Belgirate, Italy, 28-30 September 2005
ERROR INDICATOR TO AUTOMATICALLY GENERATE
DYNAMIC COMPACT PARAMETRIC THERMAL MODELS
E. B. Rudnyi
1
, L. H. Feng
2
, M. Salleras
3
, S. Marco
3
, J. G. Korvink
1
1
IMTEK-Institute for Microsystem Technology, University of Freiburg, Freiburg, Germany (e-mails:
rudnyi@imtek.de, korvink@imtek.de).
2
ASIC & System State-key Laboratory, Microelectronics Department, Fudan University, Shanghai,
China. (e-mail: lhfeng@fudan.edu.cn).
3
Departament d'Electronica, Universitat de Barcelona, Barcelona, Spain (e-mails: msallera@el.ub.es,
santi@el.ub.es).
ABSTRACT
During recent years, several groups have shown that
parametric model reduction is possible in general. In
particular, it has been shown that it allows us to generate
compact thermal models while preserving film
coefficients as parameters. Unfortunately, in order to run
algorithms, a user should specify how many moments
and what type to generate in advance and there were no
formal rules to this end but a "trial and error" approach.
We present an approach based on a local error in the
transfer function and show that it can automate the
process to build a dynamic compact parametric thermal
model in much greater extent. We demonstrate that our
algorithm can preserve three film coefficients for the first
thermal model and material properties such as heat
conductivity and heat capacity for the second model.
1. INTRODUCTION
Model reduction is a rapidly developing area of
mathematics [1]. It allows us to take a high-dimensional
finite element model developed at device level simulation
and convert it efficiently into a low-dimensional
approximation for system level simulation [2]. Model
reduction approaches have been successfully applied to a
thermal problem to automatically generate a dynamics
compact thermal model [3][4][5][6][22].
However, in its original form model reduction does
not allow us to preserve parameters in the system matrices
that naturally arise in many applications. Fortunately, a
new development, that is, parametric model reduction,
allows us to overcome this limit.
In our knowledge, the first work on parametric model
reduction has been presented by Weile et al [7] in 1999
and applied to describe frequency depended surfaces in
[8]. This approach has been generalized from two to many
parameters in [9] and in parallel re-discovered in
[10][11][12]. We have suggested an empirical solution to
a similar problem in [13] and an alternative algorithm in
[14]. Note that different authors use different names for
the same method: multiparameter model reduction in [9],
multidimensional model reduction in [10][11] and
multivariate model reduction in [12]. Our choice in this
respect is parametric model reduction as it allows us to
preserve parameters in system matrices in the symbolic
form.
In [12][14][15], this approach has been successfully
applied to a thermal problem when film coefficients have
been preserved as symbols in a reduced model. Although
these works have demonstrated that this is the right way
to go, an important practical question remains
unanswered. That is, how to choose moments to include
into the reduced model. A straightforward approach to
choose some order and then generate all the moments up
to this order does not scale well with the number of
parameters [9]. For example, if we choose to preserve four
film coefficients then a reduced model made from all first
derivatives has the dimension of 6, a reduced model made
from all second derivatives has the dimension of 21, and
a reduced model from all third derivatives already has the
dimension of 56 (see Appendix F in [9]). At the same
time, we may need derivatives of higher order than three
to describe accurately transient behavior of the original
model.
The explosion in the dimension of a reduced model
is due to mixed moments (mixed derivatives). In [15] the
authors have observed that one can actually ignore mixed
moments in the case of a thermal problem and proved this
by numerical simulation. However, even in this case it is
unclear how to choose the number of moments along each
parameter automatically. Although time in the form of the
Laplace variable formally looks like the film coefficient in
the transfer function, we may need more moments along
the time axis. In [15] the authors have limited themselves
E. B. Rudnyi, L. H. Feng, M. Salleras, S. Marco, J. G. Korvink
Error Indicator to Automatically Generate Dynamic Compact Parametric Thermal Models
to a stationary problem and have not researched this
problem further.
The use of local error estimators has been researched
in [16][17] (see also discussion in [2]). Error indicators
for Arnoldi-based model reduction have been suggested in
[18]. In the present paper, we use these results as
inspiration for a heuristic procedure suited for parametric
model reduction. We suggest an approach that controls
the dimension of the reduced model automatically based
on local error control. We apply the approach to two
thermal models and report our numerical observations.
First is a thermal model of a microthruster unit [19] (see
also [13][14]) where the goal is to preserve three film
coefficients. Second is a thermopile based IR detector [21]
when a compact thermal model should preserves material
properties of the gas in the symbolic form.
2. OVERVIEW OF PARAMETRIC MODEL
REDUCTION
Let us briefly review the application of parametric model
reduction to a thermal problem. The discretization in
space by the finite element/volume/difference method of
the heat transfer equation leads to a system of ordinary
differential equations as follows
?
E+q
i
E
i
i
?
( )
dT(t)
dt
+K+p
i
K
i
i
?
( )
T(t)=Bu(t)
y(t)=CT(t)
, (1)
where
?
T(t)is the vector of unknown temperatures at the
nodes.
E and
?
K are the heat capacity and heat
conductivity system matrices,
?
B is the input matrix, and
?
C is the output matrix. The vector
?
u comprises input
functions such as heat sources. The output matrix
specifies particular linear combinations of temperatures
that of interest to an engineer. Our goal is to preserve the
parameters
?
q
i
and
?
p
i
in the symbolic form in the reduced
model (a film coefficient or material property). A
parameter contributes to the global system matrix by
means of the matrix
?
E
i
or
?
K
i
.
The transfer function of (1) can be expressed as
follows
?
H(s)=C{s(E+q
i
E
i
i
?
)+Kp
i
K
i
i
?
)}
?1
B
, (2)
and in addition to the Laplace variable s it contains the
parameters
?
q
i
and
?
p
i
.
Model reduction is based on an assumption that there
exists a low-dimensional subspace
?
V that accurately
enough captures the dynamics of the state vector
?
T(t):
?
T?Vz. (3)
In order to find such a subspace
?
V that does not
depend on parameters to preserve, the transfer function (2)
can be treated as a function in many variables (
?
s,
q
i
and
?
p
i
) and one can perform its multivariate expansion. Then
V is taken as a subspace that spans multivariate moments
of (2) (see [9]-[15]). This way,
?
V does not depend on
parameters in (1) and (2).
Provided
?
V is known, one obtain a low-dimensional
model by projecting (2) on
?
V as follows
?
{V
T
EV+q
i
V
T
E
i
V
i
?
}
dz(t)
dt
+
{V
T
KV+p
i
V
T
K
i
V
i
}z(t)=V
T
Bu(t)
. (4)
Eq (4) preserves the original parameters in the
symbolic form and as a result we refer to this approach as
parametric model reduction.
3. LOCAL ERROR CONTROL
We have limited ourselves to the Single-Input-Single-
Output case when the transfer function (2) is a scalar, the
input matrix
?
B is a vector and the output matrix
?
C is a
single row. Another simplification is that we ignore
mixed moments following the observation in [15]. As a
result, the model reduction algorithm is practically
equivalent to that described in [15] except that we take
into consideration the Laplace variable as well.
We assume that a user specifies the range of interest
for the frequency and parameters:
?
s
min