Mesh Convergence
How to determine when your mesh is fine enough to trust the results.
Mesh convergence answers the question: is my mesh fine enough that the results don't change meaningfully if I refine it further?
Basic strategy
- Compute with the initial mesh
- Refine locally or globally (typically by a factor of 2 in element size)
- Recalculate and compare key results
- Repeat until changes fall below your threshold
Convergence criteria
| Quantity | Acceptable change on refinement | | --- | --- | | Maximum stress | < 5% | | Maximum displacement | < 2% | | Reaction forces | Stable (< 1%) |
Methods
h-refinement — systematically reduce element size (factor of 2 is standard). Simple and reliable.
p-refinement — increase the polynomial degree of the shape functions. Less common but efficient for smooth problems.
Adaptive meshing — the solver automatically refines in high-gradient regions. Useful for complex geometries.
Richardson extrapolation — extrapolate from multiple mesh levels to estimate the ideal fine-mesh value. Gives a quantitative error bound.
Accuracy vs effort
| Goal | Acceptable error | Typical effort | | --- | --- | --- | | Concept study | ±20–30% | Single coarse run | | Design validation | ±5–10% | 2–3 mesh levels | | Certification | < 1% | Full convergence study + validation |
Singularities never converge — stress at a sharp corner keeps growing with refinement. Run convergence studies away from singularities, or add fillets to remove the singularity first.