README.md

---
dataset_info:
- config_name: geometry
  features:
  - name: node_coordinates_x
    list: float64
  - name: node_coordinates_y
    list: float64
  - name: connectivity
    list:
      list: int32
  splits:
  - name: default
    num_bytes: 48693600
    num_examples: 200
  download_size: 11977292
  dataset_size: 48693600
- config_name: parameters
  features:
  - name: omega2_length
    dtype: float64
  - name: peclet
    dtype: float64
  splits:
  - name: default
    num_bytes: 3200
    num_examples: 200
  download_size: 4933
  dataset_size: 3200
- config_name: snapshots
  features:
  - name: temperature
    list: float64
  splits:
  - name: default
    num_bytes: 8256800
    num_examples: 200
  download_size: 8181284
  dataset_size: 8256800
configs:
- config_name: geometry
  data_files:
  - split: default
    path: geometry/default-*
- config_name: parameters
  data_files:
  - split: default
    path: parameters/default-*
- config_name: snapshots
  data_files:
  - 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.