NACA Airfoil Cross Field

Cross field smoothing on a triangulated airfoil domain

Configure the airfoil and domain, then explore the cross field. Click vertices to constrain/release them. Toggle boundary constraints with the checkboxes. Drag to pan, scroll to zoom.

Airfoil NACA Thickness: 12% Camber: 0% Camber pos: 0

Chord: 1.0

Domain AoA: 0°

Shape: Size: 1.0

Spacing: 0.10c

Airfoil refine: 1x

Field Constrain airfoil Constrain domain

Init:

Iters: 100

Quad Method: Target h: 0.10

Display Mesh Cross: 8 Streamlines Length: 10

Dirichlet energy per iteration

How it works

Mesh generation: points are placed on the airfoil surface (cosine-spaced), the domain boundary, and a regular interior grid. Points inside the airfoil or outside the domain are rejected. Delaunator computes the Delaunay triangulation; triangles whose centroid falls inside the airfoil or outside the domain are removed.
Cross field: each vertex carries an angle. The 4-RoSy Laplacian smoother averages cos 4θ / sin 4θ over mesh neighbors — the same representation used in frame field papers.
Constrain airfoil: pins airfoil-surface vertices to the local tangent direction.
Constrain domain: pins far-field boundary vertices to the local tangent direction.
Click to constrain: click any vertex to pin/release it at its current angle.
Run smoother: executes the specified number of Jacobi iterations and records the Dirichlet energy at each step.
Energy chart: shows E = Σ_edges (1 − cos 4(θ_i − θ_j)) per iteration. Red dot = initial energy, green dot = final. The curve should decay monotonically toward the smooth equilibrium.
Iterations slider goes up to 2000 for observing long-tail convergence.
Detect singularities: computes the Poincaré index (angle defect) around each triangle. Non-zero index means a singularity: +1 (3-valent, red) or -1 (5-valent, purple).
Trace separatrices: from each singularity, traces streamlines outward along cross field arms. These are the red curves that form potential block partition boundaries.
Build motorgraph: constructs a graph whose nodes are singularities and boundary intersection points, and whose edges are the separatrix arcs connecting them. This is the topological skeleton of the block decomposition.
Extract blocks: walks the motorgraph to identify closed regions (blocks) and fills them with distinct colors. Each block is a candidate for structured (transfinite interpolation) meshing.
Geometry changes (NACA code, AoA, domain, spacing) regenerate the mesh and reset the field. Click Run smoother again to re-solve.