Optimized Fish Locomotion using Design-by-Morphing and Bayesian Optimization

This study introduces a computational framework combining Design-by-Morphing and Bayesian optimization to generate undulatory swimming profiles that achieve 16%–35% higher propulsive efficiency than traditional anguilliform and carangiform modes by optimizing kinematic parameters and redistributing energy through favorable surface stress distributions.

The Gist

Imagine you are trying to teach a robot fish how to swim as fast and as efficiently as possible. You could just copy a real fish, like an eel or a mackerel, but what if the perfect swimming style isn't exactly like any fish that exists in nature? What if the best way to swim is a weird, hybrid motion that nature hasn't discovered yet?

This paper is about finding that "super-swim" using a clever mix of computer magic and smart guessing. Here is the story of how they did it, broken down into simple parts.

1. The Problem: The "Swim Suit" Dilemma

Think of a fish's body shape and its wiggling motion as a custom-tailored swimsuit.

• Traditional approach: Scientists usually pick a few standard "swimsuits" (like the "Eel Style" or the "Tuna Style") and just tweak the speed or the wiggle size. It's like trying to find the perfect fit by only adjusting the waistband on a few pre-made suits.
• The limitation: You might miss a completely new style that is way more efficient because you never tried mixing the fabrics together.

2. The Solution: "Design-by-Morphing" (The Blender)

The researchers invented a digital blender called Design-by-Morphing (DbM).

• Imagine you have five different "base swimsuits": two realistic fish styles (Eel and Mackerel) and three weird, abstract shapes that don't look like any real fish.
• Instead of picking just one, the computer blends them together. It takes 20% of the Eel, 30% of the Mackerel, and 50% of a "Weird Shape" to create a brand new, unique swimming profile.
• This creates a massive playground of possibilities, allowing the computer to invent swimming styles that no human would ever think to draw.

3. The Search Engine: "Bayesian Optimization" (The Smart Detective)

Now, they have millions of possible swimming styles. Testing them all by running a full physics simulation is like trying to taste every single soup in a giant kitchen to find the best one. It would take forever and burn a lot of energy.

So, they used Bayesian Optimization, which acts like a super-smart detective.

• Instead of tasting every soup, the detective tastes a few, makes a guess about what the best soup might taste like, and then only tastes the most promising ones next.
• It learns from every test, getting smarter with every step, quickly zeroing in on the "Goldilocks" swimming style without wasting time on the bad ones.

4. The Result: The "Super-Swimmer"

The computer found a winner!

• The Score: The new, computer-invented swimmer was 16% to 35% more efficient than the best natural fish styles (like the eel or mackerel).
• The Secret Sauce: The perfect swimmer looked a lot like an eel, but with a twist: its head moved in the opposite direction of its tail at certain moments.
  • Analogy: Imagine rowing a boat. Usually, you pull the oars back. This new swimmer figured out that if you push your head forward slightly while your tail pushes back, you create a "slingshot" effect that grabs more water and pushes you forward harder.

5. Why It Works: The Energy Economy

The researchers looked under the hood to see why this new style was so good. They found two main tricks:

1. Better Grip: The new shape created stronger, more organized swirls of water (vortices) behind it, like a perfect spiral staircase of water that pushes the fish forward.
2. Recycling Energy: In the front part of the fish, the new shape used less energy to move. In the back part, it was really good at "catching" energy from the water that had already been disturbed, recycling it to help push the fish forward. It was like a hybrid car that recaptures energy when braking.

The Big Picture

This study proves that by letting computers invent new shapes (morphing) and smartly search for the best ones (Bayesian optimization), we can design underwater robots that are far more efficient than anything nature has evolved so far.

In a nutshell: They didn't just copy a fish; they used math to invent a "super-fish" that swims with the efficiency of a sports car, using less fuel (energy) to go faster. This could lead to better underwater drones for exploring the ocean, delivering medicine, or monitoring the environment.

Technical Summary

Here is a detailed technical summary of the paper "Optimized Fish Locomotion Using Design-by-Morphing and Bayesian Optimization" by Farooq et al.

1. Problem Statement

The paper addresses the challenge of optimizing the swimming profiles of self-propelling, fish-like robotic systems to maximize propulsive efficiency. While traditional Computational Fluid Dynamics (CFD) studies often rely on fixed baseline geometries (e.g., standard anguilliform or carangiform modes) or localized parametric variations, these approaches limit the exploration of the broader design space. The authors identify a gap in the literature regarding the simultaneous optimization of both:

• Geometric parameters: The shape of the swimming profile (morphing weights).
• Kinematic parameters: Oscillation frequency ($f$) and non-dimensional wavelength ($\lambda^*$).

The goal is to discover unconventional, high-performance swimming gaits that outperform natural biological modes in terms of energy conversion and thrust generation under realistic hydrodynamic loading.

2. Methodology

The study employs a tightly coupled Design-by-Morphing (DbM) and Bayesian Optimization (BO) framework integrated with high-fidelity 2D CFD simulations.

A. Design-by-Morphing (DbM)

Instead of defining a single shape, the swimming profile $A(x^*)$ is modeled as a linear combination of five baseline shapes:

1. Anguilliform (eel-like, whole-body undulation).
2. Carangiform (trout-like, posterior undulation).
3. Horizontal (constant amplitude).
4. Third-degree "weird" shape (unconventional polynomial).
5. Fourth-degree "weird" shape (unconventional polynomial).

The profile is mathematically defined as:
$$A(x^*) = a_n \frac{1}{\gamma} \sum_{j=1}^{5} \omega_j A_j(x^*)$$
Where $\omega_j$ are weighting coefficients (design variables) constrained to a unit hypersphere to ensure independence, and $\gamma$ is a normalization factor. This allows for continuous interpolation between natural and unconventional shapes, creating a high-dimensional, continuous design space.

B. Fluid Dynamics Simulation (CFD)

• Governing Equations: Unsteady, incompressible Navier-Stokes equations solved using the Arbitrary Lagrangian–Eulerian (ALE) formulation to handle fluid-structure interaction (FSI) for self-propelling foils.
• Numerical Scheme: Finite difference method on an 'O'-type body-fitted grid. Spatial derivatives use second-order central differences (except for convection, which uses the QUICK scheme), and temporal discretization uses a semi-implicit Crank-Nicolson/Adams-Bashforth scheme.
• Re-meshing: A Radial Basis Function (RBF) interpolation technique is used to deform the mesh dynamically, ensuring quality during large body deformations.
• Self-Propulsion: The foil moves freely; forward velocity ($U_o$) is determined dynamically by balancing hydrodynamic forces with the body's inertia ($m \frac{dU_o}{dt} = F_x$).

C. Bayesian Optimization (BO)

To navigate the computationally expensive design space (where each evaluation requires a full CFD simulation), the authors use MixMOBO (Mixed-Variable Multi-Objective Bayesian Optimization).

• Surrogate Model: Uses Gaussian Processes to predict the performance landscape.
• Strategy: Balances exploration (searching new areas) and exploitation (refining known good areas) to find the global optimum with minimal CFD evaluations (150 epochs + 50 initial samples).
• Objective: Maximize propulsive efficiency ($\eta$), defined as the ratio of time-averaged kinetic energy of forward motion to the work performed by the undulatory actuation.

3. Key Contributions

1. Novel Framework: Integration of Design-by-Morphing with Bayesian Optimization for hydrodynamic profile optimization, allowing the discovery of non-intuitive, high-efficiency gaits that deviate from natural biological modes.
2. Simultaneous Optimization: The first study to jointly optimize geometric morphing weights and kinematic parameters (frequency and wavelength) for self-propelling systems under FSI conditions.
3. Performance Benchmarking: Establishment of a rigorous baseline by comparing optimized results against standard anguilliform and carangiform modes across a wide frequency-wavelength domain.
4. Mechanistic Insight: A detailed decomposition of hydrodynamic forces and work distribution (spatial and temporal) to explain why the optimized profiles perform better, specifically focusing on energy recovery and drag reduction.

4. Key Results

• Efficiency Gains: The optimized profiles achieved peak propulsive efficiencies between 49% and 57%, representing a 16%–35% improvement over the best-performing reference anguilliform and carangiform modes (which peaked around 35%–42%).
• Optimal Configuration: The best-performing profile (1st Best) achieved 57% efficiency at a low frequency ($f=0.2$) and a wavelength of $\lambda^* = 1.3$.
• Profile Characteristics:
  • The optimal shape is a hybrid, closely resembling the anguilliform mode but with a distinct out-of-phase head motion (the head moves opposite to the tail in certain phases).
  • This head motion creates stronger, more coherent vortex roll-up in the mid-to-posterior region, enhancing momentum transfer.
• Force and Work Decomposition:
  • Force: The optimal profile generates higher axial and lateral forces with minimal phase lag. It relies more on push-type thrust (pressure-driven) in the anterior and posterior regions compared to the pull-dominated anguilliform mode.
  • Energy: While the optimal profile consumes slightly more total work, it achieves significantly higher forward speed. Crucially, it exhibits enhanced energy recovery (negative work) in the posterior region, where the fluid does work on the foil.
  • Regional Analysis: The anterior region requires less mechanical effort, and the posterior region maximizes energy reutilization, leading to a superior net energy balance.

5. Significance and Implications

• Bio-Inspired Robotics: The study provides a powerful design tool for creating Autonomous Underwater Vehicles (AUVs) and microrobots that are significantly more energy-efficient than current bio-inspired designs.
• Beyond Nature: It demonstrates that strictly mimicking natural swimming modes is not always optimal; "unnatural" morphing strategies guided by data-driven optimization can yield superior hydrodynamic performance.
• Design Philosophy: The work validates the use of surrogate-assisted optimization (BO) combined with morphing techniques as a computationally feasible route to solving complex, high-dimensional fluid-structure interaction problems.
• Future Applications: The findings have direct implications for the design of next-generation underwater propulsion systems, flow-energy harvesting devices, and targeted drug delivery microrobots where energy efficiency is critical.

Limitations: The study is based on 2D simulations. While 2D is a standard approximation for capturing essential hydrodynamic mechanisms, 3D effects (such as spanwise flow and fin interactions) are not accounted for and may influence the final optimal characteristics in real-world applications. Additionally, the optimized kinematics assume idealized flexibility, which may be constrained by the material properties of physical robots.