Interdisciplinary Applied Mathematics

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FIGURE 17.12. Geometry of a cross channel used in transient analysis.

weighting. The minimal weighting is set to be 1/6. The steepness parameter is chosen to be 8.5 in this case.

Figure 17.13(b) shows the U-velocity prediction in the main channel, and both methods (standard and weighted KL techniques) produce almost identical    results    at    steady    state,    though    the    reduced-order    model    using

weighted KL basis gives slightly better accuracy at the beginning of the simulation. In Figure 17.14(a), we compare the performance of weighted and standard KL techniques by fixing the number of snapshots and investigating the number of basis functions required with each technique to get comparable accuracy. We use 60 snapshots in each approach. For the weighted KL technique we use 4 basis functions for the ж-component of the velocity, 4 basis functions for the у-component of the velocity, and 3 basis functions for the pressure (referred to as (4U+4V+3P) in Figure 17.14(a)). To reproduce the results obtained by the weighted KL technique, 6 basis functions had to be    used    for    the    ж-velocity,    6 basis    functions    for    the    y-

velocity, and 6 basis functions for the pressure (denoted by (6U+6V+6P)). For comparable accuracy, we need almost twice the number of basis functions in a standard KL approach than in the weighted KL technique. This result indicates that when the number of snapshots is fixed, a weighted KL technique needs fewer basis functions than a standard KL technique to reproduce full simulation results. The use of fewer basis functions leads to a computationally more efficient approach. We also observed that a technique with (4U+4V+3P) basis functions is almost twice as fast as the technique with (6U+6V+6P) basis functions; i.e., a reduced-order model based on the weighted KL technique can be twice as fast as the reduced-order model based on the standard KL technique while achieving essentially the same

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