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
   

2. Verify interface is resolved: dx = 5 μm, interface width = 10 μm → 2 points per interface width.
3. 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