Unstructured Grid

An unstructured grid (unstructured mesh) is a grid that explicitly stores the connectivity relationships between cells as a table. It allows different cell shapes like tetrahedra (tet), hexahedra (hex), triangular prisms (prism/wedge), and pyramids to be freely mixed. Its biggest advantage is that it enables automatic mesh generation for complex 3D shapes.

Pretty much. Modern CFD meshers like Fluent Meshing, STAR-CCM+, and snappyHexMesh can automatically generate volume meshes just by inputting an STL surface. This is the reason why unstructured grids have become overwhelmingly popular in industry.

The most basic one is Delaunay triangulation. For a given set of points, it constructs triangles such that no other point lies inside the circumcircle of any triangle.

Writing the Delaunay condition mathematically, for a triangle $T$ and its circumcircle $C(T)$:

Satisfying this condition maximizes the minimum angle among the triangles. In other words, you get a mesh that is as close as possible to "equilateral triangles".

In 3D, it becomes Delaunay tetrahedralization. The circumcircle is replaced by a circumsphere. However, in 3D, quality guarantees are not as effective as in 2D, and sliver tetrahedra (flat tetrahedra) tend to occur. Sliver removal is an important challenge in 3D automatic mesh generation.

Another major algorithm is the Advancing Front method. It "extrudes" elements from the boundary surface towards the interior.

1. Initialize the surface mesh on the boundary as the "front"

2. Select a face on the front, generate a new point, and create an element

3. Update the front (remove old faces, add new faces)

4. Repeat until the front disappears

The Advancing Front method has high boundary conformity and tends to produce good mesh quality near boundaries. On the other hand, the Delaunay method is more robust and easier to implement. In actual tools, hybrid algorithms combining both are often used.

We use the Finite Volume Method (FVM). It applies the conservation law in integral form to each cell.

The flux $\mathbf{F} \cdot d\mathbf{S}$ passing through the cell face is evaluated numerically. Unlike structured grids, the grid is irregular, so methods like Gauss-Green or least squares are needed to reconstruct gradients between cell centers.

The finite volume method on unstructured grids is basically standard second-order accuracy. This is because values at faces are linearly interpolated from cell center values and gradients. If the grid non-orthogonality is large, a non-orthogonal correction term is needed, which affects both convergence and accuracy.

Let's compare the main cell shapes used in CFD.

Tetrahedra have fewer faces, resulting in lower diversity of direction vectors from the cell center to each face. This reduces gradient approximation accuracy and increases numerical diffusion when evaluating fluxes in oblique directions. For the same number of cells, hexahedra are often 2–3 times more accurate.

Exactly. There are three main methods.

For meshes generated by Fluent Meshing, Green-Gauss Node-Based or Least-Squares is recommended. For tetrahedral meshes, Least-Squares is often more stable.

When calculating the diffusion term flux in the finite volume method, it's simple if the vector between cell centers $\mathbf{d}$ and the face normal vector $\mathbf{S}$ are parallel (orthogonal grid), but for non-orthogonal cases, a correction term is needed.

The second term is the non-orthogonal correction term. Treating this explicitly worsens convergence, and if the grid non-orthogonality exceeds 70 degrees, the risk of divergence increases. In OpenFOAM, the nonOrthogonalCorrectors parameter sets the number of iterations for this correction.

Yes. As a general guideline, non-orthogonality should be kept below 70 degrees, ideally below 40 degrees. Especially for segregated solvers like OpenFOAM, they are sensitive to non-orthogonality.