# Improving electric motor efficiency via shape optimization

December 22, 2015In our competitive global society, successful and economical design of automotive and industrial structures is crucial. Optimizing the geometry of individual pieces of complex machines improves performance and efficiency of the entire device.

To achieve this, the automotive and aeronautic industries often rely on shape optimization, an approach that uses modeling to create a framework for making devices as smooth and efficient 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 mathematician Ulrich Langer. Langer, along with Peter Gangl, Antoine Laurain, Houcine Meftahi, and Kevin Sturm, co-authored a paper publishing tomorrow in the

*SIAM Journal on Scientific Computing*that utilizes shape optimization techniques to enhance the performance of an electric motor. "By means of shape optimization methods, optimal motor geometries which could not be imagined beforehand can now be determined," says Langer.

Shape optimization problems are typically solved by minimizing the cost function, a mathematical formula that predicts the losses (or 'cost') corresponding with a process; the end goal is the creation of an optimal shape, one that minimizes the cost function while meeting certain constraints. Langer and his coauthors apply optimization techniques to an interior permanent magnet (IPM) brushless electric motor, the kind sometimes used in washing machines, computer cooling fans, and assembly tools. The motor's inner rotor contains an iron core and permanent magnets. Because not all parts of the rotor's geometry are able to be altered, the authors identify a modifiable design subregion in the rotor's iron core on which to apply shape optimization. Their objective is to improve the workings of the rotor, thus resulting in a smoother, more desirable rotation pattern.

"Differentiating with respect to the shape is more complicated than differentiating a function," says 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 optimization." 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 allows the authors to improve the rotation pattern. The authors' optimization procedures stem from Lagrangian methods for approaching nonlinear problems, and demonstrate an efficient, exact means of calculating the shape derivative of the cost function. This simple and comprehensive method allows for the treatment of nonlinear partial differential equations (PDEs) and general cost functions, rather than only linear PDEs. Ultimately, their optimization procedures are able to achieve a 27 percent decrease in the cost functional of an IPM brushless electric motor, the particular example explored in the paper.

Now that they have effectively utilized a shape derivative in optimization, the authors hope to experiment further in the future. "We are currently working on the derivation of the so-called 'topological derivative' for the same nonlinear optimization problem," says Peter Gangl. "This quantity indicates regions where a local change of the material would lead to a decrease of the objective function." For now, their application of mathematics in the form of a shape-Lagrangian method adapted for nonlinear PDEs results in a shape that improves the electric rotor's rotation pattern and the motor's overall performance.

-end-

Source Article: Shape Optimization of an Electric Motor subject to Nonlinear Magnetostatics *SIAM Journal on Scientific Computing*(to be published online on December 22, 2015). For an advance copy of the paper, or to interview the authors, please contact kerner@siam.org. About the authors: Peter Gangl is a PhD Student in the computational mathematics doctoral program and Ulrich Langer is a professor at the Institute of Computational Mathematics at Johannes Kepler University in Linz, Austria. Antoine Laurain and Houcine Meftahi are researchers at the Institut für Mathematik at Technische Universität Berlin. Kevin Sturm is a researcher at the University of Duisburg-Essen, Essen.

Society for Industrial and Applied Mathematics

## Related Geometry Articles from Brightsurf:

Six years in 120 pages: Researchers shed light on Ricci flows

Researchers from University of Science and Technology of China (USTC) proved two core conjectures in geometric analysis: Hamilton-Tian conjecture and the Partial C0-conjecture.

Biophysics - geometry supersedes simulations

Ludwig-Maximilians-Universitaet (LMU) in Munich physicists have introduced a new method that allows biological pattern-forming systems to be systematically characterized with the aid of mathematical analysis.

Memory in a metal, enabled by quantum geometry

Berkeley researchers led by Professor Xiang Zhang in collaboration with a Stanford University team invented a new data storage method by making odd numbered layers slide relative to even-number layers in tungsten ditelluride, which is only 3nm thick.

"Inchworm" pattern of Indonesian earthquake rupture powered seismic "boom"

A sonic boom-like seismic phenomenon of supershear rupture occurred during the 2018 Palu earthquake in Indonesia.

Order from noise: How randomness and collective dynamics define a stem cell

Without stem cells, human life would not exist. Due to them, a lump of cells becomes an organ, and a fertilized egg develops into a baby.

Geometry of intricately fabricated glass makes light trap itself

Laser light traveling through ornately microfabricated glass has been shown to interact with itself to form self-sustaining wave patterns called solitons.

22,000 tiny tremblors illustrate 3D fault geometry and earthquake swarm evolution

By mapping the more than 22,000 tremblors, researchers composed a detailed, three-dimensional image of the complex fault structure below southern California's Cahuilla Valley.

NUI Galway mathematician publishes article in world's top mathematics journal

An Irish mathematician, Dr Martin Kerin, from the School of Mathematics, Statistics and Applied Mathematics at NUI Galway, has had a research article published in the Annals of Mathematics, widely regarded as the top journal for pure mathematics in the world.

Geometry guided construction of earliest known temple, built 6,000 years before Stonehenge

Researchers at Tel Aviv University and the Israel Antiquities Authority have now used architectural analysis to discover that geometry informed the layout of GĂ¶bekli Tepe's impressive round stone structures and enormous assembly of limestone pillars, which they say were initially planned as a single structure.

How to break new records in the 200 metres?

Usain Bolt's 200m record has not been beaten for ten years and Florence Griffith Joyner's for more than thirty years.

Read More: Geometry News and Geometry Current Events

Researchers from University of Science and Technology of China (USTC) proved two core conjectures in geometric analysis: Hamilton-Tian conjecture and the Partial C0-conjecture.

Biophysics - geometry supersedes simulations

Ludwig-Maximilians-Universitaet (LMU) in Munich physicists have introduced a new method that allows biological pattern-forming systems to be systematically characterized with the aid of mathematical analysis.

Memory in a metal, enabled by quantum geometry

Berkeley researchers led by Professor Xiang Zhang in collaboration with a Stanford University team invented a new data storage method by making odd numbered layers slide relative to even-number layers in tungsten ditelluride, which is only 3nm thick.

"Inchworm" pattern of Indonesian earthquake rupture powered seismic "boom"

A sonic boom-like seismic phenomenon of supershear rupture occurred during the 2018 Palu earthquake in Indonesia.

Order from noise: How randomness and collective dynamics define a stem cell

Without stem cells, human life would not exist. Due to them, a lump of cells becomes an organ, and a fertilized egg develops into a baby.

Geometry of intricately fabricated glass makes light trap itself

Laser light traveling through ornately microfabricated glass has been shown to interact with itself to form self-sustaining wave patterns called solitons.

22,000 tiny tremblors illustrate 3D fault geometry and earthquake swarm evolution

By mapping the more than 22,000 tremblors, researchers composed a detailed, three-dimensional image of the complex fault structure below southern California's Cahuilla Valley.

NUI Galway mathematician publishes article in world's top mathematics journal

An Irish mathematician, Dr Martin Kerin, from the School of Mathematics, Statistics and Applied Mathematics at NUI Galway, has had a research article published in the Annals of Mathematics, widely regarded as the top journal for pure mathematics in the world.

Geometry guided construction of earliest known temple, built 6,000 years before Stonehenge

Researchers at Tel Aviv University and the Israel Antiquities Authority have now used architectural analysis to discover that geometry informed the layout of GĂ¶bekli Tepe's impressive round stone structures and enormous assembly of limestone pillars, which they say were initially planned as a single structure.

How to break new records in the 200 metres?

Usain Bolt's 200m record has not been beaten for ten years and Florence Griffith Joyner's for more than thirty years.

Read More: Geometry News and Geometry Current Events

Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.