Online proceedings - EDA Publishing Association

We assume that
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May 2011, Aix-en-Provence, France
does not become singular for more
(7)
than a finite set of points and
and . This is the most general form of systems that
(8)
can be treated with the proposed MOR algorithm. Spatial Fast and accurate computation of these quantities is useful
discretizations of partial differential equations like the heat for optimization loops, where the gradient of the objective
equation, Maxwell’s equations, or mechanical systems as well function would have to be approximated by finite differences
as RLC circuit equations fit into this framework.
otherwise.
The transfer function of system (1) is defined as
Another objective is the acceleration of parameter studies
(6)
with the help of parameter dependent reduced order models,
which approximate the parameter dependency of the transfer
The polynomial degree of is called order of the function of the original system (1).
system. But when speaking of model order reduction, we
III. METHODOLOGY
usually mean a reduction of the state space dimension of
the system. This is a common misconception to be found in A. Model Order Reduction
literature. It can be justified by the fact that every -th order The MOR method used in our methodology is a projection
system can be equivalently transformed into a first order based approach. It constructs (bi-)orthonormal projection
system with identical transfer function but an increased state matrices
such that the reduced system
space dimension of . In this sense, the terms order and
state space dimension relate to each other.
For systems resulting from finite element discretization, the
state space dimension of the system corresponds to the total
number of degrees of freedom (DOF). In mechanics, mostly
second order systems arise and , and resemble the
stiffness, damping, and mass matrix of the model.
A frequent property of the system (1) is reciprocity, which
is defined as the symmetry of the transfer function for all
where is defined. This is especially the case when
is symmetric for all and if a scaling function
exists such that .
Such systems will be called symmetric from now on.
First of all, we are interested in the frequency response of
the system which is obtained by evaluating for a discrete
set of frequency points , ,
. From (6) it follows that solutions of different
linear systems of equations have to be computed, which is
usually very time consuming. Time domain simulation of the
system may be even impossible for large state space
dimensions.
The second problem to be solved is the incorporation of
design parameters or manufacturing tolerances, such that the
system matrices as well as the solution vector and the
transfer function become parameter dependent. In the
sequel we assume that only depends on a parameter
. In the majority of cases the input and output matrices
and respectively are incidence matrices that pick
certain nodes of the model for excitation or measurement and
therefore do not depend on parameters. The feed through
matrix may be parameter dependent as well, but this case
is omitted for the sake of simplicity as the MOR algorithm is
not affected by the presence of a feed through matrix. Finally,
an extension to multiple parameters is straight forward and
will be demonstrated in section IV.
Furthermore, we are interested in the first order sensitivities
of the transfer function and the output respectively
w.r.t the parameter at a given nominal value , which
are given as
matrices are obtained from
(9)
(10)
(11)
(12)
Obviously, the corresponding reduced order model (ROM)
has the same number of inputs and outputs, and . For
behavior modeling and system level simulation this means that
the full order model (FOM) can be seamlessly replaced by the
ROM. The transfer function of the ROM is defined
analogously to (6). Additionally, the -th order structure of
(1) will be preserved, which prevents the costly alternative of
reducing an equivalent first order system with increased state
space dimension. Finally, the orthonormality of the projection
matrix preserves symmetry and definite properties of the
system matrices, such that passivity and stability are preserved
for the ROM as well.
A detailed description of the method to construct
is beyond the scope of this paper, but we will give a brief
outline of the essentials. First of all, we utilize multi-point
moment matching. That means that a certain amount of the
Taylor coefficients of the transfer function of the reduced
order model for certain expansion points is equal to those of
the full order transfer function. This is not to be
misinterpreted as Taylor approximation in terms of a truncated
Taylor series, but more like the approximation of a rational
function with high numerator and denominator degrees by
another rational function with lower degrees. The precise
mathematical term is Padé approximation. If the system is not
symmetrical and only a single projection matrix
is
used, we speak about Padé-type approximation. The link
between moment matching and Krylov subspaces is
thoroughly studied in [2]. A common synonym for multipoint
moment matching is rational interpolation, indicating
that the ROM interpolates the transfer function at selected
frequency points up to certain derivative orders. Basically, if
both -th Krylov subspaces associated with an expansion
point are contained in the column spaces of and
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