Data-driven low-dimensional model of a sedimenting flexible fiber (2024)

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  Figure 1

  A flexible filament of length $L$ settling under an external force ${\mathbf{F}}_g$ in a quiescent Newtonian fluid. The filament begins at an arbitrary initial inclination ${\theta }_0$ relative to the $x$ axis and evolves until it reaches a terminal shape. The fiber is modeled as a series of $N$ beads of radius $a$ connected by springs, with the center bead, which has position $\mathbf{c}\left(t\right)$, shown in red.

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  Figure 2

  The evolution of the shape of a filament settling in a quiescent Newtonian fluid from an initial orientation to a common terminal shape at $\mathcal{B}=1000$ for all initial angles of orientation within the training dataset.

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  Figure 3

  The trajectories $\mathbf{c}\left(t\right)$ of the center bead of a filament settling in a quiescent Newtonian fluid at $\mathcal{B}=1000$ for all initial angles of inclination within the training dataset. The initial positions are denoted by symbol $▾$, and the terminal positions are denoted by the symbol $\times$.

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  Figure 4

  The evolution of the shape of a filament settling in a quiescent Newtonian fluid from a common initial angle of orientation of $\pi /4$ to a terminal shape for all $\mathcal{B}$ within the training dataset.

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  Figure 5

  Block diagram for data-driven model combining the autoencoder and temporal-evolution scheme. The temporal-evolution neural network, expanded in red, can be separated into two distinct neural networks forecasting the evolution of latent representation of the shape $\mathbf{h}(t+\tau )$ and the shape-dependent change in position $\mathbf{c}$; in practice, these can be forecasted by a single neural network.

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  Figure 6

  (a)Loss over the testing data for each autoencoder architecture as a function of latent dimension. Block diagrams for the autoencoder neural network architectures: (b)No $\mathcal{B}$, (c)Encoder $\mathcal{B}$, (d)Decoder $\mathcal{B}$, and (e) Double $\mathcal{B}$.

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  Figure 7

  Evolution of the shape of a filament settling in quiescent Newtonian fluid given an initial orientation from the best and worst forecasts (red) and the true evolution (black) at a given $\mathcal{B}$. Here, $\mathcal{B}$ is within the training dataset, and the initial angles of inclination are not.

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  Figure 8

  Evolution of the shape of a filament settling in quiescent Newtonian fluid given an initial orientation from the best and worst forecasts (red) and the true evolution (black) at a given $\mathcal{B}$. Here, neither $\mathcal{B}$ nor the initial angles of inclination are within the training dataset.

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  Figure 9

  Trajectory of the center bead $c\left(t\right)$ a filament settling in quiescent Newtonian fluid given an initial orientation from the best (solid lines) and worst (dashed lines) forecasts and the true evolution at a given $\mathcal{B}$. The initial positions are denoted by symbol $▾$, and the terminal positions are denoted by the symbol $\times$, with the true and predicted positions in black and red, respectively. In (a), $\mathcal{B}$ is within the training dataset but, at initial angles of inclination, not within the training data; in (b), neither $\mathcal{B}$ nor the initial angles of inclination are within the training data.

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  Figure 10

  Ensemble-average error vs time for forecasts of the evolution of a filament settling in quiescent Newtonian fluid at each $\mathcal{B}$ (gray) and averaged over all $\mathcal{B}$ (red). In (a), $\mathcal{B}$ is within the training dataset but, at initial angles of inclination, not within the training data; in (b), neither $\mathcal{B}$ nor the initial angles of inclination are within the training data.

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