# 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 1st , 2nd 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 1st and 2nd order

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

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

Type of element Position of nodes Interpolation function
1st order Vertices Linear (1st order polynomial)
2nd order Vertices + middle of edges Quadratic (2nd order polynomial)

## Field calculation: 1st and 2nd 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
1st order Linear approximation Constant
2nd order Quadratic approximation Linear approximation
Note: more memory is needed to solve a problem meshed with second order elements but the quality of the results is better.