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In computational fluid dynamics, meshing is a discrete representation of the geometry that is involved in the problem. Essentially, it assigns cells or smaller regions over which the flow is solved. Several parts of the mesh are grouped into regions where boundary conditions may be applied to solve the problem. Moreover, the uses of meshing are not limited to computational fluid dynamics. It also finds extensive use in the analysis of geographical and cartographic data. Also, meshing can be used to solve partial differential equations using numerical techniques.

Mesh quality

The mesh quality can be conclusively determined based on the following factors.

Rate of convergence

The greater the rate of convergence, the better the mesh quality. It means that the correct solution has been achieved faster. An inferior mesh quality may leave out certain important phenomena such as the boundary layer that occurs in fluid flow. In this case the solution may not converge or the rate of convergence will be impaired.

Solution accuracy

A better mesh quality guarantees a more accurate solution. For improving the mesh quality for accurate solution, one may need to refine the mesh at certain areas of the geometry where the gradients of the field whose solution is sought is high. Also this means that, if a mesh is not sufficiently refined, the accuracy of the solution is more limited. Thus the required accuracy in turn dictates the mesh quality.

CPU time required

CPU time is a necessary yet an undesirable factor. For a highly refined mesh, where the number of cells per unit area is maximum, the CPU time required will be relatively large. In general, if the CPU time taken is more it indicates that the solution that is being generated will be of good accuracy. However, for the solution of given accuracy and rate of convergence, greater CPU time required indicates an inferior mesh quality.

Common cell shapes

Two dimensional

Basic two dimensional Cell shapes

There are two types of two dimensional cell shapes that are commonly used. These are the Triangle and the Quadrilateral.

Triangle

This cell shape consists of 3 sides and is one of the simplest types of mesh. A triangular surface mesh is always quick and easy to create. It is most common in unstructured grids.

Quadrilateral

This cell shape is a basic 4 sided one as shown in the figure. It is most common in structured grids.

Three dimensional

Basic three dimensional cell shapes

Pyramid

These are effectively used as transition elements between one type of elements and other in hybrid meshes and grids.

Prism with triangular base

The advantage with this type of layer is that it resolves boundary layer efficiently.

Tetrahedron

In most cases a tetrahedral volume mesh is generated automatically.

Hexahedron

For the same cell amount, the accuracy of solutions in hexahedral meshes is the highest.

Classification of grids

Structured Grid

Structured grids

Structured grids are identified by regular connectivity. The possible element choices are quadrilateral in 2D and hexahedra in 3D. This model is highly space efficient, i.e. since the neighborhood relationships are defined by storage arrangement.

Unstructured Grid

Unstructured grids

An unstructured grid is identified by irregular connectivity. It cannot easily be expressed as a two dimensional or three dimensional arrays in computer memory. This allows for any possible element that a solver might be able to use. Compared to structured meshes, this model can be highly space inefficient since it calls for explicit storage of neighborhood relationships. These grids typically employ triangles in 2D and tetrahedra in 3D.

Hybrid grids

A hybrid grid contains a mixture of structured portions and unstructured portions. It integrates the structured meshes and the unstructured meshes in an efficient manner. Those parts of the geometry that are regular can have structured grids and those that are complex can have unstructured grids. These grids can be non-conformal which means that grid lines don’t need to match at block boundaries.

Deciding the type of mesh

Skewness based on equilateral volume

If the accuracy is of the highest concern then hexahedral mesh is the most preferable one. The density of the mesh is required to be sufficiently high in order to capture all the flow features but on the same note, it should not be so high that it captures unnecessary details of the flow, thus burdening the CPU and wasting more time. Whenever a wall is present, the mesh adjacent to the wall is fine enough to resolve the boundary layer flow and generally quad, hex and prism cells are preferred over triangles, tetrahedrons and pyramids. Quad and Hex cells can be stretched where the flow is fully developed and one dimensional.

Depicts the skewness of a quadrilateral

Based on the skewness, smoothness, and aspect ratio, the suitability of the mesh can be decided.

Skewness

The skewness of a grid is an apt indicator of the mesh quality and suitability. Large skewness compromises the accuracy of the interpolated regions. There are three methods of determining the skewness of a grid.

Based on equilateral volume

This method is applicable to triangles and tetrahedral only and is the default method.

$\text{ Skewness }=\frac{\text{ optimal cell size - cell size }}{\text{optimal cell size}}$

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Smooth and large jump change

Based on the deviation from normalized equilateral triangle

This method applies to all cell and face shapes and is almost always used for prisms and pyramids

$\text{ Skewness ( for a quad ) }=max[\frac{{\theta }_{max}-90}{90},\frac{90-{\theta }_{min}}{90}]$

Equiangular skew

Another common measure of quality is based on equiangular skew.

$\text{ Equiangle Skew }=max[\frac{{\theta }_{max}-{\theta }_e}{180-{\theta }_e},\frac{{\theta }_e-{\theta }_{min}}{{\theta }_e}]$

where:

${\theta }_{max}$ is the largest angle in a face or cell,

${\theta }_{min}$ is the smallest angle in a face or cell,

${\theta }_e$ is the angle for equi-angular face or cell i.e. 60 for a triangle and 90 for a square.

A skewness' of 0 is the best possible one and a skewness of one is almost never preferred. For Hex and quad cells, skewness should not exceed 0.85 to obtain a fairly accurate solution.

Depicts the changes in aspect ratio

For Triangular cells, skewness should not exceed 0.85 and for Tetragonal cells, skewness should not exceed 0.9.

Smoothness

The change in size should also be smooth. There should not be sudden jumps in the size of the cell because this may cause erroneous results at nearby nodes.

Aspect ratio

It is the ratio of longest to the shortest side in a cell. Ideally it should be equal to 1 to ensure best results. For multidimensional flow, it should be near to one. Also local variations in cell size should be minimal, i.e. adjacent cell sizes should not vary by more than 20%. Having a large aspect ratio can result in an interpolation error of unacceptable magnitude.

Mesh improvement

In two dimensions, flipping and smoothing are powerful tools for adapting a poor mesh into a good mesh. Flipping involves combining two triangles to form a quadrilateral, then splitting the quadrilateral in the other direction to produce two new triangles. Flipping is used to improve quality measures of a triangle such as skewness. Mesh smoothing enhances element shapes and overall mesh quality by adjusting the location of mesh vertices. In mesh smoothing, core features such as non-zero pattern of the linear system are preserved as the topology of the mesh remains invariant. Laplacian smoothing is the most commonly used smoothing technique.

References

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