# Parametric shape optimization of flagellated microswimmers using Bayesian techniques

Lucas Palazzolo, Mickael Binois, Luca Berti, Laëtitia Giraldi

► To cite this version:

Lucas Palazzolo, Mickael Binois, Luca Berti, Laëtitia Giraldi. Parametric shape optimization of flagellated microswimmers using Bayesian techniques. SAMO 2025 - International Conference on Sensitivity Analysis of Model Output, Apr 2025, Grenoble, France. hal-05010013

HAL Id: hal-05010013

https://hal.science/hal-05010013v1

Submitted on 28 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Understanding and optimizing the design of helical microswimmers is crucial for advancing their application in various fields. The Boundary Element Method (BEM) is used to simulate the dynamic [1]. Due to the complexity of our model, we focus on parametric shape optimization. We employ the Free-Form Deformation (FFD) technique [2], which provides a sufficiently complex admissible shape space. This is integrated with the Scalable Constrained Bayesian Optimization (SCBO) method [3], ideal for high-dimensional constrained optimization problems.

<figure><img src="image_2.png" /></figure>

Published in Physical Review Fluids.

## Flagella geometry of microswimmers

The swimmer, denoted by $S$, is composed of a head $H$ and a set of $n_f$ of flagella, i.e. $S = H \cup_{i=1}^{n_F} F_i$. Each flagellum $F_i$ is a tube of radius $r$ with a centerline of total length $L$.

<figure><img src="image_3.png" /><figcaption>Figure 1. Schematic of microswimmers represented by an arbitrary flagellum.</figcaption></figure>

<table><caption>Table 1. Fixed and optimized parameters of flagella.</caption><thead><tr><th>Fixed parameters</th><th>Optimized parameters</th></tr></thead><tbody><tr><td>r</td><td>Radius of the flagella</td><td>Amplitude</td></tr><tr><td>L</td><td>Total length</td><td>Wavelength</td></tr><tr><td>l</td><td>Junction distance</td><td>α, β, γ, δ Angles for position and orientation</td></tr></tbody></table>

## Head geometry of microswimmers

The head is obtained as the image of the **Free-Form-Deformation** (FFD) map $T$ applied to the unit sphere, where **control points** $P \in \mathbb{R}^{3M}$ are shifted by a vector $\mu \in \mathbb{R}^{3M}$,

$$T: D \times \mathbb{R}^{3 \times M} \rightarrow D(\mu) \\ (\Theta, \mu) \mapsto (\psi^{-1} \circ \hat{T} \circ \psi)(\Theta, \mu),$$

where $\hat{T}(\cdot) = \sum_{i=0}^{J} \sum_{j=0}^{J} \sum_{k=0}^{K} b_{i,j,k}^{I,J,K}(\cdot) (P + \mu)_{i,j,k}$ and $b_{i,j,k}^{I,J,K}(\cdot)$ is the tensor products of **Bernstein polynomials**.

<figure><img src="image_4.png" /><figcaption>Figure 2. Illustration of the FFD method deforming a domain $D$ containing a $\Theta$ shape via a $\mu$ vector using the $T$ map.</figcaption></figure>

## Dynamic of microswimmers

Due to the **low Reynolds number** in microswimmer dynamics, the fluid behavior is governed by the **Stokes** equations given by

$$\left\{ \begin{array}{l} -\mu \Delta u + \nabla p = 0 \\ \nabla \cdot u = 0 \\ u = U + \Omega \wedge (x - x^H) \\ u = U + \Omega \wedge (x - x^H) + \omega e_1^{F_i} \wedge (x - x^{F_i}) \\ \|u\|, p \to 0 \end{array} \right. \quad \begin{array}{l} \text{in } \mathbb{R}^3 \setminus S, \\ \text{in } \mathbb{R}^3 \setminus S, \\ \text{on } \partial H, \\ \text{on } \partial F_i, \\ \text{as } x \to +\infty, \end{array}$$

for all $i \in \{1, \dots, n_F\}$.

<table><caption>Table 2. Parameters of the Stokes PDE.</caption><thead><tr><th>Parameter</th><th>Description</th></tr></thead><tbody><tr><td>u, p</td><td>Velocity and pressure of the fluid</td></tr><tr><td>U, Ω</td><td>Translational and angular velocities of the swimmer</td></tr><tr><td>ω = -2πrad.s<sup>-1</sup></td><td>Angular velocity of the flagella</td></tr><tr><td>e<sub>1</sub><sup>Fi</sup></td><td>Directional axes of the flagellum F<sub>i</sub></td></tr><tr><td>x<sup>Fi</sup>, x<sup>H</sup></td><td>Junction point of the flagellum F<sub>i</sub> and head center of mass</td></tr></tbody></table>

## Boundary Element Method

1. Green tensor kernel : $u(y) = - \int_{\partial S} G(x, y) \sigma(u, p)(x) n(x) dx$ for all $y$ in $\mathbb{R}^3$.

2. Self-propulsion : $\int_{\partial S} \sigma(u, p)(x) n(x) dx = 0$, $\int_{\partial S} \sigma(u, p)(x) n(x) \wedge (x - x^H) dx = 0$.

3. $\mathbb{P}^1$ finite element space of $\partial S$.

## Shape optimization problem

* Admissible shapes : $S_{ad} = \mathcal{H}_{ad}^{\varepsilon} \times \mathcal{F}_{ad}^{P}$

* Admissible flagella : $\mathcal{F}_{ad}^{P} = \{ F \subset \mathbb{R}^3 \mid (\lambda, R^t, \alpha, \gamma, \beta, \delta) \in \mathcal{P} \text{ and } |F| = |F^0| \}$ where $\mathcal{P}$ is a compact set.

* Admissible heads : $\mathcal{H}_{ad}^{\varepsilon} = \{ H = T(H^0, \mu) \mid \mu \in \mathcal{V} \text{ and } \|H| - |H^0| \le \varepsilon \}$ where $\mathcal{V}$ is a sufficiently small closed set around the control points to **prevent mesh collapse**.

* Cost functions : $J_1$ for the **mean velocity** and $J_2$ for the **mean efficiency** along $e_1$,

$$J_1(S) = - \frac{\bar{U}_1(S)}{\bar{U}_1^0} \quad \text{and} \quad J_2(S) = - \frac{\bar{U}_1(S)}{\bar{U}_1^0} \times \frac{\bar{P}^0}{\bar{P}(S)}.$$

Finally, the optimization problem for $i \in \{1, 2\}$ is formulated as $\inf_{S \in S_{ad}} J_i(S)$.

## High dimensional Bayesian optimization method

Gaussian Process (GP) is used to model costly functions from a given dataset $D_n$.

$$\text{Prior: } J \sim \mathcal{GP}(0, k) \quad \text{with} \quad \left\{ \begin{array}{l} \mu(x) = K(x, X_n)K_n^{-1}Y_n, \\ \Sigma(x) = K(x, x) - K(x, X_n)K_n^{-1}K(X_n, x), \\ (K(X, X'))_{i,j} = k(X_i, X_j') \text{ and } K_n = K(X_n, X_n). \end{array} \right.$$

The SCBO method is based on the **trust region** approach, which outperforms conventional Bayesian Optimization (BO).

<figure><img src="image_5.png" /><figcaption>Figure 3. Illustration of the Scalable Constrained Bayesian Optimization (SCBO) method developed in [3].</figcaption></figure>

## Results

<figure><img src="image_6.png" /><figcaption>Figure 4. (a) Reference swimmer and the best swimmers. (b) Center of mass trajectories.</figcaption></figure>

## Video

<figure><img src="image_7.png" /></figure>

## References

[1] H. Shum, "Microswimmer propulsion by two steadily rotating helical flagella," *Micromachines*, vol. 10, no. 1, 2019.

[2] T. W. Sederberg and S. R. Parry, "Free-form deformation of solid geometric models," in *Proceedings of the 13th annual conference on Computer graphics and interactive techniques*, pp. 151–160, 1986.

[3] D. Eriksson and M. Poloczek, "Scalable Constrained Bayesian Optimization," in *AISTATS*, pp. 730–738, PMLR, 2021.