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Swarmlet: a GPU-resident reformulated vortex particle method for coupled multirotor aerodynamics and flight dynamics

Wyatt James Horne

arXiv:2610.29867v1math.NAcs.NAphysics.flu-dyn

TL;DR

Swarmlet addresses the challenge of predicting interacting multirotor wakes by coupling a GPU-resident rVPM wake solver to rotor models, vehicle dynamics, and flight control. Its wake impulse is retained across rotor wakes and canonical rings, while validation is limited to hovering-rotor mean loads and the formation demonstration is not a validation.

  • Problem

    Predicting multirotor wake interactions requires an unsteady wake model coupled to the rotor loads that create the wakes and to vehicle motion.

    Source: §1 · p. 1

  • Method

    Swarmlet couples an rVPM particle wake, three rotor-model fidelities, six-degree-of-freedom vehicles, and flight control in a GPU-resident solver.

    Source: §1 · p. 2

  • Results

    Impulse ratios for rotor wakes stay between 0.97 and 1.01, while canonical rings retain at least 99.9% of their impulse.

    Source: §6 · p. 14

  • Takeaways & Limitations

    The results indicate that treatment of aged vorticity, including fixed-filter-width closure and global corrections, is central to retaining wake impulse.

    Source: §6 · p. 14

    Validation compares mean loads of hovering rotors, uses calibrated small-rotor polars and modelled DJI polars, and treats the 100-vehicle formation as a demonstration rather than validation.

    Source: §6 · p. 14

Abstract

from arXiv · show

We present Swarmlet, a GPU solver for the unsteady aerodynamics and flight dynamics of one or many multirotor vehicles. The wake is a meshless, evolving-core vortex particle field in the reformulated (rVPM) large-eddy formulation, coupled to rotor models of three fidelities, six-degree-of-freedom vehicles and flight control. Particles carry a smooth core of compact support whose field outside four core radii is exactly the singular Biot-Savart field, so the multipole far field of the treecode carries no regularization error. The subfilter closure is a dynamic eddy viscosity on a fixed filter width, applied by core spreading with a Lagrangian exchange of its local excess under an explicit stability bound, and the strengths are relaxed by projection; global corrections make both the exchange and the relaxation conserve total vorticity and linear impulse. All rotor models shed connected bound circulation, the disks along the arcs swept in each step however long, so the wake receives the lift impulse. The solver agrees with direct summation to $5\times 10^{-7}$ and converges at third order in time. Without calibration, an actuator-line model that includes its own near sheet predicts the hover thrust of a DJI 9443 propeller within 2%. With calibrated section polars it reproduces a Crazyflie propeller's thrust and torque within 3.2% at three speeds and the near wake of a hovering quadrotor with relative $L^2$ differences of 0.08-0.21 against particle image velocimetry, while the wake keeps the delivered impulse. A closed-loop formation of 100 quadrotors, run at a coarser setting on one GPU, illustrates the coupled method at swarm scale.

1. Introduction

Interacting multirotor wakes alter thrust and moments, calling for an unsteady wake model coupled to rotor loads and vehicle motion. Swarmlet uses rVPM to address turbulent-wake instability and couples the wake, rotors, vehicles, and flight control in a GPU solver.

1. Introduction

  • Wake interactions change thrust and moments, so prediction requires coupling an unsteady wake model to rotor loads and vehicle motion.The cited examples include a vehicle flying in a neighbour’s jet and rotors ingesting one another’s tip vortices.Source: §1 · p. 1
  • Blade-element momentum theory assumes isolated rotors in steady, axisymmetric inflow, while mesh-based LES is costly and requires a volume mesh that follows each vehicle.The passage describes mesh-based LES as expensive for one rotorcraft and prohibitive for a formation.Source: §1 · p. 1
  • Existing particle-wake approaches become unstable in turbulence as stretching inflates particle strengths while cores remain fixed; rVPM shares stretching between strength and core size and adds an SFS model.The passage characterizes this as yielding a meshless large-eddy simulation.Source: §1 · pp. 1–2
  • Swarmlet combines rVPM, three rotor-model fidelities, six-degree-of-freedom vehicles, and flight control in a GPU solver for one or many multirotors.The same method and wake numerics are used on one or many GPUs and across the validation cases.Source: §1 · p. 2
  • Its contributions include conservative subfilter closure, connected circulation shedding, a GPU-resident treecode, canonical verification, three experimental validations, and a 100-quadrotor formation demonstration.The closure corrections conserve total vorticity and linear impulse; the formation is a demonstration rather than a validation.Source: §1 · p. 2

2. Method

The method represents free vorticity with evolving vortex particles and uses rVPM to share stretching between particle strength and core size. A compact-support core retains the singular field outside its support, enabling exact exterior interactions and finite-range direct interactions.

2.1. Filtered vorticity transport and vortex particles

  • Free vorticity is carried by particles with positions, vector strengths, core sizes, and material volumes, while their radially symmetric unit-integral cores define the represented vorticity field.The induced velocity gradient is evaluated analytically, and rVPM shares stretching along a particle’s own axis between its strength and core.Source: §2.1 · p. 2

2.2. A smooth core of compact support

  • The compact core is constructed by subtracting a first-order Taylor expansion from the Gaussian profile and renormalizing; its profile and derivative vanish at R = 4.For a core of radius Rσ, the figure compares the compact and Gaussian profiles and shows the compact core’s relative kernel difference vanishing beyond ρ = R.Source: §2.2 · p. 3, Figure 1 · p. 3
  • Outside Rσ, a radially symmetric unit-integral compact core induces the same velocity and velocity gradient as a singular particle.The compact core has a 2.3% higher peak and a 2.2% smaller root-mean-square radius than a Gaussian core of the same σ.Source: §2.2 · p. 3
  • Because the exterior field is exact, multipole expansions outside particle supports carry no regularization error, while directly summed interactions have finite range.A thin ring translates 0.44% faster with the compact core than with a Gaussian core.Source: §2.2 · p. 3

2.3. Subfilter closure

The closure uses dynamic eddy viscosity at a fixed filter width, applying its global component through core spreading and each particle’s local excess through Lagrangian exchange. Stability limiting and global corrections preserve the closure’s total vorticity and linear impulse.

2.3. Subfilter closure

  • The closure models unresolved scales with dynamic eddy-viscosity diffusion at a fixed filter width rather than one that shrinks with particle cores.A core-following filter would switch off the model for particles whose cores shrink under stretching.Source: §2.3 · p. 3
  • The global viscosity is applied by core spreading, which grows cores while leaving strengths and positions untouched and exactly conserving total vorticity and linear impulse.The passage reports a median core size of 1.17σs for particles 24–48 revolutions old in the hover wake.Source: §2.3 · p. 4
  • A Lagrangian-averaged particle coefficient supplies excess viscosity for particle-strength exchange, which reduces to core spreading in laminar or young flow and cancels opposite-sign strength in developed turbulence.The coefficient is obtained by relaxing products of each particle’s leading-order tensors along its path.Source: §2.3 · p. 4
  • The explicit exchange limits each particle’s viscosity using its operator diagonal so that the step stays within a stated stability bound.The bound keeps ΔtDq ≤ (1 + κ)/8; no mode changes sign while κ < 3.Source: §2.3 · p. 4
  • Although exchange pairs conserve total vorticity, their redistribution changes linear impulse, so a global correction makes the closure conserve both quantities exactly.The passage notes that this impulse change can drain a wake’s momentum when cores are comparable to the rotor radius.Source: §2.3 · p. 4

2.4. Relaxation

The method relaxes particle strengths toward the local vorticity direction and corrects global changes to preserve total vorticity and linear impulse. Rotor models range from a performance-map disk to an actuator line, with connected shedding designed to deliver the lift impulse to the wake.

2.4. Relaxation

  • Strength projection reduces misaligned particle strength while preserving its magnitude, but both relaxation and the rVPM correction can alter linear impulse without a global correction.The method removes net changes in total vorticity and impulse over the whole wake.Source: §2.4 · p. 5

2.5. Particle lifecycle

  • Particle lifecycle rules remove states outside specified strength or core bounds, redistribute eligible removed strength to nearby particles, and permit nonconservative oldest-particle removal in modest-fidelity swarm runs.Redistribution preserves total vorticity and linear impulse; memory-bounded removal is not conservative.Source: §2.5 · p. 5

2.6. Rotor models and wake coupling

  • The performance-map disk, blade-element disk, and actuator line represent rotor loading with progressively resolved rotor geometry and load points.The performance-map disk uses uniform annular loading; the blade-element disk uses weighted rows with blade chord; the actuator line resolves the blades.Source: §2.6 · p. 5, Figure 2 · p. 6
  • The actuator line includes induction from its own nearby trailing sheet, while the blade-element disk corrects annulus-averaged tip induction using Prandtl’s tip loss.The disk correction raises only the rotor’s own inflow, not external inflow such as another vehicle’s wake.Source: §2.6 · pp. 5–6
  • All rotor representations shed connected circulation, and disk shedding follows swept arcs even across long steps so the wake receives the lift impulse.The actuator-line balance has a relative residual of 5 × 10^-7 in single precision; disk binning departs from arc impulse by less than 10^-3 over steps of several turns.Source: §2.6 · pp. 6–7
  • Actuator lines require at least 18 steps per revolution for chord coverage, whereas disk arc gathering departs from swept-arc impulse by less than 10^-9 in double precision when the step resolves rotation.The wake’s impulse-to-delivered-thrust-impulse ratio measures how well it retains the momentum it receives.Source: §2.6 · p. 7

2.7. Vehicle dynamics and partitioned time integration

Vehicles are modeled as rigid bodies coupled to rotor loads, drag, and wake-responsive flight control, with an outer vehicle-control step and wake substeps. The GPU treecode evaluates particle interactions using far-field expansions and direct compact-core near-field sums, with all vehicle and controller state resident on the GPU.

2.7. Vehicle dynamics and partitioned time integration

  • Vehicles follow Newton–Euler rigid-body dynamics with rotor gyroscopic moments, quaternion attitude, rotor loads, and parasitic drag, while cascaded control allocates commands to motors using current aerodynamic wrenches.The airframe is not otherwise resolved, and the wake and vehicle integration are coupled first order in the outer step.Source: §2.7 · p. 7
  • The wake advances with third-order Runge–Kutta substeps while the vehicle and bound circulation remain fixed at accepted outer-step values.The vehicles use a third-order strong-stability-preserving scheme; refining wake substeps isolates wake temporal error.Source: §2.7 · p. 7

2.8. Particle solver and GPU implementation

  • A k-d tree treecode uses degree-16 solid-harmonic expansions for accepted far-field interactions and direct compact-core sums for other interactions.The stated evaluation cost is O(NP^2 log N + N n̄), where n̄ is the mean number of neighbors within core support.Source: §2.8 · p. 7
  • All state, including vehicles, controllers, and motor allocation, stays resident on the GPU; Table 1 gives common wake parameters for rotor cases, while the swarm uses a coarser setting.On one RTX 4080, right-hand-side evaluation costs 100–140 ns per particle across 6.6 × 10^4 to 4.2 × 10^6 uniformly distributed particles.Source: §2.8 · pp. 7–8, Table 1 · p. 8

3. Verification

The particle solver closely matches direct summation, and vortex-ring tests show rapid spatial convergence and third-order temporal convergence. The subfilter closure largely preserves laminar-ring impulse and, with projection, preserves leapfrogging-pair impulse to four digits.

3.1. Accuracy of the particle solver

  • At P = 16, single-precision production errors are (2.9–5.0) × 10−7 for velocity and gradient, with truncation and rounding of the same order.Raising the expansion degree to 24 lowers errors to the single-precision rounding level, (1.1–3.3) × 10−7.Source: §3.1 · p. 8, Figure 3 · p. 9

3.2. Vortex rings: convergence in space and time

  • Position-error convergence for leapfrogging rings is third order, while translation-speed difference falls to 1.2 × 10−7 at h/σ = 0.5.The reported position-error orders are 2.90, 2.97, and 2.99; speed difference then remains at the solver's truncation level.Source: §3.2 · p. 8, Figure 4 · p. 9

3.3. Subfilter closure on canonical rings

  • The closure retains 99.90% of laminar-ring impulse at both resolutions; with projection, leapfrogging-pair impulse is unchanged to four digits at 6 s and 10 s.The laminar ring's mean core grows to 0.103 m and travels 14% less far than with PSE; the PSE variant retains 98.5% impulse for the laminar ring and 97.5% and 95.6% for the pair at 6 s and 10 s.Source: §3.3 · p. 9, Figure 5 · p. 10

4. Validation

Validation evaluates rotor thrust and torque and a hovering quadrotor near wake against measurements, using revolution-mean loads and declared averaging windows. Results include close thrust and torque agreement, while wake-velocity differences vary by measurement station.

4. Validation

  • Validation loads are reported as revolution means over declared windows because loads fluctuate by several percent between revolutions.The DJI 9443 case uses revolutions 11–20 for its mean thrust comparison.Source: §4 · p. 10, Table 2 · p. 11

4.1. DJI 9443 propeller in hover

  • The actuator line predicts DJI 9443 hover thrust within 2%, while the blade-element disk is 4.5% high.The actuator-line result is not fitted to the measurement; Table 2 reports mean thrust against 2.317 N.Source: §4.1 · p. 10, Table 2 · p. 11

4.2. Crazyflie propeller at three speeds

  • Thrust and torque agree with the Crazyflie specimen mean within 3.2% at all three speeds.The section polars were calibrated on the measurements; at 10,000 rpm, thrust falls from −2.2% in the first half of the window to −4.1% in the second.Source: §4.2 · p. 11, Table 3 · p. 11

4.3. Near wake of a hovering quadrotor

  • Relative L2 differences against PIV range from 0.08 to 0.21 across stations from z/l = 2 to 12 for the hovering quadrotor near wake.The comparison averages revolutions 53–64 and reports the station-specific differences in Figure 7.Source: §4.3 · p. 12, Figure 7 · p. 12

5. Demonstration: a formation of 100 quadrotors

A closed-loop formation of 100 quadrotors is simulated through a hold and translation using a coarser wake setting. Motor-speed changes differ across the formation as wakes interact with vehicles during holding, acceleration, and braking.

5. Demonstration: a formation of 100 quadrotors

  • The 100-vehicle closed-loop formation holds on a 1.5 m-radius sphere for 1 s, then translates 3 m along x over 3 s at a coarser setting.The wake store is capped at 7.2 × 10^6 particles, with wake older than about 0.45 s removed nonconservatively.Source: §5 · p. 13
  • During peak translation at 1.9 m/s, vehicles need 2–6% more motor speed to tilt and accelerate; during braking at 0.7 m/s, the median runs 0.5–0.7% below hover speed.Four vehicles in neighbours’ wakes still need up to 6% more motor speed while braking.Source: §5 · pp. 13–14

6. Discussion and limitations

Wake impulse retention depends on how aged vorticity is treated: the fixed-filter-width approach preserves rotor-wake impulse, while a core-following filter retains about a third. Validation is limited to mean loads of hovering rotors, with calibrated small-rotor polars and modeled DJI polars.

6. Discussion and limitations

  • 0.97–1.01 is the impulse ratio for every rotor wake with fixed-filter-width core spreading and Lagrangian exchange, versus about one third with a core-following filter.Canonical rings retain at least 99.9% of their impulse with the fixed-filter-width approach.Source: §6 · p. 14
  • Global corrections remove impulse changes from exchange, relaxation, and the rVPM correction while preserving their local effects.Source: §6 · p. 14
  • Validation compares mean loads of hovering rotors; small-rotor polars are calibrated, DJI polars are modeled, and the blade-element disk over-predicts DJI thrust by 4.5%.The vehicle’s motors, arms, and body are not represented; the 100-vehicle formation is a demonstration, not a validation.Source: §6 · p. 14

7. Conclusions

Swarmlet combines a GPU-resident rVPM wake solver with rotor models, rigid-body vehicles, and flight control. The paper reports close agreement with direct summation, third-order time convergence, rotor and wake validation results, and a 100-quadrotor formation demonstration.

7. Conclusions

  • Within 2% without calibration, the actuator line predicts reference-propeller hover thrust; with calibrated polars, small-propeller thrust and torque are reproduced within 3.2% across a factor of two in speed.The hovering-quadrotor wake has relative differences of 0.08–0.21 from particle image velocimetry, from one to six rotor spacings below the rotors.Source: §7 · p. 14
  • The particle solver agrees with direct summation to a few parts in 10^7, and the particle equations converge at third order in time.Source: §7 · p. 14
  • At a coarser setting, Swarmlet runs a closed-loop formation of 100 quadrotors on one GPU.Source: §7 · p. 14

Code and data availability

The experimental data used in this work come from cited sources.

Code and data availability

CRediT authorship contribution statement

Wyatt Horne is credited across the study’s conceptual, technical, analytical, investigative, and writing work, as well as data curation and visualization.

CRediT authorship contribution statement

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