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
  :ref:`sphx_glr_gallery_verification_example_verify_plate_with_hole.py`
  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 :math:`\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 :doc:`/reference/theory/isoparametric` chapter walks the
Jacobian's role in the assembly pipeline.  In practice:

* Visualise :math:`\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 :math:`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.

Cross-link
----------

The per-element technical sheets in
:doc:`/reference/elements/index` carry element-specific
mesh-quality guidance — restrictions sections list the
geometry classes each element handles cleanly and the ones
that need a different element.