A Recursive Acceleration Technique for Static Potential Green's Functions of a Rectangular Cavity Combining Image and Modal Series
ABSTRACT A hybrid acceleration algorithm for the computation of the static potential Green's functions of a rectangular cavity is proposed. Similarly to Ewald's method, it combines the series expansions in terms of images and modes. The main particularity with respect to Ewald resides in the fact that it does not need the evaluation of a nonalgebraic function such as the complementary error function (erfc) while maintaining the rapid convergence of the Ewald technique. Finally, the method requires the computation of eight terms (original source plus seven images) and of several modal series corresponding to bigger cavities, which can be efficiently performed.
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ABSTRACT: A compact representation is given of the electric- and magnetic-type dyadic Green's functions for plane-stratified, multilayered, uniaxial media based on the transmission-line network analog along the aids normal to the stratification. Furthermore, mixed-potential integral equations are derived within the framework of this transmission-line formalism for arbitrarily shaped, conducting or penetrable objects embedded in the multilayered medium. The development emphasizes laterally unbounded environments, but an extension to the case of a medium enclosed by a rectangular shield is also includedIEEE Transactions on Antennas and Propagation 04/1997; · 2.33 Impact Factor
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ABSTRACT: This paper describes the application of the boundary integral-resonant-mode expansion (BI-RME) method to the modeling of rectangular waveguides with metal insets. It extends to more complicated radially symmetric insets, a method recently introduced by the authors, for the simple case of a cylindric post. In this extension, a self-consistent new theory is presented, which fully exploits the peculiarities of the considered class of structures, thus straightforwardly leading to the system equations. The efficiency of the BI-RME method, already demonstrated in the wide-band modeling of arbitrarily shaped waveguide components, is further enhanced in the particular application considered in this paper because the currents on the waveguide walls are not involved in the calculation. For this reason, the method-of-moments discretization of the field equations leads to a mathematical model of order much smaller than in the general BI-RME approach and in other boundary integral methods. Due to the state-space formulation of this model, a wide-band representation of the generalized admittance matrix of the structure is easily found in the form of a pole expansion in the S-domain by the calculation of a reasonably small number of eigensolutions of a matrix eigenvalue problem. The method is very fast and reliable, and permits the realization of a very efficient software, well suited for inclusion in computer-aided design tools for microwave circuit design. Some examples show the efficiency of the method, including the application to multiple and slanting insets and to the modeling of an evanescent-mode filter.IEEE Transactions on Microwave Theory and Techniques 05/2005; · 2.23 Impact Factor
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ABSTRACT: In this work, the groundwork is laid out for the realization of efficient integral-equation/moment-method techniques, with arbitrary types of basis functions, for the computer-aided design (CAD) of geometrically complex packaged microwave and millimeter-wave integrated circuits (MMIC's). The proposed methodology is based on an accelerated evaluation of the Green's functions in a shielded rectangular cavity. Since the acceleration procedure is introduced at the Green's function level, it becomes possible to construct efficient shielded moment method techniques with arbitrary types of basis-functions. As an example, a Method of Moments (MoM) is implemented based on the mixed potential integral equation formulation with a rectangular, but nonuniform and nonfixed, mesh. The entire procedure can be extended to multilayer substratesIEEE Transactions on Microwave Theory and Techniques 01/1997; · 2.23 Impact Factor
542 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 3, MARCH 2011
A Recursive Acceleration Technique for Static
Potential Green’s Functions of a Rectangular
Cavity Combining Image and Modal Series
José M. Tamayo, Sergio López-Peña, Michael Mattes, Alexander Heldring,
Juan M. Rius, and Juan R. Mosig, Fellow, IEEE
Abstract—A hybrid acceleration algorithm for the computation
of the static potential Green’s functions of a rectangular cavity is
proposed. Similarly to Ewald’s method, it combines the series ex-
pansions in terms of images and modes. The main particularity
with respect to Ewald resides in the fact that it does not need the
evaluation of a nonalgebraic function such as the complementary
error function (erfc) while maintaining the rapid convergence of
the Ewald technique. Finally, the method requires the computa-
tion of eight terms (original source plus seven images) and of sev-
eral modal series corresponding to bigger cavities, which can be
Numerical results are provided to verify the feasibility of the al-
gorithm, which appears as a promising alternative to the existing
methods in the literature.
Index Terms—Cavities, Green’s functions (GFs), method of mo-
ments (MoM), numerical methods, series acceleration.
and closed ones –. In the framework of shielded structures
[monolithic microwave integrated circuit (MMIC)] and hollow
waveguide applications like filters and multiplexers, a key-ele-
ment in the electromagnetic model is the Green’s function (GF)
on which is based the integral equation. Its efficient and accu-
rate evaluation is therefore mandatory for a fast simulation al-
gorithm, especially if it is used inside a full-wave synthesis tool
with optimizing capabilities.
In the case of rectangular cavities, the GF can be expressed
as a triple infinite series of cavity modes or images. Both series
HE integral-equation technique is a popular approach to
model and to simulate microwave problems, both open
Manuscript received September 01, 2010; revised November 26, 2010; ac-
cepted December 05, 2010. Date of publication January 20, 2011; date of cur-
rent version March 16, 2011. This work was supported by the Spanish Inter-
ministerial Commission on Science and Technology (CICYT) under Project
TEC2010-20841-C04-02, Project TEC2007-66698-C04-01, and Project CON-
the Programa de Formación del Profesorado Universitario (FPU) Fellowship
J. M. Tamayo, A. Heldring, and J. M. Rius are with the Antenna Laboratory,
Department of Signal Processing and Telecommunications, Universitat Politec-
nica de Catalunya, 08034 Barcelona, Spain (e-mail: jose.maria.tamayo@tsc.
upc.edu; firstname.lastname@example.org; email@example.com).
S. López-Peña, M. Mattes, and J. R. Mosig are with the Laboratoire
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (e-mail:
firstname.lastname@example.org; email@example.com; firstname.lastname@example.org).
Digital Object Identifier 10.1109/TMTT.2010.2103088
(LEMA), Ecole Polytechnique
have their advantages and disadvantages. While the modal se-
ries satisfies the boundary conditions and converges fast for far
interactions, it is unable to catch the singular behavior of the GF
when source and observer are close to each other. On the other
hand, the image series works best in the latter case, but con-
verges slowly if the source and the observer are far away. Fur-
thermore, it satisfies the boundary conditions only in the limit.
be accelerated. For this, various techniques can be found in the
literature – that are often general approaches not taking
into account the physics behind the problem.
Another approach is Ewald’s technique –, which is
one of the most popular ones in the framework of Cartesian co-
ordinates due to its high efficiency and precision. It is based on
a hybridization of both modal and images series developments
combining the “best” of them. Unfortunately, the terms in the
images part inside Ewald’s method requires the evaluation of
the complementary error function (erfc), which is computation-
quency dependent problems. To mitigate this problem, the GF
could be split into a static and dynamic part using Kummer’s
transform ,  or considering only its real part . But
still, the static or real part contains the evaluation of the erfc
with real-valued arguments. In this context of hybrid accelera-
tion techniques, we propose another combination of modal and
image series, whose terms only need the evaluation of algebraic
for our method to work is that the free-space GF of the problem
tends to zero when the distance tends to infinity. Different to
Ewald’s method, the image part boils down to a single evalua-
tion of a finite sum containing only algebraic expressions. The
posed in  and  to reduce the 3-D to a 2-D modal series.
the best of the aforementioned series, i.e., images and modes.
Section II briefly states the problem considered. The method
is explained in Section III using the example of a 1-D GF. The
extension to three dimensions (3-D) is given in the Appendix.
Section IV provides a physical interpretation of the method,
this time in two dimensions (2-D). Both Sections III and IV
are meant to be self-consistent and can be read independently.
Section V shows some actual implementation issues in order
0018-9480/$26.00 © 2011 IEEE
TAMAYO et al.: RECURSIVE ACCELERATION TECHNIQUE FOR STATIC POTENTIAL GFs543
BIS PARAMETERS ASSOCIATED WITH ?
POTENTIAL GREEN’S FUNCTIONS
Section VI includes a brief remainder of an accelerated modal
series representation as they are used inside our technique. Fi-
nally, some numerical results are performed to show and eval-
uate the validity of the method, together with some guidelines
to select the parameters of the method.
II. STATEMENT OF THE PROBLEM
Let us consider the problem of finding an efficient and highly
andinside a PEC rectangular cavity
. For the sake of simplicity, the cavity
is placed with one vertex in the origin and all the others in the
positive part of the axes.
Any potential GF inside a rectangular cavity can be obtained
by a straightforward application of the images method. Let us
start by considering the so-called “Basic Image Set” (BIS) that
by the three cavity walls occupying the three coordinate planes.
The positions and signs of these images are detailed in Table I
for every type of potential Green function. It can be now easily
demonstrated  that a rectangular cavity potential GF can be
always written as a triply infinite series, representing the peri-
odic translation of the original BIS along the three coordinates
and with space periods equal to twice the cavity dimensions.
Thus, for a generic GF evaluated at a specific frequency defined
by the wavenumber , we write
with the BIS GF given by
where the coefficients
the images in the BIS of the corresponding potential GF, as de-
fined in Table I.
corresponds to the different distances
between the fixed source point
images of the observation point
stand for the signs of
and the cavity wall’s
where the distance terms
image in the BIS, as shown in Table I.
applications related to microwave filters. Usually, these devices
, , and are defined for every
are simulated in the frequency domain. Their simulation thus
involves a frequency sweep. A common way to accelerate the
GF computation is to split off the static part. The singular-at-
the-source behavior of cavity GFs is fully concentrated in the
static part and adequately represented by an images expansion.
Therefore, we will develop our strategy for the static part only.
The remainder in the GF, once thestatic part has been extracted,
is a dynamic term that can be better evaluated by modal expan-
sion. For the static part, the image expansion (1) remains valid.
The corresponding BIS GF is simply obtained by setting
in (2) with the result
III. SUBDIVISION OF 1-D PERIODIC SERIES INTO 1-D
PERIODIC SERIES WITH LARGER PERIODS
For the sake of simplicity, we will first describe the proposed
acceleration technique in the 1-D case. The generalization to
3-D is given in the Appendix. The 1-D case corresponds phys-
ically to a parallel-plate situation and the BIS is just a couple
of sources: the original one and its image respect to the plane
. The static term of this BIS GF, called here
plicity, is given by
of the parallel plate BIS GF
considering the two basic termsand as fol-
The first series
but with period
kind of subdivision, keeping the other series as it is. We can
proceed in this manner, always subdividing the first series into
iterations, the following expression:
is now equivalent to the original one,
instead ofso we can apply again the same
544 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 3, MARCH 2011
where the functions involved are defined as
Equation (10) holds for every natural number
we can take the limit of the aforementioned equation when
tends to infinity
Interchanging limit and summation in the first term of the last
expression, the only term remaining is the one with
as all the others would represent the evaluation of the function
at . But this function tends to zero for large numbers.
finally yielding the following decomposition:
Summarizing, we have subdivided the original periodic se-
ries, into one single function evaluation
how to take benefit of this decomposition.
Without entering into the details of the development for the
3-D series (see the Appendix), it is worthwhile to mention that
we would finally arrive to a subdivision of the original series in
(1) as follows:
and a sum of
. Section V will show
where now the single function evaluation is transformed into
the evaluation of the eight terms in the BIS and for each level
we have seven series with periods
of one single series. To simplify the notation,
IV. PHYSICAL INTERPRETATION (BIGGER CAVITIES)
For the sake of simplicity, we are going to consider in this
section a 2-D case of the algorithm. The inclusion of the third
dimension is immediate. The complete set of images of a cavity
problem can be subdivided into four disjoint groups of images,
as shown in Fig. 1, where the different sets are represented with
different gray tones.
The key point resides in the fact that each subset (points of
same gray tones) can, in turn, be interpreted as the images of
a point inside a bigger cavity of double size (light gray cavity
in Fig. 1) containing the original smaller cavity, nested inside
Fig. 1. Images distribution between the different problems. The whole set of
circles represent the images of the black source point inside the small cavity in
the center (1?1 squares filled with dark gray). The different gray tones of the
images stand for the set of images produced by each of the images in the new
bigger cavity (2?2 squares filled with soft gray).
Fig. 2. Subdivision of the zeroth cavity problem at level ?, ?
of four new double sized cavity problems at level ? ? ?. Although it is a 2-D
representation for a better understandability. The 3-D cavity is subdivided into
eight cavity problems instead of four. The signs depend on the potential we are
into the sum
like a matryoshka doll (dark gray in Fig. 1). A schematic rep-
resentation of this decomposition is shown in Fig. 2, where the
original problem (small cavity) is equivalent to the sum of con-
tributions coming from the bigger cavities. Note that the prob-
, , and
puted via a 2-D modal series (see Section VI). The remaining
is again a near interaction, but now inside
a bigger cavity. This problem can be subdivided again in the
same manner, whereas the others are directly computed using
the modal series representation.
This process can be repeated infinitely until we have for each
level three cavities the sizes of which are
cavity size, respectively, and a remaining problem of infinite di-
mension. This consists of an infinite space containing the orig-
inal source plus the BIS. Hence, the solution of the last problem
will be the BIS of the original problem.
Finally, the problem is solved using a BIS plus the contribu-
tion of an infinite set of problems with source and observation
points progressively further away with increasing cavity size. It
in the same figure can be con-
times the original
TAMAYO et al.: RECURSIVE ACCELERATION TECHNIQUE FOR STATIC POTENTIAL GFs545
means that the contribution of these problems to the final sum
decreases very quickly when we go to bigger cavities and only
the series up to a certain level
Regarding the connection with the last section, it can be
easily proved that each
in the periodic series subdivi-
sion is the solution to the cavity problem with dimensions
, with the source point at
, as defined in Table III, where
are the original dimensions of the cavity and
are the original source and field points. Note that the
are not exactly the
as in (32) because we need the image corresponding to the new
actually need to be com-
and the field
V. ALGORITHM IMPLEMENTATION
Our aim is to efficiently compute the function
in (1) given a cavity with dimensions
and inside the cavity, so-called source and
field points, respectively. Without loss of generality, let us as-
sume that the replicated point in the series images is the one
closest to the walls and that it is included in the octant closest to
the origin. The other cases are outlined at the end of this section
explaining how to transform them to the basic situation. We as-
sume also that the two points are not very far away from each
other. This is the case where this method will work efficiently.
For large distances, it is better to go directly to a 2-D reduced
modal series (see Section VI).
Following the aforementioned procedure, we firstly compute
alent to the BIS
and two points
a 2-D reduced modal series truncated after
indeed compute them using modes because they are still peri-
odic series or because they belong actually to other cavity prob-
lems. Note that the new problems have a larger relative distance
between source and observation point with respect to the cavity
size. It means that when grows, the value of
nentially, implying that a small number of levels is sufficient to
achieve a desired relative error. Furthermore, at each level , we
need a decreasing number of modes (see Section VII).
The following algorithm description summarizes what has
modes. We can
1:Rearrange source and field points
Fig. 3. Equivalentproblems,showingthe kindoftransformationswecan make
without affecting the GF result.
5: (with modes)
So finally we will need to compute eight square roots (term
of modes in each one of the series at level .
When the relation between the distance from source point
to observation point and the cavity size is larger than a certain
value, still to be determined (see Section VIII), we compute di-
rectly the original problem using a 2-D reduced modal series.
In order to achieve the aforementioned configuration, there
are different transformations we can perform without affecting
the result: source and field points can be interchanged; we can
perform symmetries from any coordinate plane passing through
the center of the cavity. Playing with these two properties, it is
straightforward to set the samples in the proper way. It is nec-
essary because we need to be sure that the BIS we are keeping
(remember our BIS is always done from the origin) is the one
where all the replicas are closest to the walls.
To clarify this preprocessing, let us consider an example.
Fig. 3 starts with an original source and observation points
configuration (left of Fig. 3). Different transformations are
applied in order to reach the final configuration (right of Fig. 3)
to which the algorithm is applied. In the first step, source and
observation points are interchanged because we need the field
point closer to the walls. Secondly, symmetry with respect
to the vertical line crossing the center is performed. Finally,
symmetry with respect to the horizontal line crossing the center
is used, yielding the desired configuration.
Note these kind of transformations need to be applied only
when source and observation point are close to each other.
VI. REDUCED MODAL SERIES
Within the framework of this method, it is evident that an
efficient evaluation of the modal series is required since the
computation of seven series of this type is demanded in each
level. Therefore, the classical expansion based on a 3-D sum of
modes  is dismissed because of its slow quadratic conver-
gence rate. Nevertheless, the convergence of the modal expan-
sion can be enhanced by summing the series in one dimension
, , which can be arbitrarily chosen between
. Consequently, a reduced 2-D series exhibiting a faster expo-
nential convergence rate associated to this reduced dimension
is achieved. Different strategies can be employed to obtain this
2-D series. For instance, in , the mathematical relations in
, , and
546 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 3, MARCH 2011
 are utilized in the 3-D sum, whereas in , the reduced
series is straightforwardly calculated by applying image theory
and the Poisson summation formula to a short-circuited infinite
parallel plate waveguide GF. As evinced in  and , the
2-D series is not devoid of drawbacks inasmuch as an arbitrary
reduction does not always guarantee a fast exponential conver-
gence rate for any
. An easy way to avoid this problem
consists of choosing the coordinate to apply the reduction in ac-
cordance with a convergence criterion based on
instead of selecting it arbitrarily.
A possible algorithm, which is based on this concept and en-
sures the best exponential convergence rate in each case, is as
follows: we first introduce the variable
As will be shown at the end of this section, the best choice of
the coordinate to reduce,
The possible 2-D series expansions of any GF involved in this
study can be derived from the set of functions (21)–(23)
which is only involved in the evaluation of
Dirichlet boundary conditions in all directions
since it fulfills
whose reduced dimension satisfies Neumann boundary condi-
and satisfy Dirichlet boundary conditions. It
is required in the computation of the reduced component
that fulfills Neumann boundary conditionswith respect to
Dirichlet boundary conditions with regard to
. This function is associated to the calculation of
the components transverse to
and the reduced
Fig. 4. Convergence of the three reduced versions of ?
for the algorithm described in Section VI.
. Permutation stands
where the new parameters
To compute the actual potential GFs
, we proceed as follows:
are the two coordinate variables
is the remaining coordinate different from
It can be easily inferred from an expansion in terms of expo-
nentials of (24) that the bigger
nential convergence rate is. Therefore, it justifies the choice of
the reduced variable
As a matter of completeness, Fig. 4 shows as an example the
for a case where the dominant term in (19)
. As expected, the
poor convergence, while the permutation algorithm follows au-
tomatically the best possible choice, which is here the -reduc-
is, the faster the expo-
andreduced versions exhibit
VII. NUMERICAL RESULTS
To show the feasibility of the method, we have chosen a rect-
angular cavity of dimensions 1 m
vation and source points at
m, respectively. This is a case where the points
are close to each other with respect to the cavity size (1%). As
they are placed in the center of the cavity, it can be considered
theworstcase for thehybrid method we havepresentedhere be-
cause source and observation points in the new bigger cavities
are placed the closest they can.
To have a good reference, we have pushed Ewald’s and our
method to the limits until having an agreement up to a relative
error of 10
. The considered reference solution equals
1 m1 m and obser-
TAMAYO et al.: RECURSIVE ACCELERATION TECHNIQUE FOR STATIC POTENTIAL GFs547
Fig. 5. Comparison of the evolution of the relative error with the total number
of computed terms in the series for the different methods. The number of terms
in our method is the summation of the total number of modes per level plus the
eight final images.
Fig. 6. Comparison of the evolution of the relative error with the CPU time for
the different methods.
As both Ewald’s and our method have a steep convergence of
the series elements, they can reach the machine precision (see
Figs. 5 and 6). However, as the terms in the modal series reduce
slowly, it is more affected by roundoff errors and only a relative
Figs. 5 and 6 show the comparison between our approach and
been obtained with the commonly used value for the parameter
considered somehow a reasonable choice in general. On the
contrary, in the hybrid method presented here, as there is not
a systematic procedure to obtain the number of modes at each
level yet (see Section VIII), we have optimized these numbers
for this particular case in order to have an idea of the power of
the method. Hence, the comparison is not completely fair, but it
shows the theoretical behavior. Moreover, as will be shown in
NUMBER OF MODES PER LEVEL USED IN OUR METHOD TO
OBTAIN A RELATIVE ERROR OF 10
CASE EXPLAINED IN SECTION VII
Section VIII, the impact of not choosing the optimum param-
eters for our method is almost negligible. This is different in
Ewald’s method since the balance between images and modal
series is not maintained.
cedure. We start with a number of levels and a number of modes
per level large enough to stabilize up to machine precision and
this is our reference value for the GF. We then sequentially re-
duce the number of levels, keeping the number of modes per
level, until the obtained relative error is larger than a fixed de-
the simulation before the last, which still had a relative error
smaller than the prefixed value. Afterwards, we proceed in the
same manner reducing the number of modes per level, starting
from the upper level and going down, until the number of nec-
essary modes for the first level has been set.
The presented results have been obtained utilizing MATLAB
R2009b on a PC with an Intel Core 2 Duo CPU at 3.16 GHz,
with Windows XP Professional x64 Edition. Under these con-
ditions, machine precision can be reached with Ewald’s method
and our approach in about 1 ms (see Fig. 6).
For a maximum runtime performance of the Ewald method,
the erfc has been computed by a piecewise rational approxima-
tion  to avoid its expensive evaluation. As Fig. 5 shows, our
specific implementation details need to be addressed like care-
fully choosing how to implement the erfc for an efficient evalu-
Table II shows the optimal number of modes per level used
in our method to have a relative error of 10
how the number of modes is decreasing with increasing level
and it is zero beyond a certain maximum level.
The results of Figs. 5 and 6 have been obtained for the dis-
tance between source and observer mentioned at the beginning
of this section. If it is further increased, the curves associated
with Ewald’s and our method move further right in Figs. 5 and
6, meaning a larger number of summation terms are necessary,
whereas the modal series curve moves left, becoming more ef-
ficient. This means beyond a certain distance between source
and observer the 2-D modal series will be more efficient than
Ewald’s or our method. When the distance is maintained, but
the two points are closer to the walls or a corner, the curve of
. The table shows
548 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 3, MARCH 2011
Fig. 7. CPU timein termsof the relativedistance with respect to the cavity size
with a relative error fixed to 10
our method moves to the left, improving the efficiency. This be-
havior confirms that the worst case for our method occurs when
source and observation point are located in the cavity center.
part is completely algebraic
method, which needs the evaluation of the erfc. If we consider
the further goal of computing the method of moments (MoM)
matrix elements in the framework of integral equations, both
Ewald’s and our method have the advantage that the modal part
ever, unlike Ewald, the image part in our method is of simpler
form since algebraic and chances to obtain analytic expressions
for certain MoM integrals are higher, e.g., see –.
in opposition to Ewald’s
VIII. DETERMINATION OF METHOD PARAMETERS
The first parameter to be considered is the limiting relative
distance between source and observation points, defined as
at which it is better to switch to the computation of 2-D reduced
modal series. For this, the following experiment has been per-
servation point in the center of the cavity
and the source point moving along the -direction. Fig. 7 shows
the necessary CPU time of the different methods in terms of the
distance between the points, relative to the cavity size. We can
conclude that it is worth to use Ewald or the presented method
when the relative distance is smaller than 10%, whereas a 2-D
reduced modal series should be used for larger distances.
Secondly, we need a systematic approach to determine the
number of levels and the number of modes per level that we
need to use without having to optimize for every case. To ob-
tain a desired relative error, the number of levels and modes
grow slightly with the distance between the points. Therefore,
we could consider to precompute the optimal parameters for the
limiting case for a 10% relative distance and utilize these pa-
rameters for every combination of points. The additional cost is
Fig. 8. Comparison of the evolution of the relative error with the CPU time
negligible, as we only need to perform this once per cavity and
then it will be tabulated for each desired relative error.
Fig. 8 shows the comparison for the case studied in
Section VII (1% distance) between utilizing the optimum
parameters for this particular case or using the precomputed
parameters for the 10% distance case. They behave practically
Overall, with the suboptimal, though efficient, determination
of parameters presented here, we can obtain a behavior very
close to the optimal case. The relative error can be set to any
maximum value, which is a very interesting property for the ap-
for whatever ratio “distance/cavity size” our method is able to
obtain machine precision.
As a last general remark, special treatment is necessary if
the original rectangular cavity has a distorted shape, i.e., with
different , , and
Not only the number of modes must be set differently for each
iteration or level , but also the number of modes for each one of
the seven modal summations inside a particular iteration must
differ. The main reason is that the distance between the source
and observation points in any of the large problems present at
iteration depends onthe actual dimensions of thecavity. To set
an example, if the cavity
in Fig. 2 had a vertical dimension
shorter than the horizontal one, the samples in
be closer than in
. Consequently, the number of utilized
modes should be larger for the
In this paper, a novel technique to compute efficiently and
accurately the static part of the potential GFs inside a rectan-
gular cavity has been presented. Similar to the Ewald accelera-
tion method, it hybridizes images and modal series representa-
tions, extracting the best of them. The main particularity resides
in the fact that it can be physically interpreted as the decompo-
sitionofthe originalcavityprobleminto biggercavityproblems
with better convergence rates. Furthermore, the images part in
our method is completely algebraic and does not need the eval-
uation of another function such as the erfc in Ewald’s method.
TAMAYO et al.: RECURSIVE ACCELERATION TECHNIQUE FOR STATIC POTENTIAL GFs549
A complete theoretical analysis has been introduced, based
upon the decomposition of periodic series into a sum of series
with larger periods. The numerical results presented here high-
light the proposed method as a good alternative for the acceler-
ation of cavity GFs.
The algorithm’s efficiency depends on several parameters,
mainly the number of levels and the number of modes per level
and the limiting relative distance that decides whether the GF
is better directly computed using a 2-D reduced modal series or
the proposed method. A straightforward technique for the cal-
culation of an appropriate value of these parameters has been
SUBDIVISION OF 3-D PERIODIC SERIES INTO 3-D
PERIODIC SERIES WITH LARGER PERIODS
The 3-D case can be treated analogously to the 1-D series in
Starting from (1), it is possible to subdivide it in eight new
series with double periods in each dimension as follows:
The first series
double periods. Consequently, it can be subdivided in the same
manner keeping the rest of the terms untouched. The procedure
can be iterated with the following recursion:
is equivalent to the original one, but with
, AND ?
, AND ?
INITIAL POINTS (?
IN (32). ?
) IN THE PERIODIC REPETITION SERIES
ARE THE NEW OBSERVATION POINTS
INSIDE THE BIGGER CAVITY PROBLEMS, EQUIVALENTS
TO THE PERIODIC SERIES SUBDIVISION
iterations of this process, we get
can be extracted from Table III.
The last expression holds for any
as in the 1-D case, of applying the limit when the number of
tends to infinity
, giving us the possibility,
, it follows from (30) that
,oris different from zero in the first series
and therefore, the only term that remains in the first series in
which represents the desired decomposition.
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José M. Tamayo was born in Barcelona, Spain,
on October 23, 1982. He received the Mathematics
degree and Telecommunications Engineering degree
Barcelona, Spain, both in 2006, and is currently
working toward the Ph.D. degree in telecommunica-
tions from UPC.
In 2004, he joined the Telecommunications
Department, Universitat Politècnica de Catalunya
ated numerical methods for solving electromagnetic
Sergio López-Peña was born in Barcelona, Spain,
in 1976. He received the Ingeniero de Telecomuni-
cación degree from the Universitat Politècnica de
Catalunya (UPC), Barcelona, Spain, in 2003, and the
Ph.D. degree from the Ecole Polytecnique Fédérale
de Lausanne (EPFL), Lausanne, Switzerland, in
Since February 2005, he has been with the Lab-
oratory of Electromagnetics and Acoustics (LEMA),
Assistant. He has been involved in several European
Space Agency (ESA) projects and collaborations with other European institu-
tions. His research interests include electromagnetic (EM) theory, numerical
methods, cavity backed antennas, and microwave passive devices.
Michael Mattes received the Diplom-Ingenieur
degree from the University of Ulm, Ulm, Germany,
in 1996, and the Ph.D. degree from the Ecole Poly-
technique Fédérale de Lausanne (EPFL), Lausanne,
Switzerland, in 2003.
Following one year as a Research Fellow with the
Department of Microwave Techniques, University of
Ulm, in September 1997, he joined the Laboratory
of Electromagnetism and Acoustics (LEMA), Ecole
Polytechnique Fédérale de Lausanne (EPFL). He
was responsible for the development and implemen-
tation of the Full-wave Electromagnetic Simulation Tool (FEST), version 3.0,
within the framework of the European Space Agency (ESA) project Integrated
computer-aided design (CAD) tool for waveguide components (ESA/European
Space Research and Technology Centre (ESTEC) 12 465/97/NL/NB). He is
currently involved in several European Commission and ESA projects. His
research interest include electromagnetic theory, the coupling of electromag-
netism with other physical problems, numerical techniques, and microwave
TAMAYO et al.: RECURSIVE ACCELERATION TECHNIQUE FOR STATIC POTENTIAL GFs551
Alexander Heldring was born in Amsterdam, The
Netherlands, on December 12, 1966. He received
the M.S. degree in applied physics and Ph.D. degree
in electrical engineering from the Delft University
of Technology, Delft, The Netherlands, in 1993 and
He is currently an Assistant Professor with
the Telecommunications Department, Universitat
Politècnica de Catalunya, Barcelona, Spain. His
research interests include integral-equation methods
for electromagnetic problems and wire antenna
Juan M. Rius received the Ingeniero de Telecomu-
nicación degree and Doctor Ingeniero degree from
the Universitat Politècnica de Catalunya (UPC),
Barcelona, Spain, in 1987 and 1991, respectively.
In 1985, he joined the Electromagnetic and
Photonic Engineering Group, Department of Signal
Theory and Communications (TSC), UPC, where
he is currently a Catedrático (equivalent to a Full
Professor). From 1985 to 1988, he developed a
new inverse scattering algorithm for microwave
tomography in cylindrical geometry systems. Since
1989, he has been engaged in research for new and efficient methods for
numerical computation of electromagnetic scattering and radiation. He is the
developer of the graphical electromagnetic computation (GRECO) approach
for high-frequency RCS computation, the integral-equation formulation of the
measured equation of invariance (IE-MEI) and the multilevel matrix decom-
position algorithm (MLMDA) in 3-D. He has been a Visiting Professor with
the EPFL, Lausanne, Switzerland, from May 1996 to October 1996, a Visiting
Fellow with the City University of Hong Kong, Hong Kong, from January 1997
to February 1997, the CLUSTER Chair with the EPFL from December 1997
to January 1998, and a Visiting Professor with the EPFL from April 2001 to
June 2001. He has authored or coauthored over 50 papers either published or
accepted in refereed international journals (28 in IEEE TRANSACTIONS) and
over 150 in international conference proceedings. His current interests are the
numerical simulation of electrically large antennas and scatterers.
Juan R. Mosig (S’76–M’87–SM’94–F’99) was
born in Cádiz, Spain. He received the Electrical
Engineer degree from the Universidad Politécnica
de Madrid, Madrid, Spain, in 1973, and the Ph.D.
degree from the Ecole Polytechnique Fédérale de
Lausanne (EPFL), Lausanne, Switzerland, in 1983.
In 1976, he joined the Laboratory of Electromag-
netics and Acoustics, EPFL. Since 1991, he has been
a Professor with EPFL, and since 2000, he has been
the Head of the Laboratory of Electromagnetics and
Research Associate with the Rochester Institute of Technology, Rochester, NY,
the Technical University of Denmark, Lyngby, Denmark, and the University of
Colorado at Boulder. He is currently the Chairman of the EPFL Space Center
and is responsible for many Swiss research projects for the European Space
Agency (ESA). He has authored five book chapters on microstrip antennas and
circuits and over 100 reviewed papers. His research interests include electro-
magnetic (EM) theory, numerical methods, and planar antennas.
Dr. Mosig has been a member of the Swiss Federal Commission for Space
Applications. He is currently a member of the Board of the Applied Computa-
tional Electromagnetics Society (ACES), the chairman of the European COST
Project on Antennas ASSIST (2007–2011) and a founding member and acting
chair of the European Association and the European Conference on Antennas
and Propagation (EurAAP and EuCAP).