# Reduction of the study domain: symmetries and periodicities

## The main ideas: decrease the study domain

In most cases, a preliminary analysis of the device highlights the presence of repetitive patterns (periodicities) or symmetry planes.

Under these conditions, it is possible to reduce the study domain as follows:

- representation of a fraction of the device
- assignment of appropriate boundary conditions on the model boundaries that reflect the periodicity property or symmetry conditions.

## Interest

The consequences of a reduction of the device model are as follows:

- a simplification of the geometrical description
- a reduction of the finite element problem size (and thus the file size).

The rationale for reducing the problem size is the reduction of the computation time. The computation time is roughly proportional to the square of the number of unknowns.

Example: If a problem comprises N unknowns, but after reduction of the model only N/2 unknowns, the global computation time will be reduced by a factor of 4.

## Reduction of study domain and boundary conditions

It is possible to simplify the device model if it has geometrical and physical periodicities and/or symmetries at the same time.

In other words, it is possible to simplify the device model, when specific conditions applied on the state variable (potential) allow the representation of a fraction of the device.

The boundary conditions are physical concepts.These concepts are briefly illustrated through a magnetic example (magneto-static, transient magnetic or magneto-harmonic application).

## Example: presentation

The modeled device is a magnetic levitation device. It consists of a group of coils, a magnetic flux concentrator and a plate.

Problem analysis:

This device can be described as a group of repetitive linear patterns: a succession of coils in opposition.

- from the geometrical point of view, the base pattern includes only one coil
- from the physical point of view, the base pattern includes two coils in opposition.

## Example: different models

The authorized subdivisions of the model depend on the various types of boundary conditions set on the model boundaries.

The various possible models are shown in the figures below. The boundary conditions set on the boundaries in these different configurations are explained in the following paragraphs.