Parametric Shape Optimization of Flagellated Micro-Swimmers Using Bayesian Techniques

TL;DR

The paper tackles the problem of designing efficient flagellated microswimmers at very low Reynolds numbers by integrating Free-Form Deformation (FFD) for head shaping with Scalable Constrained Bayesian Optimization (SCBO) to explore a high-dimensional shape space. By formulating the optimization around two cost functions, $J_1$ and $J_2$, and solving the boundary-value problem with Boundary Element Method (BEM), the authors identify novel morphologies (e.g., water-drop head and bullhead head) that outperform ellipsoidal benchmarks for both monoflagellated and biflagellated swimmers. Key contributions include a flexible head-shape framework, a constrained BO workflow with a trust-region strategy, and quantitative demonstrations of speed and efficiency gains alongside insightful trade-offs between flagellar amplitude, wavelength, and power. This work advances microswimmer design and microrobotics by revealing morphologies that balance propulsion performance and energy expenditure, and suggests future extensions to non-Newtonian environments and richer shape representations.

Abstract

Understanding and optimizing the design of helical micro-swimmers is crucial for advancing their application in various fields. This study presents an innovative approach combining Free-Form Deformation with Bayesian Optimization to enhance the shape of these swimmers. Our method facilitates the computation of generic swimmer shapes that achieve optimal average speed and efficiency. Applied to both monoflagellated and biflagellated swimmers, our optimization framework has led to the identification of new optimal shapes. These shapes are compared with biological counterparts, highlighting a diverse range of swimmers, including both pushers and pullers.

Figures (23)

• Figure 1: Schematic of microswimmers represented by an arbitrary flagellum $F_i$ with parameters $(\alpha_i, \beta_i, \gamma_i, \delta_i, \lambda_i, R_i^t, r_i, l_i)$ in the frame $(e_1, e_3)$ (left) and in the frame $(e_1, e_2)$ (right).
• Figure 2: Discretization of the reference swimmer $S^0$ used to obtain values in \ref{['table:S0_res']}.
• Figure 3: Illustration of the FFD method deforming a domain $D$ containing a $\Theta$ shape via a $\mu$ vector using the $T$ application described by \ref{['eq:ffd']}.
• Figure 4: Gaussian process on a toy function. The true function (red line) is approximated by the GP mean (blue line) according the observations (black point). The Gaussian posterior distribution for $x=3.3$ is depicted (green). The associated point-wise confidence intervals at level 95% are displayed (blue). A set of $100$ GP samples is displayed (gray).
• Figure 5: SCBO representation: (1) Initial shapes are taken from $\Omega$ (represented by crosses), and the trust zone is defined (a grid with the central point surrounded and the edge of the zone defined by a square). (2a) The Gaussian process models associated with the objective $J$ and the constraints $c$ are constructed. (2b) $r$ random shapes (represented by triangles) are taken from the trust region. (2c) A batch of $q$ realizations $\{\hat{J}(S_i)\hat{c}(S_i)^T\}_{i=1}^r$ is calculated. For each realization, the $x$ axis corresponds to the $S$ shapes in $\Omega$ and is discretized by the $\{S_i\}_{i=1}^r$ (the triangles) chosen previously. The $y$ axis shows the realizations. For simplicity, a single graph is drawn for the objective and the constraints. For each realization, we keep the best shape (completely filled triangle). (2d) For the $q$ best shapes $\{S_i'\}_{i=1}^q$, we calculate the real value of the objective and the constraints. (2e) These $q$ new shapes are added to the previous observations. (2f) The trust region is readjusted by modifying its center and length.