# Preliminary ConceptsΒΆ

A mesh (sometimes also called a *grid*), denoted by \(\mathcal{M}(\Omega)\), provides a discrete
represenation of a geometric domain of interest, \(\Omega\), on which, the
underlying *mathematical model* is evaluated. The mathematical model
is typically defined by a system of governing *Partial Differential Equations
(PDEs)* and associated boundary and initial conditions. The solution
to the governing PDE predicts a physical process that occurs and evolves on
\(\Omega\) over time. For example, consider the flow around an aircraft,
turbulence modeling, blast wave propagation over complex terrains, or,
heat transfer in contacting objects, to name a few.
Evolving the mathematical model to predict such a physical process is typically
done numerically, which requires discretizing the governing PDE by a numerical
scheme, such as a Finite Difference (FD), Finite Volume (FV), or, the
Finite Element Method (FEM), chief among them.

Discretization of the governing PDE requires the domain to be approximated
with a mesh. For example, Fig. 3 (a) depicts a geometric
domain, \(\Omega\). The corresponding mesh, \(\mathcal{M}(\Omega)\),
is illustrated in Fig. 3 (b). The mesh approximates
the geometric domain, \(\Omega\), by a finite number of simple geometric
entities, such as *nodes* and *cells*, depicted in red in
Fig. 3 (b). These geometric entities comprising the mesh
define the discrete locations, in space and time, at which the unknown variables,
i.e., the *degrees of freedom* of the governing PDE, are evaluated, by the
numerical scheme being employed.

There are a variety of different Mesh Types one can choose from.
The type of mesh employed depends on the choice of the underlying
numerical discretization scheme. For example, a finite difference scheme
typically requires a Structured Mesh. However, the finite volume and
finite element methods may be implemented for both Structured Mesh and
Unstructured Mesh types. In contrast, *meshless* or *mesh-free* methods,
such as *Smoothed Particle Hydrodynamics (SPH)*, discretize the governing PDE
over a set of *particles* or *nodes*, using a Particle Mesh
representation.