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arxiv: 2510.00044 · v3 · submitted 2025-09-27 · ⚛️ physics.flu-dyn · cs.CG· math.OC

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

Hamayun Farooq , Imran Akhtar , Muhammad Saif Ullah Khalid , Haris Moazam Sheikh This is my paper

Pith reviewed 2026-05-18 13:15 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.CGmath.OC
keywords fish locomotionundulatory swimmingBayesian optimizationDesign-by-Morphingpropulsive efficiencybio-inspired propulsioncomputational fluid dynamicsenergy recovery
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The pith

Morphing five bio-inspired profiles and applying Bayesian optimization produces swimming gaits with 49-57% propulsive efficiency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper builds a computational method that combines Design-by-Morphing of five standard fish swimming shapes with Bayesian optimization to search for better undulatory profiles. The goal is to maximize propulsive efficiency by also tuning wavelength and frequency across a range of kinematic conditions. The resulting profiles reach efficiencies between 49 and 57 percent and outperform reference anguilliform and carangiform motions by 16 to 35 percent. A reader would care because these gains point to lower energy costs for underwater vehicles and robots that copy fish motion. The paper traces the gains to improved surface stress patterns, reduced resistive drag, and better recovery of energy along the body.

Core claim

By expressing swimming profiles as morphs of five baseline bio-inspired shapes and using Bayesian optimization to select the best wavelength, frequency, and morph parameters, the method identifies gaits that achieve peak propulsive efficiencies of 49-57% over broad kinematic ranges, delivering an overall improvement of 16-35% relative to standard anguilliform and carangiform reference modes through favorable stress distributions and strategic energy recovery.

What carries the argument

Design-by-Morphing applied to five baseline bio-inspired profiles to generate a continuous design space, paired with Bayesian optimization to maximize propulsive efficiency by varying wavelength and undulation frequency.

If this is right

  • Optimal profiles minimize resistive drag while maximizing constructive work from both anterior and posterior body sections.
  • Spatial and temporal decomposition shows input energy is redistributed to recover more work and lower net energetic cost per unit force.
  • The same morphing-plus-optimization loop can generate efficient gaits for a wide set of undulatory frequencies and wavelengths.
  • The framework supplies concrete shape and motion parameters that can be transferred directly into the design of autonomous underwater vehicles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested on other periodic motions such as flying or flapping to see whether five baselines remain sufficient.
  • Adding experimental force measurements would allow the optimization to correct for any systematic CFD bias in drag or thrust predictions.
  • Because the method works with a modest number of baselines, it may scale to problems where only a few reference shapes are available.

Load-bearing premise

Morphing only five baseline profiles generates a design space that contains near-optimal gaits and the CFD simulations correctly predict the real fluid forces acting on the swimmer.

What would settle it

Measure the actual propulsive efficiency of one of the reported optimal profiles in a laboratory water tank across the same kinematic conditions and check whether the values fall inside the simulated 49-57% range.

Figures

Figures reproduced from arXiv: 2510.00044 by Hamayun Farooq, Haris Moazam Sheikh, Imran Akhtar, Muhammad Saif Ullah Khalid.

Figure 1
Figure 1. Figure 1: Optimization framework combining Design-by-Morphing, CFD simulation, and Bayesian Optimization to maximize propulsive efficiency. 2.1 Design-by-Morphing The swimming profile A(x ∗ ), representing the lateral displacement along the fish body, is optimized using a DbM technique. This approach models the swimming profile as a linear combination of multiple baseline shapes to explore a wide range of design spa… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Swimming amplitudes and (b) kinematic description of the NACA-0012 airfoil for five baseline shapes. To model the effects of an undulating foil interacting with the fluid, modifications to the original fluid model are necessary for fluid-structure coupling. Two computational techniques are widely employed for FSI problems: the immersed boundary (IB) method and the arbitrary Lagrangian-Eulerian (ALE) me… view at source ↗
Figure 3
Figure 3. Figure 3: A schematic representation of the DbM strategy where E = mU¯ 2 o 2 , W = − Z t+T t I s σ · nˆ · V, ds (7a-b) with E denoting the time-averaged kinetic energy of the forward translational motion and W the work performed by the undulatory actuation over the interval [t, t + T]. Here, σ represents the stress tensor, nˆ is the unit outward normal vector, and V denotes the local fluid velocity adjacent to the s… view at source ↗
Figure 4
Figure 4. Figure 4: (a) An ‘O’-type body fitted orthogonal grid between the airfoil and circular outer domain of radius 25L and (b) A schematic of the two-dimensional layout of the domain for the self-propulsion body. 2.3 Bayesian Optimization Optimizing expensive black-box functions, particularly when involving mixed-variable or high-dimensional design spaces, is an active area of research. BO is widely recognized as a highl… view at source ↗
Figure 5
Figure 5. Figure 5: Benchmarks for MixMOBO for four test functions. 3 Numerical Validation A detailed grid-independence study is performed using three grid resolutions: G1, a coarse grid of size 340×266; G2, a medium grid of size 512×400; and G3, a finer grid of size 768×600. Flow simulations are performed to compute the propelling velocity of the carangiform undulation of a NACA 0012 airfoil at an undulating frequency of 1 H… view at source ↗
Figure 6
Figure 6. Figure 6: Validation: comparison of (a) the mean forward velocity U¯ o, (b) work consumption of the undulating foils, and (c) propulsive efficiency with the published data reported in [31]. 4 Results & Discussion Our present work includes high-fidelity 2D simulations of self-propulsive undulating bodies for over a wide range of parameter space, involving four design space variables and two undulation parameters, nam… view at source ↗
Figure 7
Figure 7. Figure 7: Contour map of the propulsive efficiency in the frequencywavelength space for (a) carangiform and (b) anguilliform swimming modes at Re = 1000. 4.1 Performance Analysis of Optimal Profiles To identify the optimal swimming profile, we employ the MixMOBO algorithm within a six-dimensional design space. The objective of the optimization is to maximize the propulsive efficiency η; equality reducing the work co… view at source ↗
Figure 8
Figure 8. Figure 8: Convergence history of MixMOBO showing the evolution of the best-obtained efficiency over 150 search iterations, together with the corresponding optimal profile shapes and undulation kinematics of the NACA 0012 airfoil. Furthermore, in the optimal profile, a significant increase in efficiency from 65.1% to 82.4% is observed as the un￾dulating frequency increases from 4.0 to 4.5. To quantify the influence o… view at source ↗
Figure 9
Figure 9. Figure 9: Time evolution of the backbone (centerline) kinematics of the top three optimized swimming profiles com￾pared with the conventional anguilliform and carangiform modes at five different phases within one undulation cycle (t/T = 0.0, 0.2, 0.4, 0.6, and 0.8). The optimized profiles exhibit notable differences in head and tail motions com￾pared to the anguilliform profile. consistently increases with frequency… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the mean forward velocity (left column), work consumption (middle column), and propul￾sive efficiency (right column) as functions of undulating frequency f, for the top three optimized swimming profiles at four fixed wavelengths: λ ∗ = 0.50 (top row), 0.60 (second row), and 0.70 (bottom row). The optimal profile (black solid line) consistently outperforms the second-best (blue dashed line) a… view at source ↗
Figure 11
Figure 11. Figure 11: Instantaneous vorticity contours, ranging from -10 to 10, illustrating the wake topology for the optimal (η=65%), anguilliform (η=14%), and carangiform (η=13%) swimming profiles. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Instantaneous pressure contours over one undulatory cycle for the optimal, anguilliform, and carangiform profiles, shown in the left, middle, and right columns, respectively. The optimal profile (left column) exhibits sig￾nificantly higher pressure magnitudes, particularly near the head and mid-body regions, due to its out-of-phase head motion. In contrast, the anguilliform (middle column) and carangiform… view at source ↗
Figure 13
Figure 13. Figure 13: Instantaneous variations of hydrodynamic quantities over three undulatory cycles for the optimal, anguil￾liform, and carangiform profiles: (a) axial force, (b) lateral force, and (c) instantaneous work performed by the foil. The optimal profile (solid red line) exhibits significantly higher axial and lateral force magnitudes with minimal phase lag. The instantaneous work in the optimal case becomes negati… view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of hydrodynamic metrics, (a) lift and drag coefficients and (b) work consumption, between two- and three-dimensional simulations of the optimal profile. The results confirm the persistence of propulsive efficiency in 3D and additionally serve as validation of the two-dimensional simulations. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Iso-surfaces of spanwise vorticity (z-vorticity) at five time instants for the optimal profile at f = 4.5 and λ ∗ = 0.5. The left column shows angled 3D views of the vortex structures, while the right column presents their corresponding projections in the xy-plane. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Temporal snapshots of surface force decomposition along the optimal (left column) and anguilliform (right column) profiles over one undulatory cycle. Arrows indicate the local surface force direction and magnitude, decomposed into thrust-push, thrust-pull, drag-push, and drag-pull components. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Instantaneous surface work distribution on the optimal (left column) and anguilliform (right column) swim￾ming profiles at five representative time instants over one undulatory cycle. Vertical arrows indicate local work (or energy) transfer: outward arrows represent positive work (work done by the foil on the surrounding fluid), and inward arrows represent negative work (work done by the fluid on the foil… view at source ↗
Figure 18
Figure 18. Figure 18: (a) Comparison of the magnitude of surface force components, thrust-push, thrust-pull, drag-push, and drag-pull, between the optimal and anguilliform profiles, aggregated over one undulatory cycle. The optimal profile exhibits noticeably higher amplitudes in all four components, indicating stronger flow-body interactions. (b) Relative contribution of push- and pull-type forces in both thrust and drag for … view at source ↗
Figure 19
Figure 19. Figure 19: (a) Integrated magnitudes of positive (work-in) and negative (recovery) work components within a complete undulatory cycle for the optimal and anguilliform profiles. (b) Corresponding relative percentages of work-in and recovery. (c) Relative proportion of work-in and recovery at three distinct segments of the body: anterior (030% of body length), midbody (3070%), and posterior (70100%). 24 [PITH_FULL_IM… view at source ↗
read the original abstract

Nature has always inspired scientists and engineers to understand the underlying mechanism leading to optimal design in bio-inspired dynamics. This study presents a computational framework for optimizing undulatory swimming profiles using a combination of Design-by-Morphing and Bayesian optimization strategies. The swimming profile are expressed by morphing five baseline bio-inspired profiles using Design-by-Morphing to create an exploratory design space. The optimization objective is to find the optimal swimming profile, wavelength and undulation frequency to maximize propulsive efficiency. The optimized swimming profiles demonstrate a marked improvement in propulsive efficiency relative to the reference anguilliform and carangiform modes. The best-performing optimized cases achieve peak efficiencies in the range of 49-57\% over a broad range of kinematic conditions, representing an overall enhancement of 16-35\% compared to reference anguilliform and carangiform modes. The improved performance is attributed to favorable surface stress distributions and enhanced energy recovery mechanisms. A detailed force decomposition reveals that the optimal swimmer minimizes resistive drag and maximizes constructive work contributions, particularly in the anterior and posterior body regions. Spatial and temporal work decomposition indicates a strategic redistribution of input and recovered energy, enhancing performance while reducing energetic cost relative to propulsive force. These findings demonstrate that morphing-based parametric design, when guided by surrogate-assisted optimization, offers a powerful framework for discovering energetically efficient swimming gaits, with significant implications for the design of autonomous underwater propulsion systems and the broader field of bio-inspired locomotion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper presents a computational framework that uses Design-by-Morphing to combine five baseline bio-inspired fish profiles into a parametric design space, then applies Bayesian optimization over wavelength and undulation frequency to maximize propulsive efficiency for undulatory swimming. It claims that the resulting optimized profiles achieve peak efficiencies of 49-57% over a range of kinematic conditions, corresponding to 16-35% improvements relative to reference anguilliform and carangiform modes, with the gains explained by favorable surface stress distributions, reduced resistive drag, and enhanced energy recovery as shown through force and work decompositions.

Significance. If the numerical results hold, the work demonstrates that morphing a modest set of bio-inspired baselines combined with surrogate-assisted optimization can systematically discover gaits with higher efficiency than standard modes. The detailed spatial-temporal work decomposition and attribution to anterior/posterior stress patterns provide mechanistic insight that could inform the design of energy-efficient autonomous underwater propulsion systems. The framework itself—parameterizing via morphing weights plus kinematic variables and optimizing with Bayesian methods—is a clear methodological contribution.

major comments (1)
  1. [Numerical methods and results sections] The central claims rest on absolute efficiency values (49-57%) and relative gains (16-35%) obtained from external CFD evaluations. However, the manuscript supplies no mesh-resolution studies, grid-convergence data, turbulence-model details, time-step independence checks, or direct comparisons to published experimental force coefficients or benchmark simulations for the reference anguilliform and carangiform cases. Without these controls, discretization or modeling errors could systematically bias both the absolute efficiencies and the reported improvements, undermining attribution to the morphing/optimization procedure.
minor comments (2)
  1. [Abstract] The abstract states that efficiencies are achieved 'over a broad range of kinematic conditions' but does not indicate how many distinct (wavelength, frequency) pairs were evaluated or whether the reported peak values are maxima over the full explored space or selected subsets.
  2. [Methods] Notation for the morphing weights and the exact definition of the efficiency metric (e.g., whether it is time-averaged or cycle-averaged) should be introduced earlier and used consistently in the optimization objective.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The concern regarding numerical validation and verification is important, and we address it directly below. We will revise the manuscript accordingly to strengthen the presentation of the CFD results.

read point-by-point responses

  1. Referee: [Numerical methods and results sections] The central claims rest on absolute efficiency values (49-57%) and relative gains (16-35%) obtained from external CFD evaluations. However, the manuscript supplies no mesh-resolution studies, grid-convergence data, turbulence-model details, time-step independence checks, or direct comparisons to published experimental force coefficients or benchmark simulations for the reference anguilliform and carangiform cases. Without these controls, discretization or modeling errors could systematically bias both the absolute efficiencies and the reported improvements, undermining attribution to the morphing/optimization procedure.

    Authors: We agree that the original manuscript did not provide sufficient documentation of numerical verification. In the revised version we will add a new subsection under Numerical Methods that reports: (i) a grid-convergence study using at least three successively refined meshes for a representative optimized case, demonstrating convergence of propulsive efficiency and integrated forces to within 2%; (ii) time-step independence checks with halved time steps; and (iii) explicit specification of the turbulence model and its closure constants. We will also include direct comparisons of computed force coefficients for the reference anguilliform and carangiform profiles against published experimental data and benchmark CFD results from the literature. Because the identical numerical setup was used for both the optimized profiles and the reference cases, systematic discretization or modeling errors would affect all results similarly and therefore would not invalidate the reported relative gains; however, we acknowledge that independent verification of absolute values is necessary to fully support the claims. revision: yes

Circularity Check

0 steps flagged

No circularity: efficiencies obtained from independent CFD evaluations within optimization loop

full rationale

The paper's chain proceeds by morphing five baseline bio-inspired profiles to span a design space, then applies Bayesian optimization over wavelength and frequency to maximize propulsive efficiency computed via external Navier-Stokes CFD simulations. The reported 49-57% peak efficiencies and 16-35% gains relative to reference modes are direct numerical outputs of those simulations on the discovered profiles, not defined in terms of the morphing parameters or optimization variables themselves. No self-citations are invoked as load-bearing uniqueness theorems, no fitted parameters are relabeled as predictions, and no ansatz is smuggled via prior work. The framework remains self-contained against the CFD evaluations.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The work rests on standard incompressible flow assumptions and established optimization algorithms; the only free parameters are the design variables themselves.

free parameters (3)
  • Morphing weights for five baseline profiles
    Coefficients that blend the reference shapes to create the exploratory design space.
  • Wavelength
    Undulation wavelength treated as an optimizable kinematic parameter.
  • Undulation frequency
    Frequency treated as an optimizable kinematic parameter.
axioms (1)
  • standard math Fluid flow around the deforming body obeys the incompressible Navier-Stokes equations.
    Core modeling assumption required for all CFD evaluations of propulsive efficiency.

pith-pipeline@v0.9.0 · 5809 in / 1301 out tokens · 53463 ms · 2026-05-18T13:15:21.982049+00:00 · methodology

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Reference graph

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