Types of mesh

A mesh is a representation of a larger geometric domain by smaller discrete cells. Meshes

are commonly used to compute solutions of partial differential equations and render
computer graphics, and to analyze geographical and cartographic data. A mesh partitions
space into elements (or cells or zones) over which the equations can be solved, which then
approximates the solution over the larger domain. Element boundaries may be constrained
to lie on internal or external boundaries within a model. Higher-quality (better-shaped)
elements have better numerical properties, where what constitutes a "better" element
depends on the general governing equations and the particular solution to the model
instance.

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. Computationally poor elements will have sharp internal
angles or short edges or both.

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. Quadrilateral elements are usually excluded from being or becoming concave.

Three-dimensional
Basic three-dimensional cell shapes
The basic 3-dimensional element are the tetrahedron, quadrilateral pyramid, triangular
prism, and hexahedron. They all have triangular and quadrilateral faces. Extruded 2-
dimensional models may be represented entirely by the prisms and hexahedra as extruded
triangles and quadrilaterals. In general, quadrilateral faces in 3-dimensions may not be
perfectly planar. A nonplanar quadrilateral face can be considered a thin tetrahedral volume
that is shared by two neighboring elements.

Tetrahedron
A tetrahedron has 4 vertices, 6 edges, and is bounded by 4 triangular faces. In most cases a
tetrahedral volume mesh can be generated automatically.

Pyramid
A quadrilaterally-based pyramid has 5 vertices, 8 edges, bounded by 4 triangular and 1
quadrilateral face. These are effectively used as transition elements between square and
triangular faced elements and other in hybrid meshes and grids.
Triangular prism
A triangular prism has 6 vertices, 9 edges, bounded by 2 triangular and 3 quadrilateral faces.
The advantage with this type of layer is that it resolves boundary layer efficiently.
Hexahedron
A hexahedron, a topological cube, has 8 vertices, 12 edges, bounded by 6 quadrilateral
faces. It is also called a hex or a brick. For the same cell amount, the accuracy of solutions in
hexahedral meshes is the highest. The pyramid and triangular prism zones can be
considered computationally as degenerate hexahedrons, where some edges have been
reduced to zero. Other degenerate forms of a hexahedron may also be represented.
Classification of grids

Structured grids
Structured grids are identified by regular connectivity. The possible element choices are
quadrilateral in 2D and hexahedral in 3D. This model is highly space efficient, since the
neighbourhood relationships are defined by storage arrangement. Some other advantages
of structured grid over unstructured are better convergence and higher resolution

Unstructured grids
An unstructured grid is identified by irregular connectivity. It cannot easily be expressed as a
two-dimensional or three-dimensional array 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 neighbourhood
relationships. These grids typically employ triangles in 2D and tetrahedral 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.
Mesh quality
A mesh is considered to have higher quality if a more accurate solution is calculated more
quickly. Accuracy and speed are in tension. Decreasing the mesh size always increases the
accuracy but also increases computational cost.
Accuracy depends on both discretization error and solution error. For discretization error, a
given mesh is a discrete approximation of the space, and so can only provide an
approximate solution, even when equations are solved exactly. (In computer graphics ray
tracing, the number of rays fired is another source of discretization error.) For solution
error, for PDEs many iterations over the entire mesh are required. The calculation is
terminated early, before the equations are solved exactly. The choice of mesh element type
affects both discretization and solution error.
Accuracy depends on both the total number of elements, and the shape of individual
elements. The speed of each iteration grows (linearly) with the number of elements, and the
number of iterations needed depends on the local solution value and gradient compared to
the shape and size of local elements.
Solution precision
A coarse mesh may provide an accurate solution if the solution is a constant, so the
precision depends on the particular problem instance. One can selectively refine the mesh
in areas where the solution gradients are high, thus increasing fidelity there. Accuracy,
including interpolated values within an element, depends on the element type and shape.
Rate of convergence
Each iteration reduces the error between the calculated and true solution. A faster rate of
convergence means smaller error with fewer iterations. A mesh of inferior quality may leave
out important features such as the boundary layer for fluid flow. The discretization error will
be large and the rate of convergence will be impaired; the solution may not converge at all.
Grid independence
A solution is considered grid-independent if the discretization and solution error are small
enough given sufficient iterations. This is essential to know for comparative results. A mesh
convergence study consists of refining elements and comparing the refined solutions to the
coarse solutions. If further refinement (or other changes) does not significantly change the
solution, the mesh is an "Independent Grid."
Deciding the type of mesh
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. 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. For triangular cells, skewness should not exceed 0.85 and for
quadrilateral cells, skewness should not exceed 0.9.

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 Adaption
Mesh adaptation, often referred to as Adaptive Mesh Refinement (AMR), refers to the
modification of an existing mesh so as to accurately capture flow features. Generally, the
goal of these modifications is to improve resolution of flow features without excessive
increase in computational effort.
Mesh adaptation strategies can usually be classified as one of three general types: r-
refinement, h-refinement, or p-refinement.
r-Refinement
r-refinement is the modification of mesh resolution without changing the number of nodes
or cells present in a mesh or the connectivity of a mesh. The increase in resolution is made
by moving the grid points into regions of activity, which results in a greater clustering of
points in those regions.
h-Refinement
h-refinement is the modification of mesh resolution by changing the mesh connectivity.
Depending upon the technique used, this may not result in a change in the overall number
of grid cells or grid points. The simplest strategy for this type of refinement subdivides cells
The advantage of such a procedure is that the overall mesh topology remains the same
(with the child cells taking the place of the parent cell in the connectivity arrangement). The
subdivision process is similar for a triangular parent cell, as shown below. It is easy to see
that the subdivision process increases both the number of points and the number of cells.

p-Refinement
A very popular tool in Finite Element Modelling (FEM) rather than in Finite Volume
Modelling (FVM), it achieves increased resolution by increasing the order of accuracy of the
polynomial in each element (or cell).
Discretization methods
Finite volume method
The finite volume method (FVM) is a common approach used in CFD codes, as it has an
advantage in memory usage and solution speed, especially for large problems, high
Reynolds number turbulent flows, and source term dominated flows (like combustion).
In the finite volume method, the governing partial differential equations (typically the
Navier-Stokes equations, the mass and energy conservation equations, and the turbulence
equations) are recast in a conservative form, and then solved over discrete control volumes.
This discretization guarantees the conservation of fluxes through a particular control
volume. The finite volume equation yields governing equations in the form,

where Q is the vector of conserved variables, F is the vector of fluxes V is the volume of the
control volume element, and A is the surface area of the control volume element.

Finite element method

The finite element method (FEM) is used in structural analysis of solids, but is also applicable
to fluids. However, the FEM formulation requires special care to ensure a conservative
solution. The FEM formulation has been adapted for use with fluid dynamics governing
equations.[citation needed] Although FEM must be carefully formulated to be conservative,
it is much more stable than the finite volume approach. However, FEM can require more
memory and has slower solution times than the FVM.
In this method, a weighted residual equation is formed:

Finite difference method

The finite difference method (FDM) has historical importance and is simple to program. It is
currently only used in few specialized codes, which handle complex geometry with high
accuracy and efficiency by using embedded boundaries or overlapping grids (with the
solution interpolated across each grid)
FDM VS FVM VS FEM
A finite difference method discretization is based upon the differential form of the PDE to be
solved. FDM is a differential scheme, that is approximation of a taylor series expansion. The
computational domain is usually divided into hexahedral cells and the solution will be
obtained at each nodal point. You try to write the first order, second order differential in terms
of difference between two point values. This may bring in some approximation error. Differential
schemes are faster and error can be minimized if you have regular/simple grids

Finite Volume method is an integral scheme. Since you are integrating across the area, chance
of error is minimized. Finite volume method is a method for representing and evaluating
partial differential equations in the form of algebraic equations. Useful for problems with
body-fitted coordinate systems. Integral schemes are slightly slower but score over FDM when
the mesh is irregular. Most of the commercial software like FLUENT (FVM) are either FVM or
FEM (Deform)

Finite element method is a numerical technique for finding approximate solutions to

boundary value problems for partial differential equations. The computational domain is
divided up into smaller domains (finite elements) and the solution in each element is
constructed from the basic functions. The actual equations that are solved are typically
obtained by restating the conservation equation in weak form

The Conservative Vs Non Conservative form of governing equations

The conservative and non-conservative forms of the Naiver-Stokes equations are
mathematically the same. The difference comes when discretizing the equations. Notice that
both the non-conservative and conservative forms come from the conservation equations
(i.e. conservation of mass and conservation of momentum). So, you can have both a
conservative form of the conservation equations, and a non-conservative form of the
conservation equations.
Now, more to the point, when do you want to use each scheme? If your solution is
expected to be smooth, then non-conservative may work. For fluids, this is shock-free
flows. If you have shocks, or chemical reactions, or any other sharp interfaces, then you
want to use the conservative form.