Mesh quality#

A well-built model with poor mesh quality gives worse answers than the same model on a coarser-but-cleaner grid. Three metrics to watch and concrete thresholds to gate against.

Aspect ratio#

Aspect ratio = longest element edge / shortest element edge. For HEX8-EAS, the comfortable upper bound depends on the geometry being modelled:

  • Bulk solid (uniform stress field, no bending): aspect ratio up to 20 is fine — the answer is insensitive at this ratio.

  • Bending-dominated (cantilever, plate, shell-like solid): keep aspect ratio under 5 in the bending plane. The EAS modes recover Bernoulli kinematics exactly only when the cell-shape isn’t too distorted.

  • Steep stress gradient (notch, hole edge, contact): refine aggressively to bring the local aspect ratio under 2 at the gradient. The Plate with a circular hole — Kirsch stress concentration benchmark shows the convergence rate degradation when the bore-edge mesh isn’t refined enough.

For TET10 the same rules apply but with looser thresholds — quadratic shape functions on a tetrahedral element tolerate aspect ratios up to ~50 in the bulk before stiffness errors exceed 1 %.

Jacobian determinant#

The isoparametric mapping \(\mathbf{J}(\xi, \eta, \zeta)\) must have positive determinant at every Gauss point. A negative determinant means the cell is geometrically inverted; zero means it’s degenerate (a face has collapsed to a line). Either case produces a singular element stiffness.

The Isoparametric mapping chapter walks the Jacobian’s role in the assembly pipeline. In practice:

  • Visualise \(\det(\mathbf{J})\) on the mesh — most meshing tools output this directly. Anything below 0.1 × the maximum across the mesh is a candidate for refinement or remeshing.

  • Run a single-step static solve on a known-good load and watch for nodes with anomalously large displacement — those nearly always sit next to a bad-Jacobian element.

Skewness#

For HEX8 / HEX20 / TET10 / WEDGE15, skewness measures how far the cell deviates from its ideal angular geometry (right angles for hexes, equilateral faces for tets). Skewness > 0.7 is a warning, > 0.85 is a fail; refine or split the cell.

Mesh-density rules of thumb#

  • Bending dominated — at least 3 layers through the thickness of any plate / shell-like solid being modelled with HEX8. Two layers under-resolve the bending strain; more than three is overkill.

  • Concentration features — start with 8 cells around any hole circumference; refine if the recovered stress hasn’t converged at that density.

  • Modal analysis — wavelength of the highest mode you care about must span ≥ 8 elements. For a cantilever with first-mode wavelength ≈ 4 L (the beam is a quarter- wavelength), mode 5 would have wavelength \(L/2\), so at least 16 elements along the span — comfortably below the 80×3×3 mesh the modal-survey tutorial uses.

When to upgrade element order#

  • Concentrated stress + curved geometry: HEX8 → HEX20. Quadratic shape functions on a 20-node hex follow curved surfaces (cylindrical pressure vessels, fillets) without needing the linear-hex’s mesh-refinement crutch.

  • Unstructured meshes (CAD geometries that won’t accept a hex sweep): use TET10 from the start. TET4 is too stiff in bending and rarely earns its place except as a near-singular gap filler.

  • Thin shell behaviour: don’t try to model it with a solid — use QUAD4_SHELL. Five solid layers through the thickness costs 5× the DOFs and still under-resolves the bending strain.