mesh-generation

Mesh Generation

Goal

Provide a consistent workflow for selecting mesh resolution and checking mesh quality for PDE simulations.

Requirements

• Python 3.8+
• No external dependencies (uses stdlib)

Inputs to Gather

| Input | Description | Example |

|-------|-------------|---------|

| Domain size | Physical dimensions | 1.0 × 1.0 m |

| Feature size | Smallest feature to resolve | 0.01 m |

| Points per feature | Resolution requirement | 10 points |

| Aspect ratio limit | Maximum dx/dy ratio | 5:1 |

| Quality threshold | Skewness limit | < 0.8 |

Decision Guidance

Resolution Selection

What is the smallest feature size?
├── Interface width → dx ≤ width / 5
├── Boundary layer → dx ≤ layer_thickness / 10
├── Wave length → dx ≤ lambda / 20
└── Diffusion length → dx ≤ sqrt(D × dt) / 2

Mesh Type Selection

| Problem | Recommended Mesh |

|---------|------------------|

| Simple geometry, uniform | Structured Cartesian |

| Complex geometry | Unstructured triangular/tetrahedral |

| Boundary layers | Hybrid (structured near walls) |

| Adaptive refinement | Quadtree/Octree or AMR |

Script Outputs (JSON Fields)

| Script | Key Outputs |

|--------|-------------|

| scripts/grid_sizing.py | dx, nx, ny, nz, notes |

| scripts/mesh_quality.py | aspect_ratio, skewness, quality_flags |

Workflow

1. Estimate resolution - From physics scales
2. Compute grid sizing - Run scripts/grid_sizing.py
3. Check quality metrics - Run scripts/mesh_quality.py
4. Adjust if needed - Fix aspect ratios, reduce skewness
5. Validate - Mesh convergence study

Conversational Workflow Example

User: I need to mesh a 1mm × 1mm domain for a phase-field simulation with interface width of 10 μm.

Agent workflow:

1. Compute grid sizing:

   python3 scripts/grid_sizing.py --length 0.001 --resolution 200 --json

1. Verify interface is resolved: dx = 5 μm, interface width = 10 μm → 2 points per interface width.
2. Recommend: Increase to 500 points (dx = 2 μm) for 5 points across interface.

Pre-Mesh Checklist

• [ ] Define target resolution per feature/interface
• [ ] Ensure dx meets stability constraints (see numerical-stability)
• [ ] Check aspect ratio < limit (typically 5:1)
• [ ] Check skewness < threshold (typically 0.8)
• [ ] Validate mesh convergence with refinement study

CLI Examples

# Compute grid sizing for 1D domain
python3 scripts/grid_sizing.py --length 1.0 --resolution 200 --json

# Check mesh quality
python3 scripts/mesh_quality.py --dx 1.0 --dy 0.5 --dz 0.5 --json

# High aspect ratio check
python3 scripts/mesh_quality.py --dx 1.0 --dy 0.1 --json

Error Handling

| Error | Cause | Resolution |

|-------|-------|------------|

| length must be positive | Invalid domain size | Use positive value |

| resolution must be > 1 | Insufficient points | Use at least 2 |

| dx, dy must be positive | Invalid spacing | Use positive values |

Interpretation Guidance

Aspect Ratio

| Aspect Ratio | Quality | Impact |

|--------------|---------|--------|

| 1:1 | Excellent | Optimal accuracy |

| 1:1 - 3:1 | Good | Acceptable |

| 3:1 - 5:1 | Fair | May affect accuracy |

| > 5:1 | Poor | Solver issues likely |

Skewness

| Skewness | Quality | Impact |

|----------|---------|--------|

| 0 - 0.25 | Excellent | Optimal |

| 0.25 - 0.50 | Good | Acceptable |

| 0.50 - 0.80 | Fair | May affect accuracy |

| > 0.80 | Poor | Likely problems |

Resolution Guidelines

| Application | Points per Feature |

|-------------|-------------------|

| Phase-field interface | 5-10 |

| Boundary layer | 10-20 |

| Shock | 3-5 (with capturing) |

| Wave propagation | 10-20 per wavelength |

| Smooth gradients | 5-10 |

Limitations

• 2D/3D only: No unstructured mesh generation
• Quality metrics: Basic aspect ratio and skewness only
• No mesh generation: Sizing recommendations only

References

• references/mesh_types.md - Structured vs unstructured
• references/quality_metrics.md - Aspect ratio/skewness thresholds

Version History

• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 2 mesh quality scripts