# Finite element computation, approximation functions

## Introduction

- order of the approximation functions: first order / second order
- type of the approximation functions: edge/nodal

## Order of elements and approximation functions

- different types of finite elements mesh: 1
^{st}order mesh or 2^{nd}order mesh - different types of approximation functions: linear functions (first order) or quadratic functions (second order)

Mesh | Position of nodes | Approximation function |
---|---|---|

1st order | Vertexes | Linear (polynomial of first order) |

2nd order | Vertexes + middle of edges | Quadratic (polynomial of second order) |

^{st}order mesh, the software automatically imposes a 2

^{nd}order mesh (necessary for a computation with approximation functions of the second order).

## Nodal or edge approximation function

Flux provides to the user:

- the edge approximation functions (computation on the edges of finite element and storage of information in the middle node of edges)
- the nodal approximation functions (computation on the nodes of the finite element)

The edge functions are provided only for the electric vector potential **T**. The
approximation functions for the scalar magnetic potential **Φ** are of the nodal
type.

Therefore, the use of the edge functions is applied to the regions of the solid conductor type (Transient Magnetic and Steady state AC Magnetic applications).

## Caution

- solid conductors
- presence of induced currents tangent to the device edges. These edges are the edges of the inside corners of the device.

Example: computation of the eddy currents in a solid conductor of the thin plate type in the presence of cracks.

## Modifying the default choices

The user does not have to modify the order or the type of the approximation functions. The default choices are adapted to the most situations. However, these choices can be brought to a change under certain particular conditions.

- automatic mode: default choice
- manual mode: user’s choice

## Automatic mode

- if the mesh is of the 1
^{st}order, the nodal approximation functions are of the first order - if the mesh is of the 2
^{nd}order, the nodal approximation functions are of the second order

If there are regions of the solid conductor type (formulation in electric vector
potential), the approximation functions are the edge functions that require a
2^{nd} order mesh. The generation of 2^{nd} order mesh is then
automatically carried out.

- Magnetic scalar potential: nodal approximation functions (2
^{nd}order) - Electric vector potential: edge approximation functions

## In manual mode

In manual mode, the user can modify the order (1^{st} order / 2^{nd}
order) and the type of approximation functions (nodal type/edge type for the electric
vector potential). These modifications are carried out in the dialog box of the
application definition as presented in the figure below.

## Advice

For a huge project, if the memory is not enough during solving process, it is possible to decrease the memory by choosing the vector variables approximation order of first order. It allows earning solving time and memory, despite a light results accuracy degradation.