Numerical Anisotropy in Unstructured Meshes

#+CATEGORY: cfd

Numerical Anisotropy in Unstructured Meshes :: Directional Dependence of Discretization Error

Numerical anisotropy describes the phenomenon where the truncation error of a discretization scheme depends on direction relative to the mesh structure. In structured Cartesian meshes, the error is isotropic (the same in all directions) for symmetric stencils. In unstructured meshes (polyhedral, tetrahedral, hexahedral), the error is inherently anisotropic. The stencil relies on cell connectivity. Connectivity is direction-dependent.

This is a critical but often overlooked error source in CFD simulations. Two meshes with identical cell counts but different topologies produce solutions with different error levels. The error difference can differ by an order of magnitude in quantities of interest. These quantities include drag, heat transfer coefficient, and mass flow rate.

What Is Numerical Anisotropy?

Numerical anisotropy is the directional dependence of: 1. Numerical dissipation (truncation error diffusivity) 2. Numerical dispersion (truncation error oscillatory component) 3. Gradient reconstruction accuracy (accuracy of computed gradients) 4. Face-interpolation accuracy (accuracy of face values from cell centers)

For a discretized PDE, the modified equation equals the original PDE plus truncation error terms. In structured grids, the truncation error terms have coefficients that are direction-independent for symmetric stencils on Cartesian grids with central differencing.

In unstructured grids, the coefficients depend on local mesh topology. The arrangement of neighbors determines the coefficients. The error varies with the angle between flow direction and mesh alignment.

Anisotropy in Different Mesh Types

Tetrahedral Meshes and Numerical Anisotropy

A tetrahedral cell has only 4 faces. A polyhedral cell has 8 to 12. A hexahedral cell has 6. This minimal connectivity leads to three effects:

1. Sparse gradient reconstruction: gradient is computed from only 4 neighbors. The computation can be poorly conditioned. 2. High numerical dissipation: tetrahedral meshes are more diffusive than hexahedral or polyhedral meshes at the same cell count. 3. Directional dependence: error varies significantly between flow parallel to cell edges and flow at arbitrary angles.

Numerical studies show that tetrahedral meshes require 2 to 4 times more cells than polyhedral meshes to achieve the same accuracy in convection-dominated flows. The numerical diffusivity in tetrahedral meshes is higher. The leading-order truncation error coefficient is larger.

Polyhedral Meshes and Reduced Anisotropy

Polyhedral meshes have more faces per cell (typically 8 to 12). They have more neighbors. This yields three advantages:

1. Improved conditioning of gradient reconstruction (more data points give a better least-squares fit). 2. Reduced numerical dissipation (smaller leading-order truncation error). 3. Reduced directional dependence (more symmetric connectivity is closer to isotropic).

ANSYS Fluent and Star-CCM+ both use polyhedral meshes by default because of these advantages. OpenFOAM supports polyhedral meshes natively. All face-based fluxes work with polyhedral cells. This is why snappyHexMesh produces hex-dominant meshes with tet fallback instead of pure tetrahedral meshing.

Wall-Prism Layers and Anisotropy

Prism layers in boundary layers are strongly anisotropic. They are highly stretched in the wall-normal direction. This leads to:

1. Gradient reconstruction error: the wall-normal velocity gradient is reconstructed from only 1 to 2 prism layers in the wall-normal direction. 2. Anisotropic numerical diffusion: diffusion in the wall-normal direction is over-predicted compared to the tangential direction. 3. Wall shear stress errors: inaccurate wall shear stress tau_w equals mu dU over dy arises from errors in dU over dy.

The prism layer aspect ratio affects this anisotropy. Typical RANS runs use 10 to 50. LES and DNS runs use less than 5. As the aspect ratio increases, the wall-normal gradient reconstruction becomes more sensitive to mesh quality. Non-orthogonality and skewness are the key sensitivities.

Impact on Quantities of Interest (QoI)

Managing Numerical Anisotropy

1. Use polyhedral meshes instead of tetrahedral meshes. This reduces anisotropy by increasing face density per cell. 2. Refine boundary layers. Add more prism layers for accurate wall-normal gradient reconstruction. 3. Reduce non-orthogonality. Aim for less than 60 degrees for RANS and less than 45 degrees for LES. 4. Use higher-order schemes. Gradient reconstruction is more accurate with second-order upwind or linearUpwind. 5. Perform mesh sensitivity study. Run the same simulation on two meshes with different topologies at the same cell count and compare results.

Quantifying Numerical Anisotropy

In practice, numerical anisotropy is quantified by running the same simulation on two meshes:

• Mesh one has high anisotropy. Use tetrahedral cells with few prism layers.
• Mesh two has low anisotropy. Use polyhedral cells with more prism layers.

If the quantity of interest differs by more than 5 to 10%, numerical anisotropy is likely a significant error source. For formal quantification, compute the anisotropy index as maximum absolute difference between directional errors tau_i and average error tau-bar, divided by tau-bar. This index comes from modified equation analysis for a given mesh and discretization scheme.

Summary

1. Numerical anisotropy is the directional dependence of discretization error inherent in unstructured meshes 2. Tetrahedral meshes are the most anisotropic. Polyhedral and structured meshes are less anisotropic. 3. Wall-prism layers are highly anisotropic. They cause errors in wall shear stress and heat transfer. 4. Numerical anisotropy can cause quantity-of-interest differences of 10% to 50% between different mesh types at the same cell count. 5. Polyhedral meshes and higher-order schemes reduce numerical anisotropy.

See Also

• Numerical Dispersion
• Numerical Dissipation
• Numerical Schemes
• OpenFOAM Mesh Generation
• Mesh Quality