shiwen0710/Wave-Gauss-Circularshape

Dataset Description

Overview

This dataset contains solutions to the two-dimensional wave equation within a circular domain. The wave equation is a fundamental linear hyperbolic partial differential equation (PDE) given by:

$$\frac{{\mathrm{\partial }}^{2}u}{{\mathrm{\partial }}^{2}t}=c^2\mathrm{\Delta }u,\text{in }\mathcal{D}\times [0,T],\backslash frac\{\backslash partial^2\; u\}\{\backslash partial^2\; t\}\; =\; c^2\; \backslash Delta\; u,\; \backslash quad\; \backslash text\{in\; \}\; \backslash mathcal\{D\}\; \backslash times\; [0,\; T],$$

with initial conditions defined as:

$$u_0(x,y)=\frac{\pi }{K^2}\sum _{i,j}{a}_{ij}{(i^2+j^2)}^{-r}\sin (\pi ix)\sin (\pi jy)\mathrm{.}u\_0(x,\; y)\; =\; \backslash frac\{\backslash pi\}\{K^2\}\; \backslash sum\_\{i,\; j\}\; a\_\{i\; j\}\backslash left(i^2+j^2\backslash right)^\{-r\}\; \backslash sin\; (\backslash pi\; i\; x)\; \backslash sin\; (\backslash pi\; j\; y).$$

In our experiments, the parameters are set as follows:

• $K = 10$
• $c = 2.0$
• $r = 0.5$
• $a \in \mathbb{R}^{K \times K}$, with values uniformly sampled from the range $[-1, 1]$.

The solution domain is a circle centered at $[0.5, 0.5]$ with radius 1. We enforce zero Dirichlet boundary conditions across the domain.

Dataset Specifications

The dataset was generated using a finite element method (FEM) on a triangular mesh in a circular domain. The mesh is generated using the Delaunay algorithm with the following specifications:

• Number of points (nodes): 16431
• Mesh elements:
  • Triangles: 32441

Temporal discretization is performed with a time step $\Delta t = 0.001$, and each trajectory is simulated for 100 time steps.

Data Structure

Each dataset file contains the following arrays:

• x: (Array of shape [16431]) The x-coordinates of the mesh points.
• y: (Array of shape [16431]) The y-coordinates of the mesh points.
• u0: (Array of shape [16431, batch_size]) The initial condition ( u_0 ) at each mesh point for each sampled configuration.
• u: (Array of shape [16431, 100, batch_size]) The solution ( u ) at each mesh point for each time step and for each sampled configuration.

Applications

This dataset is suitable for applications in:

• Machine learning for scientific computing
• Time-dependent PDE solvers
• Reduced-order modeling and surrogate modeling for hyperbolic equations

Additional Information

• The dataset is generated using PyTorch-based FEM solvers (TensorMesh) and supports integration with machine learning frameworks.
• The temporal and spatial discretization must adhere to the Courant-Friedrichs-Levy (CFL) condition for stability. For the finite element method (FEM) solution, the Crank-Nicolson time-stepping scheme is utilized to ensure energy conservation during the simulation.
• The circular domain and multi-frequency initial conditions allow the dataset to cover a range of spatial scales and complexities in wave propagation.