Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
be numerically less stable than implicit moment matching with
Now, the sensitivity of w.r.t. can be explicitly
help of the projection approach as described in section A. computed by deriving (23) and both – the system matrix for
Furthermore, the error for the transfer function sensitivity is the nominal system and its sensitivity for
– are
proportional to the error of the state vector of the nominal represented as matrix polynomials w.r.t. analogously to (2).
system [10]. This guarantees the convergence of
Hence, the MOR method described in sections A-C is
towards for arbitrary parameters. As a consequence, a applicable.
single run of AMPXT for the nominal system is sufficient to
E. Parametric Model Order Reduction
allow rapid computation of sensitivities of the transfer
function for a whole series of different parameters . The main use of fast sensitivity computations is accelerating
At this point, we did not put any assumption on the structure optimization loops. Of course, one could estimate the
of the sensitivity of the system matrix . This will be variation of the transfer function for a parameter sweep. But
addressed in the next section.
as we are only able to efficiently compute first order
sensitivities, the parameter dependence of the frequency
D. System Interpolation
response cannot be accurately computed for a broad parameter
The analytical dependence of the system matrices on
geometrical or material parameters is not always available in
practice. Most tools like finite element simulators behave like
black boxes and thus only provide system matrices for fixed
parameter values.
For linear material properties like the Young’s modulus in
mechanics or permittivity and permeability in electromagnetics,
the dependency of on can be easily
reconstructed such that the sensitivity of the system matrix
can be explicitly computed.
For geometrical parameters it has been shown that the
parameter dependency can be obtained explicitly as well [9].
But this would require manual extension of the source code of
a finite element simulator because at the time of writing, no
commercially available tool is able to provide .
This is why we decided to use polynomial interpolants of
the system matrices based on a series of system matrices for
fixed parameter values from the neighborhood of the nominal
value . The parameter dependent system matrix is then
represented as a multivariate polynomial
(23)
with and for at least one
.
To be more precise, we generate an initial set of systems
for equally spaced parameter values within the neighborhood
of the nominal value . Starting with 2 interpolation
points, we successively add more from the desired
neighborhood of the nominal value using Chebychev
spacing, until the following relative error for the interpolated
system matrices is below a given threshold:
(24)
In (24),
denotes the system matrix related to
as defined in (2) and is a fixed parameter value from the set
of systems. corresponds to the interpolant as defined
in (23) and is evaluated for all and all where
.
range with this method. This is why we will focus on
parameter dependent ROMs in this section.
The topic of parametric MOR is very complex and a
multitude of methods has been proposed to provide parameter
dependent ROMs, see [11]-[14] and references therein. While
[11] is the most general one, providing simultaneous multipoint
multiple-moment matching w.r.t. both – the complex
frequency and multiple parameters – it matches
an equal number of moments for the complex frequency and
the parameters, which results in larger ROMs, the more
parameters are used. Therefore, we restrict to multi-point
multiple moment matching w.r.t. to and match only the
zeroth and the first moment w.r.t. the parameters. In the next
section we will show that resulting parametric ROMs still
capture the parameter dependency of the transfer function in a
satisfactory way.
Given the projection matrices
for the nominal
system, the reduced system matrices can be obtained
simply by projection of (23) analogously to (9). This method
is denoted with PMORnom from now on and instead of
computing the sensitivity w.r.t. the nominal value via
(21) and (22), the sensitivity of the parametric ROM can be
computed directly by derivation of the reduced order system
matrix w.r.t. and substitution in to (21).
An alternative method called PMORinterp in the sequel is to
generate ROMs for each of the previously generated
FOMs for fixed parameter values and then generate a
parametric ROM via polynomial interpolation. Obviously,
PMORinterp takes more computation time because
projection matrices
have to be computed. But
these extra costs lead to better accuracy for a broad parameter
range as will be demonstrated in the next section. Another
advantage over PMORnom is the fact that the initial FOMs are
not required to have the same state space dimension. Hence,
different discretizations or finite element meshes can be used
for the initial FOMs. In order to assure that the states of the
different ROMs used for interpolation match the same
physical quantities, we apply a state transform prior to the
interpolation step as proposed in [12].
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