# Free-shape optimization improvements

## Overview

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 図 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 図 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: 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 図 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 図 4.
Finally, to set the optimization problem up, the Objective function is set as the Average of the torque response, as shown in 図 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 図 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 図 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 表 1.

Initial design Final design Difference
Mean torque (N.m) 12.4 13.6 +9.4%
Torque ripple (N.m) 18.4 9.7 -47.1%
Volume (m3) 6.4e-5 5.1e-5 -19.7%