θ φ θ φ θ
φ φ
φ φ
+
+
+
+
= + −
= + −
= − +
= − +

is the electric surface current obtained from the solution of CBFM, while *M*

ɵ _{a}

*M* = − ×*n E*

_______________________________________________

________________________________________________________________________________

**Contribution of magnetic current on patches interior and external to the aperture **

both the electric that the magnetic one. edure, while the magnetic current is given by solving the local problem. This can be summarized by the following expressions:

(4.6)

(4.7)

derive from: (4.8)

________________________________________________________________________________

**4.6**

**4.6**

** Numerical result **

**Numerical result**

A numerical example will be now presented to show the validity of the procedure We will show a few examples of PEC objects with aperture antennas. The results are compared with FEKO, to prove the correctness of the method.

**4.6.1 Electrically large cylinder with a slot **

The first example is a 10

### λ

long PEC cylinder with a 1### λ

radius, which has an axial slot in the middle. The slot spans 10° along the φ direction and from2
*D*
*z*= − to
2
*D*
*z*= along the
z axis, with

*D*

### =

### λ

. In particular, it extends from 85°to 95°. The slot is excited by an electric field*Ea*

### ( )

*M*. In this example we assume that the magnetic current densities

*M*is represented

by half-sine distributions, as follows:

ˆ

cos( / ) when* _{z}* / 2 / 2, 85 95

*M*=

### π

*z D i*−

*D*< <

*z*

*D*< <

### φ

_{(4.9) }

Operating frequency is 30 GHz. The cylinder is divided in 3 blocks, as shown in Figure 4.17.
*Blocks 1 and 3 have z dimension of *

### 3

### λ

, whereas block 2 is### 4λ

long. The total number of unknowns is 23493, 8106 for both extended blocks 1 and 3, while for the block 2 are 12361. The number of CBFs are 255 for the first one and the third, 370 for the second one.Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ 88

In Figure 4.18 it can observed the normalized magnitude (a) and the phase (b) of co-polar component of the electric field far field, for the XY plane.

a)

b)

**Figure 4.18 a ) normalized magnitude and b) phase of co-polar component of the electric field far field, for the **
**XY plane. **

________________________________________________________________________________

**4.6.2 Slot on a large platform. **

A 6 λ square PEC platform, which has 1 axial slot is now presented. The slot is 0.4 × 0.9 inc and it is excited through a WR90 waveguide by an TE 01, as shown in Figure 4.19. Specifically, we assume that the operating frequency is 9 GHz.

**Figure 4.19 A 6**

### λ

**squared PEC plate fed by a WR90 waveguide.**

As reference solution, we solve the problem by the use of a MoM based code (FEKO), which requires 20579 number of unknowns. With the same code we evaluate the field on the aperture

*a*

*E* . Next, we solve the problem described above by using the CBMoM. Towards this end, we
divide the geometry into 3 blocks, where: blocks 1 and 3 have 3711, whereas block 2 is 7603.
The slot is confined entirely within block 2, as shown in Figure 4.20. A 0.5 λ block overlap
with tapering is used to suppress the fictitious edge effects resulting from the decomposition of
the original object, which was carried out in the process of defining the blocks. Concerning
CBMoM, the total number of unknowns is 15025. The number of unknowns for the extended
blocks are: 4957 and 4952 for extended block 1 and 3, 9981 for extended block 2. The CBMoM
leads to 147 CBFs for block 1 and 3, and 177 CBFs for block 2. The total time for the
generation of the CBFs is 71 and 72 sec for block 1 and 3, and 324 sec CBFs for block 2. The
time for filling the reduced matrix is 423.68 sec, and the total simulation time is 2468 sec.
Figure 4.21 a) shows the far field calculated by employing the proposed procedure and FEKO
in the XZ plane and YZ plane in b).

Chapter _________________________________

________________________________________________________________________________

**Figure 4.20 Partitioning of the 6**

**Figure 4.21 Comparison of the E far field calculated on: a) **

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________

**Partitioning of the 6**

### λ

**squared PEC plate into 3 blocks. The aperture is filled with PEC**

a)

b)

**Comparison of the E far field calculated on: a) **φ** component XZ plane, b) **θ

**plane **

_______________________________________________

________________________________________________________________________________

**squared PEC plate into 3 blocks. The aperture is filled with PEC**

_________________________________

**4.6.3 Slot antenna array onboard a PEC missile**

In this example is presented a 2-model of missile, shows in Figure about

### λ

. Each slot is approximately The hybrid MM/FEM-SD [31-33distribution over the aperture (first 24 modes were used). The missile has been divided in 17 blocks

a)

**Figure 4.22 a) Model of missile with a slot array antenna**

In the next figure (Figure 4.23) principal plane, for the φ componen

________________________________________________________________________________

**Slot antenna array onboard a PEC missile **

-by-2 slot antenna array mounted on the fuselage of a PEC Figure 4.22. The missile is slightly longer than 30

### λ

. Each slot is approximately

### 0.72

### λ

### ×

### 0.34

### λ

and excited with the fundamental mode. 33] has been employed to reconstruct the magnetic current distribution over the aperture (first 24 modes were used). The missile has been divided in 17b)

**Model of missile with a slot array antenna, b) detail triangular mesh of array**

) we can observe the magnitude of electric far field on two component (continuous line) and θ component (dotted line).

_______________________________________________

2 slot antenna array mounted on the fuselage of a PEC

### λ

and its radius is and excited with the fundamental mode. ] has been employed to reconstruct the magnetic current distribution over the aperture (first 24 modes were used). The missile has been divided in 17**, b) detail triangular mesh of array **

we can observe the magnitude of electric far field on two component (dotted line).

Chapter 4 Improvement of the CBFM ________________________________________________________________________________ ________________________________________________________________________________ 92 a) b)

**Figure 4.23 E far field calculated for both component **φ** and **θ ** : a) XZ plane, b) YZ plane **

In Figure 4.24 it displayed a detail of normalized electric current on metalized slot and on a missile part.

________________________________________________________________________________

**4.6.4 Standard WR10 waveguide with flange **

In this last example, it is presented the analysis at 94 GHz of a standard WR10 waveguide with a flange. For simplicity, we have choose PEC as material for the waveguide. The WR10 is been simulated with FEKO, to obtain the electric far field to compare with that derived with the CBMoM. The flange dimensions are shown in Figure 4.25 and can be found at [34,35], while the diameter of the flange is 19.05 mm and the whole length of object (see Figure 4.26) is 50 mm.

Chapter _________________________________

________________________________________________________________________________

**Figure 4.26 PEC model of WR10 waveguide with flange**

Then, always with FEKO, it was simulated a simple waveguide

same aperture dimension of the WR10, to obtain the electric near field on the ap

electric field will be used how excitation for the only flange with CBMoM. Indeed, using the procedure presented above, we don't need to simulate the whole object to obtain the electric far field, but it is necessary only the flange

**Figure 4.27 Model of waveguide with the same aperture of WR10**

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________

**PEC model of WR10 waveguide with flange **

Then, always with FEKO, it was simulated a simple waveguide (see Figure same aperture dimension of the WR10, to obtain the electric near field on the ap

electric field will be used how excitation for the only flange with CBMoM. Indeed, using the procedure presented above, we don't need to simulate the whole object to obtain the electric far field, but it is necessary only the flange (see Figure 4.28).

**Model of waveguide with the same aperture of WR10 **

_______________________________________________

________________________________________________________________________________ Figure 4.27), with the same aperture dimension of the WR10, to obtain the electric near field on the aperture. This electric field will be used how excitation for the only flange with CBMoM. Indeed, using the procedure presented above, we don't need to simulate the whole object to obtain the electric far

_________________________________

**Figure 4.28 Transparency of the only WR10 flange mesh**

In Figure 4.29 it shown the normalized electric field for the co plane (a) and on YZ plane (b). We can notice th

direction, but some difference in the back direction. This difference can be due to the different model used to simulate the same problem.

________________________________________________________________________________

**Transparency of the only WR10 flange mesh **

it shown the normalized electric field for the co-polar component (

YZ plane (b). We can notice the excellent superposition in the radiation fference in the back direction. This difference can be due to the different model used to simulate the same problem.

a)

_______________________________________________

polar component (θ) on the XY e excellent superposition in the radiation fference in the back direction. This difference can be due to the different

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ 96

b)

**Figure 4.29 Co-polar component (**

### θ

**) of normalized electric field [V/m] on the: a) XY plane, b) YZ plane**

The number of unknowns needed to solve the whole problem is 39549 (simulation time 44.24' ), to solve the only waveguide is 14918 (simulation time 4.41' ), both with FEKO, and to solve the flange problem with CBMoM is 15516 (152 CBFs - simulation time 14.06' ). In the TABLE 5 we can observe the unknowns, the total time and the saving time (percent time gain) using CBFM (simulation of waveguide with FEKO plus simulation of flange with CBMoM) compared to the full model with FEKO. The simulations were carried out on an Intel(R) Core (TM) i7 @3.07GHz, with 64 bit operative system.

**TABLE 5 Comparison of simulation time for WR10 waveguide **

**FEKO ** **CBMoM **

Unknowns 39549 30434 Total Time 44.24 ' 18.47 ' Saving Time - 41.7%

A possible application of this work concern the analysis of radiated field in presence of a body [36,37], as shown in fig. At 94 GHz, we can assume the human body made of metallic surface, using the IBC approximation. With this approximation, we can use the CBMoM.

_________________________________

**Figure 4.30 Application of aperture antenna with CBFM for on**

**4.7**

**4.7**

** CBMoM GUI **

**CBMoM GUI**

To make easier to use CBMoM was developed a graphic user interface (GUI), shown in 4.31. The GUI allows the user to

works, generating the input file wit the GUI can be summarized as follows

• choose work directory: directory where results are saved; • insert name project;

• choose mesh directory: directory where the mesh is saved;

• frequency sweep: insert the frequency start, stop and step. If the simulation is an only frequency, star and stop must be the same and step will be equal to 1. It can check the UCBFs box;

• ground: if this box is checked, a ground is considered in the simulation;

• excitation kind: at this moment the interface provides three kind of problem, i.e. monostatic RCS, bistatic RCS and delta gap. In a short time, will be insert the aperture antenna. For each exitation the user have to compile the relative box to provide all information on the simulation;

• output: the user have to define the

monostatic RCS case this section is not enabled; • new: generate a new input file;

• save: write the information on the input file stored named as project work directory;

________________________________________________________________________________

**Application of aperture antenna with CBFM for on body communicati**

To make easier to use CBMoM was developed a graphic user interface (GUI), shown in The GUI allows the user to utilize the code without having to know exactly

, generating the input file with all the information on the simulation. The operation of the GUI can be summarized as follows:

choose work directory: directory where results are saved;

choose mesh directory: directory where the mesh is saved;

the frequency start, stop and step. If the simulation is an only frequency, star and stop must be the same and step will be equal to 1. It can check the

ground: if this box is checked, a ground is considered in the simulation;

at this moment the interface provides three kind of problem, i.e. monostatic RCS, bistatic RCS and delta gap. In a short time, will be insert the aperture antenna. For each exitation the user have to compile the relative box to provide all

the simulation;

output: the user have to define the planes of observation for the electric field. In the monostatic RCS case this section is not enabled;

new: generate a new input file;

save: write the information on the input file stored named as project name file in the _______________________________________________

**body communications **

To make easier to use CBMoM was developed a graphic user interface (GUI), shown in Figure the code without having to know exactly how it The operation of

the frequency start, stop and step. If the simulation is an only frequency, star and stop must be the same and step will be equal to 1. It can check the

at this moment the interface provides three kind of problem, i.e. monostatic RCS, bistatic RCS and delta gap. In a short time, will be insert the aperture antenna. For each exitation the user have to compile the relative box to provide all

planes of observation for the electric field. In the

Chapter 4 Improvement of the CBFM

________________________________________________________________________________

________________________________________________________________________________ 98

• run: starts the simulation that uses the input file created; • exit: exit to GUI.

Without this interface, the user should write a text file, inserting the right information in the right position. To develop the GUI it was used JAVA.