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mesh-generation

Plan and evaluate mesh generation for numerical simulations — estimate grid resolution from physics scales (interface width, boundary layers, wavelengths), check aspect ratios and skewness against quality thresholds, choose between structured, unstructured, and adaptive mesh refinement strategies, and compute grid sizing for 1D/2D/3D domains. Use when setting up a new mesh, diagnosing poor solver convergence caused by mesh quality, deciding how many points to place across a phase-field interface or boundary layer, or preparing a mesh convergence study, even if the user only asks "what resolution do I need" or "why is my solver failing."

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Comando de instalación

npx skills add https://github.com/HeshamFS/materials-simulation-skills --skill mesh-generation

Copia y pega este comando en Claude Code para instalar la habilidad

Fuente

HeshamFS/materials-simulation-skills

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Actualizado18 de mayo de 2026, 08:58

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SKILL.md

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convergence-study

HeshamFS/materials-simulation-skills

Perform spatial and temporal convergence analysis for solution verification — compute observed convergence orders from grid or timestep refinement studies, apply Richardson extrapolation to estimate discretization error, and calculate the Grid Convergence Index (GCI) per ASME V&V 20 standards. Use when verifying that a numerical solution converges at the expected rate, estimating the error on the finest mesh, checking whether grids are in the asymptotic range, or preparing formal verification reports, even if the user only asks "is my mesh fine enough" or "how accurate is my solution."

2026-05-1840

differentiation-schemes

HeshamFS/materials-simulation-skills

Select and apply numerical differentiation schemes for PDE and ODE discretization — generate finite-difference stencils at arbitrary order and accuracy, choose between central, upwind, compact (Pade), and spectral methods, handle boundary stencils, and estimate truncation error scaling. Use when discretizing spatial derivatives, picking a scheme for advection- or diffusion-dominated problems, building custom stencils for nonstandard operators, or comparing dispersion and dissipation properties of candidate schemes, even if the user just says "how do I approximate this derivative" or "my solution is too diffusive."

2026-05-1840

linear-solvers

HeshamFS/materials-simulation-skills

Select and configure linear solvers for Ax=b systems arising in numerical simulations — choose between direct (LU, Cholesky) and iterative (CG, GMRES, BiCGSTAB, MINRES) methods, analyze sparsity patterns and matrix conditioning, recommend preconditioners (AMG, ILU, IC), apply row/column scaling, and diagnose convergence stagnation from residual histories. Use when setting up a linear solve for FEM/FVM assembly, debugging slow or stalled Krylov iterations, choosing a preconditioner for SPD or nonsymmetric systems, or investigating ill-conditioning, even if the user only says "my solver is slow" or "GMRES won't converge."

2026-05-1840

nonlinear-solvers

HeshamFS/materials-simulation-skills

Select and configure nonlinear solvers for root-finding f(x)=0, optimization min F(x), and least-squares problems — choose among Newton, Newton-Krylov, quasi-Newton (BFGS, L-BFGS), Broyden, Anderson acceleration, and Levenberg-Marquardt methods, configure line search or trust-region globalization, diagnose convergence rate (quadratic, linear, stagnated), and assess Jacobian quality and conditioning. Use when a Newton solver converges slowly or diverges, choosing between line search and trust region, debugging nonlinear iteration failures in FEM or phase-field codes, or selecting a solver for large-scale unconstrained optimization, even if the user only says "my Newton iterations aren't converging."

2026-05-1840

numerical-integration

HeshamFS/materials-simulation-skills

Select and configure time integration methods for ODE and PDE simulations — choose among explicit Runge-Kutta, BDF, Rosenbrock, and Adams families, set relative and absolute error tolerances, implement adaptive step-size control with I/PI/PID controllers, plan IMEX operator splitting for mixed stiff and non-stiff terms, and estimate splitting errors. Use when picking an integrator for a new simulation, diagnosing step rejections or tolerance failures, setting up operator splitting for phase-field or reaction-diffusion problems, or deciding between explicit and implicit time marching, even if the user only says "my solver keeps rejecting steps" or "which ODE method should I use."

2026-05-1840

numerical-stability

HeshamFS/materials-simulation-skills

Analyze numerical stability for time-dependent PDE simulations — check CFL and Fourier criteria, perform von Neumann stability analysis, detect stiffness, evaluate matrix conditioning, and recommend explicit vs implicit time-stepping schemes. Use when selecting time steps, diagnosing numerical blow-up or solver divergence, checking convergence criteria, or evaluating scheme stability for advection, diffusion, or reaction problems, even if the user doesn't explicitly mention "stability" or "CFL."

2026-05-1840

Fuente

HeshamFS

HeshamFS/materials-simulation-skills

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Desarrolladores de softwareOcupaciones informáticas y matemáticas15-1252L4

name	mesh-generation
description	Plan and evaluate mesh generation for numerical simulations — estimate grid resolution from physics scales (interface width, boundary layers, wavelengths), check aspect ratios and skewness against quality thresholds, choose between structured, unstructured, and adaptive mesh refinement strategies, and compute grid sizing for 1D/2D/3D domains. Use when setting up a new mesh, diagnosing poor solver convergence caused by mesh quality, deciding how many points to place across a phase-field interface or boundary layer, or preparing a mesh convergence study, even if the user only asks "what resolution do I need" or "why is my solver failing."
allowed-tools	Read, Write, Grep, Glob
metadata	{"author":"HeshamFS","version":"1.1.0","security_tier":"medium","security_reviewed":true,"tested_with":["claude-code","gemini-cli","vs-code-copilot"],"eval_cases":4,"last_reviewed":"2026-03-26"}

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)

Script	Key Outputs
`scripts/grid_sizing.py`	`dx`, `nx`, `ny`, `nz`, `notes`
`scripts/mesh_quality.py`	`aspect_ratio`, `skewness`, `quality_flags`

Workflow

Estimate resolution - From physics scales
Compute grid sizing - Run scripts/grid_sizing.py
Check quality metrics - Run scripts/mesh_quality.py
Adjust if needed - Fix aspect ratios, reduce skewness
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:

Compute grid sizing:

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

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

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: 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