What is a hex-dominant mesh?

A mesh that fills most of the domain with hexahedra, using tetrahedra, pyramids and prisms only near complex geometry and in transition zones where hexes are hard to fit. It aims for the low cell count and accuracy of hexahedra without the effort of a fully structured hex mesh.

How do cell counts compare with tetrahedral meshes?

Dividing a 1 m cube at 10 mm gives 1 million hexahedra. Splitting each into 6 tetrahedra gives 6 million cells, and faces rise from about 3 million to about 12 million. A hex-dominant mesh with 85% of volume in hexes and the rest in tets has 1.75 million cells, about 30% of the all-tet count.

Why are hexahedra more accurate?

Their faces can align with the flow. First-order upwind false diffusion grows as flow crosses the grid obliquely: at 10 mm cells and 1 m/s, a 45° misalignment gives 3.5×10⁻³ m²/s, 236 times air's kinematic viscosity. Flow-aligned hexes give zero. The difference is greatest in strongly directional flows such as boundary layers.

Hexadominant Mesh

Category: Fluid Analysis – Meshing | Revised 2026-10-01

CAE visualization for hex dominant theory - technical simulation diagram

Hex-Dominant Mesh

Theory: cell count and flow alignment

Overview

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Tet meshes are easy to generate automatically. Why bother with hexahedra?

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Two reasons. First, cell count: at the same resolution hexes need a fraction of the tets. Second, accuracy: hex faces can align with the flow. When flow crosses the grid obliquely, discretisation error appears as artificial diffusion that blurs velocity and temperature profiles. Filling complex geometry entirely with hexes is laborious, though, so hex-dominant meshing fills most of it automatically with hexes and patches the remaining gaps with tets and similar cells.

Estimating cell count

$$ N_{hex} = \frac{V}{h^3},\qquad N_{tet} \approx 6\,N_{hex},\qquad N_{hexdom} = f\,N_{hex} + 6\,(1-f)\,N_{hex} $$

V is volume, h cell size and f the fraction of volume filled with hexes. The tet count assumes each hex is split into six tets.

False diffusion of first-order upwind

$$ \Gamma_{false} = \frac{\rho U\,\Delta x\,\Delta y\,\sin 2\theta}{4\left(\Delta y\sin^3\theta + \Delta x\cos^3\theta\right)} $$

θ is the angle between flow and grid lines. It is zero at θ = 0 (aligned) and largest at 45°.

Coffee Break Trivia

Blurring by oblique flow

In the 1970s de Vahl Davis and Mallinson derived the false diffusion created by first-order upwind differencing when flow crosses the grid obliquely. The result spurred both higher-order schemes and efforts to align grids with flow. Second-order schemes are now standard and false diffusion is much smaller, but grid orientation still matters, and resolving boundary and shear layers with flow-aligned hexes or prisms remains fundamental.

Worked examples

Example 1: 1 m cube at 10 mm cells

Mesh	Cells	Faces	Note
Hex only	1.0 M	~3 M	6 faces per cell
Tet (hex split in 6)	6.0 M	~12 M	4 faces per cell
Regular tet (10 mm edge)	8.5 M	—	Volume h³/(6√2)
Polyhedral (tet dual)	~1.0 M	~7 M	~14 faces per cell
Hex-dominant (85%)	1.75 M	—	About 30% of tet

Example 2: flow angle and false diffusion (10 mm cells, 1 m/s)

Flow-grid angle	False diffusion	vs air ν	vs water ν
0°	0	0	0
15°	1.4×10⁻³ m²/s	91	1,361
30°	2.8×10⁻³ m²/s	186	2,795
45°	3.5×10⁻³ m²/s	236	3,536

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False diffusion over 200 times air's viscosity — can the results be trusted?

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That's first order; the second-order schemes used in practice give much less. Still, error remains larger wherever flow crosses the grid obliquely. Tetrahedra are oblique to flow in every direction, so they need finer grids for the same accuracy, while flow-aligned hexes keep error down with fewer cells. In hex-dominant meshing, put the hex region where the main flow is, and align near-wall flow with prism layers. Pyramids appear at hex–tet interfaces and tend to degrade quality, so keep those interfaces away from important flow regions.

Meshing tips

Add prism (inflation) layers on walls first.
Fill the interior with hexes, patching only near complex geometry with tets.
Align hexes with the main flow direction.
Check quality (skewness, volume ratio) of pyramids and transitions.
Check mesh independence by varying cell count.

Coffee Break Trivia

“Pipe pressure drop that wouldn't match”

Pressure drop in a pipe with many bends came out about 15% above measurement on a tet mesh. Adding cells barely closed the gap, and the count reached tens of millions. Remeshing as hex-dominant — flow-aligned hexes in straight runs, tets only in bends — cut cells to about a third and the discrepancy to a few percent. The likely cause was false diffusion smearing the velocity profile in straight runs on the tet mesh.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
No prism layers on walls	Wrong wall friction and heat transfer	Inflation layers
Hexes oblique to flow	No accuracy gain	Align with main flow
Interface in key region	Errors from pyramids	Move the interface
Comparing by cell count only	Face count and cost overlooked	Check face counts too
No mesh-independence check	Unknown error	Compare 3 levels

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