1
1
2
 An
Effective Interpolation Scheme for Multi-Resonant Antenna
Responses: Generalized Stoer-Bulirsch Algorithm with Adaptive Sampling
3
4
Yasser El-Kahlout, Member, IEEE and Gullu Kiziltas, Member, IEEE
5
Abstract— Multiple analysis calls and the high computational demand for each analysis present
6
themselves as important bottlenecks in conducting practical electromagnetic design optimization studies
7
particularly for complex material designs in metamaterial studies. To enable efficient reanalysis in such
8
studies, in this paper we develop an algorithm for rational interpolation by generalizing the Stoer-Bulirsch
9
algorithm to allow for non-diagonal Neville paths rather than the known standard diagonal path to
10
enhance its interpolation capability. The algorithm is then integrated to an adaptive sampling strategy that
11
exploits the non-diagonality and finds an optimum Neville path that provides more efficient and reliable
12
fittings for multi-resonance antenna response functions. The resulting technique is applied to return loss
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responses of antenna models with textured material substrates and complex conductor topologies.
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Interpolation results are compared to the performance of a standard Stoer-Bulirsch algorithm. Results
15
show that the proposed generalized scheme outperforms the existing Stoer-Bulirsch technique in terms of
16
computational accuracy by detecting resonances while still maintaining minimum number of support
17
points. To demonstrate the capability of the proposed generalized algorithm, it is adapted to a large-scale
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antenna design optimization example achieving significant bandwidth performance enhancements for a
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novel antenna structure within practical timespans.
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Index Terms—Approximation theory, frequency response, microstrip antennas, genetic algorithms, StoerBulirsch algorithm, adaptive sampling.
.
Y. El-Kahlout was with the Mechtronics Program, Sabanci University, Orhanli 34956, Tuzla, Istanbul, Turkey but is currently with the Electrical Engineering
Department, Istanbul Aydin University, Sefakoy 34295, Kucukcekmece, Istanbul, Turkey, tel.: +90 212 425 6151, ( e-mail: yasser@aydin.edu.tr)
G. Kiziltas is with the Mechtronics Program, Sabanci University, Orhanli 34956, Tuzla, Istanbul, Turkey, tel.: +90 216 4839584, fax: +90 216 4839550. ( email: gkiziltas@sabanciuniv.edu)
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I. INTRODUCTION
1
2
DESIGN optimization has been a difficult, demanding but necessary task for the development of new
3
wireless applications. It is reasonable to expect that designs resulting from global design optimization
4
studies that allow for full design space exploration will lead to novel configurations with enhanced
5
performance. For instance, such designs include antenna shape, size, feed location and material variations
6
in 3D. However, global large-scale volumetric antenna synthesis via heuristic techniques is a challenging
7
task due to the need for fast and accurate re-analysis and proper definition of the optimization model. Such
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a synthesis framework is applied to novel metamaterial designs with complex material variations and
9
conductor topologies particularly. Therefore, unless antenna design studies are limited to only a few
10
number of design variables [1] such as in conventional substrate designs, global large-scale studies can
11
become impractical, hence novel designs cannot be achieved. To address this issue, an approximation
12
scheme suitable for frequency response functions of electromagnetic systems such as return loss curves of
13
multi-resonance antennas with novel material and conductor topologies is proposed in this paper.
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Rational functions offer an attractive solution for providing a global approximation taking into account
15
the entire band of the frequency response into consideration. In addition, they are well suited for
16
approximating resonances due to their inherent pole predicting behavior. As a result, their use has resulted
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in various representations of resonance type curves with reasonable number of support points [2-11].
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Solving for the coefficients of the rational function is known as the Cauchy method and is first introduced
19
in [12]. It has been employed for the extraction of a circuit model [13], the response of which fits the
20
microwave device reflection and transfer functions of non-lossy systems. Lamperez et al. [14] extended the
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Cauchy method in [13] to reduce the model order of systems which may be lossy. In addition to the
22
complicated mathematics needed for deducing the resulting expression or equivalent circuits, this technique
23
is restricted to electromagnetic optimization of devices such as filters and multiplexers. Moreover, the
24
solution within a standard Cauchy based method with techniques such as direct inversion becomes more
3
1
prone to numerical errors as the number of support points increases (by producing a system of linear
2
equations which is ill-conditioned).
3
A solution to overcome ill-conditioned systems via the Cauchy method is presented in [9] This technique
4
has the potential of accurately predicting wide-band response utilizing narrow-band information via both
5
interpolation and extrapolation. The computations have been automated later in [8]. In both of these
6
studies, the order of the optimum rational function is chosen such that the number of unknown coefficients
7
is less than or equal to the rank of the corresponding stiffness matrix for interpolation. However, a multi-
8
resonance frequency response cannot be approximated via the rank constrained strategy as it limits the
9
order of the rational model. Also, since for each newly added point the system is re-solved, it becomes
10
computationally expensive when integrated with adaptive sampling techniques. An alternative approach to
11
overcome these issues is a support point recursive technique initially introduced by Stoer and Bulirsch [15].
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Stoer-Bulirsch (S-B) technique is a recursive method that adds one support point at a time and solves for
13
the unique rational interpolator that passes through all existing support points. This recursive strategy in
14
turn enhances its suitability to adaptive sampling techniques where each new support point is determined
15
based on a certain error norm and its interpolator is found using the available data set. When compared with
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direct inversion solutions of rational interpolations, Stoer-Bulirsch technique is significantly less prone to
17
numerical errors and is not constrained by the rank of the system.
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It is known that the data set used for interpolation determines the quality of the resulting fitted curve.
19
Therefore, adaptive sampling of a frequency response constitutes a key aspect in interpolation, hence,
20
careful selection of these informative support points to serve as input data to the interpolation technique
21
should result in a more successful interpolation [2, 3, 16, 17]. In addition, different numbering of the
22
support points can define different interpolators with different accuracies [10]. One widely used
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interpolation strategy integrated with adaptive sampling is an iterative least-squares pole-residue modeling
24
technique known as Vector Fitting [18]. It starts with an initial set of poles and in successive steps, the
25
poles are relocated and the residues are calculated to optimize the fit. Each stage of the algorithm reduces to
4
1
a linear problem. Nevertheless, the solution requires matrix inversion which is time consuming and
2
depends on piecewise interpolation [19] for wide broadband modeling. In the method proposed here
3
frequency data sets have been adaptively constructed and integrated to the proper choice of the rational
4
function within Stoer-Bulirsch technique to deliver an efficient interpolation scheme as was applied in [20,
5
21] with standard Stoer-Bulirsch. Yan et al. [20] proposed a similar adaptive sampling through rational
6
functions based on alternative diagonal Neville paths created by incrementing the numerator and
7
denominator by one after a specific switching grid is chosen, where these paths remained essentially
8
parallel to the main diagonal. Thus, existing Stoer-Bulirsch interpolations constructed based on a pre-
9
determined standard diagonal path (where the order of numerator and denominator of the interpolator are
10
equal or different by one) are not guaranteed to be optimum, and therefore resulting interpolations even if
11
adaptively constructed could be further improved for accuracy and computational efficiency.
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In this paper, we present the detailed steps of how to improve the standard Stoer-Bulirsch algorithm by
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generalizing the Neville algorithm that allows for an off-diagonal path as suggested in [22] and integrate it
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with a novel adaptive sampling technique. This technique not only selects the support points but also
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determines the optimized path used in constructing the interpolator. Consequently it enhances the potential
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of detecting resonances and minimizing the interpolation error. Also, to further enhance the performance of
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an adaptive Neville path for multi resonance wide band responses, the proposed algorithm is integrated to
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an initial sampling scheme. The resulting Generalized Stoer-Bulirsch algorithm with Adaptive Sampling
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(G-SBAS) and initial sampling (G-SBAIS) are analyzed and compared with the standard Stoer-Bulirsch (S-
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B) technique. Results demonstrate the capability of the optimized method G-SBAIS to overcome common
21
problems of existing methods such as premature convergence and missing significant resonances. As a
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result, the proposed algorithm leads to approximations with enhanced accuracy norms and less number of
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support points. The capability of the method is demonstrated by integrating it to a large scale design
24
optimization problem of a microstrip patch antenna with a complex topology.
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1
II. INTERPOLATION SCHEME: GENERALIZED STOER-BULIRSCH TECHNIQUE WITH ADAPTIVE SAMPLING
2
Occurrence of nulls/poles in the frequency response of antennas such as return loss, gain, and efficiency
3
motivate the use of rational functions to approximate these quantities. In its general form, a rational
4
function can be described via a fractional polynomial with numerator and denominator of orders N and D ,
5
respectively, as
6
n
a  a1 f    a N f N a0  n1 a n f
S f   0

D
1  b1 f    bD f D
1  d 1 bd f d
N
(1)
7
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The solution of coefficients a0  a N  and b1 bD  of a frequency response function in the above form
9
(1), known as the Cauchy method, requires k  N  D  1 conditions in the form of function values and/or
10
their derivatives.
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Instead of standard known techniques such as direct inversion and total least squares, which suffer from
12
disadvantages as addressed in the introduction, here we utilize an alternative technique; Stoer-Bulirsch
13
algorithm. In what follows we introduce the Stoer-Bulirsch technique and update it to follow a generalized
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Neville path that is in its most general form not diagonal.
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Letting xi , yi  denote the i th support point and  s , x   Ps x  / Qs x  define a rational function with
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s , xi   yi for i  s, s  1, s  2,  , s     where Ps and Qs x  are polynomials of degrees  and  ,
17
respectively. The Neville path is used recursively to generate rational interpolators according to the
18
following formulas
19
 s , x    s,1 1 x  
s
 s   
 s,1 1 x    s , 1 x 
  s,1 1 x    s , 1 x  
1   , 1

 1, 1
  s 1 x    s 1 x  
(2)
6
1

 ,
s
x   
 1,
s 1
x  
s
 s   
 s11, x    s 1, x 
  s11, x    s 1, x  
1   1,

 1, 1
  s 1 x    s 1 x  
(3)
2
3
where  i  x  xi . For details and derivation the reader is referred to [15].
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By following a pre-determined diagonal path as shown in Fig. 1, calculation of successive interpolators,
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s , , required for each newly added support point becomes straightforward as shown by diagonal branches
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of the interpolator construction tree in Fig. 2. As an example, consider the addition of the third support
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point; the algorithm starts by assigning the support point functional value y 3 to  30 , 0 (a constant
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interpolator of order 0,0 passing through the third support point), followed by 20,1 using (2) (order 0,1
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passing through support points 2 and 3), and finally 11,1 using (3) (order 1,1 passing through the support
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points 1, 2, and 3) as demonstrated graphically in Fig. 2. All required s , with negative  and/or  used
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in the calculation are assigned zero value. Newly constructed interpolators at the third level,  30 , 0 and 20,1 ,
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in addition to interpolators at lower levels are available for the calculation of the last interpolator 11,1 as a
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result of following the diagonal Neville path. However, if the interpolators are formed following a non-
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diagonal path, (e.g. dashed line in Fig. 1), the calculation of the final rational function becomes a difficult
15
task since necessary interpolators may not be available. More specifically, based on the recursive scheme as
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suggested by formulas (2) and (3), additional intermediate interpolators, ‘intermediates’1, are needed which
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can be evaluated recursively via the use of old support points. It is noted that these intermediates also
18
depend on how far the path is from the diagonal.
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1
μ,ν
In this context, we refer to intermediates as any interpolator different from the final one 𝛷1 that passes through all available support points
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Fig. 1 Neville path used in constructing a rational function of order   5 and   5 following diagonal
(solid) and non-diagonal (dashed) paths
1
2
Once Stoer-Bulirsch technique attains an M , N order interpolator with k  M  N  1 sample points
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depending on a specified error criterion, the new direction is chosen as either M  1, N  or M , N  1 .
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Consequently, a new sample point is added and the recursive iteration of the newly added point continues.
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Starting with s  M  N  2 ,   0 and   0 the adaptive path is calculated until the criteria of s  1,
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  M and   N 1 , or alternatively s  1,   M  1 and   N are met. It is noted that according to
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recursive algorithm stated by (2) and (3), the calculation of the last interpolation function 1M 1, N x  or
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1M , N 1 x  requires all intermediate interpolation functions such as 1M , N , 2M , N , and ( 2M , N 1 or 2M 1, N ) to
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be known a priori. Regardless of the path chosen, 1M , N and 2M , N are always available but 2M , N 1 and/or
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2M 1, N are in general not available unless a diagonal path is followed. The path norm r , which is the
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maximum allowed search distance from the minimum of  and  , used in determining these intermediates
12
is defined as
13
r  max ,   min , 
14
(4)
8
Φ10,0
Φ10,1
Φ11,1
Φ20,0
oint
o rt p
upp lators
s
4
rp o
inte
th
Φ11,2
Φ20,1
Φ30,0
oint
o rt p rs
p
p
u
o
3 s e rp o l a t
int
rd
Φ21,1
Φ30,1
Φ40,0
Fig. 2 Construction tree for interpolators s , based on diagonal Neville path algorithm; each branch
corresponds to a newly added support point.
1
2
The missing intermediates are evaluated at the recursive iteration level s  M  N  1 by evaluating the
3
terms 2M , N 1 , 3M , N  2 ,  , rM , N r 1 using (3) if 2M , N 1 is part of the adaptive path or evaluating 2M 1, N ,
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3M 2, N ,  , rM r 1, N using (2) if 2M 1, N is on the path. As the search distance r increases the search space
5
for the adaptive path increases but gives rise to an increase in the number of intermediate computations as
6
well. To achieve a tradeoff between these two effects, the value of r was restricted to 4 according to
7
computational experiments based on trial and error. The intermediates calculated are shown graphically in
8
Fig. 3 for the special case of r  2 . Following such an arbitrary trajectory of the Neville path leads to
9
numerical difficulties not encountered when a conventional diagonal path is followed. Along the horizontal
10
axis  at   0 (increase of numerator order) or along the vertical axis  at   0 (increase of
11
denominator order), pure polynomial Neville’s formulas, given in [15], are employed with support points
12
 xi , y i 
13
intermediates with a comparatively high order numerator and low order denominator (e.g. for   8 and
14
  2 ) fittings of pure polynomials presents a challenge for fitting sharp resonances. This leads to
15
degradation of accuracy when high and low intermediates are being subtracted in (2) and (3). Therefore, a
16
computationally optimized path is followed throughout instead of a purely adaptive path for calculating the
17
intermediates. This computationally optimized path is constructed by following a diagonal path until the
18
minimum value of numerator or denominator order is attained and then a horizontal or vertical line
and
xi ,1/ yi 
in order to avoid this type
of instability. Also, during the calculation of the
9
1
movement to the final desired interpolator is used. It is noted that this computationally optimized path is
2
not considered in determining the order of the final rational function and hence is not subject to the path
3
norm constraint.
4
Fig. 3 Representative interpolators and orders necessary for following alternative paths with path norm
r  2 . Paths to be followed are associated with points and corresponding interpolators with the same type
(dashed or solid)
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6
In the following two sections we discuss an adaptive sampling technique using the proposed generalized
Stoer-Bulirsch technique.
7
8
A. Initial Sampling Strategy
9
Starting with good initial points enhances the convergence of the fitting scheme [19]. In this section we
10
introduce the initial sampling algorithm employed within the generalized S-B algorithm. The algorithm
11
divides the frequency domain into sub-regions based on information related to the occurrence of
12
pronounced resonances measured via relative error norms of successive interpolator with the goal of
13
confining each likely resonance in one region. During the initial sampling strategy pure polynomial
14
interpolations are used since they tend to diversify the sampling until convergence based on a chosen error
15
norm,  Pol , is satisfied.
16
17
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The flowchart, shown in Fig. 4, demonstrates this initialization procedure. Here, the normalized error
norm of two successive interpolators  x  and  x  is defined as
10
e1 
  x    x 
x
(5)
max   x   min   x 
1
2
 M 1, N x 
Where  x    1M , N 1 ,  x   1M , N x  , and M and N are the numerator and the denominator order,
x 
1
3
respectively. This formulation is proposed by [23] and proved to give a good mature stopping criterion.
4
Here, the iteration number or the number of support points K is M  N  1 . The normalized error norm e1
5
basically measures how much the new interpolator   differs from the previous one  and is therefore a
6
relative convergence metric of the adaptive interpolation process.
7
8
Also, we select the next support point to be the argument that maximizes the second error norm e2 which
is a function of x (which could correspond to frequency) as follows
9
arg max e2 x 
x
(6)
10
11
where e2 x    x    x  . Hence, the next sampling point is chosen where the error of successive
12
interpolator functions is maximum, possibly where a major resonance is likely to occur.
13
The algorithm starts with pure polynomial interpolation until the normalized error norm e1 drops below a
14
specified threshold (  Pol ) that should guide towards diversity of initial support points. This threshold is a
15
parameter that is tuned in order to sample diversely over the wideband range and captures resonances. It is
16
noted that an increase of the chosen  Pol value has no significant effect on the number of total support
17
points of the resulting interpolators but a decreasing effect on the overall interpolation RMS error at first up
18
until values of 150 and an increasing effect afterwards. Therefore  Pol is specified as 150. A suggested
19
tuning range for  Pol when fitting multi-resonance curves with at least 3 resonances below -2 dB is
11
1
50   Pol  250 .
2
Fit a polynomial of
all support points
Start with f1 as a
support point
Fit a polynomial
(Constant)
Next support
point at f with
max e2
Count=0
e1<εPol
No
Yes
Count=Count+1
Add f2 as the
second support
point
No
Count=3
Yes
Count=0
STEP II
Fig. 4 Step I: Initial diverse sampling algorithm
3
4
B. Adaptive Sampling Strategy
5
The generalized interpolation algorithm that accounts for the non-diagonal Neville path was introduced in
6
the beginning of Section 2. Having established the initial sampling scheme, this section focuses on a novel
7
technique for integrating frequency adaptive sampling to the generalized Stoer-Bulirsch algorithm. It is
8
noted that once a resonance is correctly interpolated, the proposed hybrid method aims to drive the search
9
towards sampling data in regions which are likely to contain a new resonance.
10
The flowchart of the generalized Stoer-Bulirsch technique is shown in Fig. 5. The algorithm starts by
11
constructing interpolators of previously found support points in Step I by following a diagonal path of the
12
Stoer-Bulirsch technique. A diagonal path simply corresponds to a rational function the numerator order of
13
which is equal to the denominator order or one less (first block of step II in Fig. 5). The next sampling
14
point, similar to Step I, is always selected according to (6), i.e. points maximizing the error e2 are selected.
12
Fig. 5 Step II: Adaptive sampling with generalized Neville path
1
2
Regarding the generalized path selection, there are two alternative rational functions to be continued with
3
along the non-diagonal Neville path. More specifically, the normalized error norms e1 given by (5) of two
4
rational functions   of order M  1, N  and M , N  1 passing through K  M  N  2 points are
5
compared against each other. The choice determines the next branch of the generalized path and
6
corresponds to the one that minimizes the normalized error norm e1 between successive interpolators. This
7
minimum error strategy has the effect of speeding up the interpolation convergence and consequently
8
decreasing the required total number of support points towards meeting the criterion e1   Rat . Meeting the
9
condition e1   Rat successively more than once helps preventing local premature convergence. Here, the
10
stopping criterion is reached when the condition e1   Rat is met three times successively. Nevertheless, this
13
1
convergence is still local, i.e. it predicts a region possibly encapsulating a major resonance to minimize the
2
error, causing additional resonances of the overall response to be left out in the final interpolator. Following
3
a maximum error path drives towards sampling at out-of-local regions that potentially possess resonances,
4
Therefore, the proposed sampling strategy relies on a hybrid optimized route that is expected to effectively
5
capture maximum number of resonances by allowing for jumps from regions where a local convergence is
6
reached first by following a path that minimizes the normalized error 𝑒1 to other regions by following a
7
path that maximizes the error e1 . The convergence criterion for the maximum error path is reached when
8
the error norm criterion e1   Rat is satisfied for three successive interpolations. The value of  Rat chosen as
9
4 based on trial and error study for this path. It is noted that within a typical range of possible  Rat values of
10
3-10 a tradeoff exists between the effect of  Rat on maximum accuracy and minimum number of support
11
points. Therefore value of  Rat is specified as the optimal value of 4 in the algorithm.
12
III. RESULTS
13
14
In this section, the proposed generalized S-B is first tested on return loss responses of various
15
representative complex conductor topologies supported by heterogeneous material substrates as discussed
16
in Section A. Then, in section B, a large scale design optimization study that relies on the proposed G-
17
SBAIS technique of a complex antenna structure is presented.
18
A. Comparison of Interpolation Schemes
19
Here, we present and compare results of the proposed interpolation strategy with two of its characteristic
20
features, namely the initial sampling (G-SBAIS) and adaptive sampling strategy (G-SBAS) within
21
generalized S-B algorithm as introduced in Section II and the standard S-B. The error thresholds of all three
22
methods are the same as given in Section II and corresponding flowcharts. Their short descriptions are
23
given below.
14
1
2
3
4
5
1. Stoer-Bulirsch (S-B): Standard Stoer-Bulirsch algorithm following a diagonal Neville path is
implemented.
2. Generalized Stoer-Bulirsch with Adaptive Sampling (G-SBAS): The adaptive sampling, explained in
Section II B, utilizing the generalized Stoer-Bulirsch algorithm is implemented.
6
3. Generalized Stoer-Bulirsch with Adaptive and Initial Sampling (G-SBAIS): The adaptive sampling
7
(Section II B) and the initial sampling (Section II A) are integrated with the generalized Stoer-Bulirsch
8
algorithm.
9
10
Above three strategies were used to construct interpolations to approximate the return loss response of
11
complex microstrip patch antennas with arbitrary material distribution and patch conductor topologies.
12
Comparative interpolation results for the return loss response of five antennas with different conductor and
13
material topologies are shown in Fig. 6.
14
Each interpolation curve is compared to the original return loss response which is numerically simulated
15
using a fine sampling rate of 1 MHz, i.e. 1001 uniformly distributed frequency points are sampled between
16
1-2 GHz. Since finer sampling does not improve the response further, it can be accepted as the original
17
antenna simulation response.
18
To compare the performance of interpolations with respect to their capability in detecting resonances
19
with minimum number of support points for return loss curves belonging to complex antenna designs, the
20
proposed interpolations were compared for a set of return loss curves of five different design candidates
21
with at least 3 resonances, where the standard Stoer-Bulirsch algorithm failed to provide a successful
22
interpolation by missing at least one major resonance below -2dB. Fig. 6 (a-c) depicts interpolation results
23
with the same stopping criterion threshold of  Rat  4 for one of the chosen sample curves according to the
24
three interpolation strategies. According to the results shown in Fig. 6 (a), when a standard Stoer-Bulirsch
25
algorithm is used, only a few major resonance of the response could be captured. Results adopting the
15
proposed G-SBAS depicted in Fig. 6 (b) show that additional major resonances (< -2dB) are successfully
2
captured. Moreover, with the G-SBAIS technique, as shown in Fig. 6 (c), all resonances are successfully
3
captured. A full comparison of the three interpolation strategies for the five antenna design samples is
4
conducted and provided graphically using the root mean square error (RMS) in Fig. 6 (d). Based on the
5
resulting accuracy norms, proposed G-SBAS and G-SBAIS techniques provide a better performance for all
6
of the design cases when compared with the standard S-B technique. In order to evaluate each interpolation
7
strategy’s overall performance, the average sum of error norms of all five designs is plotted in Error!
8
Reference source not found..
9
has dropped by 29.3% and 67.3% when compared with the proposed G-SBAS and G-SBAIS methods,
10
respectively. In order to compare the computational time performances of three interpolation strategies, an
11
additional interpolation study is carried out to determine the average number of support points required to
12
reach the same reference accuracy norm. Specifically, the number of support points to reach the same RMS
13
error of that in G-SBAIS (Fig. 6 d) is determined and tabulated in Error! Reference source not found.. While
14
the results of S-B and G-SBAS are almost identical they point towards a reduction in the number of support
15
points by 13.9% when compared with G-SBAIS proving that the proposed G-SBAIS method outperforms
16
existing S-B technique both in terms of accuracy and number of support points.
Results show that the error in using the naive diagonal Stoer-Bulirsch algorithm
0
0
-5
-5
-10
-10
-15
-20
S11 (dB)
S11 (dB)
1
Original
S-B
Support points
-20
-25
1
-15
Original
G-SBAS
Support points
-25
1.2
1.4
1.6
Frequency (GHz)
1.8
2
(a) Method 1: Standard Stoer-Bulirsch
technique
1
1.2
1.4
1.6
Frequency (GHz)
1.8
2
(b) Method 2: Generalized Stoer-Bulirsch
technique
16
0
2.5
-5
Root mean square (dB)
S11 (dB)
-10
-15
-20
S-B
G-SBAS
G-SBAIS
2
Original
G-SBAIS
Support points
1.5
1
0.5
-25
1
1.2
1.4
1.6
Frequency (GHz)
1.8
2
0
1
2
3
Dessign number
4
5
(d) Method 4: Average Error norm (Root mean
square) of the three interpolation strategies for
five different antenna design configurations
Fig. 6 Interpolation results via various interpolation strategies (a-c) and their corresponding average
error norm
(c) Method 3: Generalized Stoer-Bulirsch
technique with initial sampling
1
TABLE I
Average sum of root mean square error of five
designs obtained using three different interpolation
strategies
Average RMS (dB)
S-B
1.8275
G-SBAS
1.2920
G-SBAIS
0.5974
TABLE II
Average total number of support points of five
designs obtained using three different interpolation
strategies
Average number of
support points
S-B
55.5
G-SBAS
56.3
G-SBAIS
47.8
2
3
B. Large Scale Design Optimization via G-SBAIS
4
The goal of the chosen design problem here is to improve the bandwidth of an initially unmatched
5
miniaturized antenna structure via searching for its novel optimal material and conductor distributions in
6
three dimensions. The problem is chosen to be symmetric around perpendicular axes passing through the
7
center of the antenna. This symmetry is chosen merely to reduce the number of unknowns in the Genetic
8
Algorithm (GA) encoding. The proposed generalized G-SBAIS algorithm is not only applicable to GA but
9
can as well be used to speed-up optimization problems solved using other type of optimization solvers. The
10
design volume corresponds to a substrate with a specified size of 0.318 cm  2.5 cm  2.5 cm and a
17
1
conductor surface printed on the top of the substrate. The irregularly shaped conductor is probe-fed at
2
1.875 cm 1.25 cm from the lower left corner of the substrate. The substrate is divided into 3 layers with
3
1.06 mm thickness. Each substrate layer of the antenna is discretized into 20  20  400 design cells with
4
800 finite triangular prism elements used in the Finite Element Analysis simulations. Each cell’s relative
5
permittivity and permeability can range from 1 to 30 allowing for novel material distributions and is
6
represented by a design variable in the design space. Also, the radiating conductor’s topology can vary via
7
the on or off (conductor or no conductor) nature of design cells allowing for non-intuitive radiating
8
conductor shapes. Consequently, another 400 cells are added to determine the optimal topology of the
9
conductor. Therefore, total number of design variables N in the design example can be evaluated via
10
400 / 4  number of layers  400 / 4  number of layers  400 / 4 turning the design problem into a large scale
11
one that could be solved using a micro-GA Algorithm typically running for 100 generations with an
12
average population of 40 individuals. The final optimum result that the optimizer has converged to is a
13
random mixture consisting of certain number of material shades that were specified in the design process
14
for the design variables to be selected from. The number of material shades specified here is 30
15
corresponding to the integer values of 1-30 for permittivity and permeability values. Similar antenna design
16
studies with complex topologies were conducted in literature using gradient based techniques [24, 25] and
17
possible fabrication strategies for complex antenna substrates were also proposed [26] demonstrating the
18
potential impact and realization possibilities of the complex antenna designs presented in this paper.
19
Computational time requirement for a single layer geometry using a full wave finite element analysis such
20
as the Fast Spectral Domain Analysis (FSDA) algorithm [27] and standard linear interpolation through 101
21
uniformly distributed frequency points within 1-2 GHz corresponds to approximately 1 week using a
22
micro-GA with a population size of 40 individuals evolving within 100 generations. A GA based
23
optimization tool is used here in order to avoid local optima. More specifically, the total computational
24
design time required can be calculated using below formula:
18
1
Max # of generation s  population size  # support points  computatio nal time
 100  40  101 3  12000 s/layer  1 week/laye r
2
Here, the computational time of the design process becomes prohibitively long as the scale of the
3
problem increases with added number of layers and the increase of number of support points.
4
The antenna model created in FSDA constitutes of 800 triangular prism elements per layer. It uses finite
5
element-boundary integral (FE-BI) technique for analyzing three-dimensionally inhomogeneous doubly
6
periodic structures with in-layer periodicities. The boundary integral employs the FSDA for efficient
7
analysis. The user supplies the triangular surface mesh modeling the different layer surfaces of the array
8
unit cell. The program then grows the volume mesh along the surface normals. The building block of the
9
resulting volume mesh is the triangular prism. Periodic boundary conditions are imposed on the vertical
10
walls of the patch antenna and hence an infinite array of the analyzed antenna is simulated. The solver used
11
here is BiConjugate Gradient. The default relative accuracy of 0.01 of the solver is used as a termination
12
criterion.
13
The fitness function corresponds to the return loss response and is simulated using the proposed G-
14
SBAIS interpolation technique integrated to the full wave FSDA solver. The design study converged to the
15
return loss curve plotted in Fig. 7 with a bandwidth performance of 11.4%. The interpolated return loss
16
response using the proposed G-SBAIS technique is compared to the original response sampled via 101
17
support frequency points proving an almost exact match with correctly predicted bandwidth performance.
18
However, the return loss response interpolated using the standard S-B interpolation technique is not able to
19
capture the original response as depicted in Fig. 7.
20
19
0
S11 (dB)
-5
Original
G-SBAIS
G-SBAIS
support points
S-B
S-B
support points
-10
-15
1
1.2
1.4
1.6
Frequency (GHz)
1.8
2
Fig. 7 Return loss curve of the optimum design interpolated using G-SBAIS and S-B
1
In addition to the miscalculated response of the final design’s return loss curve in Fig. 7 due to missing
2
resonances of fittest individuals, it is noted that the standard S-B was not able to deliver an improvement
3
within 100 generations because of erroneous bandwidth calculations during the design iterations. The
4
design study using G-SBAIS and the micro-GA converges to the optimum design with an overall
5
improvement from 3.4% to 11.4% in 100 generations in about 10 days. With standard linear interpolation
6
this example would not be practically feasible. The optimal conductor topology and three-dimensional
7
material distributions in terms of permittivity and permeability distributions within each layer are given in
8
Fig. 8 (a), (b), and (c) respectively representing a non-intuitive complex topology. The average number of
9
support points used by the G-SBAIS technique during the optimization process corresponds to 50 support
10
points per individual. As expected, complexity of the design problem and the range of the design variables
11
affect the behavior of the return loss curves in terms of number and depths of resonances within the design
12
search and consequently influences the number of support points required for the entire process [22].
13
20
(b)
(a)
(c)
Fig. 8 Distribution of (a) conductor (white), (b) Permittivity and (c) permeability of lowermost to
uppermost layers
1
IV. DISCUSSION AND CONCLUSION
2
In this paper a generalized Stoer-Bulirsch approach, namely G-SBAIS, for adaptively interpolating
3
complex multi-resonant antenna response curves with reduced number of support points and improved
4
accuracy was presented. The method relies on generalizing the standard Stoer-Bulirsch technique with a
5
non-diagonal Neville path and makes use of this relaxation in shaping the best path that minimizes error
6
and number of data samples needed. A hybrid technique following a path that minimizes the square error
7
norm, e1 given by (5), allowing for reduced number of support points followed by a path maximizing e1
8
that enhances accuracy ensures a compromise between these conflicting merits and outperforms
9
conventional adaptive sampling techniques. The proposed method has been implemented on return loss
10
responses of various complex conductor shaped antennas with textured material substrates and resulted in
11
an overall accuracy increase of 67.3% when compared with the naive Stoer-Bulirsch interpolation
21
1
technique. A design optimization example to maximize bandwidth of a complex antenna with
2
inhomogeneous material and conductor topologies was successfully integrated with the proposed G-SBAIS
3
technique and resulted in a bandwidth improvement from 3.4% to 11.4% with an average number of 50
4
support points. The 3-fold bandwidth improvement achieved using the proposed G-SBAIS algorithm in this
5
large scale design optimization example of a complex antenna motivates its use for novel antenna structures
6
made of non-intuitive conductor and material compositions. By allowing for remarkable antenna
7
performance improvements not possible via intuitive designs, this capability is expected to make a
8
significant impact in many critical antenna applications.
9
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