Curvature factor

acoustics.mesh.curvature_factor Step 08 · Mesh

The element size on a curved surface as a fraction of its local radius of curvature. A smaller value gives more elements around pipes and port rims. On real mufflers this setting, more than the wavelength, decides how fine the acoustic mesh is.

Last updated 2026-11-03

Theory

A pipe of radius $r$ meshed with elements of size $h$ is a polygon of about $2\pi r/h$ sides. With too few sides the cross-section area is too small and the duct modes and resonances shift (a 25 mm tube mouth meshed 4.5 % too small moved a chamber's quarter-wave notch 7 % low). The curvature rule keeps $h \approx f_c\,r$ , equation (1). The program passes it to Gmsh as a number of elements per $2\pi$ of curvature, Mesh.MeshSizeFromCurvature, equation (2) (Gmsh reference manual, 2025). The mid-edge nodes of the elements are then placed on the CAD surface (second order), which brings the curved area much closer to the CAD than the straight-sided polygon.

The resulting size is bounded by the maximum and minimum element sizes. Because the curvature rule refines the surface, and the growth rule carries the surface size into the volume, it determines most of the mesh of a compact muffler: on case V1 (a 2.0 m multi-chamber muffler with a perforated pipe and three absorptive packings) at 500 Hz, curvature factors 0.52, 0.4, 0.3 and 0.2 give 21k, 41k, 75k and 199k tetrahedra.

The reference pipe of the insertion loss uses the same settings except this one, which never goes below 0.524 there (12 elements around): the curvature size applies along the whole 3 m length of the pipe, and 12 elements per circle already match the pipe's analytic radiated power to $3 \times 10^{-7}$ dB.

Equations

h(\mathbf{x}) \approx f_c\,r(\mathbf{x})

(1)

n_{2\pi} = \max\!\left(3,\ \mathrm{round}\!\left(\frac{2\pi}{f_c}\right)\right)

(2)

where	meaning	unit
$h$	Element size on a curved surface	m
$f_c$	Curvature factor	-
$r$	Local radius of curvature of the surface	m
$n_{2\pi}$	Elements per $2\pi$ of curvature given to Gmsh (`Mesh.MeshSizeFromCurvature`)	-

Default and rule

Default

auto: 0.3 (about 21 elements around a circle)Computed by the programMeasured

Rule

Auto = 0.3, i.e. 21 elements around a circle (0.4: 16 elements, faster).

Source

Validation results, transmission loss against a reference finite-element solution on an extremely fine mesh:

Case V1 (20-500 Hz, three absorptive packings, 14 rigid internal-wall faces): curvature factors 0.52 / 0.4 / 0.3 give 21.3k / 40.5k / 75.3k tetrahedra and an rms difference of 1.02 / 0.50 / 0.16 dB (maximum 3.1 / 1.6 / 0.49 dB), every frequency converging monotonically onto the reference.
Case V4 (a 0.3 m expansion chamber with an extended inlet tube): 0.3 lowers the rms difference to the reference from 0.82 to 0.12 dB.
Case V2 (a two-inlet single-chamber muffler with internal inlet and outlet tubes): the difference grows slightly as the mesh is refined (median 0.74 to 0.89 dB, largest at sharp 65 dB notches); that reference curve comes from a coarser mesh of unknown convergence.

The Gmsh option is described in its manual (Gmsh reference manual, 2025).

Valid range 0.05 < f_c <= 1.5

Values

Above 0.05 and at most 1.5 (checked when the setting is saved).

Cost

On V1, factors 0.5 and 0.3 give 43k and 120k unknowns; with the current solver one frequency takes 0.50 s and 1.64 s (4-core laptop).

Model fidelity

After meshing, the model fidelity check compares every meshed face area and the fluid volume with the CAD (ports warn at 0.5 %, walls at 1 %, volume at 0.5 %; above 5 % or 3 % the run stops). Two curved internal faces of V1 read 2-4 % off at every factor; the cause has not been identified.

References 1

Geuzaine, C. and Remacle, J.-F. (2025). Gmsh reference manual, version 4.15. gmsh.info. Web page

Theory

Equations

Default and rule

Valid range 0.05 < f_c <= 1.5

Related 3

References 1