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100%
0.1%
1e-04%
1e-07%
1e-10%
1e-13%
10 -2 10 0 10 2 10 4 10 6
Frequency (Hz)
Fig. 2. Progress of the AMPXT error indicator
0.01%
1e-04%
1e-06%
err for H(s)
1e-08%
err for H (s) E
err for H (s) θ
1e-10%
10 2 10 3 10 4 10 5 10 6
Frequency (Hz)
11-13
May 2011, Aix-en-Provence, France
TABLE III
RUNTIMES AND MAXIMUM RELATIVE ERRORS FOR PARAMETER SWEEP
Fig. 3. Maximum relative errors of nominal ROM transfer function and
Maximum relative error for parameter sweep w.r.t. 0.08%
sensitivities according to (16)
dimension of 72. Using the FOM reference solutions, we neighborhood of the nominal value.
computed the exact relative errors of the ROM transfer For PMORinterp, a parameter sweep for the magnitude and
function and its sensitivities w.r.t. and shown in phase of the ROM transfer function is plotted for selected
Figure 3. The plots have a maximum of 0.06% and thus input-output-pairs in Figures 5 and 6. Compared to ,
clearly demonstrate that the error indicator must not be clearly affects a broader portion of the frequency range.
misinterpreted as an error bound, but is sufficient to monitor
V. CONCLUSIONS AND OUTLOOK
the convergence of ROM.
In summary, the computation of 500 frequency samples of In this paper we presented an extension of existing MOR
the transfer function and its sensitivities w.r.t. and takes methods for rapid and accurate computation of the frequency
roughly 51 minutes for the FOM opposed to a total of 75 response and its sensitivities w.r.t. arbitrary parameters for
seconds for generating and evaluating the parametric ROMs large scale finite element models. Optimization tasks with an
with PMORnom. Hence, we obtained a speedup factor of objective function dependent on the transfer function will
40.6.
greatly benefit from the obtained speedup factor of more than
For parameter sweeps, the speedup through model order 40.
reduction is even more extreme. We considered an increment The lack of tools for the generation of parameter dependent
of m for the suspension beam thickness , such that 21 full order models has been overcome with a system
parameter values are obtained in the interval of m.
interpolation approach that proved to be practical. For the
sake of simplicity, we focused on interpolation of a single
This results in a variation of rougly w.r.t the nominal
parameter at a time, but the method can be easily extended to
value of . For the Young’s modulus we considered 21
multivariate polynomial interpolation involving cross terms
parameter values in the interval of
GPa, yielding
for the interpolation polynomial. However, the more
a variation of w.r.t the nominal value of . This sums parameters are involved, the more expensive the
up to 41 distinct transfer function evaluations for 500
10%
frequency points each.
Table III lists the corresponding runtimes, speedup factors, 1%
and the maximum relative transfer function errors as defined
in (16) taken over all 500 frequency points for all 41
0.1%
parameter values. Note that we did not use interpolated 0.01%
parametric FOMs for the computation of the full order transfer
0.001%
functions which would have increased computation time even
130 140 150 160 170 180 190
further. Instead, we generated 41 single FOMs for fixed
E (GPa)
PMORnom
parameter values and the time to generate these FOMs and
PMORinterp
export them to MATLAB ®
0.001%
was not accounted for the time
10%
measurements.
1%
Figure 4 shows a parameter dependent comparison of the
0.1%
maximum transfer function errors obtained with method
PMORnom and PMORinterp over the frequency range of
0.01%
0.001%
interest. Clearly, PMORinterp provides an accurate
2.8 3 3.2
approximation of the transfer function over the whole
θ (µ m)
3.4 3.6 3.8
parameter range while PMORnom is only accurate in the Fig. 4. Maximum relative transfer function errors for parameter sweep
FOM
PMORnom
PMORinterp
Total costs for transfer function parameter sweep for 500
frequency points and 41 parameter values
25.1h
Total costs for generation of param. ROMs w.r.t. and 74.9s
Total costs for parameter sweep w.r.t. and 27.2s
Speedup factor for parameter sweep
885x
Maximum relative error for parameter sweep w.r.t. 1.07%
Maximum relative error for parameter sweep w.r.t. 78.93%
Total costs for generation of parametric ROM w.r.t. 156.2s
Total costs for generation of parametric ROM w.r.t. 371.8s
Total costs for parameter sweep w.r.t. and 27.6s
Speedup factor for parameter sweep
163x
Maximum relative error for parameter sweep w.r.t. 0.84%
70