# Periodicity: about

## Periodicity attached to the study domain

If the model device is characterized by possible periodicity, this one can be attached to the study domain.

The boundary conditions on the corresponding boundaries are imposed in the Physical module.

The boundary conditions of cyclic or anti-cyclic type on the borders of repetitive pattern will be correctly considered, only if the corresponding lines (2D domain) / faces (3D domain) have been meshed with a linked mesh generator (same mesh on lines / faces placed face to face).

## Planes of periodicity

The periodicity can be linear or circular. The modeled part is the part delimited by two planes that are:

- two planes parallel with the main planes in the case of linear periodicity
- two planes in rotation (one with respect to the other) around one of the principal axes in the case of circular periodicity

## Linear periodicity

The linear periodicity is defined by the translation of one of the main planes (YOZ, ZOX or XOY):

- the first plane (P
_{1}) is defined by its position in X (Y or Z respectively): offset position (X_{1}) - the second plane (P
_{2}) is defined by a translation of plane P_{1}along X (Y or Z respectively): displacement along the X-axis (X_{2}- X_{1})

2D domain | 3D domain |
---|---|

## Circular periodicity

The circular periodicity is defined by the rotation of one of principal planes (YOZ, ZOX or XOY):

- the first plane (P
_{1}) is defined by its angular position with respect to the principal plane ZOX (XOY, YOZ respectively): offset angle with respect to the ZOX plane (θ_{1}) - the second plane (P
_{2}) is defined by a rotation of plane P_{1}around the Z (X, Y respectively) axis: included angle of the domain (θ_{2}- θ_{1})

Where the rotation angle around the axis is

and n is the number of repetitive patterns (repetitions)

2D domain | 3D domain |
---|---|

## Periodicity and infinite box

It is possible to combine infinite box and circular periodicity. In this case, the geometry of the infinite box (points and lines) automatically follows the periodicity attached to the study domain.

The rules to be respected are presented in the table below.

Domain | Rule |
---|---|

2D | the box infinite must be of disc type and periodicity must be of the circular type around the axis of this disc |

3D | the box infinite must be of cylinder type and periodicity must be of the circular type around the axis of this cylinder |

## Examples

The examples of the infinite box geometry for the study domain with and without periodicity attached to the domain are presented in the table below.

2D domain | |
---|---|

no periodicity | periodicity |

Complete infinite box of disc type | A portion of infinite box of disc type following the circular periodicity |

3D domain | |
---|---|

no periodicity | periodicity |

Complete infinite box of Z-axis cylinder type | A portion of infinite box of cylinder type following the circular periodicity around the Z-axis |

## Parameter setting

The length, the position and the angles of periodicity can be defined using algebraic expression. The algebraic expression can contain:

- constants
- geometric parameters (created beforehand)
- basic mathematical functions using operators : +, -, *, /, ( )
- usual mathematical functions admitted by Fortran.

The mathematical functions are described in section Functions.