Multiobjective optimization for flapping foil hydrodynamics with a multitask and multifidelity approach

We develop a multitask and multifidelity Gaussian process (MMGP) model to accurately predict and optimize the multiobjective performance of a flapping foil while minimizing the cost of high-fidelity data. Through a comparison of three kernels, we have selected and applied the spectral mixture kernel and validated the robustness and effectiveness of a multiacquisition function. To effectively incorporate data with varying levels of fidelity, we have adopted a linear prior formula-based multifidelity framework. Additionally, Bayesian optimization with a multiacquisition function is adopted by the MMGP model to enable multitask active learning. The results unequivocally demonstrate that the MMGP model serves as a highly capable and efficient framework for effectively addressing the multiobjective challenges associated with flapping foils.

Figure 1

Schematics of (a) pitching motion, (b) heaving motion, and (c) combined-heaving-and-pitching motion of a foil.

Figure 2

Sketch of the calculation domain.

Figure 3

Composition and workflow of the MMGP framework. The MMGP combines the multifidelity GP based on the Hadamard multitask GP and the independent multitask GP with multi-AF. The Hadamard multitask GP and the independent multitask GP belong to the multitask GP, which is extended from the SF-GP. The multifidelity GP works with low-fidelity data sourced from interacting with the environment and high-fidelity data derived from the high-fidelity ergodic database. The database is divided into the training set and the test set based on whether the data are used by the MMGP model.

Figure 4

The isosurfaces under $\mathrm{St}$ corresponding to the $max{\widehat{f}}_{\mathrm{ct}}$ and $max{\widehat{f}}_{\eta }$. Inside the yellow-contour, we present the thrust curve and the instantaneous flow field diagram under the $max{\widehat{f}}_{\mathrm{ct}}$ case. Subplots (a)–(h) represent the instantaneous vorticity fields corresponding to the specific instants (a)–(h) on the thrust curve, as denoted by the normalized time parameter $\tau =\omega t/2\pi$.

Figure 5

The ${\mathcal{M}}_{\mathrm{ct}}$ of different kernels during iterations.

Figure 6

The three-dimensional view of low-fidelity $C_T$ comparison between the uniformly sampled result (left) and the GP model with 600 points (right).

Figure 7

The training process of the MMGP model. We fuse two fidelities of the multitask GP model in the MMGP model. The low-fidelity model provides trend and gradient information with 1200 points. The high-fidelity model initially resembles the low-fidelity model and gradually converges to the raw high-fidelity database during the training process. The color bars correspond to the values represented by the isosurfaces in the respective plots, with different color maps being employed for $f_{\mathrm{ct}}$ and $f_{\eta }$ for clearer distinction.

Figure 8

(a) The prediction loss of multiobjective function $L_i$ during the training process. (b)–(d) The instantaneous vorticity diagram's optimization shown in the early, middle, and late stages of training are also marked on the curve. (b) For $[\mathrm{St},y_0,{\theta }_0,\psi ]=[0.26,0.2,15,150]$, $[C_T,\eta ,P_2]=[-1.06,0.1,-0.38]$. (c) For $[\mathrm{St},y_0,{\theta }_0,\psi ]=[0.26,0.1,15,30]$, $[C_T,\eta ,P_2]=[6.17,0.049,5.3]$. (d) For $[\mathrm{St},y_0,{\theta }_0,\psi ]=[0.3,0.4,20,0]$, $[C_T,\eta ,P_2]=[21.3,0.052,17.49]$. The color bar below represents the vorticity magnitude in panels (b)–(d).

Figure 9

The Pareto front in the objective space. The square, triangle, and pentagon points indicate the selected optimal points corresponding to tasks $P_1$, $P_2$ and $P_3$, where $P_1$ is for $[\mathrm{St},y_0,{\theta }_0,\psi ]=[0.22,0.5,15,90]$, $P_2$ is for $[\mathrm{St},y_0,{\theta }_0,\psi ]=[0.22,0.3,35,0]$, and $P_3$ is for $[\mathrm{St},y_0,{\theta }_0,\psi ]=[0.26,0.4,20,60]$.

Figure 10

Compare the two models' global optimization ability and convergence by predicting the two tasks. (a) The prediction of maximum ${\widehat{f}}_{\mathrm{ct}}$ during the training process. (b) The prediction of maximum ${\widehat{f}}_{\eta }$ during the training process. (c) The ${\mathcal{M}}_{\mathrm{ct}}$ of ${\widehat{f}}_{\mathrm{ct}}$ during the training process. (d) The ${\mathcal{M}}_{\eta }$ of ${\widehat{f}}_{\eta }$ during the training process.

Figure 11

Compare the performance of multi-AF with different basic acquisition functions. (a) The prediction of maximum ${\widehat{f}}_{\mathrm{ct}}$ during the training process. (b) The ${\mathcal{M}}_{\mathrm{ct}}$ of ${\widehat{f}}_{\mathrm{ct}}$ during the training process.