Update README.md

README.md

@@ -51,3 +51,88 @@ configs:
   - split: default
     path: snapshots/default-*
 ---
+
+# Graetz Problem Dataset
+## Dataset Description
+This dataset contains thermal simulations of the Graetz problem with varying geometric and flow parameters.
+
+### Dataset Summary
+The Graetz dataset provides numerical simulations of heat transfer in a channel with developing thermal boundary layer. The problem is characterized by varying channel geometry (omega2_length) and Péclet number, making it suitable for parametric reduced-order modeling and thermal-fluid analysis.
+
+## Dataset Structure
+
+### Data Instances
+The dataset consists of three configurations:
+
+- **geometry**: Mesh information (nodes and connectivity) - varies with geometric parameter
+- **snapshots**: Temperature field solutions
+- **parameters**: Geometric and flow parameters for each simulation
+
+### Data Fields
+
+#### Geometry Configuration
+- `node_coordinates_x`: Sequence of x-coordinates of mesh nodes (float64)
+- `node_coordinates_y`: Sequence of y-coordinates of mesh nodes (float64)
+- `connectivity`: Sequence of element connectivity (triangular elements, int32)
+
+#### Snapshots Configuration
+- `temperature`: Temperature field at each node (float64)
+
+#### Parameters Configuration
+- `omega2_length`: Geometric parameter defining the channel configuration (float64)
+- `peclet`: Péclet number characterizing the heat transfer regime (float64)
+
+### Data Splits
+
+- `default`: Contains all simulations with varying parameters
+
+## Dataset Creation
+
+### Source Data
+
+The dataset was generated using finite element simulations of the convection-diffusion equation representing the Graetz problem. The mesh geometry varies with the omega2_length parameter, making this a parametrized geometry problem.
+
+### Preprocessing
+
+Each simulation has its own mesh corresponding to the geometric parameter value. Solutions are stored as 1D arrays corresponding to the nodal values on each respective mesh.
+
+## Usage
+
+```python
+from datasets import load_dataset
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.tri as mtri
+
+# Load geometry
+ds_geom = load_dataset("SISSAmathLab/graetz", name="geometry")
+
+# Load snapshots
+ds_data = load_dataset("SISSAmathLab/graetz", name="snapshots")
+
+# Load parameters
+ds_params = load_dataset("SISSAmathLab/graetz", name="parameters")
+
+# Visualize temperature distribution for simulation 16
+idx = 16
+pts_x = np.asarray(ds_geom['default']['node_coordinates_x'][idx]).flatten()
+pts_y = np.asarray(ds_geom['default']['node_coordinates_y'][idx]).flatten()
+connectivity = ds_geom['default']['connectivity'][idx]
+temperature = ds_data['default']['temperature'][idx]
+
+omega2_length = ds_params['default']['omega2_length'][idx]
+peclet = ds_params['default']['peclet'][idx]
+
+triang = mtri.Triangulation(pts_x, pts_y, connectivity)
+plt.tripcolor(triang, temperature, cmap='coolwarm')
+plt.colorbar(label='Temperature')
+plt.title(f'Graetz Problem (ω₂={omega2_length:.2f}, Pe={peclet:.1f})')
+plt.xlabel('x')
+plt.ylabel('y')
+plt.axis('equal')
+plt.show()
+```
+
+## Contact
+
+For questions or issues, please contact SISSA mathLab.