Mathematical approach could improve electric motor efficiency
22 December 2015
According to an Austrian group, optimising the geometry of individual parts of complex machines using mathematical functions can improve the performance and efficiency of the entire machine.
To achieve this, the automotive and aeronautic industries often rely on 'shape optimisation', an approach that uses mathematical modelling to create a framework for making devices run as smoothly and efficiently as possible.
"A smoother rotation of the rotor can increase the energy efficiency of the motor, and at the same time reduce unwanted side effects like noise and vibrations," says Ulrich Langer, a professor at the Institute of Computational Mathematics, Johannes Kepler University in Linz, Austria.
Langer and colleagues have puplished a paper describing the use of shape optimisation techniques to enhance the performance of an electric motor. "By means of shape optimisation methods, optimal motor geometries, which could not be imagined beforehand, can now be determined," claims Langer.
Shape optimisation problems are typically solved by minimising the cost function, a mathematical formula that predicts the losses (or 'cost') corresponding to a process; the end goal is the creation of an optimal shape, one that minimises the cost function while meeting certain constraints.
Langer and his colleagues have applied optimisation techniques to a permanent magnet (IPM) brushless electric motor. Because not all parts of the rotor's geometry are able to be altered, the authors identified a modifiable design sub-region in the rotor's iron core on which to apply shape optimisation, their objective being to improve the workings of the rotor to achieve a smoother, more desirable rotation pattern.
"Differentiating with respect to the shape is more complicated than differentiating a function," says Langer's co-worker, Antoine Laurain. "In fact, there are many ways to define shape perturbations and differentiation with respect to shapes. The so-called 'shape derivative' is one incarnation of these possibilities. It allows us to explore a wide range of possible geometries for the optimisation."
Unlike the topological derivative, which generates a shape with uneven contours, the shape derivative employs a smooth alteration of the boundary. Implementing the obtained shape derivative in a numerical algorithm provides a shape that allowed Langer's team to improve the rotation pattern.
Langer's team included Peter Gangl, Antoine Laurain, Houcine Meftahi, and Kevin Sturm. Their paper is published in SIAM Journal on Scientific Computing.