Post on 17-Aug-2020
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Evaluation of Green’s Function Integrals inConducting Media
Swagato Chakraborty and Vikram Jandhyala
{swagato,jandhyala}@ee.washington.edu
Dept of EE, University of Washington Seattle WA, 98195-2500
UWEE Technical Report Number UWEETR-2002-0016 August 2, 2002 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu
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Evaluation of Green’s Function Integrals in Conducting Media
Swagato Chakraborty and Vikram Jandhyala, Member IEEE
Department of Electrical Engineering University of Washington
Seattle WA 98195 Phone: 206-543-2186
Fax: 206-543-3842 Email: jandhyala@ee.washington.edu
Abstract
This paper presents an accurate integration method for computing Green’s function
operators related to lossy conducting media. The presented approach is ultra-wideband
i.e. the integration schemes cover the entire range of frequency behavior, from high
frequencies where skin current is prevalent to low frequencies where volume current flow
dominates. The scheme is a step towards permitting exact ultra-wideband frequency
domain surface-only-based integral-equation simulation of arbitrarily-shaped 3D
conductors, and towards obviating the need for volume-based explicit frequency-
dependent skin effect modeling. This work deals specifically with the computation of
Green’s functions and not with the unrelated but important low-frequency conditioning
issue associated with the standard electric field integral equation.
1. Introduction
Surface and volumetric integral equation techniques are powerful paradigms for
modeling electromagnetic (EM) interactions in integrated circuit (IC) and packaging
problems. While coupled electromagnetic and circuit analyses have been successfully
realized through the popular volumetric partial element equivalent circuit (PEEC)
3
approach [1,2], the search for more general approaches, especially for modeling
frequency-dependent skin effects and for arbitrarily-shaped structures, has led to circuit-
coupled surface-based electric field integral equation (EFIE) formulations [3,4]. In these
and other works [5-11] it has been shown that surface integral equations and method of
moment (MoM) formulations can be interpreted and applied as generalizations of
volumetric EFIE - based PEEC. In particular, Rao-Wilton-Glisson (RWG)-function [6]
based triangular surface tessellations permit modeling of arbitrarily-shaped structures and
arbitrarily-directed equivalent surface currents. These forms of modeling are particularly
useful for package and system-on-chip simulation and can also enable coupled circuit and
electromagnetic simulation [3].
Surface integral equation formulations are desirable for simulating packaging and
interconnect structures due to the related ease in modeling arbitrary geometries and
equivalent current flow. Also, at high frequencies, surface impedance approximations are
sufficiently accurate to model losses and inductive behavior caused by skin effects.
However, at lower frequencies, where cross sections of conductors are smaller than the
skin depth, standard surface impedance approximations are invalid. Therefore, for
broadband simulation as necessitated in digital or ultra-wideband systems, a volumetric
formulation is typically required at low frequencies. In a volumetric formulation, the skin
effect needs to be modeled explicitly. This modeling requires fine and frequency-
dependent volume meshing (Fig. 1). It is noted that some recent efforts have been aimed
at obtaining new surface impedance approximations that might be valid at low
frequencies. These are typically restricted to cases of assumed or uniform cross sections
[7], as opposed to more general 3D structures, such as packages and on-chip inductors.
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Handling a mix of full-wave and skin-like effects with a surface-only formulation
is desirable since frequency-dependent effects can be tracked without changing geometric
discretization and without taking recourse to a special volume formulation at low
frequencies. This is particularly true for small microelectronic structures where geometry
detail and not wavelength is the guiding factor in mesh discretization. To accomplish a
surface-only formulation valid for realistic conductors over a broad range of frequencies,
the interior lossy medium EM problem must be addressed and coupled to the external
medium model.
This paper presents an exact formulation and accurate numerical quadrature
scheme to efficiently compute highly damped Green’s functions in lossy conductors. The
presented method is general in terms of geometries, frequencies, material parameters, and
relative separation and orientation of source and observer regions, and potentially forms
an important step towards the realization of a surface-only ultra-wideband integral
equation formulation. The motivation behind the presented lossy medium Green’s
function quadrature is, as discussed in the previous paragraph, that a coupled integral
equation formulation, linking an exterior homogeneous medium problem to an interior
lossy medium problem is required in order to correctly predict electromagnetic behavior
of realistic conductors in specific frequency bands. As line widths of interconnects
reduce, and as progressively smaller devices and structures are integrated at the package
and chip levels, the variation in the frequency behavior becomes larger and these cannot
be handled in an ad hoc manner by mixing surface and volume formulations.
It should be noted that the low frequency-dependence and modeling issue being
addressed here is distinct from the classical low frequency ill-conditioning of an EFIE
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formulation; the problem discussed here is unrelated and not due to the relative strength
of vector and scalar potentials. In fact, depending on the conductance involved, the issue
discussed here can arise at much larger frequencies than those where the EFIE is
inherently ill-conditioned. The treatment here is complementary to advances in
improving EFIE conditioning [8] at low frequencies.
This paper presents a new integration formulation and quadrature scheme for
modeling lossy medium Green’s functions with RWG functions, including scalar and
vector (and its curl) potentials, in an accurate manner. The quadrature is initially
facilitated by transforming the Green’s function computation associated with RWG
functions into polar coordinates. Subsequently, the proper order of integration results in
one analytic integration along one coordinate. Finally, a remaining one-dimensional
integral is computed as a summation of several superposed integrals over different bands
in the integration coordinate. Each such integral is computed with an efficient quadrature
scheme described herein.
Section 2 of this paper presents the two-region formulation that utilizes the
integrals that are the subject of this paper. Existing quadrature schemes are discussed in
Section 3. The specific frequency dependence of the integrals under study is outlined in
Section 4. Section 5 presents the polar-coordinate-based integration schemes. In Section
6 the adaptive quadrature rule designed to carry out a complex one-dimensional
integration is discussed. Numerical results including behavior of the resultant one-
dimensional integrands and their sampling, the self-consistency checks and comparisons
with other techniques are detailed in Section 7, and Section 8 presents conclusions and
continuing work.
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2. Formulation and Resultant Integrals
Consider the two-region problem depicted in Fig. 2, with the two regions being a
homogeneous lossless medium, typically free space or a lossless dielectric, and the
interior of a realistic conductor. The exterior equivalent problem utilizes the
homogeneous medium Green’s function, while the lossy medium Green’s function is
required for the interior equivalent problem. For the electric field integral equation
(EFIE), scalar and vector potential integrals will be necessitated, while for the magnetic
field integral equation (MFIE), an integrand that represents the curl of the vector potential
is required. In general, for PMCHW [9] and combined field integral equation (CFIE)
formulations, all three types of integrands need to be computed.
Typically for a region characterized with arbitrary material properties, the electric
and magnetic field E and H can be represented by the equations
FAEE ×∇−∇−−= φωjinc (2.1a)
AFHH ×∇+∇−−= ψωjinc (2.1b)
where incE and incH are the incident electric and magnetic fields in the region, A and
F are the magnetic and electric vector potentials, φand ψ represent the electric and
magnetic scalar potentials, and fπω 2= where f is the frequency of operation.
The scalar and vector potentials can be written in terms of the Green’s function G and the
electric and magnetic current density, J and M as :
∫′
′′′=S
sdG )(),(4
)( rJrrrAπµ (2.2 a)
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∫′
′′′=S
sdG )(),(4
)( rMrrrFπε (2.2 b)
∫′
′′⋅∇′=S
sdGj )(),(4
)( rJrrrωεπ
φ (2.2c)
∫′
′′⋅∇′=S
sdGj )(),(4
)( rMrrrωµπ
ψ (2.2d)
where εµ , represents the material permittivity, and permeability, respectively, and
),( rr ′G for a source point r′ located in the source region S ′ , and an observation point
r is
rr
rrrr
′−=′
′−− kjeG ),( (2.3)
where k is the wave number at a frequency ω , for a material with σ , rµ , rε as the
conductivity, relative permeability and permittivity , and is given by
)(0
00 εωσεεµµω
jk rr += (2.4)
Two auxiliary potentials Π , and Γ are introduced to represent the four potentials in Eqn.
(2.2) as, ΠA µ= , ΠF ε= ; ε
φ Γ= ,µ
ψ Γ= , where
∫′
′′′=S
sdG )(),(41)( rXrrrΠπ
(2.5a)
∫′
′′⋅∇′=ΓS
sdGj )(),(4
)( rXrrrωπ
(2.5 b)
Additionally, the curl operators in Eqn. (2.1) are represented as
∫′
′′×′∇ ′−=×∇S
sdG )(),(41)( rXrrrΠπ
(2.5c)
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where Χ represents the electric or magnetic current density. The popular triangle-pair-
based Rao-Wilton-Glisson (RWG) functions [6] are used to represent )(rΧ ′ , wherein
current is modeled by edge-based piecewise linear vector functions, and the divergence
of current is represented by piecewise constant scalar functions as ±
±′=
Al
2)( ρrΧ , and
±=⋅∇Al)(rΧ [6] where ±′ρ represents the vector joining the node opposite to the edge
in question to (from) the source point r′ in the positive (negative) triangle, ±A denotes
the area of the positive(negative) triangle, and l is the length of the edge.
The generalized potential integrals Eqn. (2.5) can be written for RWG sources as
)(8
1)( scalc
vect MA
ρΜrΠ +=π
(2.6a)
scalMA
jπω4
)( =Γ r (2.6b)
)]([8
1)( scalc
vecti N
AρNRrΠ +×=×∇
π (2.6c)
where
sdR
e
T
Rjk
vect ′= ∫∫−
ρM (2.7a)
sdR
eMT
jkR
scal ′= ∫∫−
(2.7b)
sdR
jkRe
T
jkR
vect ′+= ∫∫−
3
)1(ρN (2.7c)
sdR
jkReNT
jkR
scal ′+= ∫∫−
3
)1( (2.7d)
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and iR represents the vector joining the vertex of the source triangular region T (Fig. 4)
opposite to the edge in question to the observation point, cρ is the vector from the same
vertex to the projection of the observation point onto the plane of T, and ρ is the vector
from the projection of the observation point r on the plane of T to a source point r′ on T.
rr ′−=R , is the radial distance between the source and the observation point.
3. Existing Analytical and Numerical Quadrature Schemes for Green’s Functions
In the extant literature, evaluation of the potential integral Eqn. (2.5) in free-space and
low-loss media has been done by a variety of numerical schemes. For near-field terms,
singularity extraction of the kernels in Eqn. (2.5a-b) is performed analytically [10-11] to
leave a function that can be integrated numerically with a low-order quadrature rule [12].
Recently, methods based on the Duffy transform have emerged, wherein the triangular
integration region is transformed to a rectangle with a subsequent cancellation of the
singularity. The integral in Eqn. (2.5c) has been evaluated in free space [13], and in
lossless dielectrics [9] .
When the medium is conducting, even the singularity-extracted part may exhibit a
rapid spatial decay, i.e. the extracted integral appears nearly singular when the
observation point is sufficiently close to the source triangle. Hence, standard singularity
extraction [10] fails to evaluate the integral accurately.
A suitable approach to Green’s function computation in lossy media is polar
coordinate integration, which can render the non-essential singularity cancelled through
the Jacobian of transformation. Such methods are discussed previously in [14-15] for
lossless media and in [16] for lossy media, for the restricted case of the scalar Green’s
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function in Eqn. (2.7b). However these methods are not sufficiently general for the
integrals in Eqns. (2.7a,c,d) that are related to the vector potential or its curl. Another
polar coordinate approach is proposed in [17] to evaluate the vector integral for the
specific case of self-term integration. The method is extendable to the case when the
observation point is located anywhere in the plane of the source triangle itself. This
precludes the important case of observation at a near-singular point located above or
below the source triangle, as occurs in thin conductors.
In Section 5 we propose a general method for evaluating scalar, vector, and
gradient Green’s functions in lossy media with RWG basis functions. The presented
technique works for all frequencies and for all relative positions between source triangles
and observation points. The next section discusses the frequency-dependent behavior of
the generalized potential integrals in conducting media that necessitates the specialized
quadrature presented in later sections.
4. Frequency Dependence of Green’s Functions in Conducting Media
The behavior of the Green’s functions in Eqns. (2.7) in conducting media is highly
dependent on frequency, as shown in Fig. 3. Consider a Method of Moments (MoM) [6]
matrix created for interactions between RWG functions for the interior medium
equivalent, which uses the conducting medium Green’s functions. At high frequencies,
the MoM matrix is nearly diagonal because of a very rapid exponential spatial decay of
the conducting medium Green’s function owing to the large imaginary part of the
wavenumber in Eqn. (2.4).
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At lower frequencies, the interactions between non-overlapping RWG functions
are not neglible; and the MoM matrix becomes progressively less sparse but has sections
which are numerically sparse (e.g. in double precision arithmetic) due to large
exponential decays. As the frequency is further lowered the MoM matrix is completely
full while showing a weak exponential decay with distance. Eventually, the MoM matrix
is full and the exponential decay is very weak or absent.
To summarize, at intermediate frequencies, between sharp fall-off and no fall-off
regimes, special numerical treatment is required; the integrands presented by the lossy
medium Green’s function have sharp radial decay, and non-self interactions are also
prominent. Depending on the frequency, the entire MoM matrix might be numerically
significant. Low-order Gaussian quadrature rules in [12], that are popular in RWG-based
MoM implementations will not provide accurate answers at such frequencies, owing to
rapid decays of the Green’s functions over finite distances.
5. Computation of Lossy Medium Generalized Potential Integrals
The generalized potential integrals Eqn. (2.6) for RWG sources are constituted by the
four terms in Eqn. (2.7), which can be transformed into polar coordinates as,
θρθρ
ρθρθρ
ρ ρρ
ddd
eddd
e
T
djk
T
djk
vect sinˆcosˆ22
2
22
2 2222
∫∫∫∫ ++
+=
+−+−
yxM (5.1a)
θρρ
ρ ρ
ddd
eMT
djk
scal ∫∫ +=
+−
22
22
(5.1b)
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θρθρρρ
θρθρρρ
ρ
ρ
ddd
edjk
ddd
edjk
T
djk
T
djk
vect
sin)(
)1(ˆ
cos)(
)1(ˆ
322
222
322
222
22
22
∫∫
∫∫
+
++
++
++=
+−
+−
y
xN
(5.1c)
θρρ
ρρ ρ
ddd
edjkN
T
djk
scal ∫∫+
++=
+−
322
22
)(
)1(22
(5.1d)
In the above equations, the x and y coordinates are local to the source triangle T (Fig. 4)
and define the plane in which T lies. Also, d is the perpendicular distance of the
observation point from the plane of T, and ( θρ, ) is the polar coordinate of a source point
in T, with the projection of the observation point onto the plane of T as the origin. The
scalar integrals in Eqns. (5.1b,d) and the scalar components of the vector integrals in
Eqns. (5.1a,c) can be written in a generalized form, as
∫ ∫=ρ θ
ϕχ θρθχρϕ ddI )()( (5.2)
where ϕ is one of vectM ,ϕ , scalM ,ϕ , vectN ,ϕ , scalN ,ϕ defined below as
22
2
,
22
)(d
e djk
vectM+
=+−
ρρρϕ
ρ
(5.3a)
22,
22
)(d
e djk
scalM+
=+−
ρρρϕ
ρ
(5.3b)
322
222
,)(
)1()(
22
d
edjk djk
vectN+
++=
+−
ρρρ
ρϕρ
(5.3c)
322
22
,)(
)1()(
22
d
edjk djk
scalN+
++=
+−
ρρρ
ρϕρ
(5.3d)
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Also χ is one of cχ , sχ , 0χ defined below as
θθχ cos)( =c (5.4a)
θθχ sin)( =s (5.4b)
1)(0 =θχ . (5.4c)
Owing to the simple closed form expressions for the integral of )(θχ , the integral ϕχI can
be recast as a function of ρ as
∫∫ ∫==T
dddImax
min
)()()()(ρ
ρϕχ ρρξρϕθρθχρϕ (5.5)
where minρ and maxρ are the extremal ρ for which TP ∈∋ℑ ),( θρθ , ),( θρP denotes a
point having coordinate ),( θρ (Fig. 4), and ξ is one of cξ , sξ , 0ξ with
( ) ( )
−
+−
−
=
−=
=
=
∑∑∑ ∫∑====
)()(
)(cos)(cos
)(sin)(sin
cossin
1sincos
)(
)(
)(
)()()(
minmax
minmax
minmax)(
1
)(
)(
)(
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)(
)(0
10
max
min
max
min ρθρθρθρθ
ρθρθ
θθ
θθθ
θ
ρξρξρξ
ρξρξρξ
ρ
ρθ
ρθ
ρρ ρθ
ρθ
ρ
ii
ii
ii
K
i
K
i
K
ii
is
icK
is
c
i
i
i
i
d
(5.6)
Also, )(ρK is the number of intervals (Fig. 4) in θ , πθ 20 <≤ , for which ),( θρP lies
in T, and imaxθ and i
minθ are the limits on θ for the thi interval. The values of )(max ρθ i
and )(min ρθ i for each section are computed by obtaining the intersection of T and the
circle of radius ρ centered at the projection of the observation point onto the plane of T.
If the circle with radius ρ lies entirely in T, 1)( =ρK , πθ 21max= ,and 01
min =θ .Hence
=
πρξρξρξ
200
)()()(
0
s
c
(5.7)
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Alternatively, if for a given ρ , if the circle is completely outside T then the integral
contributions are all zero. Consequently, the constituents of the generalized potential
integrals Eqn. (5.1) can be computed using Eqn. (5.3-5.6) as
ρρξρϕρρξρϕρ
ρ
ρ
ρ
dd svectMcvectMvect )()(ˆ)()(ˆmax
min
max
min
,, ∫∫ += yxM (5.8a)
ρρξρϕρ
ρ
dM scalMscal )()(max
min
0,∫= (5.8b)
ρρξρϕρρξρϕρ
ρ
ρ
ρ
dd svectNcvectNvect )()(ˆ)()(ˆmax
min
max
min
,, ∫∫ += yxN (5.8c)
ρρξρϕρ
ρ
dN scalNscal )()(max
min
0,∫= (5.8d)
Section 6 discusses the adaptive integration rule that has been designed in order to
perform the integration in Eqns. (5.5 and 5.8) efficiently.
6. Generalized Adaptive Integration Rule for Piecewise Smooth Functions
In the one-dimensional integral Eqn. (5.5), the function )(ρϕ is smooth and continuous
over the integration interval, and the function )(ρξ is piecewise smooth and continuous.
Hence the over-all integrand is piecewise smooth and continuous. If the integrand
)()()(1 ρξρϕρ =+if , is smooth over the subintervals ),( 1+ii ρρ , where { } ,1,.....2,1,0 −∈ Li
min0 ρρ = , and maxρρ =K , the total integral is written as
ρρρ
ρϕχ dfI
L
ii
i
i
)(1
1
∑ ∫=
−
= (6.1)
In this manner the required quadrature scheme, outlined next, will not need to compute
integrals for functions that have first derivative discontinuities. The integration scheme
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works as follows. Initially a coarse estimate of estIϕχ is obtained by using a 5 point
Newton-Cotes formula based on the Bodé rule [12], over each of the individual L
segments. The rule computes an integral dxxfQb
aab )(∫= with the approximation
)}(7)4
3(32)2
(12)4
(32)(7{90
hafhafhafhafafhQab ++++++++≅ ( 6.2)
where h = b-a. Thus the total integral is initially estimated as
∑=
−=
L
i
estii
QI1
1ρρϕχ ( 6.3)
The quadrature is refined recursively by using an adaptive integration method. For
efficiency in terms of number of sample points, non-uniform sampling of the function
)(ρf is required within each subinterval ( )ρρ ,1−i . At a particular level of recursion, for a
given subinterval (a,b), an estimate of the integral abQ is obtained using Eqn. (6.2).
Subsequently, a binary split is performed on (a,b), and cbacab QQQ +=′ is obtained
where 2/)( bac += . The estimate of initIϕχ is dynamically refined as
)( abcbacinitinit QQQII −++← ϕχϕχ (6.4)
at each level of recursion. If the change in the integration result due to the binary split,
abcbac QQQ −+=∆ (6.5)
relative to the total integral initIϕχ is smaller than a pre-specified threshold tolerance tol1,
then the contribution of that split to the over-all integral is ignored. The exact stopping
criteria used for the convergence test are given by
initItol ϕχ⋅≤∆ 1 (6.6a )
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or
abQtol ⋅≤∆ 2 (6.6b )
where the criterion in Eqn. (6.6b) using the pre-specified tolerance tol2 is required in
order to expedite the convergence in the case of a nearly linear function segment.
The method described above is similar in approach to the existing popular
Matlab-based adaptive quadrature method quad[18], with some important differences. It
uses a different order rule to evaluate abQ in Eqn. (6.2). Also it uses an improved dynamic
refinement of the estimate of the total integral Eqn. (6.4), and leads to rules with smaller
number of samples for a given approximation error for the integrals in this paper. The
code/pseudocode for the rule is not included here but can be found online [19].
7. Numerical Results In this section, the proposed integration schemes are used to compute integrals for all the
cases in Eqn. (5.8). The resulting integrands and sampling quadrature points in polar
coordinates are shown here for different frequencies and accuracies. Comparisons with
two-dimensional Gaussian quadrature are also presented.
For purposes of illustration, and without loss of generality, the source triangle
sourceT for the presented results has nodes located at ( )0,1,1 − , ( )0,5.0,1 , ( )0,5.0,2− , and
the observation point obsP lies outside the plane of sourceT , at ( )1,0,0 , with all distances
measured in mm. The conductivity of the medium is that of copper 7108.5 × S/m.
The one dimensional integral in Eqn. (5.5) is evaluated using the adaptive
quadrature rule described in Section 6. Behavior of the six related integrands in Eqn.
(5.8) are shown in Figs. 5-7, for the source triangle sourceT and observation point obsP .
17
The operating frequency is 1 MHz. The adaptively sampled locations for such integrands
are also pictured. The integrands comprise piecewise smooth functions over all relevant
intervals as discussed in Section 5.
The effect on integrands of lowering frequency is depicted in Figure 8.
Specifically, the integrands )()(, ρξρφ svectM in Eqn. (5.8a) and )()( 0, ρξρφ scalM in Eqn.
(5.8b) are shown for a lower frequency 1KHz, corresponding to the higher frequency
plots in Figs. 5 and 6. It can be noted that with reduction in frequency the integrand
decays slower in distance ρ , and exhibits less oscillations, as both the imaginary and the
real parts of the wave-number k in Eqn. (2.4) becomes smaller.
A feature of the quadrature scheme outlined in this paper is the ability to vary the
tolerance and correspondingly the number of samples in order to achieve a trade-off
between efficiency and accuracy. The effect on the sample points of relaxing the
convergence criterion is shown in Fig. 9. The integrands with sample points
corresponding to higher convergence tolerances for )()(, ρξρφ svectM and )()( 0, ρξρφ scalM
are shown at 1MHz and 1 KHz respectively. The corresponding plots for lower tolerances
are presented in Figs. 5 and 8, and the difference in location and density of sample points
is evident. In general, as can be seen from the results, the scalar integrals appear to
converge to lower errors for given convergence thresholds.
A relative accuracy comparison between the proposed scheme and 7-point
Gaussian quadrature with singularity extraction is demonstrated in Figs. 10 and 11. At
low frequencies, the Green’s functions in lossy media exhibit slow decay over distance
and hence a 7-point Gaussian quadrature scheme [12] works adequately, and the relative
difference between the two methods is small. As the frequency is increased, the details
18
of the decay in the Green’s functions due to the increased imaginary part of the wave-
number (Eqn. 2.4) are not captured by the low-order Gaussian rule and the proposed
methodology of this paper is required. The fact that the discrepancy between the results
from the proposed method and from 7-point Gaussian quadrature is due to the Gaussian
quadrature becoming inaccurate is further evident from comparisons with a higher order
Gaussian quadrature rule using 25 points on a triangle. In this case the frequency at which
the 25 point quadrature breaks down increases compared to the 7 point quadrature. In
general, for any order of Gaussian quadrature, there is a frequency point beyond which
the quadrature will be inaccurate due to insufficient sampling of the details in the decay
of the Green’s function. The presented method accurately models the decay through an
analytic integration and is therefore accurate at any frequency. This is seen in both the
vector integrals (Eqns. 5.8a,c ; Fig. 10) and the scalar integrals (Eqns. 5.8 b,d; Fig. 11).
While the main aim of this work is the formulation and development of the
quadrature rules themselves, one example of the behavior of the rules when included in a
complete two-region PMCHW formulation is shown next. Figure 12 compares the
extracted resistance using a coupled circuit-EM formulation [3] and the quadrature
scheme presented in this paper, with the analytic quasi-static resistance at low frequency.
It also demonstrates the importance of radiation loss at a higher frequency, which is
modeled by a full-wave formulation. Figure 13 shows the agreement in the extracted
resistance obtained by a PMCHW formulation using the standard Gauss quadrature rule
and the proposed method at low frequency. The result matches with an impedance
boundary formulation [19] at high frequencies; the impedance boundary condition is
inaccurate at low frequencies relative to skin depth, and fails to capture the leveling off of
19
the resistance at low frequency. Conversely, the Gauss quadrature scheme becomes
inaccurate at high frequencies, which is demonstrated in Fig. 14. At such frequencies the
proposed quadrature scheme produces same result as the impedance boundary condition
formulation, while at low frequencies the two quadrature schemes produce the same
result.
8. Conclusions In this paper, a new approach to evaluate the Green’s function operators for RWG
functions in conducting media is presented. The method works for arbitrarily located
sources and observers for any frequency. This technique has been incorporated into a
broadband two-region surface formulation for accurate computation of frequency-
dependent parameters, and shows the potential to obviate the need to switch to volumetric
formulations at frequencies where volumetric current flow is dominant.
9. Acknowledgements This work was partially supported by DARPA-MTO NeoCAD grant N66001-01-1-8920,
NSF-CAREER grant ECS-0093102, NSF-SRC Mixed-Signal Initiative grant CCR-
0120371, and by a grant from Ansoft Corporation.
References
[1] H. Heeb and A.E. Ruehli, “Three-dimensional interconnect analysis using partial element equivalent circuits,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 39(11), pp. 974-982, Nov. 1992. [2] A. Rong and A.C.Cangellaris, “Generalized PEEC models for three-dimensional interconnect structures and integrated passives of arbitrary shapes,” Electrical Performance of Electronic Packaging, pp. 225 –228, Oct. 2001.
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[3] V. Jandhyala, W.Yong, D.Gope, and R. Shi, “Coupled electromagnetic-circuit simulation of arbitrarily-shaped conducting structures using triangular meshes,” Proceedings International Symposium on Quality Electronic Design, pp. 38-42, Mar. 2002. [4] S. Ponnapalli, A. Deutsch, and R. Bertin, “A package analysis tool based on a method of moments surface formulation,” IEEE Transactions on Components, Hybrids, and Manufacturing Technology, vol. 16(8), pp. 884-892, Dec. 1993. [5] J. Wang, J. Tausch, and J. White, “A wide frequency range surface integral formulation for 3-D RLC extraction,” Digest of Technical Papers International Conference on Computer-Aided Design, pp. 453-457, 1999. [6] S.M. Rao, D.R.Wilton, and A.W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Transactions on Antennas Propagation, vol. 30, pp. 409-418, 1982. [7] J.-S. Zhao and W.C. Chew, “Accurate and efficient simulation of crosstalks,” Proceedings of the Progress in Electromagnetics Research Symposium, pg. 396, Boston, July 2002. [8] S. Chen, J. S. Zhao, and W.C. Chew, “ Analyzing low-frequency electromagnetic scattering from a composite object,” IEEE Transactions on Geoscience and Remote Sensing, vol. 40(2), pp. 426-433 , Feb. 2002. [9] K.Umashankar, A.Taflove, and S.M.Rao, “Electromagnetic scattering by arbitrary shaped three dimensional homogeneous lossy dielectric objects,” IEEE Transactions on Antennas and Propagation, Vol 34(6) , pp. 758-766, June 1986. [10] R.D. Graglia, “On the numerical integration of the linear shape function times the 3-D Green’s function or its gradient on a planar triangle,” IEEE Transactions on Antennas and Propagation, vol. 41, pp. 1448-1455, 1993. [11] D.R.Wilton, S.M.Rao, A.W.Glisson, D.H.Schaubert, O.M. Al-Bundak, and C.M.Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domain,” IEEE Transactions on Antennas and Propagation, vol. AP-32, pp. 276-281, March 1984. [12] M. Abramowitz and I. Stegun, Chapter 25, Handbook of Mathematical Functions, Dover, New York, 1970. [13] R.E.Hodges and Y.Rahmat Samii, “The evaluation of MFIE integrals with the use of vector triangle basis function,” Microwave and Optical Technology Letters, vol. 14 (1), pp. 9-14, Jan.1997.
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[14] M. Gimersky, S. Amari, and J. Bornemann, “Numerical evaluation of the two-dimensional generalized exponential integral,” IEEE Transactions on Antennas and Propagation, vol. 44, pp. 1422-1425, 1996. [15] J.K.H. Gamage, “Efficient method of moments for compact large planar scatterers in homogeneous medium,” Proceedings of 11th International Conference on Antennas and Propagation, no. 480, pp. 741-744, 2001. [16] Z. Zhu, J. Huang, B. Song, and J. White, “Improving the robustness of a surface integral formulation for wideband impedance extraction of 3D structures,” Proceedings of International Conference on Computer Aided Design, pp. 592-597, 2001. [17] L. Rossi and P.J. Cullen, “On the fully numerical evaluation of the linear-shape function times the 3-D Green’s function on a planar triangle,” IEEE Transactions on Microwave Theory and Techniques, vol. 47, pp. 398-402, 1999. [18] W.Gander and W.Gautschi, “Adaptive quadrature - revisited,” Technical Report 306, Department Informatik, ETH Zürich, August 1998. [19] S. Chakraborty and V. Jandhyala, “A modified adaptive quadrature scheme,” online at http://www.ee.washington.edu/faculty/jandhyala/adaptive_quadrature, unpublished. [20] A.W.Glisson, “Electromagnetic scattering by arbitrarily shaped surfaces with impedance boundary conditions,” Radio Science, vol. 27(6) , pp. 935-943, Nov. 1992.
22
Figure 1: Volumetric gridding as a function of frequency: (top) low frequency, volumetric current flow, (middle) intermediate frequency skin effect modeling, (bottom) high frequency surface current flow. Figure 2: Classical equivalence principle modeling of a two region problem, with one region being the interior of a conductor. The subscripts denote interior and exterior parameters. Figure 3: Behavior of the MoM matrix in a lossy conducting medium. At high frequencies, surface impedance approximations are accurate and lead to a nearly diagonal matrix. As the frequency is reduced, the matrix becomes less sparse but can still have numerically zero regions where exponential decays are very large. As frequency continues to reduce, the matrix becomes dense and eventually has very little exponential decay. Note: The “structure” of the matrix above assumes that basis functions near each other are located next to each other in the matrix order. This structure is only used as a schematic guide to explain the frequency-dependent nature of the matrix for lossy media. Figure 4: Region of integration is shown for ( 321 ,, ρρρρ = ), for a triangular region T , for the projection of the observation point on the plane of triangle O . Gray sections denote intervals of θ where the source point ),( θρP lies within the triangle. Figure 5: Behavior of )()(, ρξρϕ cvectM (top) and )()(, ρξρϕ svectM (bottom) in Eqn. (5.8a) and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
1=α mm, at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in a relative integration error of 4105.1 −× . Figure 6: Behavior of )()( 0, ρξρϕ scalM (top) and )()(, ρξρϕ cvectN (bottom) in Eqns. (5.8b,c)and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
1=α mm, at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in relative integration errors of 8105.3 −× and 41028.1 −× , respectively. Figure 7: Behavior of )()(, ρξρϕ svectN (top) and )()( 0, ρξρϕ scalN (bottom) in Eqn. (5.8c,d) and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
1=α mm, at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in relative integration errors of 41028.1 −× and 13109.1 −× , respectively. Figure 8: Behavior of )()(, ρξρϕ svectM (top) and )()( 0, ρξρϕ scalM (bottom) in Eqn. (5.8a,b) and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
23
1=α mm, at a frequency of 1KHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in relative integration errors of 5105.2 −× and 121087.8 −× , respectively. Figure 9: Behavior of )()(, ρξρϕ svectM (top) and )()( 0, ρξρϕ scalM (bottom) in Eqn. (5.8a,b) and adaptive sampling for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
1=α mm, at a frequency of 1MHz and 1KHz, respectively, with 7108.5 ×=σ S/m. The stopping threshold resulted in relative integration errors of 3103.3 −× and 8106.2 −× , respectively. Figure 10: Comparison between 2D Gaussian rules with singularity extraction and proposed method for evaluation of the integral vectN and vectM in Eqns. (2.7c,2.7a) for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
1=α mm , with 7108.5 ×=σ S/m. Figure 11: Comparison between 2D Gaussian rules with singularity extraction and proposed method for evaluation of the integral scalN and scalM in Eqns. (2.7c,2.7a) for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
1=α mm , with 7108.5 ×=σ S/m. Figure 12: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm, using PMCHW formulation with the proposed quadrature scheme, for a full-wave and a quasi-static formulation and the analytic value of resistance using skin-depth. Figure 13: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm, using PMCHW formulation with the proposed quadrature scheme, Gaussian quadrature, and impedance boundary condition. Figure 14: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm using a two region PMCHW formulation with the standard Gaussian quadrature method and the method proposed in this paper, and an impedance boundary condition formulation.
24
Figure 1: Volumetric gridding as a function of frequency: (top) lowfrequency, volumetric current flow, (middle) intermediate frequency skineffect modeling, (bottom) high frequency surface current flow.
25
σint,ε int,µ int
εext,µext
Lossy Conducting Medium External Homogeneous Medium
Interior Exterior
Figure 2: Classical equivalence principle modeling of a two region problem, withone region being the interior of a conductor. The subscripts denote interior andexterior parameters.
σint,ε int,µ int
σint,ε int,µ int
εext,µext
εext,µext
26
Decreasing Frequency
Surface Impedance Nearly Diagonal
Rapid Exponential Decay Partially Sparse
Exponential Decay Dense
Slow Decay Dense
Figure 3: Behavior of the MoM matrix in a lossy conducting medium. At high frequencies, surfaceimpedance approximations are accurate and lead to a nearly diagonal matrix. As the frequency isreduced, the matrix becomes less sparse but can still have numerically zero regions whereexponential decays are very large. As frequency continues to reduce, the matrix becomes dense andeventually has very little exponential decay. Note: The “structure” of the matrix above assumes thatbasis functions near each other are located next to each other in the matrix order. This structure isonly used as a schematic guide to explain the frequency-dependent nature of the matrix for lossymedia.
27
Figure 4: Region of integration is shown for ( 321 ,, ρρρρ = ), for a triangular region T ,for the projection of the observation point on the plane of triangle O . Gray sections denote intervals of θ where the source point ),( θρP lies within the triangle.
1ρ
2ρ
3ρO
ρdT),( θρP
θ
ρ
x
y
28
Figure 5: Behavior of )()(, ρξρϕ cvectM (top) and )()(, ρξρϕ svectM (bottom) in Eqn. (5.8a) andadaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where 1=α mm,at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in a relativeintegration error of 4105.1 −× .
0.5 1 1.5 2x 10-3
-8
-6
-4
-2
0
2
4x 10-12
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
0.5 1 1.5 2x 10-3
-2
-1
0
1
2
3
4
5x 10-12
ρ(m)
Inte
gran
d X Sampled Real Part Sampled Imaginary Part
29
Figure 6: Behavior of )()( 0, ρξρϕ scalM (top) and )()(, ρξρϕ cvectN (bottom) in Eqns. (5.8b,c)andadaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where 1=α mm,at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping threshold resulted in relativeintegration errors of 8105.3 −× and 41028.1 −× , respectively.
0 0.5 1 1.5 2 2.5x 10-3
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5 x 10-7
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
0.5 1 1.5 2x 10-3
-1
-0.5
0
0.5
1
1.5
2x 10-4
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
30
Figure 7: Behavior of )()(, ρξρϕ svectN (top) and )()( 0, ρξρϕ scalN (bottom) in Eqn. (5.8c,d)and adaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
1=α mm, at a frequency of 1MHz, with 7108.5 ×=σ S/m. The stopping thresholdresulted in relative integration errors of 41028.1 −× and 13109.1 −× , respectively.
0.5 1 1.5 2x 10-3
-8
-6
-4
-2
0
2
4
6
8x 10-5
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
0 0.5 1 1.5 2 2.5x 10-3
-5
-4
-3
-2
-1
0
1
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
31
Figure 8: Behavior of )()(, ρξρϕ svectM (top) and )()( 0, ρξρϕ scalM (bottom) in Eqns. (5.8a,b) andadaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where
1=α mm, at a frequency of 1KHz, with 7108.5 ×=σ S/m. The stopping threshold resultedin relative integration errors of 5105.2 −× and 121087.8 −× , respectively.
0.5 1 1.5 2x 10-3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5x 10-4
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
0 0.5 1 1.5 2 2.5x 10-3
-1
-0.5
0
0.5
1
1.5
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
32
Figure 9: Behavior of )()(, ρξρϕ svectM (top) and )()( 0, ρξρϕ scalM (bottom) in Eqns. (5.8a,b) andadaptive sampling for the non-self-term integral, for a triangle with vertices( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), and observation point located at ),0,0( α ,where 1=α mm,at a frequency of 1MHz and 1KHz, respectively, with 7108.5 ×=σ S/m. The stopping thresholdresulted in relative integration errors of 3103.3 −× and 8106.2 −× , respectively.
0.5 1 1.5 2x 10-3
-2
-1
0
1
2
3
4
5x 10-12
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
0 0.5 1 1.5 2x 10-3
-1
-0.5
0
0.5
1
1.5
ρ(m)
Inte
gran
d
X Sampled Real Part Sampled Imaginary Part
33
Figure 10: Comparison between 2D Gaussian rules with singularity extraction and proposedmethod for evaluation of the integral vectN and vectM in Eqns. (2.7c,2.7a) for the non-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ), andobservation point located at ),0,0( α ,where 1=α mm , with 7108.5 ×=σ S/m.
101 102 103 104 105 10610-4
10-2
100
102
104
Frequency(Hz)
Rel
ativ
e D
iffer
ence
X
vectlN -25 point rule vectM -25 point rule
vectN - 7 point rule vectM - 7 point rule
34
101 102 103 104 105 10610-4
10-3
10-2
10-1
100
101
102
Frequency(Hz)
Rel
ativ
e D
iffer
ence
X
scalN - 25 point rule scalM - 25 point rule scalN - 7 point rule scalM -7 point rule
Figure 11: Comparison between 2D Gaussian rules with singularity extraction andproposed method for evaluation of the integral scalN and scalM in Eqns. (2.7c,a) for thenon-self-term integral, for a triangle with vertices ( 0,, αα − ),( 0,2/,αα ),( 0,2/,2 αα− ),and observation point located at ),0,0( α ,where 1=α mm , with 7108.5 ×=σ S/m.
35
107 108 1090
0.5
1
1.5
2
Frequency (Hz)
Res
ista
nce
(Ohm
)Full-Wave with New Quadrature Quasi-Static with New Quadrature Analytic
Figure 12: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm, usingPMCHW formulation with the proposed quadrature scheme, for a full-wave and a quasi-staticformulation and the analytic value of resistance using skin-depth.
36
Figure 13: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm,using PMCHW formulation with the proposed quadrature scheme, Gaussianquadrature, and impedance boundary condition.
107 108 1090
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Frequency (Hz)
Res
ista
nce(
Ohm
) Impedance Boundary ConditionPMCHW with the New Quadrature PMCHW with Gaussian Quadrature
37
Figure 14: Extracted resistance of a cylinder with radius 0.5mm and length 5 mm using atwo region PMCHW formulation with the standard Gaussian quadrature method and themethod proposed in this paper, and an impedance boundary condition formulation.
108 109 1010
0
20
40
60
80
100
120
140
160
180
Frequency (Hz)
Res
ista
nce
(Ohm
)
Proposed Quadrature SchemeExisting Gauss Quadrature SchemeImpedance Boundary Condition