Mesh Generators

Overview

FrontIntrinsicOps.jl provides analytical mesh generators for standard geometries. All generators produce consistently-oriented, outward-normal meshes suitable for convergence studies.

sample_circle — regular polygon

mesh = sample_circle(R, N)

Generates a regular $N$-gon inscribed in a circle of radius $R$ in the $xy$-plane.

Vertices:

\[p_k = \left(R \cos\!\frac{2\pi k}{N},\; R \sin\!\frac{2\pi k}{N}\right), \quad k = 0, \ldots, N-1\]

Convergence:

• Length $\to 2\pi R$ as $N \to \infty$ with rate $O(1/N^2)$.
• Enclosed area $\to \pi R^2$ with rate $O(1/N^2)$.
• Curvature $\to 1/R$ with rate $O(1/N^2)$.

generate_uvsphere — UV sphere

mesh = generate_uvsphere(R, nphi, ntheta)

Generates a latitude-longitude sphere of radius $R$ with nphi meridians and ntheta latitude bands.

Parameterization:

\[p(\phi, \theta) = \left(R\cos\phi\sin\theta,\; R\sin\phi\sin\theta,\; R\cos\theta\right)\]

• \[\phi \in [0, 2\pi)\]

  — longitude (azimuth), nphi values

• \[\theta \in (0, \pi)\]

  — colatitude (polar angle), ntheta values

Notes:

• Pole vertices are singular (high-valence, poor triangle quality).
• Not recommended for convergence studies — use generate_icosphere instead.
• Useful when longitude/latitude alignment is required.

generate_icosphere — icosphere (recommended)

mesh = generate_icosphere(R, level)

Generates a sphere of radius $R$ by recursively subdividing a regular icosahedron.

Algorithm:

1. Start with a regular icosahedron (12 vertices, 20 faces).
2. At each subdivision level, replace each triangle by 4 sub-triangles (midpoint subdivision).
3. Project all vertices to radius $R$.

Vertex counts by level:

Properties:

• All triangles have approximately equal area: ideal for convergence studies.
• No singular vertices (uniform valence ≈ 6 for interior vertices).
• Gaussian curvature convergence: $O(h^2)$.

generate_torus — standard torus

mesh = generate_torus(R, r, ntheta, nphi)

Generates a torus with major radius $R$ (distance from centre to tube centre) and minor radius $r$ (tube radius).

Parameterization:

\[p(\theta, \phi) = \left((R + r\cos\phi)\cos\theta,\; (R + r\cos\phi)\sin\theta,\; r\sin\phi\right)\]

• \[\theta \in [0, 2\pi)\]

  — toroidal angle (ntheta values)

• \[\phi \in [0, 2\pi)\]

  — poloidal angle (nphi values)

Topology:

• Genus 1: $\chi = 0$, $\int_\Gamma K \, dA = 0$.
• Two independent cycles: toroidal ($\theta$) and poloidal ($\phi$).

Convergence reference: The torus is the canonical non-simply-connected closed surface for testing Hodge decomposition and transport convergence.

generate_ellipsoid — axis-aligned ellipsoid

mesh = generate_ellipsoid(a, b, c, nphi, ntheta)

Generates an ellipsoid with semi-axes $a$ (along $x$), $b$ (along $y$), $c$ (along $z$).

Parameterization:

\[p(\phi, \theta) = \left(a\cos\phi\sin\theta,\; b\sin\phi\sin\theta,\; c\cos\theta\right)\]

When $a = b = c = R$: reduces to a UV sphere of radius $R$.

Curvature formulas:

Mean curvature at the poles $(0, 0, \pm c)$: $H = (a^2 + b^2 - c^2) / (a^2 b c)$ (approximate; exact only for a sphere of revolution with $a = b$).

Use in convergence studies: The ellipsoid geometry breaks spherical symmetry while remaining smooth and simply-connected.

generate_perturbed_sphere — "bumpy sphere"

mesh = generate_perturbed_sphere(R, ε, k, nphi, ntheta)

Generates a radially perturbed sphere with equation:

\[r(\phi, \theta) = R\bigl(1 + \varepsilon \cos(k\phi)\cos(k\theta)\bigr)\]

in spherical coordinates, giving vertices:

\[p(\phi, \theta) = r(\phi, \theta) \cdot \left(\cos\phi\sin\theta,\; \sin\phi\sin\theta,\; \cos\theta\right)\]

Parameters:

• \[R\]

  — base radius

• \[\varepsilon\]

  — perturbation amplitude (typically $0.05$ to $0.3$)

• \[k\]

  — mode number (typically $2$–$4$; odd $k$ breaks anti-podal symmetry)

When $\varepsilon = 0$: reduces to a UV sphere.

Use in convergence studies: The bumpy sphere retains spherical topology but is distinctly non-spherical, making it ideal for verifying that PDE solvers do not exploit sphere symmetry.

Mesh quality diagnostics

mesh = generate_icosphere(1.0, 3)
info = check_mesh(mesh)
println(info)
# (n_vertices=642, n_edges=1920, n_faces=1280, closed=true,
#  manifold=true, consistent_orientation=true, euler_characteristic=2)

All built-in generators produce closed, consistently-oriented manifold meshes with the correct Euler characteristic.

Summary table

See also

• Mesh types and data structures
• Topology and incidence matrices