**By R. Belmans**

**Devices have a three-dimensional geometry.** Very often it is a complicated shape or the device contains moving parts. It is possible to reduce the geometrical dimensions to build a FEM model with sufficient accuracy. To obtain a model, symmetries in the device can be used or transformations on the geometry can be performed in order to describe the problem with a simpler discretisation.

**Fig. 9.2. Systematic to reduce the geometrical dimensions of numerical models.**

### Reduction of the geometrical dimensions

**Two-dimensional Cartesian model:**

**If the device owns the following properties:**

**1.** the dimensions of the geometry compared to those of its cross-section are long,

**2.** the shape of the cross-section remains the same along the length of the model, and

**3.** it can be assumed that all flux lines are present in the cross-section,a two-dimensional Cartesian model (Fig. 9.3) can approximate the geometry. Neglecting the end-region effects, typical examples for such models are cylindrical electrical machines or long inductors.

**Fig. 9.3. Geometrical reduction from 3D to 2D Cartesian.**

**Two-dimensional axis-symmetrical model:**

If the properties of the device’s geometry are true,

**1.** the geometry has a cylinder symmetry, and

**2.** the field is axis-symmetrical and is not periodic,an axis-symmetrical model (Fig. 9.4) can approximate the geometry of the device.

**Fig. 9.4. Geometrical reduction from 3D to 2D axis-symmetry.**

**Entire three-dimensional model:**

**If it is not possible to reduce** the geometrical dimension of the model, the field problem must be solved using a three-dimensional model. Fig. 9.5 shows, as an example, the geometry of an electrostatic micro motor. An axis-symmetrical model can not approximate this geometry because the field is periodic due to the voltage excitation at the stator electrodes. With known periodicity and employing appropriate boundary conditions the 3D model can be reduced.