41 Reduced Order Models for Systems with Disparate Spatial and Temporal Scales 451 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Forcing Amplitude, q0 Displacement, x30 Fig. 41.2 Bifurcation diagram for full scale nonlinear system for harmonic forcing with! D7:97Hz 41.4 Results The aim is to reproduce the results of the full-scale model using the ROMs. It is needed to have a ROM that can account for a relatively wide range of amplitudes; otherwise MOR would be pointless. Therefore, we excited the system by the white noise with 12 different amplitudes to get 12 independent cases. For each case, the data matrices were recorded as explained in the previous section. Following the data analysis using the four methods in Sects. 41.2.1 and 41.2.2, the resultant modes were mixed using the method in [3, 39]. The obtained modes are then used for ROMs. The 120-dimensional full-scale model of the system can be reduced down to a five-dimensional full POD-based ROM. The full POD-based ROMs were not stable for the dimensions lower than five. Yet, the full-scale model of the system is reduced down to a six-dimensional full POD-based ROM, since five-dimensional models did not correlate well with the fullscale model. For full SOD, four and higher dimensional models were stable and had a good correlation with the full-scale model. The results from the full POD- and SOD-based ROM simulations are depicted in the left columns in Figs. 41.3 and 41.4. These results correspond to the harmonic excitation amplitudes of Œ0:05; 0:12; 0:22; 0:35 . While both POD and SOD have a reasonable accuracy, the four-dimensional full SOD-based model has a slightly better performance than the six-dimensional full POD-based model. This is in consistent with the previous works that were done on SOD-based MOR [3, 39], where SOD-based ROMs were shown to capture the dynamics of the nonlinear systems using lower dimensional models. An interesting result is the effect of separating the data matrices on stabilizing the low-dimensional POD-based ROMs. Now not only the separated POD-based ROMs are stable for all dimensions, but also the performance of the models is improved. This is illustrated in Fig. 41.3, where six-dimensional POD-based and six-dimensional separated POD-based ROMS are compared. The figure shows that the ROM simulation results are slightly improved. The performance of separated SOD-based ROMs is also investigated and reported via Fig. 41.4. The figure shows that separated SOD-based ROM does not improve the results. In some cases the results do not change much and in some cases, e.g. q0 D0:12, the performance of ROM reduces. With a more than 50 times faster ROM, we are able to reproduce the bifurcation diagrams in more details. Figure 41.5 shows the bifurcation diagrams obtained for the four methods. It can be observed that there is a slight improvement in the diagram of the separated POD-based ROM compared to that of the full POD-based ROM. For instance, there is better match for the forcing amplitude in the range of 0:2 to 0:25 for the separated POD diagram with that of the full-scale model in Fig. 41.2. Although this improvement is not significant, it shows the improving nature of the separated POD-based reduced order modeling. The bifurcation diagram of four-dimensional full SOD-based ROM is consistent with that of the full scale model. However, it is little shrunk around the forcing amplitude of 0:31. Also, the diagram of four-dimensional separated SODbased ROM is slightly corrupted for higher amplitudes.
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