# Two-way Coupling of Fluid-Structure Interaction for Elastic Magneto-Swimmers: A Finite Element ALE Approach

Christophe Prud'Homme, Vincent Chabannes, Laetitia Giraldi, Agathe Chouippe, Céline Van Landeghem

► To cite this version:

Christophe Prud'Homme, Vincent Chabannes, Laetitia Giraldi, Agathe Chouippe, Céline Van Landeghem. Two-way Coupling of Fluid-Structure Interaction for Elastic Magneto-Swimmers: A Finite Element ALE Approach. 2026. [https://arxiv.org/abs/2001.05482](https://arxiv.org/abs/2001.05482)

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# Graphical Abstract

## Two-way Coupling of Fluid-Structure Interaction for Elastic Magneto-Swimmers: A Finite Element ALE Approach

Christophe Prud'homme, Vincent Chabannes, Laetitia Giraldi, Agathe Chouippe, Céline Van Landeghem

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

## Two-way Coupling of Fluid-Structure Interaction for Elastic Magneto-Swimmers: A Finite Element ALE Approach

Christophe Prud'homme, Vincent Chabannes, Laetitia Giraldi, Agathe Chouippe, Céline Van Landeghem

* Finite element ALE framework for deformable magneto-swimmers in confined fluid domains.

* Full-order model with two-way coupling of fluid-elasto-magneto-swimmer interactions.

* Advanced conforming mesh with adaptive remeshing procedures for accurate swimmer tracking.

* Parallel scalable implementation with the open source Feel++ finite element library.

* Validation benchmarks in 2D and 3D against experimental and theoretical results.

* Platform for digital twin development and biomedical applications.
Two-way Coupling of Fluid–Structure Interaction for
Elastic Magneto-Swimmers: A Finite Element ALE
Approach

Christophe Prud'homme<sup>a</sup>, Vincent Chabannes<sup>a</sup>, Laetitia Giraldi<sup>b</sup>, Agathe Chouippe<sup>c</sup>, Céline Van Landeghem<sup>a</sup>

<sup>a</sup>IRMA, University of Strasbourg & CEMOSIS, 7 rue René Descartes, 67084 Strasbourg, France

<sup>b</sup>Inria Sophia Antipolis Méditerranée, MCUBE Team, 2004 route des Lucioles, 06902 Sophia Antipolis, France

<sup>c</sup>Université de Strasbourg, CNRS, ICUBE UMR 7357, 67000 Strasbourg, France

Abstract

Artificial micro-swimmers actuated by external magnetic fields hold signif-
icant promise for targeted biomedical applications, including drug delivery
and micro-robot-assisted therapy. However, their dynamics remain chal-
lenging to control due to the complex nonlinear coupling between magnetic
actuation, elastic deformations, and fluid interactions in confined biological
environments. Numerical modeling is therefore essential to better under-
stand, predict, and optimize their behavior for practical applications. In this
work, we present a comprehensive finite element framework based on the
Arbitrary Lagrangian–Eulerian formulation to simulate deformable elastic
micro-swimmers in confined fluid domains. The method employs a full-order
model that resolves the complete fluid dynamics while simultaneously track-
ing swimmer deformation and global displacement on conforming meshes.
Numerical experiments are performed with the open-source finite element
library Feel++, demonstrating excellent agreement with experimental data
from the literature. The validation benchmarks in both two and three di-
mensions confirm the accuracy, robustness, and computational efficiency of

Email addresses: christophe.prudhomme@cemosis.fr (Christophe Prud'homme), vincent.chabannes@cemosis.fr (Vincent Chabannes), laetitia.giraldi@inria.fr (Laetitia Giraldi), chouippe@unistra.fr (Agathe Chouippe), celine.van-landeghem@unistra.fr (Céline Van Landeghem)

Preprint submitted to Journal of Fluids and Structures

January 28, 2026
the proposed framework, representing a foundational step toward developing
digital twins of magneto-swimmers for biomedical applications.

Keywords: Fluid-structure interaction, Finite element method, Arbitrary Lagrangian–Eulerian framework, Two-way coupling, Elasto-magneto swimmer, Feel++

1. Introduction

In recent years, significant attention has been devoted to the development
of artificial micro-swimmers for biomedical applications, such as targeted
drug delivery or minimally invasive diagnostics [1]. One promising approach
involves actuating these micro-swimmers using external magnetic fields, of-
ten generated by MRI technology [2], to enable wireless control inside the
human body. However, controlling flagellated swimmers in such conditions
remains a challenging task. These systems are typically non-homologous,
and their dynamics are often poorly controllable [3, 4] due to the complex
coupling between magnetic actuation, elastic deformations, and hydrody-
namic interactions at low Reynolds number. This complexity motivates the
development of accurate numerical models capable of capturing the coupled
fluid-elasto-magneto interactions governing their dynamics.

Modeling the motion of micro-swimmers in fluids involves the coupling
of fluid dynamics with rigid and, eventually, elastic body mechanics, as well
as complex fluid-structure interactions. Various numerical approaches have
been developed to address this problem, each offering a trade-off between
computational efficiency and modeling accuracy.

At low Reynolds numbers, where inertial effects are negligible, simplified
models, such as the *Resistive Force Theory* [5, 6, 7] and the *Slender Body
Theory* [8, 9, 10, 11], provide computationally efficient approximations. These
methods are particularly useful for slender filaments, but they rely on strong
geometric assumptions and do not fully capture near-field hydrodynamic
interactions or swimmer-boundary effects.

To improve accuracy, *Boundary Integral Methods* [12, 13] have been intro-
duced, offering a more refined description of fluid-structure coupling without
discretization of the entire fluid domain [14]. These methods rely on Green’s
functions of the Stokes equations [15], which are available in closed form
and allow the velocity of the fluid to be expressed as an integral over the
swimmer’s surface. However, this approach is restricted to the Stokes regime
and cannot be generalized to inertial flows. In addition, the singular behavior
of the Green functions near the boundary requires regularization techniques
to ensure numerical stability and precision [16, 17, 18].

When more complex geometries, large deformations, or interactions with
boundaries are involved, full discretization of both fluid and swimmer domains
becomes necessary. In such cases, the *Finite Element Method* provides a
versatile framework, particularly when combined with techniques like the
*Immersed Boundary Method* [19, 20, 21, 22, 23] or *CutFEM* [24, 25, 26].
These approaches allow the simulation of moving rigid swimmers immersed in
a fluid, but they introduce additional challenges, such as force interpolation
errors and numerical instabilities near the fluid-structure interface. Overall,
while existing models provide valuable insight into micro-swimmer dynamics,
many approaches that include both fluid- and elastic-solid interactions are
limited to configurations where the solid undergoes only passive deformations,
without rigid-body motion. As a result, these models do not capture the full
dynamics of a self-propelled swimmer, particularly in scenarios where elastic
deformation allows the swimmer to push the fluid, enabling its displacement.

In this work, we introduce a finite element method based on the *Arbitrary Lagrangian–Eulerian* framework to compute the displacement of deformable swimmers immersed in viscous flows within geometrically complex domains [27, 28, 29, 30, 31]. This approach resolves the full fluid dynamics while tracking both swimmer deformation and global motion on a conforming mesh. Compared to immersed boundary methods, the explicit tracking of the interface during the computation allows for precise evaluation of all physical quantities at the fluid-structure boundary. Thus, it provides an accurate description of the fluid-structure interface and can be extended to complex fluids relevant to biological applications.

The paper is organized as follows. In Section 2 we introduce the mathemat-
ical models, starting with the fluid equations, then describing the swimmer
through rigid-body motion and elasticity, and finally presenting the coupled
fluid-structure interaction problem. Section 3 details the spatial and temporal
discretization of all these equations. Section 4 focuses on computational strate-
gies, including the remeshing procedure, and the preconditioning technique.
Section 5 presents the numerical results, consisting of validation tests, all
performed using the open-source finite element library Feel++ [32]. Finally,
section 6 provides conclusions and perspectives.
## 2. Mathematical modeling

This section introduces the mathematical model for the fluid-magneto-swimmer interaction problem. We first describe the hydrodynamics of incompressible Newtonian fluids and the dynamics of elastic bodies, governed by hyper-elasticity equations. We then present the coupling between these two components, taking into account the rigid motion of the swimmer.

### 2.1. Fluid model

Let $\mathcal{F}^t \subset \mathbb{R}^d$, with $d = 2, 3$ denoting the spatial dimension, be the region occupied by the fluid at time $t \in ]0, T]$, where $T > 0$ is the final time. In what follows, we use the notation with superscript $t$ to emphasize that the corresponding quantity is time-dependent. The physical properties of the incompressible Newtonian fluid are its dynamic viscosity $\mu_{\mathcal{F}} \in \mathbb{R}^+$ and its density $\rho_{\mathcal{F}} \in \mathbb{R}^+$. Their hydrodynamics are described by the non-linear Navier-Stokes equations. These equations are formulated in the Eulerian framework and describe the time evolution of the velocity field $u^t : ]0, T] \times \mathcal{F}^t \to \mathbb{R}^d$, and the pressure field $p^t : ]0, T] \times \mathcal{F}^t \to \mathbb{R}$ of the fluid.

The Navier-Stokes equations are given at time $t > 0$ by

$$ 
\left\{ 
\begin{array}{ll} 
\rho_{\mathcal{F}}(\partial_t u^t + (u^t \cdot \nabla)u^t) - \nabla \cdot \sigma(u^t, p^t) = 0_{\mathbb{R}^d} & \text{in } \mathcal{F}^t, \\ 
\nabla \cdot u^t = 0 & \text{in } \mathcal{F}^t, \\ 
u^t = 0_{\mathbb{R}^d} & \text{on } \partial \mathcal{F}_D^t, \\ 
\sigma(u^t, p^t)n_{\mathcal{F}}^t = 0_{\mathbb{R}^d} & \text{on } \partial \mathcal{F}_N^t. 
\end{array} 
\right. 
$$

The boundary conditions are either homogeneous Dirichlet boundary conditions, imposed on $\partial \mathcal{F}_D^t$, or homogeneous Neumann boundary conditions, imposed on $\partial \mathcal{F}_N^t$, such that $\partial \mathcal{F}^t = \partial \mathcal{F}_D^t \cup \partial \mathcal{F}_N^t$. Here, $n_{\mathcal{F}}^t$ is the outward unit normal vector to $\partial \mathcal{F}_N^t$. The stress tensor is defined as

$$ \sigma(u^t, p^t) = -p^t \mathbb{I}_d + \mu_{\mathcal{F}}(\nabla u^t + (\nabla u^t)^T), $$

where $\mathbb{I}_d$ is the identity matrix of size $d$.

### 2.2. Elasto-magneto-swimmer model

The swimmer $\mathcal{S}$ consists of its magnetic head $\mathcal{S}_{\text{head}}$ and its elastic tail $\mathcal{S}_{\text{tail}}$, such that $\mathcal{S} = \mathcal{S}_{\text{head}} \cup \mathcal{S}_{\text{tail}}$. We introduce the following notation: $\mathcal{S}^*$ denotes the reference domain of the swimmer, i.e., the initial domain at each
remeshing step, while $S^t$ denotes the domain at the current time. The same notation is used for the fluid. The notations are shown in Fig. 1. The swimmer is described by the position of its center of mass $x_{cm}^t$ and the configuration of its structure, i.e., its deformation. These two quantities are determined by its time-dependent translational velocity $U^t$, angular velocity $\omega^t$, and the elastic displacement $\eta^t$ of the flagella. This elastic displacement is actuated by an external magnetic field $B^t : ]0, T] \to \mathbb{R}^d$.

<figure><img src="image_3.png" /><figcaption>Figure 1: Configuration of the magneto-swimmer at time $t$ inside a fluid domain $\mathcal{F}^t$. The magnetic moment $M^t$ tends to align with the external magnetic field $B^t$, and $\theta^t$ represents the swimmer's head orientation.</figcaption></figure>

The linear velocity $U^t : ]0, T] \to \mathbb{R}^d$ is described by the Newton's second law

$$m^t \mathrm{d}_t U^t = f_H^t, \qquad (1)$$

and the angular velocity $\omega^t : ]0, T] \to \mathbb{R}^{d^*}$, with $d^* = 1$ in two dimensions and $d^* = 3$ in three dimensions, is described by the Euler's equation

$$\mathrm{d}_t[R(\theta^t)J^t R(\theta^t)^T \omega^t] = T_H^t + T_m^t. \qquad (2)$$

In (1), $m^t \in \mathbb{R}^+$ describes the mass of the swimmer, which is computed from its density $\rho_S \in \mathbb{R}^+$. In (2), $J^t \in S_{++}^{d^*}$ represents its inertia tensor, $R$ the
rotation matrix, and $\theta^t : ]0, T] \to \Theta$ the rotation angle of the swimmer, which is derived from

$$d_t \theta^t = \omega^t.$$

In two dimensions, the interval $\Theta = [-\pi, \pi]$, whereas in three dimensions it is given by $\Theta = [-\pi, \pi] \times [0, \pi] \times [0, \pi/2]$. In (1) and (2), $f_H^t : ]0, T] \times \partial S^t \to \mathbb{R}^d$ and $T_H^t : ]0, T] \times \partial S^t \to \mathbb{R}^{d*}$ describe the hydrodynamical forces and torques

$$f_H^t = \int_{\partial S^t} \sigma(u^t, p^t) n_S^t, \quad T_H^t = \int_{\partial S^t} (x^t - x_{cm}^t) \times (\sigma(u^t, p^t) n_S^t), \quad (3)$$

where $n_S^t$ is the current outward unit normal vector to $\partial S^t$. The swimmer is also subjected to the magnetic torque $T_m^t : ]0, T] \times S_{\text{head}}^* \to \mathbb{R}^{d*}$ defined as

$$T_m^t = m R(\theta^t) M^t \times B^t,$$

where $M^t : ]0, T] \to \mathbb{R}^d$ is its magnetic moment and $m \in \mathbb{R}^+$ its magnetization.

Moreover, the elastic displacement $\eta^t : ]0, T] \times S^* \to \mathbb{R}^d$ of the flagella is governed by the hyper-elasticity equations

$$\left\{ \begin{array}{ll} \rho_S \partial_{tt} \eta^t - \nabla \cdot (F(\eta^t) \Sigma(\eta^t)) = 0_{\mathbb{R}^d} & \text{in } S^*, \\ \eta^t = R(\theta^t)(x^* - x_{cm}^*) - (x^* - x_{cm}^*) & \text{on } S_{\text{head}}^*, \\ F(\eta^t) \Sigma(\eta^t) n_S^* = 0_{\mathbb{R}^d} & \text{on } \partial S_{\text{tail}}^* \end{array} \right. \quad (4)$$

here $x_{cm}^*$ is the center of the swimmer's head $S_{\text{head}}^*$. To describe the deformation of the hyper-elastic flagella, we consider the Saint-Venant-Kirchhoff model. In this framework, the inertial and surface loads are defined in terms of the deformation gradient

$$F(\eta^t) = I_d + \nabla \eta^t,$$

and the second Piola-Kirchhoff tensor

$$\Sigma(\eta^t) = \lambda \mathrm{Tr}(\epsilon(\eta^t)) I_d + 2\mu\epsilon(\eta^t),$$

with $\lambda, \mu$ the Lamé coefficients and $\epsilon(\eta^t)$ the Green-Lagrange tensor.

### 2.3. Arbitrary Lagrangian-Eulerian framework

The Navier-Stokes equations are formulated in an Eulerian reference frame, where they describe the evolution of hydrodynamics within a fixed computational domain. However, the dynamics of the swimmers are described in a Lagrangian reference frame. To account for the geometric coupling
condition, we adopt the Arbitrary Lagrangian-Eulerian framework [33]. This approach allows the fluid domain to follow the motion of the fluid-structure interaction interface. Specifically, the reference frame remains Lagrangian near the swimmers but stays Eulerian farther from the fluid-structure interaction interface. Both the swimmer and the fluid are discretized, and the discretized fluid domain follows the motion of the swimmer.

We define the ALE map $A_{\mathcal{F}^*}^t : ]0, T] \times \mathcal{F}^* \to \mathcal{F}^t$ as a continuous and bijective function that gives the position of a particle in the current fluid domain $\mathcal{F}^t$ based on its position in the reference domain $\mathcal{F}^*$

$$A_{\mathcal{F}^*}^t(x^*) = x^t.$$

To construct this ALE map, we introduce the displacement of the discretized fluid domain over time, denoted as $\eta_{\mathcal{F}^*}^t : ]0, T] \times \mathcal{F}^* \to \mathcal{F}^t$. A possible definition of the ALE map is then:

$$A_{\mathcal{F}^*}^t(x^*) = x^* + \eta_{\mathcal{F}^*}^t(x^*).$$

The fluid domain displacement $\eta_{\mathcal{F}^*}^t$ can be obtained, for instance, by a harmonic extension of the swimmer's displacement at time $t$, defined on $\partial S^*$, to the interior of the fluid domain. This extension is determined by solving the following Laplace smoothing equation

$$\left\{ \begin{array}{ll} \nabla \cdot \left( (1 + \tau) \nabla \eta_{\mathcal{F}^*}^t \right) = 0_{\mathbb{R}^d} & \text{in } \mathcal{F}^*, \\ \eta_{\mathcal{F}^*}^t = \eta^t & \text{on } \partial S_i^*, \end{array} \right. \qquad (5)$$

where $\eta^t$ is the solution of (4), and $\tau$ acts as a space-dependent diffusion coefficient. It is a piecewise constant coefficient, defined on each element $e$ of the domain's discretization as

$$\tau|_e = \frac{1 - V_{\min}/V_{\max}}{V_e/V_{\max}},$$

where $V_{\max}$, $V_{\min}$, and $V_e$ are the volumes of the largest, smallest, and current elements of the domain discretization, respectively [34]. This coefficient $\tau$ ensures that mesh deformation is applied primarily to elements with larger volumes.

When transitioning from the fluid Eulerian frame to an ALE frame, an additional term appears in the Navier-Stokes equations to account for the velocity of the moving domain, denoted by $u_A^t$. The full system in the ALE
reference frame is divided into two problems. First, the fluid-rigid problem, solving the hydrodynamics and the swimmer rigid motion, is governed by the following equations

$$
\left\{
\begin{array}{ll}
\rho_{\mathcal{F}}(\partial_t u^t + ((u^t - u_{\mathcal{A}}^t) \cdot \nabla)u^t) - \nabla \cdot \sigma(u^t, p^t) = 0_{\mathbb{R}^d} & \text{in } \mathcal{F}^t, \\
\nabla \cdot u^t = 0 & \text{in } \mathcal{F}^t, \\
u^t = 0_{\mathbb{R}^d} & \text{on } \partial \mathcal{F}_D^t, \\
\sigma(u^t, p^t)n_{\mathcal{F}}^t = 0_{\mathbb{R}^d} & \text{on } \partial \mathcal{F}_N^t, \\
u^t = \tilde{u}^t & \text{on } \partial \mathcal{S}^t, \\
m^t \, d_t U^t = f_H^t, \\
d_t [R(\theta^t) J^t R(\theta^t)^T \omega^t] = T_H^t + T_m^t,
\end{array}
\right.
\quad (6)
$$

where $\tilde{u}^t = U^t + \omega^t \times (x^t - x_{cm}^t) + \partial\eta^t \circ (A_{\mathcal{F}^*}^t)^{-1}$ the velocity of the swimmer.
Then, the fluid-elastic problem computes the elastic deformations and introduces the continuity of displacements and stresses. This problem is described
by

$$
\left\{
\begin{array}{ll}
\rho_S \partial_{tt} \eta^t - \nabla \cdot (F(\eta^t) \Sigma(\eta^t)) = 0_{\mathbb{R}^d} & \text{in } S^*, \\
\eta^t = \eta_T^t + \eta_R^t & \text{on } S_{\text{head}}^*, \\
F(\eta^t) \Sigma(\eta^t) n_S^* = -(\sigma(u^t, p^t) n_{\mathcal{F}}^t) \circ A_{\mathcal{F}^*}^t & \text{on } \partial S^*,
\end{array}
\right.
\quad (7)
$$

where the translational and rotational displacements are respectively given by

$$
d_t \eta_T^t = U^t, \quad \eta_R^t = R(\theta^t)(x^* - x_{cm}^t) - (x^* - x_{cm}^t).
$$

3. Numerical discretization

For the spatial discretization of the discretized domain $\Omega_h^t = \mathcal{F}_h^t \cup S_h^t$, we use standard Lagrange finite elements. The associated discrete approximation spaces are constructed from piecewise polynomial functions of degree $N$, denoted $P_c^N(\Omega_h^t)$

$$
P_c^N(\Omega_h^t) = \{v \in C^0(\Omega_h^t) \mid v \circ \varphi_{K_e}^{\text{geo}} \in \mathbb{P}^N(\widehat{K}), \quad \forall K_e\},
$$

where $\mathbb{P}^N(\widehat{K})$ denotes the space of scalar polynomials of total degree less
than or equal to $N$ defined on the reference element $\widehat{K}$, and $\varphi_{K_e}^{\text{geo}} : \widehat{K} \rightarrow K_e$
the geometric transformation allowing to obtain each element $K_e$ from the
reference element $\widehat{K}$. Vector-valued finite element spaces are constructed as
Cartesian products of the scalar-valued spaces, denoted by $[P_c^N(\Omega_h^t)]^d$. We
also use the notation $H^m(\Omega_h^t)$ to denote the Sobolev space of order $m$, and with boundary condition $g$ on a subset $\Gamma \subset \partial\Omega_h^t$ it is given by

$$H_{(g,\Gamma)}^m(\Omega_h^t) = \{v \in H^m(\Omega_h^t) \mid v|_{\Gamma} = g\}.$$

The vector-valued versions are respectively given by $[H^m(\Omega_h^t)]^d$ and $[H_{(g,\Gamma)}^m(\Omega_h^t)]^d$.

Regarding temporal discretization, the time interval $]t_0 = 0, T = t_{N_t}]$, with $T > 0$ denoting the final time, is divided into $N_t$ time steps of size $\delta t$. The discrete time instances are denoted by $t_n$ for $n = 1, \dots, N_t$. Finally, a mesh at time $t_{n+1}$ is denoted by $\Omega_h^{n+1}$, and a discrete solution at this time is written as $u_h^{n+1}$.

We will first present the spatial and temporal discretization of the ALE map before introducing that of the *fluid-rigid* and *fluid-elastic* problems.

### 3.1. Discretization of the ALE map

The approximation spaces at time $t_{n+1}$ of the trial and test functions are respectively given by:

$$ 
\begin{aligned}
V_{A,h}^* &= \{\mathbf{v}, \mathbf{v} \in [H_{(0_{\mathbb{R}^d},\partial\mathcal{F}_h^*)}^1(\mathcal{F}_h^*)]^d \cap [P_c^1(\mathcal{F}_h^*)]^d, \mathbf{v} = \eta_{S_{h}^*,h}^{n+1} \text{ on } \partial S_h^*\}, \\
W_{A,h}^* &= \{\mathbf{v}, \mathbf{v} \in [H_{(0_{\mathbb{R}^d},\partial\mathcal{F}_h^*)}^1(\mathcal{F}_h^*)]^d \cap [P_c^1(\mathcal{F}_h^*)]^d, \mathbf{v} = 0_{\mathbb{R}^d} \text{ on } \partial S_h^*\}.
\end{aligned}
$$

Thus, the fluid domain displacement is discretized using continuous affine finite elements. The space- and time-discretized version of (5) at time $t_{n+1}$ gets

Find $\eta_{\mathcal{F}_h^*,h}^{n+1} \in V_{A,h}^*$ such that

$$ 
\begin{aligned}
\underbrace{\int_{\mathcal{F}_h^*} (1+\tau) \nabla \eta_{\mathcal{F}_h^*,h}^{n+1} : \nabla \mathbf{v}}_{:=a_{A^*}(\eta_{\mathcal{F}_h^*,h}^{n+1}, \mathbf{v})} = 0 \quad \text{for all } \mathbf{v} \in W_{A,h}^*.
\end{aligned}
\quad (8) $$

The discrete ALE map is then defined as

$$A_{\mathcal{F}_h^*,h}^{n+1}(x^*) = x^* + \eta_{\mathcal{F}_h^*,h}^{n+1}(x^*).$$

Large deformations may lead to element inversions, resulting in invalid triangulation. To avoid this, mesh quality metrics [35] are used. If the quality remains above a fixed threshold, the domain is updated using the ALE map. Otherwise, remeshing is performed before applying it. Details of these procedures are given in [31].
For the numerical resolution, we define the algebraic system associated with (8) as follows

$$A\eta_{\mathcal{F}} = 0_{\mathbb{R}^{N_{\eta_{\mathcal{F}}}}}$$

where we denote by

$$A = (a_{\mathcal{A}^*}(\xi_i, \xi_j))_{i,j} \in \mathbb{R}^{N_{\eta_{\mathcal{F}}} \times N_{\eta_{\mathcal{F}}}}, \quad \text{for } 1 \le i, j \le N_{\eta_{\mathcal{F}}}$$

and $(\xi_i)_{i, i = 1, \dots, N_{\eta_{\mathcal{F}}}}$ is the basis of the space $V_{\mathcal{A},h}^*$. The solver and preconditioner applied to this algebraic system are detailed in 4.

## 3.2. Discretization of the fluid-rigid problem

The Dirichlet boundary condition, enforcing the coupling of velocities at the interface of the swimmer is incorporated both in the function space associated with the fluid velocity and in the space of test functions. The discrete spaces for the fluid velocity and pressure at time $t_{n+1}$ are then respectively defined as

$$V_{\mathcal{F},h}^{n+1} = \left\{ \mathbf{v} : \mathcal{F}_h^{n+1} \to \mathbb{R}^d, \mathbf{v} = \tilde{\mathbf{v}} \circ (\mathcal{A}_{\mathcal{F},h}^{n+1})^{-1}, \tilde{\mathbf{v}} \in [H_{(0,\mathbb{R}^d,\mathcal{F}_{D,h}^*)}^1] (\mathcal{F}_h^*)^d \cap [P_c^M(\mathcal{F}_h^*)]^d, \right. \\ \left. \qquad \tilde{\mathbf{v}} = \tilde{\mathbf{u}} \text{ on } \partial \mathcal{S}_h^{n+1}, \tilde{\mathbf{u}} \in \mathbb{R}^d \right\},$$

$$Q_{\mathcal{F},h}^{n+1} = \left\{ \mathbf{q} : \mathcal{F}_h^{n+1} \to \mathbb{R}, \mathbf{q} = \tilde{\mathbf{q}} \circ (\mathcal{A}_{\mathcal{F},h}^{n+1})^{-1}, \tilde{\mathbf{q}} \in P_c^N(\mathcal{F}_h^*) \right\}.$$

We choose the Taylor-Hood inf-sup finite elements to satisfy the compatibility condition. The polynomial degrees are related by $N = M - 1$, and typically we set $M = 2$ and $N = 1$. Thus, the velocity and pressure are respectively discretized using continuous piecewise quadratic, and affine finite elements. Following the ideas of [36], the discretized formulation of the fluid-rigid problem (6) deriving from the variational form is stated as follows

Find the fluid velocity and pressure $(u_h^{n+1}, p_h^{n+1}) \in V_{\mathcal{F},h}^{n+1} \times Q_{\mathcal{F},h}^{n+1}$, and the swimmer's linear and angular velocities $U^{n+1}, \omega^{n+1} \in \mathbb{R}^d \times \mathbb{R}^{d*}$, such that for all $(\mathbf{v}, \mathbf{q}) \in V_{\mathcal{F},h}^{n+1} \times Q_{\mathcal{F},h}^{n+1}$, and $\mathbf{U}, \omega \in \mathbb{R}^d \times \mathbb{R}^{d*}$, one has

$$\begin{align}
a_{\mathcal{F}^t}(u_h^{n+1}, \mathbf{v}) - b_{\mathcal{F}^t}(\mathbf{v}, p_h^{n+1}) + m^{n+1} \mathrm{d}_t U^{n+1} \cdot \mathbf{U} \nonumber \\
+ \mathrm{d}_t \left[ R(\theta^{n+1}) J^{n+1} R(\theta^{n+1})^T \omega^{n+1} \right] \cdot \omega = l_{\mathcal{F}}(\mathbf{v}, \mathbf{U}, \omega), \tag{9} \\
b_{\mathcal{F}^t}(u_h^{n+1}, \mathbf{q}) = 0. \nonumber
\end{align}$$
with the bilinear and linear forms

$$a_{\mathcal{F}^t}(u_h^{n+1}, \mathbf{v}) = \rho_{\mathcal{F}} \int_{\mathcal{F}_h^{n+1}} \partial_t u_h^{n+1} \cdot \mathbf{v} + \rho_{\mathcal{F}} \int_{\mathcal{F}_h^{n+1}} ((u_h^{n+1} - u_{A,h}^{n+1}) \cdot \nabla) u_h^{n+1} \cdot \mathbf{v} \\ + \mu_{\mathcal{F}} \int_{\mathcal{F}_h^{n+1}} \nabla u_h^{n+1} : \nabla \mathbf{v},$$

$$l_{\mathcal{F}^t}(\mathbf{v}) = f_H^{n+1} \cdot \mathbf{U} + T_H^{n+1} \cdot \boldsymbol{\omega} + T_m^{n+1} \cdot \boldsymbol{\omega},$$

$$b_{\mathcal{F}^t}(u_h^{n+1}, \mathbf{q}) = - \int_{\mathcal{F}_h^{n+1}} \mathbf{q} \cdot \nabla \cdot u_h^{n+1}.$$

The time derivative of $u_h^{n+1}$ as well as the ALE velocity $u_{A,h}^{n+1}$ are time-discretized using the second-order backward differentiation formula, BDF2. For the numerical resolution, the degrees of freedom of the fluid velocity on the interface of the swimmer are treated differently since they depend only on the linear and angular velocities.

We denote the vectors $\mathbf{u} = (\mathbf{u}_1, \dots, \mathbf{u}_{N_u}) = (\mathbf{u}_I, \mathbf{u}_\Gamma)$ for the velocity, where the subscript $\Gamma$ represent the degrees of freedom associated with the interface, $\mathbf{p} = (\mathbf{p}_1, \dots, \mathbf{p}_{N_p})$ for the pressure, and $\mathbf{U} \in \mathbb{R}^d$ and $\boldsymbol{\omega} \in \mathbb{R}^{d^*}$ represent the linear and angular velocities, respectively. The resulting algebraic system of (9) is then given by

$$\begin{bmatrix} A_{II} & A_{\Gamma\Gamma} & 0 & 0 & B_I^T \\ A_{\Gamma I} & A_{\Gamma\Gamma} & 0 & 0 & B_\Gamma^T \\ 0 & 0 & m^{n+1}\mathbb{I}_d & 0 & 0 \\ 0 & 0 & 0 & R(\theta^{n+1})J^{n+1}R(\theta^{n+1})^T & 0 \\ B_I & B_\Gamma & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \mathbf{u}_I \\ \mathbf{u}_\Gamma \\ \mathbf{U} \\ \boldsymbol{\omega} \\ \mathbf{p} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ l_U \\ l_\omega \\ 0 \end{bmatrix}, \quad (10)$$

where

$$A_{JK} = (a_{\mathcal{F}^t}(\phi_{J_i}, \phi_{K_j}))_{i,j} \in \mathbb{R}^{N_{u_J} \times N_{u_K}}, \text{ for } J, K \in \{I, \Gamma\},$$

$$B_J = (b_{\mathcal{F}^t}(\phi_{J_i}, \psi_j))_{i,j} \in \mathbb{R}^{N_{u_J} \times N_{p}} \text{ for } J \in \{I, \Gamma\},$$

$$l_U = f_H^{n+1} \cdot \mathbf{U},$$

$$l_\omega = T_H^{n+1} \cdot \boldsymbol{\omega} + T_m^{n+1} \cdot \boldsymbol{\omega},$$

and $(\phi_i)_i, i = 1, \dots, N_u$, respectively $(\psi_i)_i, i = 1, \dots, N_p$, the basis of the spaces $V_{\mathcal{F},h}^{n+1}$ and $Q_{\mathcal{F},h}^{n+1}$.

Using the no-slip boundary conditions $\mathbf{u} = \tilde{\mathbf{u}} = \mathbf{U} + \boldsymbol{\omega} \times (x^{n+1} - x_{cm}^{n+1})$ on $\partial S_h^{n+1}$, we introduce the operator $\mathcal{P}$ such as

$$(\mathbf{u}_I, \mathbf{u}_\Gamma, \mathbf{U}, \boldsymbol{\omega}, \mathbf{p})^T = \mathcal{P}(\mathbf{u}_I, \mathbf{U}, \boldsymbol{\omega}, \mathbf{p}), \quad (11)$$
with

$$ \mathcal{P} = \begin{bmatrix} \mathbb{I}_d & 0 & 0 & 0 \\ 0 & \tilde{P}_U & \tilde{P}_\omega & 0 \\ 0 & \mathbb{I}_d & 0 & 0 \\ 0 & 0 & \mathbb{I}_d & 0 \\ 0 & 0 & 0 & \mathbb{I}_d \end{bmatrix}. $$

Finally, by using (11), and by multiplying by $\mathcal{P}$, one obtains the algebraic system of the fluid-swimmer problem

$$ \mathcal{P}^T \begin{bmatrix} A_{II} & A_{\Gamma I} & 0 & 0 & B_I^T \\ A_{\Gamma I} & A_{\Gamma \Gamma} & 0 & 0 & B_\Gamma^T \\ 0 & 0 & m^{n+1}\mathbb{I}_d & 0 & 0 \\ 0 & 0 & 0 & R(\theta^{n+1})J^{n+1}R(\theta^{n+1})^T & 0 \\ B_I & B_\Gamma & 0 & 0 & 0 \end{bmatrix} \mathcal{P} \begin{bmatrix} \mathbf{u}_I \\ \mathbf{U} \\ \boldsymbol{\omega} \\ \mathbf{p} \end{bmatrix} = \mathcal{P}^T \begin{bmatrix} 0 \\ 0 \\ l_U \\ l_\omega \\ 0 \end{bmatrix}. $$

### 3.3. Discretization of the fluid-elastic problem

The approximation spaces of admissible elastic displacements, and of the tests functions, at time $t_{n+1}$, are respectively defined by

$$ V_{S,h}^* = \{ \mathbf{v}, \mathbf{v} \in [H_{(\eta_h^{n+1}+\eta_R^{n+1}, \partial S_{\text{head},h}^*)}^1(S_h^*)]^d \cap [P_c^1(S_h^*)]^d \}, $$

$$ W_{S,h}^* = \{ \mathbf{v}, \mathbf{v} \in [H_{(0, \partial \mathbb{R}^d, \partial S_{\text{head},h}^*)}^1(S_h^*)]^d \cap [P_c^1(S_h^*)]^d \}. $$

The fully discretized fluid-elastic problem is stated as follows

Find the elastic displacement $\eta_h^{n+1} \in V_{S,h}^*$ such that, for all $\mathbf{v} \in W_{S,h}^*$, one has

$$ a_{S^*}(\eta_h^{n+1}, \mathbf{v}) = - \int_{\partial S_h^{n+1}} |\det F(\eta_h^{n+1})| ||(F(\eta_h^{n+1}))^{-T}||(\sigma(u_h^{n+1}, p_h^{n+1})n_{\mathcal{F}}^{n+1}) \cdot \mathbf{v}, $$

where the boundary condition is expressed on the current interface $\partial S_h^{n+1}$, since the hydrodynamic forces and torques are resolved onto the current domain, see (3). The bilinear form is defined as

$$ a_{S^*}(\eta_h^{n+1}, \mathbf{v}) = \rho_S \int_{S_h^*} \partial_t \eta_h^{n+1} \cdot \mathbf{v} + \int_{S_h^*} F(\eta_h^{n+1}) \Sigma(\eta_h^{n+1}) : \nabla \mathbf{v}. $$

Once again, we consider the BDF2 scheme to discretize the time derivative. The corresponding algebraic system is written as

$$ A \eta_S = G, \qquad (12) $$
where the stiffness matrix $A \in \mathbb{R}^{N_{\eta_S} \times N_{\eta_S}}$ is expressed in the basis $(\varphi_i)_i$, $i = 1, \dots, N_{\eta_S}$ of the discrete space $V_{S,h}^*$ as

$$A = (a_{S^*}(\varphi_i, \varphi_j))_{i,j},$$

and the right-hand side vector $G \in \mathbb{R}^{N_{\eta_S}}$ includes the boundary conditions.

# 4. Resolution strategies

## 4.1. Full algorithm

The complete coupled algorithm for the elasto-magneto-swimmer is given in Algorithm 1. At each iteration, the swimmer's displacement is updated using a relaxation method, where the relaxation parameter $t^{k+1}$ is computed using the Aitken method, as described in [37].

## 4.2. Remeshing strategy

The mesh deformation described by the ALE map may lead to significant element distortion during long-term simulations. To preserve mesh quality, a remeshing procedure is performed at a prescribed frequency, treated as a user-defined parameter depending on the simulated trajectories. At each remeshing step, the reference domain is updated accordingly, and all fields are projected onto the new mesh. The remeshing operations are carried out using the MMG tool in sequential mode, or ParMMG in parallel [38]. The latter libraries take as input the current domain, which could be of bad quality, and a scalar function defined as the remeshing metric. This metric specifies the desired characteristic size of the elements in the resulting mesh. To accurately capture the fluid-swimmer interaction, we adopt a graded remeshing strategy, meaning that the mesh size is adjusted according to the distance to the swimmer's boundary in the current fluid domain.

## 4.3. Solver and preconditioner

The numerical scheme of the magneto-swimmer model is based on solving a fixed-point algorithm which iterates between the ALE (5), fluid-rigid (6), and fluid-elastic (7) problems corresponding to the step 6, 7 and 10 of the full algorithm. Each sub-problem is solved using appropriate solvers and preconditioners, which are summarized for both two- and three-dimensional cases in Table 1. The Feel++ library interfaces with PETSc [39] for the efficient solution of large-scale linear and nonlinear systems.
**Algorithm 1 Fixed-Point algorithm for the magneto-swimmer**

1: **Input:** Solid displacements at previous time steps $\eta_h^{n-1}$ and $\eta_h^n$, tolerance tol, maximum iterations $k_{max}$.

2: **Output:** New solid displacement $\eta_h^{n+1}$.

3: Initialize iteration counter: $k = 0$.

4: Predict initial displacement using BDF2-consistent extrapolation

$$\eta_h^{n+1,0} = 2\eta_h^n - \eta_h^{n-1}.$$

5: **while** $\epsilon > \text{tol}$ and $k < k_{max}$ **do**

6: Compute ALE map $\tilde{\mathcal{A}}_{\mathcal{F}^*,h}^{n+1,k+1}$ from $\eta_h^{n+1,k}$.

7: Solve fluid-rigid problem on $\tilde{\mathcal{A}}_{\mathcal{F}^*,h}^{n+1,k+1}(\mathcal{F}_h^*)$ to obtain $\eta_T^{n+1,k+1} + \eta_R^{n+1,k+1}$.

8: Update total solid displacement on the reference interface

$$\tilde{\eta}_h^{n+1,k+1} = \eta_h^{n+1,k} + \eta_T^{n+1,k+1} + \eta_R^{n+1,k+1}.$$

9: Compute ALE map $\mathcal{A}_{\mathcal{F}^*,h}^{n+1,k+1}$ from $\tilde{\eta}_h^{n+1,k+1}$.

10: Solve fluid-elastic problem on $S_h^*$ to obtain total displacement $\tilde{\eta}_h^{n+1,k+1}$.

11: Compute error

$$\epsilon = \|\tilde{\eta}_h^{n+1,k+1} - \eta_h^{n+1,k}\|_{L^2(S_h^*)}.$$

12: **if** $\epsilon > \text{tol}$ **then**

13: Update swimmer displacement

$$\eta_h^{n+1,k+1} = t^{k+1}\tilde{\eta}_h^{n+1,k+1} + (1-t^{k+1})\eta_h^{n+1,k}.$$

14: Subtract rigid motion

$$\eta_h^{n+1,k+1} = \eta_h^{n+1,k+1} - (\eta_T^{n+1,k+1} + \eta_R^{n+1,k+1}).$$

15: Increment iteration counter $k = k + 1$.

16: **else**

17: Set $\eta_h^{n+1} = \tilde{\eta}_h^{n+1,k+1}$ and exit loop.

**5. Numerical results**

We perform different numerical tests to validate and illustrate the proposed magneto-elastic swimmer model. First, we analyze the relationship between the swimmer's net displacement and the area enclosed by its stroke in the configuration space. Second, we investigate the net displacement of
<table><caption>Table 1: Solvers and preconditioners for the magneto-swimmer model.</caption><thead><tr><th>Sub-problem</th><th>Two Dimensions</th><th>Three Dimensions</th></tr></thead><tbody><tr><td>ALE map</td><td>CG + GAMG</td><td>CG + GAMG</td></tr><tr><td><i>Fluid-rigid</i></td><td>Direct solver + LU</td><td>GMRES + Block preconditioner</td></tr><tr><td><i>Fluid-elastic</i></td><td>Direct solver + LU</td><td>GMRES + GASM</td></tr></tbody></table>

the swimmer as a function of the external magnetic field frequency in two dimensions. Finally, we extend the validation to three dimensions by simulating the full 3D magneto-swimmer and comparing the results with the 2D case and available literature. The main physical parameters used in the simulations are summarized in Table 2, these values are chosen according to [40]. We prescribe the external magnetic field and the swimmer's magnetic moment based on experimental data from [41, 40],

$$B^n = (b_x, b_y \sin(2\pi ft_n))^T, \qquad (13)$$

with $b_x = b_y = 0.005$T. Under the action of this oscillating magnetic field, the swimmer follows a straight horizontal path.

<table><caption>Table 2: Physical properties of the magneto-swimmer taken from references [41, 40].</caption><thead><tr><th>Physical parameter</th><th>Value</th></tr></thead><tbody><tr><td>Young modulus of the head</td><td>41GPa</td></tr><tr><td>Poisson coefficient of the head</td><td>0.281</td></tr><tr><td>Magnetization of the head</td><td>10<sup>5</sup>A/m</td></tr><tr><td>Density of the head</td><td>7000kg/m<sup>3</sup></td></tr><tr><td>Height of the head</td><td>0.5mm</td></tr><tr><td>Young modulus of the tail</td><td>0.1MPa</td></tr><tr><td>Poisson coefficient of the tail</td><td>0.4</td></tr><tr><td>Density of the tail</td><td>1300kg/m<sup>3</sup></td></tr><tr><td>Length of the tail</td><td>7.5mm</td></tr><tr><td>Maximal diameter of the tail</td><td>1.5mm</td></tr><tr><td>Magnetic field intensity b<sub>x</sub>, b<sub>y</sub></td><td>5mT</td></tr></tbody></table>

*Net displacement vs swimmer's strokes.* We introduce the magneto-head and magneto-tail angles $\theta^n_{\text{head}}$ and $\theta^n_{\text{tail}}$ relative to the horizontal axis as illustrated
in Fig. 2. We build their phase portrait based on their time evolution over one period as depicted in Fig. 3 to represent the swimmer's stroke in the configuration space. The obtained curve approximates an ellipse, and according to [42], when the deformation is small, the mean x-displacement per period is directly correlated to the area of this ellipse. This result confirms the theoretical prediction that the swimmer will achieve an x-net displacement after the stroke cycle.

<figure><img src="image_4.png" /><figcaption>Figure 2: The illustration of the two angles used for the phase configuration space.</figcaption></figure>

*Net displacement vs frequency.* In this case, we work with the two-dimensional model of the magneto-elastic swimmer. In Fig. 4, we vary the frequency $f$ of the magnetic field defined in equation (13) within the range $f \in [0 \text{ Hz}, 3 \text{ Hz}]$, while measuring the net displacement $\Delta x$ of the swimmer over one oscillation period of duration $1/f$. This analysis is performed for three different values of the tail's Young modulus: $E_S = 5 \times 10^4 \text{ Pa}$, $E_S = 8 \times 10^4 \text{ Pa}$, and $E_S = 2 \times 10^5 \text{ Pa}$. The corresponding results are shown in Fig. 4, which plots the net displacement obtained during the third oscillation cycle, during the time interval $[\frac{2}{f} \text{ s}, \frac{3}{f} \text{ s}]$.

We observe that the displacement per cycle increases with frequency, reaching a maximum around $f \approx 0.8 \text{ Hz}$, before decreasing as the frequency continues to rise. This non-monotonic trend is consistent with experimental and theoretical findings reported in the literature [42, 41], where an optimal frequency was also identified, corresponding to a resonance between the magnetic actuation and the elastic response of the swimmer's tail. At low frequencies, the magnetic actuation is too slow to generate significant tail deformation, resulting in limited propulsion. Conversely, at high frequencies, the head cannot follow the rapid field oscillations, leading to attenuated elastic waves along the tail, reducing the net displacement.

Furthermore, we note that a smaller Young modulus leads to larger net displacements per cycle. This is attributed to the increased flexibility of the tail, which allows for greater deformation and thus more efficient conversion of oscillatory motion into forward propulsion. This observation also agrees
<figure><img src="image_5.png" /><figcaption>Figure 3: At the top, the deformations of the swimmer with respect to the stroke cycle plotted at the bottom in the configuration space.</figcaption></figure>

with the results of [41], where flexible magnetic swimmers were shown to outperform stiffer ones under similar low-Reynolds-number conditions.

<figure><img src="image_6.png" /><figcaption>Figure 4: Displacement of the magneto-swimmer over one period as a function of the frequency of the magnetic field.</figcaption></figure>

*3D displacement of the elasto-magneto-swimmer.* We now consider the same test case using the three-dimensional model of the elasto-magneto-swimmer.
The geometry of the swimmer is shown in Fig. 5. Its head is cylindrical, with a diameter of 1.5 mm and a height of 0.5 mm, and is attached to a flexible tail of length 7.5 mm and a minimum diameter of 0.2 mm. The density and Poisson's ratio of both the head and the tail are set to typical values used in the literature for soft magnetic swimmers.

<figure><img src="image_7.png" /><figcaption>Figure 5: Discretized geometry of the three-dimensional magneto-swimmer.</figcaption></figure>

The Young's modulus is fixed to $E_s = 2 \times 10^5$ Pa, while the frequency of the external magnetic field is varied within the range [0 Hz, 3 Hz]. The external magnetic field in three dimensions is generalized by applying the same oscillatory pattern in the y-direction but keeping the z-component zero, see equations (13). The intensity of the magnetic field is maintained at $b_x = b_y = 5$ mT. The magnetization of the magneto-head remains fixed at $m = 10^5$ A/m as in the two-dimensional case and following experimental data from [41, 40].

<figure><img src="image_8.png" /><figcaption>Figure 6: Visualization of the three-dimensional magneto-swimmer at different time instants. Transparent gray surfaces represent the boundaries between domains used by the processors.</figcaption></figure>

The Fig. 7 shows the net displacement $\Delta x$ of the three-dimensional
magneto-swimmer over one period as a function of the frequency *f* of the external magnetic field. It leads to results similar to those in the two-dimensional case: the displacement increases with frequency, reaching a maximum at *f* = 0.8Hz, and then decreases. The behavior is consistent with the two-dimensional simulations and experimental observations reported in the literature [41]. Moreover, Fig. 6 illustrates the deformation and motion of the micro-swimmer at different times during a stroke cycle. The transparent gray surfaces represent the boundaries between domains used by the processors in the parallel computation. However, the three-dimensional simulations are computationally expensive, requiring long execution times. One perspective of this work is to improve the solver and preconditioner strategies, as well as to verify the parallel scalability of the model.

<figure><img src="image_9.png" /><figcaption>Figure 7: Net displacement of the three-dimensional magneto-swimmer over one period as a function of the frequency.</figcaption></figure>

**6. Perspectives and Conclusion**

This work presents a comprehensive numerical framework for simulating
fluid-structure interactions of elastic magneto-swimmers in confined domains,
representing a significant advance toward developing digital twins for biomed-
ical micro-robotics applications. The developed platform combines realistic
modeling, numerical robustness, and high-performance computing capabilities
to enable accurate simulation of magneto-swimmers.

The proposed full-order approach based on the Arbitrary Lagrangian-
Eulerian formulation provides a major advancement in the physical under-
standing of active swimmer problems. The 2D and 3D validation benchmarks
demonstrate the maturity and accuracy of the method, showing excellent agreement with experimental data from the literature. The framework successfully captures the complex nonlinear coupling between magnetic actuation, elastic deformations, and fluid interactions that characterize these systems.

Key contributions of this work include: (i) a robust finite element discretization strategy that handles large swimmer deformations through adaptive remeshing procedures, (ii) advanced coupling schemes that ensure stability and convergence of the fluid-structure interaction, and (iii) a parallel implementation within the open-source Feel++ library that leverages high-performance computing resources.

The developed platform opens numerous perspectives for biomedical applications. It incorporates advanced path-planning algorithms that allow autonomous navigation of magneto-swimmers through complex biological environments. These algorithms are essential for practical biomedical applications, where precise control and collision avoidance are critical for successful targeted delivery missions. The framework also provides a solid foundation for in silico calibration using experimental data, optimal control strategies for targeted navigation, and the development of model reduction techniques. Its capabilities in handling realistic biological environments make it particularly well-suited for applications in drug delivery and micro-robot-assisted therapy.

Future developments will focus on extending the framework to multi-swimmer configurations to study collective behavior, implementing advanced control algorithms based on machine learning techniques, and optimizing the computational strategies for exascale computing. The integration with experimental validation campaigns will further enhance the platform's predictive capabilities, ultimately enabling the transition from laboratory prototypes to clinical applications in targeted biomedical interventions.

## Acknowledgements

This work of the Interdisciplinary Thematic Institute IRMIA++, as part of the ITI 2021-2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR-10-IDEX-0002), and by SFRI-STRAT'US project (ANR-20-SFRI-0012) under the framework of the French Investments for the Future Program. The authors acknowledge the financial support of the French Agence Nationale de la Recherche (grant ANR-21-CE45-0013 project NEMO), and Cemosis. Part of this work was also funded by the
France 2030 NumPEx Exa-MA (ANR-22-EXNU-0002) project managed by
the French National Research Agency (ANR).

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