Model wave coarsening as an active, nonlinear instability—not as ordinary diffusion in a passive lattice. Start with a calibrated discrete mechanical model containing inertia, elasticity, damping and direction-dependent (non-reciprocal) active couplings. Derive its long-wavelength continuum or amplitude description, identify the unstable band and fastest-growing mode, then integrate the nonlinear equations long enough for modes to compete. Measure a correlation length or structure-factor peak while independently tracking period, amplitude and elastic energy.
In the demonstrations reported by Veenstra, Binysh, Seinen and co-authors in Nature (2026), vibrations first appeared at a short wavelength and then grew in wavelength, period and amplitude before settling into a globally synchronized time-crystalline state. The workflow below shows how to reproduce that route, determine a regime-specific coarsening exponent, include friction and boundaries that arrest growth, and validate a simulation against measured wave fields.
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What wave coarsening is, and what a model must reproduce
Wave coarsening is the progressive enlargement of a strain or displacement pattern in an active elastic medium. A useful simulation must reproduce four coupled observations: the dominant wavelength increases, the oscillation period increases, the wave amplitude grows, and several initially independent regions eventually synchronize. This is a dynamical process driven by active non-reciprocal interactions and nonlinear mode competition. A passive linear lattice can support normal modes, but it does not produce the reported self-selected wavelength growth and synchronization.
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Choose and calibrate a discrete mechanical model
Represent each unit cell with measured mechanical variables
Use one translational coordinate per mass, one angular coordinate per rotor, or a vector containing both when a cell has coupled translation and rotation. For a translational chain, a generic equation is
m ü_i = F_i^elastic(u) + F_i^active(u, u̇) − c u̇_i.For a rotational cell, replace mass m with rotational inertia I and displacement u_i with angle θ_i. A two-dimensional lattice can carry longitudinal springs, torsional springs and active non-reciprocal longitudinal bonds, as in the mechatronic architecture used by Veenstra and colleagues. The controller and sensors implement the direction-dependent force rule; the numerical model should use the same measured rule rather than an assumed symmetric spring.
Include the ingredients that permit coarsening
- Inertia: retain the second time derivative when the experiment is vibrational. Dropping it changes the dispersion and can turn a wave problem into an overdamped phase-ordering problem.
- Elasticity: include the linear stiffness measured around the operating point and nonlinear terms required by the force–displacement or torque–angle curve.
- Damping and friction: distinguish viscous damping, dry friction and other momentum-conserving-breaking losses. Their frequency dependence and spatial distribution affect both the onset and arrest of coarsening.
- Active non-reciprocity: retain the odd, direction-dependent coupling. A reciprocal replacement may match a static stiffness matrix while eliminating the instability that starts wave coarsening.
- Geometric constraints: encode the actual lattice spacing, coordination, cell mass or inertia, and boundary connections. These determine which low-wavenumber modes fit in a finite sample.
Calibrate before fitting long-time behavior
Measure the unit-cell force–displacement or torque–angle law, damping, actuator delay and sensor response. Fit the local constitutive law first, using the same amplitude and frequency range that the lattice will explore. Then identify the active coupling from directional tests: apply the same displacement or velocity pattern from opposite sides and verify that the measured forces differ as expected. Keep calibration parameters separate from coarsening observables so that the exponent is not fitted by silently changing the underlying mechanics.
| Model component | What to measure or specify | Why it matters for coarsening |
|---|---|---|
| Mass or rotational inertia | Cell mass, rotor inertia and any attached payload | Sets inertial time scales and the real part of the dispersion relation |
| Elastic elements | Linear stiffness, nonlinear force or torque curve, prestress | Sets wave speed, saturation and mode coupling |
| Damping | Viscous coefficient, frequency dependence and controller losses | Changes growth rates and can suppress long wavelengths |
| Non-reciprocal bond | Direction-dependent force law, gain, phase lag and saturation | Breaks left–right symmetry and supplies active instability |
| Friction or pinning | Static and kinetic thresholds, contact locations | Can arrest bulk coarsening and leave defects or edge waves |
| Geometry | Cell spacing, lattice connectivity and boundary type | Determines allowed wavenumbers and dimensional scaling |
Derive a continuum or amplitude description
Take the long-wavelength limit without deleting odd terms
After writing the discrete equations, expand neighboring variables about a cell position, for example u_{i±1} = u(x) ± a∂ₓu + a²∂ₓ²u/2 + …, where a is the lattice spacing. Keep enough orders to represent the unstable band and the eventual low-wavenumber dynamics. A schematic one-dimensional inertial continuum equation is
ρ uₜₜ = ∂ₓ[σ(ε)] − γ uₜ + η₁ ∂ₓu + η₂ ∂ₓuₜ + D₂ ∂ₓ²u + D₄ ∂ₓ⁴u + N(u, ∂ₓu, uₜ), ε = ∂ₓu.Here σ(ε) is the nonlinear stress, γ is ordinary damping, the η terms represent odd or non-reciprocal response, the higher-gradient terms regularize short wavelengths, and N collects nonlinear mode coupling. The exact signs and tensor structure must come from the discrete architecture. In two dimensions, replace derivatives by gradients and use the measured longitudinal and torsional constitutive tensors.
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Use an amplitude equation when only the envelope is needed
Near onset, represent a nearly periodic displacement as a slowly varying complex amplitude multiplying the fastest linear mode. An illustrative amplitude model is
∂ₜA = μA + (D + iDₒ)∇²A − g|A|²A + h∇⁴A + …The real part of μ controls onset, D controls ordinary diffusion of the envelope, and the imaginary or odd coefficient Dₒ captures phase transport that is absent from a reciprocal model. The cubic term limits amplitude; additional terms may be required for inertial oscillations, multiple polarizations or subcritical onset. Treat this equation as a reduction to be validated against the full lattice, not as a substitute for calibration.
The coarse-graining perspective developed in the University of Amsterdam thesis on non-reciprocal metamaterials defines wave coarsening as a progressive increase in the size of strain patterns. That interpretation is useful when deciding which field to coarse-grain: displacement, strain, rotation or a complex oscillation envelope can give different short-time lengths, although they should agree on the long-wavelength trend when the reduction is valid.
Check the reduction against the discrete dispersion
For each candidate continuum coefficient, compare its small-wavenumber eigenvalues with those of the discrete model. Match wave speed, damping rate, instability threshold and the location of the fastest-growing band before running nonlinear simulations. If the continuum model predicts a stable long-wave limit while the lattice has a growing low-frequency branch, a term was lost during expansion.
Find the onset mode from the complex dispersion relation
Linearize about the rest state
Write the discrete state as a small perturbation and substitute a Bloch form q_i(t) = q̂ exp(λt + iki a). For a multi-degree-of-freedom cell, solve
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det[λ²M + λC(k) + K(k)] = 0.M is the mass or inertia matrix, C(k) contains damping and active velocity couplings, and K(k) contains elastic and active displacement couplings. Because the system is non-reciprocal, these matrices need not satisfy the symmetry relations of a passive reciprocal lattice. The real part of λ(k) is the growth rate; the imaginary part gives the oscillation frequency.
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- Sample the first Brillouin zone at a resolution fine enough to resolve narrow growth-rate peaks.
- For every
k, compute all eigenvalues and retain the branch with the largest real part. - Mark the interval where
Re(λ) > 0. Its boundaries are the linear instability band. - Record
k*, the wavenumber at the maximum growth rate, andω* = Im(λ(k*)). - Use
2π/k*as the expected initial wavelength, subject to the finite-domain mode grid.
The phenomenological model in the Nature study has an unstable band. Its friction term gaps the spectrum at zero wavenumber, which is a useful diagnostic: if your zero-wavenumber mode remains neutrally translating when the experiment has friction, the dissipation model is incomplete.
Separate linear onset from nonlinear coarsening
The fastest-growing mode predicts the pattern immediately after onset, not the final synchronized state. Once amplitudes are finite, nonlinear mode coupling transfers activity toward lower wavenumbers. Do not infer a coarsening law from the linear growth curve; use it only to initialize and interpret the early-time simulation.
| Question | Linear calculation answers | Nonlinear simulation must answer |
|---|---|---|
| Which pattern appears first? | Unstable band and fastest-growing k* |
Whether defects or competing bands alter the selected pattern |
| Does the rest state destabilize? | Sign of Re(λ) |
Whether nonlinear saturation produces a bounded attractor |
| What is the frequency? | Im(λ) near onset |
How period changes as wavelength and amplitude grow |
| Will the system synchronize? | Not determined | Long-time phase locking, arrest or persistent defects |
Integrate the nonlinear equations reliably
Resolve both the fastest wave and the slow coarsening
Use a validated explicit or implicit second-order integrator. The spatial mesh must resolve the shortest unstable wavelength with several cells per wavelength; refine it until the growth rate and spectral peak stop moving appreciably. The time step must resolve the highest mechanical frequency, controller bandwidth and any stiff torsional or contact mode, not merely the slow coarsening time.
Use a staged simulation protocol
- Equilibrium check: set active gain to zero and verify that the rest state and passive normal modes match analytical or measured values.
- Linear-growth check: turn on the active coupling with perturbations small enough for linear theory and compare measured growth rates with
Re(λ(k)). - Nonlinear run: restore the calibrated nonlinear law and integrate until the characteristic length changes by many factors or reaches the domain size.
- Parameter sweep: vary active gain, damping, friction, domain size and boundary condition one at a time before exploring combined changes.
- Convergence check: repeat with smaller time step, finer mesh and a longer domain. A coarsening exponent that changes under these tests is numerical, not physical.
Choose initial conditions that reveal mode competition
Use low-amplitude broadband noise, several localized perturbations or a superposition of modes around the unstable band. A single perfectly periodic initial condition can suppress defects and falsely suggest immediate synchronization. Record the random seed and initial spectrum so runs can be reproduced.
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Track energy without assuming it is conserved
Compute kinetic energy, elastic energy, actuator work and dissipated power separately. Active work can inject energy while friction removes it; total mechanical energy therefore need not be monotonic. A stable simulation should show bounded energy after nonlinear saturation, with the injected and dissipated power balancing over long windows.
Measure the coarsening length and exponent
Use correlations or the structure factor as the primary length
Let s(x,t) be the chosen scalar field, commonly strain or a displacement component, after subtracting its spatial mean. The equal-time correlation is
C(r,t) = ⟨s(x+r,t)s(x,t)⟩ₓ.Define a characteristic length consistently, for example the first zero or the distance where C(r,t)/C(0,t) falls to a specified threshold. In Fourier space, calculate the structure factor S(k,t) = |ŝ(k,t)|². A robust spectral length is ξₛ(t) = 2π/kₚₑₐₖ(t); a first-moment length can be used when the peak is broad. State the definition because different choices differ by a constant factor and can differ during defect-rich transients.
Fit a power law only inside the scaling interval
For a coarsening interval, test
ξ(t) ∝ tⁿ, or equivalently kₚₑₐₖ(t) ∝ t⁻ⁿ.Plot log ξ against log t, identify the interval with an approximately constant local slope, and fit only that interval. Exclude startup, the crossover when the peak leaves the unstable band, and the finite-size plateau when the wavelength approaches the domain. Report the fit interval, uncertainty, spatial dimension, damping and friction, boundary condition, and whether the length came from correlations or a structure-factor moment.
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Use complementary observables
- Period: obtain the dominant temporal frequency from a local probe or temporal Fourier spectrum. Its inverse should grow as the spatial wavelength grows, but phase slips can make the two measures differ temporarily.
- Amplitude: track the envelope of strain or displacement after filtering around the dominant band. Growth that comes only from a changing filter is not physical amplitude growth.
- Low-mode energy: integrate
S(k,t)over the first few allowed modes to detect transfer toward system-scale patterns. - Defect count: count phase singularities, strain discontinuities or domain walls when the field supports them. Defects often explain departures from a clean power law.
- Elastic energy: compare the energy in the dominant band with broadband energy to distinguish coherent coarsening from simple heating.
Veenstra and colleagues use spatial correlations, amplitude spectra, low-mode amplitudes and inverse-correlation-length plots together. Reproducing the same trend in at least one length measure and two independent temporal or energetic measures is stronger evidence than fitting a single curve.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Model friction, boundaries and dimensionality explicitly
Friction and other momentum-breaking terms
Add friction where it exists physically: at contacts, bearings, supports or actuator interfaces. Dry friction can create a threshold below which a defect or domain wall stops moving; viscous friction instead changes the growth rate continuously. Sweep the loss parameter and record the arrest length, residual defect pattern and edge activity. The reported experiments show frictional arrest, stable strain defects and edge-localized waves when coarsening is interrupted.
Boundary conditions
- Periodic ring or torus: removes physical edges and is the cleanest test of bulk scaling, but restricts wavelengths to discrete modes.
- Free or compliant edge: permits boundary-localized modes and can seed coarsening from the perimeter.
- Clamped edge: reflects waves and can pin phase or create standing-wave defects.
- Mixed boundaries: useful when the experiment has supports on only part of its perimeter; compare directly with the measured fixture.
Run the same active parameters under at least two boundary conditions before attributing edge waves to a bulk instability.
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Compare one and two dimensions before extrapolating
A one-dimensional chain or ring has a single wavevector axis and a simple defect topology. A two-dimensional lattice has directional coarsening, anisotropic correlations and more ways for defects to move or pin. The 2026 demonstrations include a two-dimensional mechatronic sample with 37 unit cells and a one-dimensional elastic-ring demonstration with 24 torsional and non-reciprocal unit cells. Those counts describe specific experimental devices, not a minimum size for every simulation.
For a three-dimensional active solid, calculate correlation lengths separately in radial, in-plane and out-of-plane directions. Extended Data Fig. 7 of the Nature article reports power-law inverse-correlation-length behavior in those directions, illustrating why a single isotropic average can hide anisotropic scaling.
Validate a simulation against experiment
Match observables, not just a visually similar snapshot
- Synchronize clocks: define a common trigger, sampling rate and time origin for measured and simulated data.
- Match fields: compare the same displacement, rotation or strain component at the same sensor locations or interpolated coordinates.
- Match preprocessing: apply equivalent detrending, filtering, windowing and normalization; document every filter cutoff.
- Compare onset: test the first unstable wavelength, growth rate and oscillation frequency against the linear prediction.
- Compare coarsening: calculate the same correlation or structure-factor length, period and amplitude in both datasets.
- Compare defects and edges: map persistent strain defects, phase slips and edge-localized waves rather than discarding them as noise.
- Quantify error: report residuals or confidence intervals over time and across repeated runs, not only representative images.
Use the published geometries as checks
For the specific replication/preprint dataset described as 90 seconds averaged over 10 experimental runs, verify the averaging window and run selection against the released materials before using that number as a benchmark. It should not be generalized to every experiment. First reproduce the published figures and parameter settings, then alter one parameter at a time. This sequence distinguishes an implementation error from a genuine model improvement.
Reproduce before extending
The Zenodo replication package provides simulation code, environment information, figures and supplementary materials. Its record lists an 11.7 GB file set and 161.8 GB of total data volume across versions. Preserve the package version, software environment, commit or archive identifier, random seeds and hardware details in your own run log. Re-running a figure from the package is a practical test of your numerical pipeline before introducing a new constitutive law.
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Compare model choices with a decision framework
| Choice | Use it when | Main strength | Main risk |
|---|---|---|---|
| Discrete lattice | Cell geometry, actuator rules or defects are central | Retains architecture and direct calibration | Large systems are expensive and parameters can be numerous |
| Continuum PDE | Wavelength is many cell spacings and scaling is the goal | Clarifies long-wave terms and dimensional effects | Coarse-graining can remove lattice resonances or odd couplings |
| Amplitude equation | Dynamics stay near a well-defined onset band | Efficient analysis of envelope and phase competition | May fail at large amplitude, defects or multiple bands |
| Reciprocal coupling | Passive control case or null experiment | Provides a baseline dispersion | Cannot reproduce the active coarsening route by itself |
| Non-reciprocal coupling | Modeling the reported active instability | Captures direction-dependent growth and phase transport | Requires directional calibration and controller dynamics |
| Inertial dynamics | Vibrational metamaterials and wave propagation | Retains frequency, period and wave momentum | Stiffer and more costly time integration |
| Overdamped dynamics | Validated low-frequency limit with negligible inertia | Simple and inexpensive | Can remove the oscillatory mechanism under study |
| Periodic boundaries | Bulk scaling and clean exponent estimates | Eliminates edge contamination | Cannot predict edge waves or support-induced arrest |
| Physical friction and edges | Comparing with a finite laboratory sample | Explains arrest, defects and localization | Exponent fits may have a shorter scaling interval |
Troubleshoot common failures
No instability appears
- Check the sign and units of the active gain and verify that the computed
Re(λ)is positive somewhere. - Confirm that controller delay and sensor filtering were not omitted; either can move the unstable band.
- Ensure damping was not applied twice, once in the actuator model and once in the mechanical equation.
- Test a small perturbation with the nonlinear terms disabled to separate a coding error from nonlinear suppression.
The pattern grows but never coarsens
- Run longer and use a domain with several wavelengths of the initial fastest-growing mode.
- Inspect whether the active law saturates before modes can interact.
- Remove artificial pinning, excessive dry friction or an overly strong boundary spring in a controlled comparison.
- Check spatial resolution; numerical dispersion can lock the peak to the grid.
The fitted exponent changes whenever the fit window changes
- Plot the local log–log slope and identify whether there is a genuine plateau.
- Separate startup, crossover and finite-size regimes instead of fitting the entire trace.
- Increase domain size or simulation duration if the scaling interval is shorter than a decade.
- Report anisotropic lengths and regime parameters rather than averaging incompatible directions.
Simulation and experiment disagree only near the edges
- Replace ideal periodic boundaries with the measured support stiffness and damping.
- Include sensor and actuator placement in the observation model.
- Compare bulk and edge regions separately; an edge-localized mode is a physical result, not necessarily a failed bulk model.
A reproducible end-to-end checklist
- Define the target field, geometry, cell count and boundary condition.
- Measure and fit mass or inertia, elasticity, damping, friction and directional active coupling.
- Implement the discrete second-order equations and verify passive modes.
- Derive the continuum or amplitude reduction while retaining odd/non-reciprocal derivatives.
- Compute the complex dispersion relation, unstable band, fastest-growing wavenumber and onset frequency.
- Run a resolved nonlinear simulation with documented initial conditions, time step and mesh.
- Calculate correlations or a structure factor, then track period, amplitude, low-mode energy and elastic energy.
- Fit
ξ ∝ tⁿonly over a demonstrable scaling interval and state all regime qualifiers. - Repeat with friction, alternative boundaries and the relevant dimensionality to locate arrest and edge effects.
- Reproduce published figures from the Zenodo package before changing parameters or constitutive laws.
- Compare simulated and measured wave fields, spectra, lengths, periods, amplitudes and defects using identical preprocessing.
References and further technical reading
For the active coarsening mechanism, use Veenstra, Binysh, Seinen and co-authors, Nature (2026), including the figures on the phenomenological model and three-dimensional scaling. The University of Amsterdam thesis on coarse-graining non-reciprocal metamaterials develops the continuum interpretation of increasing strain-pattern size. Schmitz and Smith, Mechanical Vibrations: Modeling and Measurement, 2nd edition (Springer, 2020), covers modal analysis, measurement, continuous-beam models and finite-element fundamentals. Elastic Waves and Metamaterials: The Fundamentals provides a discrete-to-continuum route for structural vibration and acoustics. ETH Zurich’s mechanical-metamaterials lecture notes cover waves in solids, discrete systems, continuum limits and topological mechanical metamaterials. The Zenodo replication package supplies the code and data needed for reproducibility.
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