# Multi-material topology optimization of a rotating electrical machine with a density-based method

Théodore Cherrière, Luc Laurent, S Hlioui, François Louf, Hamid Ben Ahmed, Mohamed Gabsi

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Théodore Cherrière, Luc Laurent, S Hlioui, François Louf, Hamid Ben Ahmed, et al. Multi-material topology optimization of a rotating electrical machine with a density-based method. 8th GAMM Juniors' Summer School (SAMM 2021), Jul 2021, Gratz, Austria. 21 (S1), 2021, 10.1002/pamm.202100259. hal-03541810

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Submitted on 27 Jun 2023

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Distributed under a Creative Commons Attribution 4.0 International License
# MULTI-MATERIAL TOPOLOGY OPTIMIZATION OF A ROTATING ELECTRICAL MACHINE WITH A DENSITY-BASED METHOD

T. Cherrière¹*, L. Laurent², S. Hlioui³, F. Louf⁴, H. Ben Ahmed⁵, M. Gabsi⁶

¹Université Paris-Saclay, ENS Paris-Saclay, SATIE, Gif-sur-Yvette, France ; ²LMSSC, Cnam, Paris, France ; ³SATIE, CNRS, ENS Paris-Saclay, Cnam, Paris, France ;

⁴Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMT, Gif-sur-Yvette, France ; ⁵SATIE, ENS Rennes, CNRS, 35170 Bruz, France ;

⁶Université Paris-Saclay, ENS Paris-Saclay, SATIE, Gif-sur-Yvette, France

*theodore.cherriere@universite-paris-saclay.fr

## Context and motivations

**Topology optimization** has become a popular tool especially in mechanical engineering. Many interesting methods have been developed [1].

**Density methods** like SIMP have been introduced in electrical engineering by [2], and are now widely used to design electrical actuators. **Multi-materials** problems involving source distribution have been addressed [3,4]. However, most of the work focuses only on single parts of the machine (rotor, stator teeth), and a few papers such as [5] deal with the coil distribution within the stator.

This work presents a **density-based method** which could be used to separately optimize the rotor and the stator of a rotating machine, with the future objective to optimize both from scratch.

## Magnetostatics equation

### 2D vector potential local formulation

$$-\nabla \cdot [\nu(\mathbf{a}_z) \cdot \nabla \mathbf{a}_z] = j_z + \nabla \cdot \left( \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \cdot \mathbf{M} \right) \quad (1)$$

$\mathbf{a}_z$ is the unknown z-component of vector potential, $\nu$ the magnetic re-luctivity which depends on $a_z$ within ferromagnetic materials, $j_z$ the z-component of current density and $\mathbf{M}$ the magnetization of permanent magnets. $\nu = H/B$ is defined by material law (B-H curves).

**Anti-periodic** boundary conditions are used to model a single pole of the machine.

## Principles of density methods

Density methods rely on a bounded spatial field $\rho : x \mapsto \rho(x)$ which supports **differentiable** interpolations between material properties and allow to define a differentiable optimization problem.

<figure><img src="image_2.png" /></figure>

Intermediate materials should be eliminated during the optimization process, by the use of penalization (SIMP, RAMP [1]), thresholding, constraints, etc.

## Problem formulation

The aim of the optimization is to find the discrete density vector $\rho_{opt}$ which maximizes the average torque

### Optimization problem

$$\text{find } \rho_{opt} = \arg \max \langle C(\mathbf{a}(\rho)) \rangle \\ \text{subject to } 0 \le \rho_i \le 1 \\ \forall i \in [0, n-1]$$

With $n$ the total number of densities. The torque $C$ is computed with Arkkio's method [6]. It depends **implicitly** on $\rho$ through the discretization of equation (1) by finite elements method :

### Finite element system

$$K(\mathbf{a}, \rho) \cdot \mathbf{a} = \mathbf{b}(\rho) \quad (2)$$

## Adjoint variable method

The **adjoint variable method** was used [3] to compute efficiently the derivatives of the objective function with respect to the density :

### Sensitivity expression

$$\frac{\partial C}{\partial \rho_i} = \mathbf{p}^T \left( \frac{\partial \mathbf{b}}{\partial \rho_i} - \frac{\partial K}{\partial \rho_i} \mathbf{a} \right)$$

$\mathbf{p}$ is the so-called **adjoint state**, solution of the following linear system

### Adjoint system

$$(K')^T \mathbf{p} = \frac{\partial C}{\partial \mathbf{a}} \\ \text{with } K'_{ij} = K_{ij} + \sum_k \frac{\partial K_{ik}}{\partial a_j} a_k$$

$K'$ is the **tangent operator** of (2).

## Optimization process

<figure><img src="image_3.png" /></figure>

## Optimization of SRM rotor

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>ABB® SRM</figcaption></figure>

**Filtering** inspired by ESO procedure [7] was applied : materials which carried few flux was replaced by air. Then, **flux barriers** [8] appear clearly and correspond to industrial design like ABB® SRM :

## Optimization of PMSM stator

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

4-materials topology optimization [4] was performed on a one-phase PMSM stator with **uniform initialization**. Two densities per element were used. The convergence is slow near the airgap, so each gradient pair of elementary terms was **normalized**. As it is multiplied by a symmetric definite matrix, it remains a descent direction and leads to an **asymmetrical** design.

## Conclusion and outlook

Encouraging results with **flux barriers** were obtained for the SRM rotor optimization. The multimaterial PMSM stator optimization of the stator leads to intriguing **asymmetrical design**, which is maybe a local optimum. Further research is needed to compare this topology with established ones. It is also necessary to improve the robustness of the method especially for high **current density**, before optimizing both the rotor and the stator. **Multiphysics** considerations will be taken into account in future work.

## Acknowledgements

Many thanks to Pr. P.Duysinx and Pr. C.Geuzaine (Liège univ.) for their insightful advice and to the Moulon Mesocentre team.

## References

[1] Ole Sigmund and Kurt Maute. Topology optimization approaches: A comparative review. *Structural and Multidisciplinary Optimization*, 48(6):1031–1055, 2013.

[2] D.N. Dyck and D.A. Lowther. Automated design of magnetic devices by optimizing material distribution. *IEEE Transactions on Magnetics*, 32(3):1188–1192, 1996.

[3] J.S. Choi and J. Yoo. Structural optimization of ferromagnetic materials based on the magnetic reluctivity for magnetic field problems. *Comp. Meth. in Appl. Mech. and Eng.*, 197:4193–4206, September 2008.

[4] Y. Jung and S. Min. Material Interpolation in Multi-Material Topology Optimization for Magnetic Device Design. *IEEE Transactions on Magnetics*, 55(11):2019–2022, 2019.

[5] T. Labbé and B. Dehez. Convexity-Oriented Mapping Method for the Topology Optimization of Electromagnetic Devices Composed of Iron and Coils. *IEEE Transactions on Magnetics*, 46(5):1177–1185, 2010.

[6] A. Arkkios. Time-stepping finite element analysis of induction motors. In *Proceedings of ICEM*. IEEE, 1988.

[7] Y.M. Xie and G.P. Steven. A simple evolutionary procedure for structural optimization. *Computers & Structures*, 49(5):885–896, 1993.

[8] M. Martinovic, S. Stipetic, D. Zarko, and M. Klanac. Comparison of synchronous reluctance machines using topology optimization. In *SPEEDAM*, pages 472–477, 2020.