Chladni Plate 2D Dataset

Numerical solutions to the 2D Chladni plate vibration equation.

Equation

The Chladni plate dataset models the steady-state response of a 2D vibrating plate to various forcing patterns. The mathematical formulation involves modal decomposition using cosine basis functions:

Forcing function:

S(x,y) = Σₙ Σₘ α(n,m) cos(μₙx) cos(λₘy)

Displacement response:

Z(x,y) = Σₙ Σₘ α(n,m) Φ(n,m) cos(μₙx) cos(λₘy)

Mode factor:

Φ(n,m) = (v²/β(n,m)) × I(n,m) × (4/(LM)) × cos(μₙL/2)cos(λₘM/2)

Where:

• μₙ = nπ/L, λₘ = mπ/M are spatial wavenumbers
• β(n,m) = √(μₙ² + λₘ² + 3v² - γ⁴)
• I(n,m) is a time integral: ∫₀ᵗ sin(ω(τ-t)) exp(-γ²+v²τ) sin(β(n,m)τ) dτ

Variables

The dataset returns a dictionary with the following fields:

Coordinates

• spatial_coordinates: (numPoints², 2) - Array of (x, y) coordinate pairs
• X: (numPoints,) - 1D array of x coordinates
• Y: (numPoints,) - 1D array of y coordinates

Solution Fields

• forcing: (numPoints, numPoints) - 2D forcing function S(x,y)
• displacement: (numPoints, numPoints) - 2D displacement response Z(x,y)
• S: (numPoints²,) - Flattened forcing function
• Z: (numPoints²,) - Flattened displacement response

Model Coefficients

• alpha_coefficients: (n_range, m_range) - Random forcing coefficients α(n,m)
• alpha: (n_range × m_range,) - Flattened coefficients

Physical Parameters

• omega: Angular frequency (rad/s)
• frequency: Driving frequency (Hz)
• plate_length_x: Plate length in x-direction (m)
• plate_length_y: Plate length in y-direction (m)
• damping: Damping parameter γ
• velocity_param: Velocity parameter v
• evaluation_time: Time at which solution is evaluated

Grid Parameters

• grid_points: Number of spatial grid points per dimension
• n_modes: Number of modes in x-direction
• m_modes: Number of modes in y-direction

Dataset Parameters

• Domain: [0, L] × [0, M] where L = M = 8.75 × 0.0254 m (square plate)
• Grid points: 50 × 50 (default)
• Spatial resolution: L/(numPoints-1) ≈ 4.5 mm
• Mode range: 10 × 10 modes (default)

Physical Parameters

• Plate dimensions: L = M = 8.75 × 0.0254 m ≈ 0.222 m
• Driving frequency: ω = 55π/M ≈ 778 rad/s (≈ 124 Hz)
• Damping coefficient: γ = 0.02
• Velocity parameter: v = 0.5
• Evaluation time: t = 4 s
• Boundary conditions: Free boundaries (cosine modes)

Physical Context

This dataset simulates the vibration patterns of a Chladni plate, a thin elastic plate that exhibits complex standing wave patterns when driven by acoustic forcing. The equation models the steady-state displacement response of the plate to various spatial forcing distributions.

Chladni plates are famous for creating beautiful geometric patterns (Chladni figures) when sand or powder is placed on the vibrating surface. The sand accumulates at nodal lines where the displacement is minimal, revealing the underlying mode shapes of the plate vibration.

This dataset is relevant for:

• Structural vibration analysis
• Acoustic wave propagation studies
• Modal analysis and system identification
• Pattern formation in physical systems
• Inverse problems in vibration engineering

The forcing-response relationship captured in this dataset allows for learning the complex mapping between spatial excitation patterns and the resulting displacement fields.

Usage

from dataset import Chladni2DDataset

# Create dataset
dataset = Chladni2DDataset(numPoints=50, n_range=10, m_range=10)

# Generate a sample
sample = next(iter(dataset))

# Access solution data
spatial_coords = sample["spatial_coordinates"]
forcing = sample["forcing"]
displacement = sample["displacement"]
frequency = sample["frequency"]

Visualization

Run the plotting script to visualize samples:

python plot_sample.py      # Static visualization with imshow plots

Note: Animation is not applicable for this dataset as it generates steady-state responses rather than time evolution.

Data Generation

Generate the full dataset:

python generate_data.py

This creates train/test splits saved as chunked parquet files in the data/ directory.