In the shape optimization module, available in Beta mode, it is now possible to set
Physical constraints using response entities in an optimization problem. The user
can also minimize or maximize the surface of a set of faces while constraining other
quantities. These added features are leveraging the shape optimization to a next
level, where most costumer’s problem can be optimized.

Example of application

Let us consider the following electrical machine represented by only one quarter of
the full device in Figure 1.

The goal of this study is to design the inner-rotor parts, in order to:

Maximize the average torque value,

Constrain the torque ripple under a given threshold,

Reduce the rotor weight of 20%,

Keep the symmetry of the rotor’s design.

To run the shape optimization, a response for the Torque must be created to be
used for maximizing its average value and to constraint the torque ripple. Creating
a Torque response requires a path in the fixed part of the air gap region. A rule of
thumb for the path discretization is to set one point per degree to ensure an
accurate computation, see Figure 2.

To constrain the torque ripple, the Constraint given by a Compose function
must be used. The use of a Compose function allows custom and complex computations
based on Physical responses to be done. For this study, the actual constraint
consists in imposing C < 30% with: $C=100\frac{TorqueRipple}{\left|MeanTorque\right|}$ so maximizing the torque’s mean value brings an
additional factor to this constraint.

The specifications mention the user wants the final design to be symmetrical. This
is done using a symmetry constraint, where the origin point, and the symmetry axis
must be given as in Figure 3.
Here, the origin point is set to (0.0, 0.0) and the symmetry axis to (0.5, 0.5).

To reduce the weight of the rotor, a volume constraint must be used with an upper
bound lower to the current volume. Even if at the starting point the initial design
is not between the bounds, the optimization algorithm will deteriorate the objective
function to find a design that fit the bounds, then the algorithm will start to
optimize the design to increase or decrease the objective function. In this study,
the upper bound is set the -20% to force a volume reduction of the iron in the rotor
as shown in the Figure 4.

Finally, to set the optimization problem up, the Objective function is set as
the Average of the torque response, as shown in Figure 5 - (a). The shape
optimization is run on the flux barriers, the blue faces of the rotor, and the air
gap boundary, as illustrated in Figure 5 – (b) by the yellow lines.

The optimization process is then moving the nodes of the selected lines, and
ultimately gives a design complying with the constraints while maximizing the
torque. The optimized design is presented in the Figure 6.

The shape optimization was able to find a design that complies with every
specification, keeping the symmetry, reducing the volume of iron, and allowed a
major improvement of the torque mean value of + 9.4%, as show in Table 1.

Table 1. Comparison between the initial design and the final design
performances.