128 D. Chelidze Dynamical Consistency, ζ Subspace Dimension, k 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 POD: Harmonic Forcing POD: Random Forcing SOD: Harmonic Forcing SOD: Random Forcing 5 10 15 20 0.7 0.75 0.8 0.85 0.9 0.95 1 Subspace Dimension, k Subspace Robustness, γk POD: Harmonic Forcing POD: Random Forcing SOD: Harmonic Forcing SOD: Random Forcing 2 4 6 8 10 0.94 0.96 0.98 1 (k) Fig. 11.2 Dynamical consistency (top) and subspace robustness (bottom) estimated for different subspace dimensions (number of modes) and SOD subspaces of five or higher dimensions were consistent. While the lower-dimensional POD subspaces were robust, they can only provide a good ROM if the nonlinear extension of the first LNM is very small or negligible. The subspace robustness for the SOD subspaces scales monotonically, while for the POD it does not. In other words, the robustness always improves for SOD, but can get worse for POD with the increase in dimension. Since POD maximizes the energy in the projected signal, randomness of the driving signal distorts the dominant direction in the phase space. In contrast, since SOD maximizes the smoothness of the projection, the randomness of the excitation or initial conditions has much less effect on the identified subspaces irrespective of the dimensionality of subspaces. The variation in the forcing amplitude showed that the POD-based five-dimensional ROM follows closely the actual trajectories, but shows some divergence for certain forcing amplitudes. The SOD-based five-dimensional ROM showed a very good approximation of the actual dynamics. SOD ROM provides the most faithful representation of the actual dynamics by examining the corresponding phase portraits. Acknowledgements This paper is based upon work supported by the National Science Foundation under Grant No. 1100031.
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