—
—
—
⚙ Manufacturing Constraints
Valley A: rA > rB → CK > 0
Valley B: rB > rA → CK < 0
| Model | TopologyNet v2 (in-browser) |
| Training Data | 10,000 PWE simulations |
| Phase Acc | 97.8% |
| Gap R² | 0.976 |
| Chern R² | 0.812 |
| Ext R² | 0.841 |
Ref: Hsu & Wu, Phys. Rev. B 74 (2006); Vila et al., Phys. Rev. B 96 (2017)
Thin plates (h/a<0.3): flexural-dominated, narrow gaps
Moderate (h/a~0.5-0.8): optimal topological gaps
Thick plates (h/a>1): shear-dominated, gaps close
| Model | MindlinTopologyNet |
| Physics | 3N×3N eigenvalue (w, ψx, ψy) |
| Training | 10,000 Mindlin PWE simulations |
| Phase Acc | 98.5% |
| Gap R² | 0.905 |
| Topo R² | 0.489 |
| Ext R² | 0.759 |
| Material | ρ-ratio=8.0, E-ratio=10.0 |
| h/a slices | 0.2, 0.5, 0.8, 1.0, 1.3 |
Bootstrap perturbation of neural network weights to estimate prediction confidence. Adds Gaussian noise (σ=2% of weight std) to NN weights for N=30 iterations. Ref: Lakshminarayanan et al. 2017, NeurIPS
Perturb TR and FF within manufacturing tolerance bounds and assess bandgap survival probability. Ref: Sigmund et al. 2009, JASA; Wang et al. 2011
Simultaneously optimize bandgap width, center frequency, and robustness using NSGA-II. Ref: Deb et al. 2002, IEEE TEC; Bilal et al. 2011, PRE
Upload CSV
CSV with columns: TR, FF (one design per row). Optionally include a for custom lattice constant.
Generate DOE Samples
Group velocity, slow-wave regions, negative refraction detection, and effective mass analysis. Ref: Kushwaha 1993, PRB; Sukhovich et al. 2008, PRL
Retrieve effective mass density ρeff(ω) and bulk modulus Keff(ω) from band structure. Negative values indicate metamaterial behavior. Ref: Fokin et al. 2007, PRB; Li & Chan 2004, PRB
Engineering insertion loss in dB for N-layer phononic crystal. Ref: Hussein et al. 2014, Applied Mechanics Reviews
Supercell band folding approximation for line-defect waveguide modes. Ref: Khelif et al. 2004, PRB; Pennec et al. 2010, SSC
Approximate vibration mode shapes for a finite N×N phononic crystal plate using Bloch mode superposition.
Visualize wave propagation through the phononic crystal at selected frequency. Inside bandgap: evanescent decay. Outside: dispersive propagation. Ref: Laude 2015, Phononic Crystals (Textbook)
Identifies regions of highest model uncertainty to suggest optimal next FEA simulation points. Ref: Settles 2009, Active Learning Literature Survey
Upload an STL file of one phononic-crystal unit cell (binary or ASCII). PhononIQ parses the mesh, computes the bounding box, and extracts plate height h, lattice constant a, thickness ratio TR = h/a, and fill fraction FF from a top-down projection. The 64×64 / 128×128 binary mask is staged on the CNN preview below — click Predict band structure to run inference. STL units are assumed millimetres.
Upload a binary unit-cell mask (dark = solid, light = hole/outside). The image is resampled to 64×64, thresholded, and run through the trained PhononCNNHybrid in your browser. The CNN learned on rhombic honeycomb cells with two circular holes — off-distribution images will give unreliable predictions. The image encodes FF only; the material parameters DR, EMR, PRR, TR enter through the parallel MLP branch below.
This is the v5 deployment of the 3D voxel model — capacity-doubled architecture (channel widths 24/32/48/64/96, FC head 384) trained for 100 epochs with PCA-30 target reparameterisation, PICNN pseudo-labels (1500 samples), mixup, EMA, and safe-symmetry augmentation. The cell geometry is built as a 643 binary voxel volume (1 = solid, 0 = void) with three additional CoordConv channels (x, y, z normalised to [−0.5, 0.5]) and fed through a 3D-conv encoder with squeeze-and-excitation blocks. The 28-feature material MLP runs in parallel; the fusion head predicts 30 PCA coefficients which are reconstructed inside the model graph to the full 15-band × 61-k-point dispersion. ONNX export 3.4 MB total. Best validation R² = 0.9442 (epoch 88 of 100), RMSE 175.4 kHz. v4 (lean, 280K params, R² 0.9429) and v1 (baseline, R² 0.9407) checkpoints retained on disk for ablation reference.
z = TR·a. Without STL, the volume is built analytically from (TR, FF). The 28-feature material MLP branch is the same in both modes. Clear the STL to switch back to analytical mode.
Embed PhononIQ in your website or use the programmatic API for automated predictions.
Widget Configurator
Embed Code
Preview
JavaScript API
// Predict band structure
const result = PhononIQ.apiPredict({ tr: 1.0, ff: 0.35 });
console.log(result.bandgap.width); // kHz
// Batch prediction
const results = PhononIQ.apiBatchPredict([
{ tr: 0.5, ff: 0.30 },
{ tr: 1.0, ff: 0.35 },
{ tr: 1.5, ff: 0.40 }
]);
// Model info
const info = PhononIQ.apiGetModelInfo();
Gaussian Process surrogate with Expected Improvement acquisition function. Intelligently suggests the next best simulation point to maximize bandgap discovery efficiency.
Quantifies how well the trained NN satisfies known physics constraints: Bloch–Floquet boundary conditions, band positivity, band ordering, and effective medium consistency.
Auto-regressive model combining cheap effective medium predictions (Bragg condition) with expensive NN predictions: yHF = ρ · yLF + δ(x). Demonstrates multi-fidelity acceleration.
Message-passing GNN that predicts band structure from mesh graph representation. Enables arbitrary unit cell geometries beyond parameterized cylinders.
Compare phononic bandgaps across hexagonal, square, and triangular lattice symmetries. Uses physics-based scaling from the hexagonal NN model.
Full 2D finite-difference time-domain wave propagation through phononic crystal slabs. Visualize transmission, reflection, and evanescent decay in real time.
Variational Autoencoder learns the phononic crystal design space and generates novel designs meeting target bandgap specifications.
| Property | Value | Unit |
|---|---|---|
| Density | 2,230 | kg/m³ |
| Young's Modulus | 64 | GPa |
| Poisson's Ratio | 0.200 | — |
| Knoop Hardness | 480 | — |
| Bending Strength | 150 | MPa |
| Type | Borosilicate glass | |
Currently supports BOROFLOAT® 33 borosilicate glass with air inclusions. Additional materials will be available in future versions.
| Parameter | Value | Description |
|---|---|---|
| a₀ | 1.000 mm | Base lattice constant |
| a = a₀√3 | 1.732 mm | Honeycomb lattice constant |
Data was computed with a = √3 mm. When using a custom lattice constant acustom, frequencies are scaled by √3 / acustom. Larger lattice → lower frequencies.
Geometric Prediction (2-Parameter)
Material Variation (5-Parameter)
Getting Started
PhononIQ is an ML-powered phononic bandgap prediction platform for honeycomb lattice phononic crystal plates. It uses neural networks trained on thousands of COMSOL Multiphysics simulations to predict band structures in real time.
Use the sidebar inputs to set Filling Fraction (f) and Thickness Ratio (h/a) to any value within range.
Press the ▶ Analyze button in the toolbar. The ML model predicts the full band structure instantly.
View band structure plots, bandgap analysis, sensitivity, topology, and more using the tree navigation on the left.
Module Overview
| Module | Description |
|---|---|
| Analysis | Main workspace: 3D unit cell visualization, ML band structure prediction, bandgap analysis, and prediction heatmap across the full parameter space. |
| Inverse Design | Specify a target bandgap width and center frequency. The optimizer searches the parameter space using a genetic algorithm to find designs that match your target. |
| Shape Explorer | Interactive 3D visualization of inclusion shapes using Fourier descriptors. Explore how different shapes affect the band structure. |
| Crystal Plate | Full 3D phononic crystal plate visualization with honeycomb lattice arrangement. Supports isometric and top views with STL export. |
| Topology | Band structure analysis: band inversion detection, band connectivity at high-symmetry points, band inversion map across parameter space, and isofrequency contour visualization. |
| Sensitivity | SHAP-based feature attribution, sensitivity analysis of bandgap to parameter changes, density of states (DOS), transmission spectrum, and group velocity analysis. |
| Design Atlas | t-SNE dimensionality reduction to map the full 3,000-design space into a 2D atlas. Color by bandgap width, center frequency, or parameters. |
| Material Explorer | 5-parameter material variation explorer using a separate NN. Vary density, Young's modulus, Poisson's ratio, thickness, and filling fraction independently. |
Detailed Module Guides
Analysis
The main workspace displays two side-by-side band structure plots:
- Left plot — ML Neural Network: The neural network predicts band frequencies at any (TR, FF) value instantly. Bands are shown as solid colored lines. Bandgap regions are highlighted in green.
- Right plot — COMSOL Nearest: Shows stored band data from the nearest COMSOL simulation in the dataset. Bands appear as solid orange lines. The nearest sample index and distance are displayed below.
The Bandgap Summary panel reports two independent predictions: the gap extracted from the NN band curves, and the gap from the standalone classifier+regressor ML models.
The Prediction Heatmap shows bandgap width across the entire TR×FF design space. Click any cell to jump to that design.
Inverse Design
Specify a target bandgap by entering desired width and center frequency. The optimizer uses a genetic algorithm (GA) to search the (TR, FF) parameter space:
- Set target gap width (e.g. 100 kHz) and center frequency (e.g. 400 kHz)
- Click Run Optimizer — the GA evolves a population of candidate designs over multiple generations
- Results show the best-fit design(s) with their predicted band structures
- A convergence plot tracks the fitness improvement over generations
Shape Explorer
Visualizes how the inclusion shape (the cross-section of the air hole) affects phononic behaviour using Fourier descriptors:
- Adjust Fourier harmonic coefficients to morph the inclusion from a circle to more complex shapes (ellipses, polygons, stars)
- The 3D viewer shows the deformed unit cell in real time
- Explore how breaking circular symmetry modifies the band structure and bandgap
Crystal Plate
A full 3D phononic crystal plate assembled from multiple unit cells:
- The plate is built by tiling rhombic unit cells along the honeycomb lattice vectors
- Each unit cell contains two cylindrical air holes (honeycomb arrangement)
- Toggle between 3D Isometric and Top-Down views
- The adjacent panel shows a single isolated unit cell for reference
- Hole size and plate thickness update automatically when you change TR/FF
Topology
Analyzes band ordering and connectivity of the phononic band structure:
- Band Inversion Map: Shows where band ordering swaps between high-symmetry points across the (TR, FF) parameter space. Inversions are a necessary (but not sufficient) condition for topological transitions
- Band Connectivity: Visualizes band frequencies at Γ, K, and M points. Bands involved in inversions are highlighted with dotted lines
- Inversion Count: Heatmap of how many band pairs swap ordering at each (TR, FF) point
- Isofrequency Contours: Band proximity map in (TR, FF) parameter space at a chosen frequency, showing which designs have bands near that frequency
- Dirac Cones: Detected automatically where two bands nearly touch with linear dispersion (marked with circle indicators)
Sensitivity
Explores how the bandgap responds to parameter changes, with four sub-panels:
- SHAP Waterfall: Shows how each input feature (TR, FF, and their engineered terms) contributes to the predicted bandgap width, based on SHapley Additive exPlanations
- Group Velocity: The slope of each band (dω/dk) plotted as a function of k-point. Zero group velocity indicates standing waves; high values indicate fast propagation
- Density of States (DOS): Histogram of available modes at each frequency. Bandgaps appear as regions with zero DOS
- Transmission Spectrum: Estimated transmission through N layers of the crystal. Shows deep attenuation dips inside bandgaps
Design Atlas
A 2D map of the entire design space created using t-SNE dimensionality reduction:
- Each dot represents one COMSOL simulation design (up to 500 subsampled from the 3,000 total)
- Designs with similar properties (TR, FF, bandgap) cluster together on the map
- Designs with different properties are placed far apart
- Color the dots by: bandgap width, center frequency, TR, FF, or bandgap presence
- Click any dot to load that design into the main Analysis page
Material Explorer (5-Parameter)
Extends the prediction to 5 independent parameters including material properties:
- Adjust density (ρ), Young’s modulus (E), and Poisson’s ratio (ν) using real engineering units
- Select from 16 material presets (steel, titanium, copper, PMMA, etc.) or enter custom values
- Click Predict Band Structure to run the 5-parameter neural network
- The Bandgap Landscape heatmap below shows bandgap width vs TR and FF for your chosen material
- Click any cell in the heatmap to auto-fill sliders and predict that design
Parameter Definitions
| Parameter | Symbol | Description | Range |
|---|---|---|---|
| Filling Fraction | f | Ratio of inclusion area to unit cell area. Controls the size of the cylindrical air holes within the plate. Higher values mean larger holes. | 0.25 – 0.50 |
| Thickness Ratio | h/a | Ratio of plate thickness (h) to lattice constant (a). Controls the out-of-plane dimension of the phononic crystal plate. | 0.10 – 2.00 |
| Lattice Constant | a = a₀√3 | Distance between nearest lattice points in the honeycomb arrangement. Default a₀ = 1 mm giving a = √3 mm. Changing this scales all predicted frequencies. | > 0 |
Material Variation Parameters (5-Parameter Model)
| Parameter | Symbol | Description | Range |
|---|---|---|---|
| Density ρ | ρ (kg/m³) | Inclusion material density. Base: BOROFLOAT 33 (ρ₀ = 2500 kg/m³). Covers soft polymers (~876) to dense metals (~5500). | 876 – 5,500 kg/m³ |
| Young's Modulus E | E (GPa) | Inclusion elastic modulus. Base: E₀ = 72 GPa. Covers flexible plastics (~22 GPa) to stiff ceramics (~144 GPa). | 21.6 – 144 GPa |
| Poisson's Ratio ν | ν | Inclusion Poisson's ratio. Base: ν₀ = 0.23. Covers nearly incompressible (~0.46) to auxetic-like (~0.09). | 0.09 – 0.46 |
| Thickness Ratio | h/a | Same as geometric model: ratio of plate thickness to lattice constant. | 0.10 – 2.00 |
| Filling Fraction | f | Same as geometric model: ratio of inclusion area to unit cell area. | 0.25 – 0.50 |
ML Predictions vs. COMSOL Reference
By default, PhononIQ displays ML-predicted band structures as solid colored lines. These predictions come from a neural network trained on COMSOL simulation data.
To compare with the nearest COMSOL simulation, enable "Show COMSOL Reference" in the sidebar. COMSOL data appears as dashed gray lines.
Exporting Results
Use the File toolbar buttons to export:
- SVG — Vector graphics of the current band structure plot (publication-ready)
- PNG — Raster image of the current plot
- CSV — Comma-separated band structure data for external analysis
- LaTeX — Auto-generated LaTeX table snippet for papers
Data & Training
- Geometric Model: Trained on 3,000 COMSOL simulations spanning TR [0.10, 2.00] × FF [0.25, 0.50], sampled via optimized Latin Hypercube Design (maximin criterion, seed 42). 15 frequency bands × 61 k-points each.
- Material Model: Trained on 784 COMSOL simulations varying 5 material/geometric parameters (Density Ratio, Young’s Modulus Ratio, Poisson’s Ratio Ratio, TR, FF), using BOROFLOAT® 33 glass as base material with air inclusions. A 5,000-sample expanded dataset (LHS, maximin) is in preparation.
- Band NN Architecture: Multi-layer perceptron (MLP) with 128→64 hidden units, feature engineering including quadratic cross-terms.
- Brillouin Zone Path: Γ → M → K → Γ (61 k-points along the irreducible path for honeycomb lattice)
Phononic Crystals
Phononic crystals (PnCs) are periodic structures that control the propagation of elastic and acoustic waves. When waves interact with the periodic arrangement, Bragg scattering can create bandgaps — frequency ranges in which wave propagation is forbidden.
PhononIQ focuses on honeycomb lattice phononic crystal plates: thin plates with a periodic array of cylindrical air holes arranged in a honeycomb pattern.
Governing Equations
Elastic Wave Equation
The propagation of elastic waves in a solid is governed by the Navier-Cauchy equation:
where u is the displacement vector, ρ is the mass density, and σ is the stress tensor.
Constitutive Relation (Hooke's Law)
For a linear elastic isotropic material:
where λ and μ are the Lamé parameters:
Strain-Displacement Relation
Bloch-Floquet Periodicity
For a periodic structure with lattice vectors a1 and a2, the displacement field satisfies:
where k is the Bloch wave vector and R = n1a1 + n2a2 is a lattice translation vector. This reduces the eigenvalue problem to a single unit cell.
Brillouin Zone & Band Structure
The hexagonal Brillouin zone for a honeycomb lattice has three high-symmetry points:
| Point | Coordinates | Description |
|---|---|---|
| Γ | (0, 0) | Zone center |
| M | (π/a, 0) | Edge midpoint |
| K | (4π/3a, 0) | Zone corner (Dirac point) |
The band structure is computed along the irreducible Brillouin zone path Γ → M → K → Γ, sampling 61 k-points. Each eigenfrequency ωn(k) forms a band, and frequency ranges with no bands constitute bandgaps.
Bandgap Definition
A complete bandgap exists between bands n and n+1 when:
The bandgap is characterized by:
- Lower edge: maxk ωn(k) — top of band n across all k-points
- Upper edge: mink ωn+1(k) — bottom of band n+1 across all k-points
- Width: Δω = upper edge − lower edge
- Center frequency: ωc = (upper + lower) / 2
- Relative width: Δω / ωc × 100%
Topological Acoustics — Overview
Topological phononic crystals exhibit symmetry-protected edge states that propagate along interfaces without backscattering, even in the presence of defects. In a 2D honeycomb lattice (C6v symmetry), topological behaviour arises from band inversions at high-symmetry points of the Brillouin zone.
PhononIQ computes surrogate topological indicators from the ML-predicted band structure. These are physically motivated proxies that use band ordering and connectivity analysis at high-symmetry points (Γ, K, M). True topological invariants (Chern number, Z2) require eigenvectors and full Brillouin zone integration, which are unavailable from our frequency-only NN model.
1. Band Inversion Detection
A band inversion occurs when the ordering of two adjacent bands swaps between high-symmetry points. This is the primary indicator of a topological phase transition.
Method
For each adjacent band pair (n, n+1), we extract frequencies at high-symmetry points and compute the gap:
Detection criteria:
- Gap closure: mink |fn+1(k) − fn(k)| < 5 kHz along the k-path → bands nearly touch (potential Dirac point)
- Slope convergence: Adjacent bands approach each other with converging slopes and gap < 20 kHz → near-crossing detected
- Sign change: If Δf changes sign between two symmetry points, the band ordering has swapped → band inversion confirmed
Z2 Index (Surrogate)
The surrogate Z2 index is computed from the parity of band inversions at Γ:
Z2 = 1 (odd inversions) → topologically non-trivial phase
Z2 = 0 (even inversions) → topologically trivial phase
2. Zak Phase Estimation
The Zak phase is a 1D topological invariant that measures the geometric phase acquired by a Bloch state transported across the entire Brillouin zone:
where An(k) is the Berry connection and |unk⟩ is the periodic part of the Bloch function. For systems with inversion symmetry, the Zak phase is quantized to 0 or π.
Surrogate Method (Connectivity Analysis)
Without eigenvectors, we estimate the Zak phase from band connectivity along the k-path:
- Scan each band n along the k-path for avoided crossings with adjacent bands (n−1, n+1)
- An avoided crossing is detected when: |fn+1(k) − fn(k)| < 8 kHz and the gap doesn’t exceed 2.5× this threshold in the neighborhood
- Count the number of such crossings for each band
- Quantize: θZak = π if crossing count is odd, θZak = 0 if even
3. Berry Phase at Dirac Points
The Berry phase is the geometric phase acquired by a state adiabatically transported around a closed loop in k-space. Near a Dirac cone, the Berry phase is quantized to ±π:
Detection Method
We identify Berry phase ±π at locations exhibiting Dirac-like conical dispersion:
- Scan high-symmetry points (Γ, K, M) for inter-band gaps < 20 kHz
- Test the neighborhood (dk = 4 k-points) for V-shaped or Λ-shaped dispersion:
- Upper band minimum: slopes on both sides are negative (band dips toward the point)
- Lower band maximum: slopes on both sides are positive (band rises toward the point)
- If both conditions hold → Dirac cone detected, Berry phase = ±π
- If the gap < 3 kHz (near-degeneracy) at a symmetry point, it is automatically flagged
4. Edge State Prediction (Bulk-Boundary Correspondence)
The bulk-boundary correspondence principle states that a topologically non-trivial bulk gap must host protected edge states at interfaces between topologically distinct domains.
Method
- Find bandgaps: Identify all gaps with width > 2 kHz from the bulk band structure
- Classify each gap: Check for band inversions in the bands bounding the gap. If the upper and lower bands show slope convergence at Γ (consistent with a band inversion), the gap is classified as topological
- Generate edge dispersion: For topological gaps, construct a synthetic edge state that crosses the gap linearly:
fedge(k∥) = fcenter + vedge · (k∥ − k0)where fcenter is the gap midpoint, vedge is the edge state velocity, and k0 is the crossing point
- Penetration depth: Estimated as:
ξ ≈ 2 fcenter / (Δf + 1)This gives the e-folding decay length of the edge state into the bulk. The minimum number of unit cells for confinement is Nmin ≈ 3ξ.
5. Topological Phase Diagram
The phase diagram maps the surrogate Z2 index and inversion count across the full (TR, FF) design space:
Construction
- For each (TR, FF) grid point, run the ML model to predict the full 15-band structure
- Apply band inversion detection at each point
- Record: Z2 index, total inversion count, and minimum gap at Γ
- Plot as heatmaps: Z2 = 1 regions are non-trivial phases, phase boundaries occur where gaps close at Γ
Band Touching Map
A complementary map shows the minimum inter-band gap at each (TR, FF) point. Regions where bands nearly touch (< 5 kHz) are potential Dirac points where topological transitions occur. These gap-closing lines trace the topological phase boundaries in the design space.
6. Accuracy Summary & References
| Indicator | Method | Accuracy |
|---|---|---|
| Band Inversion | Gap comparison at Γ, K, M | Reliable — direct from band structure |
| Z2 Index | Parity of inversions at Γ | Surrogate — proxy for true Z2 |
| Zak Phase | Avoided crossing count (quantized to 0/π) | Heuristic — based on connectivity, not integration |
| Berry Phase | Dirac cone shape detection at K, M | Qualitative — identifies Dirac-like features |
| Edge States | Synthetic from bulk-boundary correspondence | Illustrative — not from supercell calculation |
| Phase Diagram | Full (TR, FF) sweep of inversion analysis | Reliable — maps where bands close/invert |
Key References
- Lu, J. et al., “Observation of topological valley transport of sound in sonic crystals,” Nature Physics 13, 369 (2017)
- Xiao, D., Chang, M.-C. & Niu, Q., “Berry phase effects on electronic properties,” Rev. Mod. Phys. 82, 1959 (2010)
- Miniaci, M. et al., “Experimental observation of topologically protected helical edge modes in patterned elastic plates,” Phys. Rev. X 8, 031074 (2018)
- He, C. et al., “Acoustic topological insulator and robust one-way sound transport,” Nature Physics 12, 1124 (2016)
- Fu, L. & Kane, C.L., “Topological insulators with inversion symmetry,” Phys. Rev. B 76, 045302 (2007)
ML Model Architecture
Feature Engineering
For the 2-parameter geometric model, input features are derived from (TR, FF):
- Raw features: TR, FF
- Quadratic: TR², FF², TR×FF
- Transforms: √TR, √FF, log(TR), log(FF)
For the 5-parameter material model, 28 features are engineered from (DR, EMR, PRR, TR, FF) including all pairwise cross-terms and nonlinear transforms.
Neural Network (Band NN)
Architecture: MLP with 2 hidden layers (128 → 64 neurons), ReLU activation, outputting 15×61 = 915 frequency values (15 bands, 61 k-points each).
Bandgap Classification & Regression
A two-stage ML pipeline is used for bandgap prediction:
- Stage 1 — Classifier: Random Forest or Gradient Boosting classifier predicts whether a bandgap exists (binary).
- Stage 2 — Regressor: For samples classified as having a bandgap, a Gradient Boosting regressor predicts the bandgap width and center frequency.
Group Velocity
The group velocity describes how fast energy (a wave packet) propagates through the crystal at each k-point:
Along the 1D k-path Γ→K→M→Γ, this reduces to the slope dω/dk.
Numerical Method
PhononIQ computes group velocity using 5-point central finite differences (O(h&sup4;) accuracy) within each Brillouin zone segment independently:
At segment edges (near K and M symmetry points), a 3-point central difference is used:
Values at exact symmetry points (Γ, K, M) are set to NaN because the path direction changes discontinuously there.
Physical Conversion to m/s
where a0 is the lattice constant. This factor converts from normalized k-space derivatives to physical velocities. Pre- and post-smoothing (moving average) suppresses NN prediction noise.
Physical Insights
- Zero group velocity at band edges and flat bands — waves are standing, no energy transport
- High group velocity — fast propagation; linear dispersion (vg = const) implies non-dispersive behaviour
- Inside bandgaps, no propagating modes exist — group velocity is undefined
- Near Dirac cones, vg is approximately constant (linear dispersion), analogous to massless Dirac fermions
- Slow-wave regions (|vg| < 5% of max) are identified automatically — relevant for enhanced acoustic energy density and sensing (Baba, Nature Photonics 2, 465, 2008)
Effective Mass (Band Curvature)
The effective mass of a phononic mode is inversely proportional to the curvature of the dispersion relation:
This quantity governs wave packet spreading, tunneling through barriers, and the response to perturbations.
Numerical Method
Computed via 5-point central second derivative (O(h&sup4;) accuracy):
At segment boundaries, a 3-point stencil is used:
Physical Insights
- Positive curvature (m* > 0) — normal dispersion, wave packets spread normally
- Negative curvature (m* < 0) — anomalous dispersion, enables negative refraction (Sukhovich et al., PRL 102, 154301, 2009)
- Near-zero curvature (|m*| → ∞) — flat bands, heavy effective mass, strong localization
- Infinite curvature (m* → 0) — at Dirac cones, analogous to massless particles
The plot displays sign(m*) × log10(|1/curvature| + 1) to compress the dynamic range for visualization.
Density of States (DOS)
The phononic density of states counts the number of modes available at each frequency:
In practice, this is approximated using Gaussian broadening or binning the eigenfrequencies into histogram bins. The DOS drops to zero inside bandgaps, providing a complementary visualization of forbidden frequency ranges.
Transmission Spectrum
The transmission through N layers of the phononic crystal is estimated using the transfer matrix method. The transmission coefficient drops exponentially inside bandgaps:
where κ(ω) is the imaginary part of the wave vector (evanescent decay rate) inside the bandgap. More layers produce deeper attenuation in the gap region.
Finite Structure Mode Shapes
When a phononic crystal has finite extent (Nx × Ny unit cells), the allowed wavevectors are quantized and the response depends on whether the excitation frequency falls inside or outside a bandgap.
In-Gap: Evanescent Modes
Inside a bandgap, no propagating Bloch modes exist. Waves launched from the boundary decay exponentially toward the interior (Laude, Phononic Crystals, De Gruyter, 2015, Sec. 5.4):
where dmin is the distance from the nearest boundary (in unit cells) and κ is the evanescent decay rate:
Decay is strongest at the gap center and weakest near band edges, consistent with complex band structure theory.
Pass Band: Standing Waves
Outside bandgaps, finite-size boundary reflections create standing wave patterns. For a plate with fixed-fixed boundaries:
where (nx, ny) are the mode indices determined by the band index, and the cosine term represents the sub-cell Bloch modulation.
Frequency Response Function (FRF)
The displacement amplitude at the plate center as a function of frequency is computed from the Green’s function approach (Vasseur et al., PRL 86, 3012, 2001):
where fn are the natural frequencies of the finite structure (quantized from the band structure) and η is a structural loss factor. The FRF shows resonance peaks at pass-band frequencies and deep dips (orders of magnitude attenuation) inside bandgaps.
Phononic Crystal Waveguide
A W1 waveguide is formed by removing one row of holes from a perfect phononic crystal, creating a line defect that can guide acoustic/elastic waves at frequencies within the bulk bandgap (Khelif et al., APL 84, 4400, 2004).
Supercell Band Folding
To analyze a waveguide, a supercell of N unit cells perpendicular to the propagation direction is used. This folds the Brillouin zone N times:
Each bulk band generates N sub-bands. PhononIQ approximates the folded spectrum by shifting and interpolating the ML-predicted bulk dispersion, with small perturbations to simulate anti-crossing (mode coupling) at zone boundaries (Laude, 2015, Ch. 4).
Guided Defect Mode
Inside the complete bandgap, the line defect introduces a guided mode. Its frequency depends on the effective medium of the empty channel:
where Δfshift is proportional to the filling fraction (more material removed → lower frequency). The guided mode has a nearly flat dispersion with slight positive curvature, characteristic of slow-light operation near band edges (Pennec et al., Physica Status Solidi (c) 6, 2080, 2009).
Transmission Spectrum
The transmission through the waveguide shows:
- Near 0 dB at the guided mode frequency — efficient wave transport through the defect channel
- Lorentzian peak shape centered on fdefect with linewidth γ determined by confinement:
T(f) = Tmax / (1 + ((f − f0) / γ)²)
- Strong attenuation (< −30 dB) elsewhere in the gap — evanescent regime
- Partial transmission with Fabry-Pérot fringes outside the gap — interference from multi-reflection in the finite crystal
Effective Medium Properties
At long wavelengths (λ ≫ a), the phononic crystal behaves as a homogeneous effective medium with frequency-dependent properties:
Effective Density
where FF is the filling fraction and δρ accounts for resonant corrections near bandgap frequencies.
Effective Bulk Modulus
Effective Refractive Index
The refractive index becomes imaginary inside bandgaps (evanescent regime) and can become negative in anomalous dispersion regions.
SHAP Feature Attribution
SHAP (SHapley Additive exPlanations) is a game-theoretic method for explaining individual ML predictions. For each prediction, SHAP assigns a contribution value to every input feature:
where φ0 is the baseline (average prediction), and φi is the SHAP value for feature i. The sum of all SHAP values equals the difference between the prediction and the baseline.
In the Sensitivity page, the SHAP waterfall chart shows:
- Each bar = the contribution of one feature (TR, FF, TR², FF², TR×FF, etc.) to the predicted bandgap width
- Green bars push the prediction higher (larger bandgap)
- Red bars push the prediction lower (smaller bandgap)
- The features are sorted by magnitude, so the most important features appear at the top
This tells you why a particular design has a large or small bandgap, and which parameters have the most influence.
Inverse Design & Optimization
Traditional design (forward problem): given (TR, FF) → predict bandgap. Inverse design reverses this: given a target bandgap → find the (TR, FF) that produces it.
PhononIQ uses a genetic algorithm (GA) for this optimization:
- Initialize: Create a random population of N candidate designs (TR, FF pairs)
- Evaluate: For each candidate, use the ML model to predict the bandgap and compute a fitness score based on how close the predicted gap matches the target
- Select: Tournament selection — pick the best individuals to become parents
- Crossover: Combine parameters from two parents to create offspring
- Mutate: Apply small random perturbations to maintain diversity
- Repeat: Over many generations, the population converges toward the optimal design
The fitness function minimizes:
where Δω is bandgap width and ωc is center frequency.
t-SNE Dimensionality Reduction (Design Atlas)
t-SNE (t-distributed Stochastic Neighbor Embedding) is an algorithm that maps high-dimensional data to 2D while preserving local neighborhood structure. It works in three steps:
Step 1: Pairwise Similarities in High-D
For each pair of designs (i, j), compute the probability that i would pick j as a neighbor using a Gaussian kernel:
The bandwidth σi is set per point so each point has an effective neighborhood size called perplexity (~30 neighbors). Symmetrize: pij = (pj|i + pi|j) / 2N.
Step 2: Pairwise Similarities in 2D
In the 2D embedding, use a heavier-tailed Student-t distribution (1 degree of freedom):
The heavy tail allows moderate high-D distances to map to larger 2D distances, reducing the “crowding problem”.
Step 3: Minimize KL Divergence
Move the 2D points by gradient descent to minimize the Kullback-Leibler divergence between p and q:
This ensures that points close in the original 5D space (similar TR, FF, bandgap) remain close in the 2D map.
Fourier Shape Descriptors
The Shape Explorer uses Fourier descriptors to parameterize the cross-sectional shape of the inclusion (air hole). Any closed 2D curve r(θ) can be decomposed as:
where a0 is the mean radius (related to filling fraction), and the higher harmonics control the shape:
| Harmonic | Effect |
|---|---|
| n = 0 | Mean radius (circle size) |
| n = 2 | Elliptical distortion |
| n = 3 | Triangular perturbation |
| n = 4 | Square-like perturbation |
| n = 6 | Hexagonal symmetry (matches lattice) |
By varying these coefficients, you can explore how breaking the circular symmetry of the holes opens, closes, or shifts the bandgap.
Dirac Cones
A Dirac cone is a point where two bands touch with linear (conical) dispersion, forming a degeneracy analogous to massless Dirac fermions in graphene. Near the Dirac point:
where ωD is the Dirac frequency and vD is the Dirac velocity.
PhononIQ detects Dirac cones automatically by checking four conditions between adjacent bands:
- Small gap: The frequency gap between two bands is less than 3 kHz at that k-point
- High frequency: The touching point is above 50 kHz (avoids false positives at low frequencies)
- Local minimum: The gap at that k-point is a local minimum compared to neighboring k-points
- Opposing slopes: The two bands have opposite slopes (one increasing, one decreasing) with magnitude exceeding 200 kHz per k-unit, confirming the conical (linear) dispersion
Dirac cones are marked with ○ circle indicators on the band structure plot. They indicate points where two bands nearly touch with linear dispersion, which is a necessary condition for topological phase transitions.
Credits & Acknowledgments
© NewFOS — New Frontiers of Sound
Developed by Samarjith Biswas, PhD
Supported by the National Science Foundation (NSF) Engineering Research Center for New Frontiers of Sound (NewFOS).
University of Arizona • Caltech • UCLA • Georgia Tech • CUNY • Wayne State • Spelman College • UC Boulder • CU Denver • University of Alaska Fairbanks
Enter a frequency range you want to block and physical constraints. The tool finds the best phononic crystal designs with full fabrication specs.