# Introduction to meshing process

## Definition

The meshing process consists of dividing the study domain into mesh elements.

The summit of a mesh element is called a node.

Mesh or elements are called:

- volume elements, for a volume domain
- surface elements, for a surface domain
- line elements, for a line domain

## Meshing and finite elements

The meshing process is a key step in the finite element method. The finite element method calculates and gives an approximation of the state variables on each node of the mesh, (scalar or vector potentials, temperature,…) and of the fields which are derived from (magnetic field and induction, electric field, flux thermal density,…).

## Mesh and results

The quality of the approximate solution depends on the mesh. Thus, the quality of the solution depends on:

- the number and the dimensions of the finite elements,
- the interpolation functions in each element, which can be 1
^{st}, 2^{nd}order polynomial functions, - the continuity conditions imposed on the sub-domain boundaries.

## Mesh elements form

The different forms of the mesh elements are presented in the table below.

Face mesh | Volume mesh | ||
---|---|---|---|

Triangle | Tetrahedron | ||

Pentahedron | |||

Rectangle | Hexahedron | ||

Pyramid |

## Structure of a mesh element

In terms of the geometry, a volume element is characterized by its vertices, edges and faces.

## Elements of 1^{st} and 2^{nd} order

Different types of finite elements are available to the user: these are called
1^{st} order elements or 2^{nd} order elements.

Specific information about these elements is presented in the following table.

Type of element | Position of nodes | Interpolation function |
---|---|---|

1^{st} order |
Vertices | Linear (1^{st} order polynomial) |

2^{nd }order |
Vertices + middle of edges | Quadratic (2^{nd} order polynomial) |

## Field calculation: 1^{st} and 2^{nd} order approach

Using 1 st order elements: the potentials are approximated linearly and the fields derived from the potentials are constant.

Using 2 nd order elements: the potentials are approximated quadratically and the fields are approximated linearly.

Element | Potentials | Field |
---|---|---|

1^{st} order |
Linear approximation | Constant |

2^{nd} order |
Quadratic approximation | Linear approximation |