Krylov Model Order Reduction of Finite Element … Model Order Reduction of Finite Element Models of...
Transcript of Krylov Model Order Reduction of Finite Element … Model Order Reduction of Finite Element Models of...
Krylov Model Order Reduction of Finite Element Models of
Electromagnetic Structures with Frequency-Dependent Material Properties
Hong Wu and Andreas Cangellaris
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
Urbana, Illnois 61801, Email: hongwu, [email protected]
Abstract— Krylov subspace-based model order reduction(MOR) of finite element models of electromagnetic structuresis not readily applicable when the electromagnetic properties ofthe materials exhibit arbitrary frequency dependence. This paperpresents a methodology for overcoming this hurdle. The proposedKrylov MOR process is demonstrated through its application tothe expedient broadband analysis of the impact of skin-effectloss on the transmission properties of a microstrip bandpassfilter and the extraction of the propagation characteristics ofa microstrip line on a dielectric substrate with frequency-dependent permittivity described by a Debye model.
Index Terms— Model order reduction, finite element method,dispersive media, frequency dependent rational function.
I. INTRODUCTION
In the context of state-space representations of linear sys-
tems of high dimension the objective of model order reduction
(MOR) is the development of a new model of the linear
system of much smaller dimension than the original one, yet
capable of describing the behavior of the original model with
acceptable accuracy over a broad frequency bandwidth. It is
this attribute of MOR that has prompted its application for
expediting solution efficiency and reducing modeling com-
plexity of the large linear circuits associated with high-speed
interconnect modeling in state-of-the art integrated electronic
systems [1]–[5].
The discretization of the spatial derivatives in Maxwell’s
time-dependent equations using finite methods results in state-
space semi-discrete approximations of the electromagnetic
system. Hence, the same techniques that have been used
successfully for model order reduction of large linear cir-
cuits can be adopted with only slight modifications for the
reduced-order modeling of electromagnetic systems. Among
the various techniques currently available for the reduced-order
modeling of passive electromagnetic devices and structures the
Asymptotic Waveform Evaluation (AWE) method [6] was the
first one to be used. This was followed by the application of
the computationally more robust class of Krylov subspace-
based methods and in particular the Pade via Lanczos (PVL)
process [1], [7]. For applications that called for the utilization
of the reduced-order model as a multiple-input multiple-output
(MIMO) matrix transfer function representation of the electro-
magnetic device, the requirement for preserving the passivity
attributes of the original system in its reduced macromodel
led to the development of passivity-preserving model order
reduction techniques. For the case of model order reduction
of finite-difference/finite-element models of electromagnetic
systems, the block Arnoldi process-based, passive reduced-
order macro-modeling algorithm (PRIMA) [2] was shown to
be particularly useful [3], [8].
The aforementioned Krylov-based model order reduction
techniques are most suitable for the expedient broadband
analysis and macromodeling of electromagnetic devices and
systems with constant material properties. When the elec-
tromagnetic properties of the materials are frequency de-
pendent, characterized by Krylov subspace methods are not
as readily applicable and alternative model order reduction
techniques are sought (see, for example, [9]). In addition to
media that naturally exhibit frequency dependence in their
electromagnetic properties, there are numerous application
of practical interest where the ability to incorporate in a
convenient manner in the electromagnetic model frequency-
dependent relationships between the fields and associated
flux quantities provides for enhanced modeling versatility and
increased computational efficiency. Most typical examples are,
a) the utilization of frequency-dependent surface impedance
boundary conditions to account for skin effect loss in metallic
volumes [10], [11]; b) the utilization of frequency-dependent
multi-port, matrix transfer functions for the macromodeling of
sub-domains of arbitrary (passive) material composition [3]; c)
the macroscopic description of the electromagnetic attributes
of artificially constructed materials (metamaterials) [12].
In this paper a systematic methodology is proposed for
extending the application of Krylov subspace techniques to the
model order reduction of passive electromagnetic structures
with material and or boundary conditions exhibiting general
frequency dependence. The proposed methodology is based
on the extension of the technique introduced in [13]–[15] for
the reduction of second-order dynamical systems. The paper
is organized as follows. First, the development of the pertinent
finite element model of the electromagnetic system is briefly
reviewed. This is followed by the mathematical formulation
of the proposed model order reduction process. The paper
concludes with the validation of the proposed methodology
through its application to the analysis of a bandpass microstrip
filter with skin effect loss taken into account and the extraction
of the propagation characteristics of a microstrip line on a
dispersive dielectric substrate.
II. FINITE ELEMENT APPROXIMATION
The methodology described in [16] is followed for the
development of the finite element approximation of the vector
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Helmholtz equation for the electric field using edge ele-
ments. For the purposes of reduced-order macromodeling it
is assumed that the device under modeling interacts with
its exterior through Np ports. Hence, the Laplace-domain
representation of the finite element model of the Np-port
system is described by the following system of equations [16]
(Y + sZ + s2Pe)x = sBI
y = LHx
(1)
where N is the number of degrees of freedom in the finite
element approximation, the vector x contains the coefficients
in the finite element approximation of the electric field, the
matrices Y, Z, Pe are in RN×N , the matrices B, L are in
CN×Np , the vectors I , y are in C
Np×1 and s is the complex
frequency. The vector I indicates unit excitation at each port.
The matrix B is dependent on the port characteristics and
is used to map the unit excitation to the excitation on state-
space variables. Similarly, the matrix L is used to sample
the calculated electric field to generate the desirable output
quantity output.
As far as the matrices Y, Z and Pe are concerned, under
the assumption of perfect electric conductors and materials
of constant permittivity and permeability, these matrices are
frequency independent. This is not the case anymore when the
electromagnetic properties of the media are frequency depen-
dent. For example, assuming media of constant permeability
but frequency-dependent permittivity ǫ(s), the elements of Pe
are given by
Peij =
∫∫∫
Ω
~wti· ǫ(s) ~wtj
dv, (2)
where Ω is computation domain and ~wt indicates the tangen-
tially continuous edge element. Furthermore, when conductor
loss is taken into account and a surface impedance, η(s), is
used for its modeling, the Z matrix is augmented by an extra
term Zsuf , with elements given by
Zsuf ij = s
∫∫
SSIBC
n × ~wti· 1
η(s)n × ~wtj
ds. (3)
Thus, equation (1) becomes
(Y + sZ + s2Pe(s) + Zsuf (s))x = sBI,
y = LHx.
(4)
III. MODEL ORDER REDUCTION WITH EQUATION
STRUCTURE PRESERVED
For simplicity and for the sake of brevity we will present
the proposed model order reduction methodology for the case
where Pe is constant and the general frequency dependence in
(4) is only due to the surface impedance of lossy conductors.
Under this assumption we can write
Zsuf (s) = H(s)Zs, (5)
where it is Zs is defined as
Zsij =
∫∫
SSIBC
n × ~wti· n × ~wtj
ds, (6)
and
H(s) =s
η(s)= s
(
(1 + j)
√
sµ
2σ
)−1
. (7)
In the above equation σ and µ are, respectively, conductivity
and permeability of the conductors. Thus, equation (4) may
be cast in the following form
(Y + sZ + s2Pe + H(s)Zs)x = sBI,
y = LHx.
(8)
Next, use is made of the vector fitting technique of [17]
to express H(s) in terms of a rational function approximation
H(s). Thus, we have,
H(s) ≈ H(s) = h0 + h1s +
K∑
i=1
ri
s − pi
. (9)
Such a rational function fit is used for all terms in the
finite element approximation that exhibit general frequency
dependence. It is worth noting that rational function represen-
tations are commonly used for the representation of dispersive
media [18].
We proceed to define the impedance matrix for the Np-port
MIMO system as follows
ZG(s) = sLH(Y + sZ + s2Pe + H(s)Zs)
−1B. (10)
The Taylor expansion of ZG(s) around an expansion complex
frequency point s0 has the form
ZG(s) = sLH
∞∑
i=0
ri(s − s0)i, (11)
where LHri is often referred to as the block moment of
ZG(s). By matching the coefficients (or moments) in the Tay-
lor’s series expansion of (10) with those in the above equation,
the following expressions are obtained for the calculation of
ri:
r0 = R,
r1 = A1r0,
r2 = A1r1 + A2r0,
rn =
n∑
i=1
Airn−i, n ≥ 3,
(12)
where R, A1,A2, ... are defined as
R = (Y + s0Z + s20Pe + H(s0)Zs)
−1B,
A1 = −(Y + s0Z + s20Pe + H(s0)Zs)
−1
(
Z + 2s0Pe +
(
h1 −K∑
i=1
ri
(s0 − pi)2
)
Zs
)
,
A2 = −(Y + s0Z + s20Pe + H(s0)Zs)
−1
(
Pe +
(
K∑
i=1
ri
(s0 − pi)3
)
Zs
)
,
An = −(Y + s0Z + s20Pe + H(s0)Zs)
−1
(
(−1)n
(
K∑
i=1
ri
(s0 − pi)n+1
)
Zs
)
, n ≥ 3.(13)
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Hence, a Krylov subspace is constructed
Kq(A1,A2, ...;R) = colsp[r0, r1, ..., rn−1, r0, ..., rl], (14)
where q is the order of the subspace, ri is ith column of the
matrix rn, n = ⌊q/Np⌋, and l = q −nNp. For the purpose of
model-order reduction, it is desired that q << N.An orthogonalization process is used to develop the or-
thonormal basis for Kq . Let
F = [F1; F2; ...; Fq ]. (15)
be the matrix with columns the generated orthonormal basis.
This matrix is used as the congruence transformation matrix
in the reduction process of the original MIMO system. The
reduced-order impedance matrix is obtained as follows
ZG(s) = sLH(Y + sZ + s2Pe + H(s)Zs)
−1B, (16)
where
Y = FHYF, Zs = F
HZsF, Z = F
HZF,
Pe = FHPeF, L = F
HL, B = F
HB.
(17)
It is evident from (13) that the number of moments that
should be kept in the Taylor series expansion is dependent on
the distance between the expansion frequency and the poles
in the rational function fit of H(s). Thus, if the distance is
sufficiently large, the Taylor series expansion of H(s) can be
safely truncated after only a few terms. In any case, since Zs is
a very sparse matrix, its multiplication with the vectors ri does
not impact adversely the computational efficiency of the model
order reduction process. Thus, higher order derivative terms in
the Taylor series expansion of H(s) can be accommodated, if
needed, at negligible additional computational cost.
IV. VALIDATION STUDIES
A. A microstrip bandpass filter
The first validation study considers the microstrip bandpass
filter depicted (in top and side view)in Fig. 1. The center
strip is, approximately, one-half wavelength long at the center
frequency of the pass band. The resonator is capacitively
coupled to the two strips at its ends. At the ends of the
input and output strips wave ports are used to drive and
terminate the structure. The computational domain used for
the finite element analysis is truncated at its top and side
(parallel to the resonator) boundaries by means of a first order
absorbing boundary condition. The observable quantities are
the scattering parameters over a frequency band centered at
the first resonant frequency.
The objective of this study is to examine the impact of
skin-effect loss on the transmission properties of the filter
for different values of the conductivity of the metallization.
The proposed model order reduction process was used for the
expedient calculation of the response (fast frequency sweep)
over the desired bandwidth. The expansion frequency was set
at 15 GHz.
The magnitude of the calculated S-parameters are depicted
in Fig. 2. For the lossless case, the first resonant frequency
is, approximately, the frequency at which the length of the
Fig. 1. Top view and side view of a single-resonator microstrip bandpassfilter.
resonator is one-half wavelength [19]. For the structure of Fig.
1 this first frequency is 17.22 GHz, as correctly predicted by
the finite element solution. The inclusion of the metal loss
causes the resonant frequency to shift to the left [20]. The
higher the loss the larger the shift. This trend is correctly
captured by the finite element solution.
1.6 1.65 1.7 1.75 1.8
x 1010
−25
−20
−15
−10
−5
0
Frequency (Hz)
|S11| and |S
12| (d
B)
Lossless
σ = 5.8e7 S/m
σ = 3.5e7 S/m
σ = 1.7e7 S/m
Fig. 2. |S11| and |S12| of the microstrip bandpass filter.
B. Microstrip Line on a Debye Substrate
The purpose of the second example study is to demonstrate
the ability of the proposed model order reduction scheme to
handle electromagnetic structures involving dispersive media.
The structure under consideration is a microstrip line of the
same cross-sectional geometry as the one depicted in Fig.
1. However, for this case the substrate is taken to be a
Debye medium with frequency-dependent relative complex
permittivity as follows:
ǫsub = ǫ∞ +A
1 + jωτ, (18)
where ǫ∞ = 2, A = 10, and τ = 5 × 109. The calculated
quantity is the input impedance of a 6 cm-long section of
the microstrip line terminated at a 60-Ohm resistive load. In
addition a point-by-point finite element solution was carried
out at 21 equidistant frequency points over the frequency
bandwidth of 2 GHz. The way the resistive load is introduced
in the finite element model is discussed in [21].
To provide for a reference solution, the input impedance of
the short-circuited microstrip line was calculated using trans-
mission line theory. For this purpose, the effective permittivity
for the microstrip at different frequencies was calculated using
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the formula in [22] with values for ǫsub obtained from (18).
In the above equation H is the substrate thickness and W is
the width of the microstrip. The calculated relative effective
permittivity, ǫeff (f), at each frequency was combined with
the characteristic impedance, Z0, of the microstrip cross
sectional geometry in free space, to calculate the frequency-
dependent characteristic impedance of the microstrip on the
Debye substrate through the formula Z0/√
ǫeff .
The frequency dependence of the real and imaginary parts
of the calculated input impedance is depicted in Fig. 3. The
results from the MOR-assisted fast frequency sweep are in
excellent agreement with the point-by-point finite element so-
lution, validating the accuracy of the proposed MOR method-
ology. The finite element solution results are also in very good
agreement with the transmission line theory results, especially
over the lower frequency range. At higher frequencies a
discrepancy is observed between the finite element solution
result and the transmission line result. This is attributed to the
fringing capacitance at the terminated end of the microstrip
which, even though present in the finite element model, is not
taken into account in the transmission line model.
0 0.5 1 1.5 2
x 109
20
30
40
50
60
Frequency (Hz)
Real(Z
in)
(Ohm
)
(a) resistance
MOR with rational function
Point by point simulation
Transmission line analysis
0 0.5 1 1.5 2
x 109
−20
−10
0
10
20
30
Frequency (Hz)
imag(Z
in)
(Ohm
)
(b) reactance
MOR with rational function
Point by point simulation
Transmission line analysis
Fig. 3. Input impedance of a microstrip line on a Debye substrate.
V. CONCLUSION
In summary, we have proposed and validated a Krylov
subspace-based model order reduction methodology for use
in conjunction with finite element approximation of electro-
magnetic systems containing dispersive materials and sub-
structures characterized in terms of frequency-dependent trans-
fer impedance matrices. The versatility of the proposed method
in handling dispersive materials and transfer functions of
arbitrary frequency dependence stems from its utilization of
a rational function fit of the frequency dependent parameters
in the structure.
VI. ACKNOWLEDGEMENT
This research was supported in part by the Air Force Office
for Scientific Research through a MURI on the “Analysis and
Design of Ultrawide-band and High-power Microwave Pulse
Interactions with Electronic Circuits and Systems.”
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