Structured Grids

A structured grid (structured mesh) is a grid that fills the computational domain with regularly arranged hexahedral (3D) or quadrilateral (2D) cells. Each cell's index can be uniquely represented by $(i,j,k)$. This means the adjacency relationships between grid points are implicitly determined, so there's no need to explicitly store connectivity information.

Exactly. In a structured grid, neighboring cells are always at $(i\pm1, j\pm1, k\pm1)$, so data access is contiguous and cache hit rates are high. For the same number of cells, memory consumption can be about half compared to an unstructured grid, and computational speed can be 2-3 times faster, which is not uncommon.

That's where the body-fitted curvilinear coordinate system comes in. It maps the physical space $(x,y,z)$ to a computational space $(\xi,\eta,\zeta)$. In the computational space, the grid becomes a uniform rectangular box, allowing finite-difference schemes to be written simply.

The Jacobian matrix of the coordinate transformation is crucial and is defined as follows.

If the determinant of this Jacobian, $|J|$, becomes zero or negative, it's evidence that the grid is collapsed or inverted, causing the computation to fail.

Typically around sharp corners or when grid lines intersect. The golden rule during grid generation is to verify $|J| > 0$ for all cells.

The Navier-Stokes equations in physical space are transformed into computational space. In conservative form, it becomes:

Here, $\mathbf{Q}$ is the vector of conservative variables, and $\hat{\mathbf{E}}, \hat{\mathbf{F}}, \hat{\mathbf{G}}$ are the transformed fluxes containing metric terms.

For example, $\hat{\mathbf{E}} = \frac{1}{J}(\xi_x \mathbf{E} + \xi_y \mathbf{F} + \xi_z \mathbf{G})$, where the partial differential coefficients of the coordinate transformation $\xi_x, \xi_y$, etc., multiply the fluxes. If these metric terms are not discretized accurately, freestream preservation breaks down, generating spurious numerical errors.

Using a geometric progression is fundamental.

Here, $\Delta s_1$ is the thickness of the first layer, and $r$ is the growth ratio. If $r = 1.2$, each layer becomes 1.2 times thicker than the previous one. The wall-side $\Delta s_1$ is determined from $y^+$ requirements, and a growth ratio $r$ of 1.1 to 1.3 is recommended.

The size ratio between adjacent cells becomes too large, increasing truncation error and worsening numerical diffusion. Especially for second-order schemes, it's desirable to keep the volume ratio of adjacent cells below 1.2.

Let me introduce three representative methods.

Given the distribution of grid points on the boundary, interior grid points are determined by interpolation. Computationally fast, but controlling grid quality for complex shapes is difficult.

Grid points are placed by solving Poisson equations.

Grid concentration near walls and orthogonality can be controlled by adjusting the control functions $P, Q, R$. Produces the highest grid quality but requires iterative computation and takes time to generate.

A method that "grows" the grid from the wall in the normal direction. Particularly effective for boundary layer mesh generation, naturally ensuring wall orthogonality.

In practice, a hybrid approach is common: first create an initial grid with TFI, then smooth it with an elliptic solver. This is the standard workflow in tools like ICEM or Pointwise.

In that case, use a multi-block structured grid. The computational domain is divided into multiple blocks, each containing a structured grid, and information is exchanged between blocks via interfaces.

There are two main types of connections between blocks.

• Conformal: Grid points match at the interface. No interpolation error.
• Non-conformal: Grid points do not match. Interpolation is required.

In terms of accuracy, yes, but for complex shapes, maintaining conformity makes topology design extremely difficult. In practice, block topology editors like ICEM CFD Hexa are invaluable. However, this topology design is the biggest bottleneck for structured grids; even an expert can take days to weeks to mesh a single car.

Because grid points are arranged regularly in structured grids, high-order accuracy finite-difference schemes can be naturally constructed. For example, 5th-order WENO schemes or compact finite-difference methods (Pade-type) truly shine on structured grids.

Achieving equivalent high-order accuracy on unstructured grids requires gradient reconstruction via least squares and handling of non-uniform spacing, which skyrockets computational cost.

Exactly. Structured grids are still mainstream for high-precision academic computations. Especially for DNS of relatively simple shapes like channel flow or pipe flow, structured grids are essentially the only choice.