Mesh Generation

Goal

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

Requirements

• Python 3.10+
• 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)

All scripts emit a top-level object with inputs (the echoed CLI values) and results (the computed fields below). Index as result["results"]["..."].

Script	results Fields
scripts/grid_sizing.py	dx, counts (list of per-dimension cell counts, length == dims), notes
scripts/mesh_quality.py	aspect_ratio, skewness, size_anisotropy, quality_flags, dims, notes

mesh_quality.py describes axis-aligned (orthogonal Cartesian) cells defined purely by edge spacings. For such cells every interior angle is 90°, so the true angular skewness is always 0.0 and high_skewness is never flagged. Cell elongation is reported separately via aspect_ratio and the redundant convenience field size_anisotropy (= 1 - 1/aspect_ratio).

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 (3D cell)
python3 scripts/mesh_quality.py --dx 1.0 --dy 0.5 --dz 0.5 --json

# High aspect ratio check (2D cell; --dz omitted is treated as 2D)
python3 scripts/mesh_quality.py --dx 1.0 --dy 0.1 --json

Error Handling

All validation errors are written to stderr and the script exits with code 2.

Error message	Cause	Resolution
length must be positive, got ...	Non-positive domain size	Use a positive value
resolution must be positive, got ...	Non-positive resolution (resolution=1 is a valid single-cell mesh)	Use a positive integer
dims must be one of (1, 2, 3), got ...	Unsupported dimension count	Use 1, 2, or 3
<name> must be a finite positive number, got ...	dx/dy/dz not finite or not positive	Use a finite positive value
<name> exceeds maximum (...), got ...	Input above the resource-exhaustion bound	Use a smaller value

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 is the angular deviation from the ideal cell shape (max(|90° - θ_i|) / 90° for quads/hexes — see references/quality_metrics.md). mesh_quality.py works from axis-aligned edge spacings, which describe orthogonal Cartesian cells whose interior angles are all exactly 90°; it therefore always reports skewness = 0.0 for these cells. The thresholds below apply when a genuine skewness value is obtained from real cell-corner geometry (e.g. from an unstructured mesh), not from dx/dy/dz spacings.

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

Note: cell elongation is not skewness. An anisotropic but orthogonal cell (e.g. a wall-aligned boundary-layer cell) has high aspect_ratio / size_anisotropy but zero skewness, and is often perfectly acceptable.

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

Verification checklist

• Recorded dx and counts from grid_sizing.py --json and confirmed the smallest physical feature gets enough points (interface ≥5×dx, boundary layer ≥10×dx, wavelength ≥20×dx per Resolution Selection above).
• For an anisotropic domain, ran grid_sizing.py once per differing edge length (or applied --dx per axis) — did NOT apply a single --length-derived count to unequal edges.
• Checked the notes field for "Grid does not fully cover length" and resolved any partial-coverage warning before trusting counts.
• Logged aspect_ratio and quality_flags from mesh_quality.py --json; confirmed high_aspect_ratio is absent OR that the elongation is intentional and physics-aligned (e.g. wall-aligned boundary-layer cell with AR≤100 along the wall).
• Confirmed the reported skewness = 0.0 is the expected orthogonal-Cartesian result, NOT a measured quality pass — for unstructured/non-orthogonal cells, obtained a real angle-based skewness from cell-corner geometry and checked it against the <0.8 threshold.
• Verified dx also satisfies the solver's stability constraint (cross-check with numerical-stability) before committing to the resolution.
• Ran a mesh convergence study (≥3 successively refined grids) and confirmed the quantity of interest changes monotonically/asymptotically before declaring the mesh adequate.

Common pitfalls & rationalizations

Tempting shortcut	Why it's wrong / what to do
"skewness came back 0.0, so the mesh quality is fine."	mesh_quality.py always returns skewness = 0.0 for axis-aligned spacings — it is a definitional property of orthogonal cells, not a measurement. Real skewness needs cell-corner angles from an unstructured mesh; don't read 0.0 as a passing quality check.
"Two grids gave nearly the same answer, so the mesh is converged."	Two grids cannot establish the observed order or the asymptotic range. Use ≥3 successively refined grids and confirm the quantity of interest is converging before quoting any result as mesh-independent.
"High aspect_ratio was flagged, so the cell is bad."	Elongation is not skewness. A wall-aligned boundary-layer cell with AR up to ~100 is acceptable when aligned with the flow/field; check size_anisotropy and the physics, not just the high_aspect_ratio flag.
"I'll set one --length and reuse the counts for all axes."	grid_sizing.py is isotropic per call — it applies the single derived count to every dimension. For unequal edges this over/under-resolves axes; run it per edge length or supply --dx per axis.
"dx = length/resolution resolves my feature because resolution is large."	Points-per-domain is not points-per-feature. A fine global dx can still place too few cells across a thin interface/layer; check feature_size / dx against the Resolution Guidelines (5-10 for interfaces, 10-20 for boundary layers).
"The mesh is fine enough, so I can ignore the time step."	Mesh resolution and temporal stability are coupled: shrinking dx tightens explicit CFL/diffusion limits. A refined mesh that violates the solver's stability constraint diverges — re-check dt against numerical-stability after any refinement.

Security

Input Validation

• All inputs (length, resolution, dx, dy, dz) are validated as finite positive numbers with upper bounds to prevent resource exhaustion
• dims is restricted to {1, 2, 3}
• argparse type parameters reject non-numeric input at the CLI boundary before any processing occurs

File Access

• Scripts read no external files; all inputs are provided via CLI arguments
• Scripts write only to stdout (JSON output); no files are created unless the agent explicitly uses the Write tool

Tool Restrictions

• Read: Used to inspect script source, references, and user configuration files
• Write: Used to save grid sizing results or mesh quality reports; writes are scoped to the user's working directory
• Grep/Glob: Used to locate relevant files and search references
• The skill's allowed-tools excludes Bash to prevent the agent from executing arbitrary commands when processing user-provided inputs

Safety Measures

• No eval(), exec(), or dynamic code generation
• All subprocess calls use explicit argument lists (no shell=True)
• Reduced tool surface (no Bash) means the agent should use Read and Write to prepare inputs and capture outputs rather than constructing shell commands from user text
• All output is deterministic JSON with no shell-interpretable content

Limitations

• 2D/3D only: No unstructured mesh generation
• Quality metrics: Aspect ratio and size anisotropy from axis-aligned spacings only; skewness is reported as 0 for these orthogonal cells (true angular skewness requires real cell-corner geometry)
• No mesh generation: Sizing recommendations only
• Isotropic per call: grid_sizing.py takes a single --length and applies the resulting count to every dimension. For an anisotropic domain (e.g. 10 cm × 5 cm), run it once per differing edge length, or compute dx from physics and apply it per axis (e.g. --length 0.10 --dx 5e-5, then --length 0.05 --dx 5e-5).

References

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

Version History

• v1.2.0 (2026-06-23): Corrected skewness science (orthogonal cells now report skewness 0), added size_anisotropy, made mesh_quality.py --dz optional (2D cells), fixed grid_sizing off-by-one for resolution-derived counts, surfaced dx-override note, corrected output/error-handling docs
• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 2 mesh quality scripts