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Multi-material topology optimization of electric machines under maximum temperature and stress constraints

Peter Gangl, Nepomuk Krenn, Herbert De Gersem

TL;DR

The paper tackles reducing losses and improving torque in electric machines by integrating electromagnetic, thermal, and mechanical considerations within a multi-material topology optimization framework. It develops a multi-material level-set method driven by topological derivatives for both purely electromagnetic and magnetoquasistatic problems, including a temperature constraint via a stationary heat equation and a Von Mises stress constraint. Key contributions include derivation of topological derivatives for both EM-thermal problems, a volume-constrained level-set update strategy, and efficient evaluation via offline precomputation and symmetry exploitation. Numerical results on a PMSM demonstrate torque gains while meeting temperature and stress limits, highlighting the method's potential for producing high-density, thermally safe machine designs. The work advances integrated free-form design of electric machines and sets the stage for extensions to two-way coupling and broader material families to further enhance performance and reliability.

Abstract

The use of topology optimization methods for the design of electric machines has become increasingly popular over the past years. Due to a desired increase in power density and a recent trend to high speed machines, thermal aspects play a more and more important role. In this work, we perform multi-material topology optimization of an electric machine, where the cost function depends on both electromagnetic fields and the temperature distribution generated by electromagnetic losses. We provide the topological derivative for this coupled multi-physics problem consisting of the magnetoquasistatic approximation to Maxwell's equations and the stationary heat equation. We use it within a multi-material level set algorithm in order to maximize the machine's average torque for a fixed volume of permanent-magnet material, while keeping the temperature below a prescribed value. Finally, in order to ensure mechanical stability, we additionally enforce a bound on mechanical stresses. Numerical results for the optimization of a permanent magnet synchronous machine are presented, showing a significantly improved performance compared to the reference design while meeting temperature and stress constraints.

Multi-material topology optimization of electric machines under maximum temperature and stress constraints

TL;DR

The paper tackles reducing losses and improving torque in electric machines by integrating electromagnetic, thermal, and mechanical considerations within a multi-material topology optimization framework. It develops a multi-material level-set method driven by topological derivatives for both purely electromagnetic and magnetoquasistatic problems, including a temperature constraint via a stationary heat equation and a Von Mises stress constraint. Key contributions include derivation of topological derivatives for both EM-thermal problems, a volume-constrained level-set update strategy, and efficient evaluation via offline precomputation and symmetry exploitation. Numerical results on a PMSM demonstrate torque gains while meeting temperature and stress limits, highlighting the method's potential for producing high-density, thermally safe machine designs. The work advances integrated free-form design of electric machines and sets the stage for extensions to two-way coupling and broader material families to further enhance performance and reliability.

Abstract

The use of topology optimization methods for the design of electric machines has become increasingly popular over the past years. Due to a desired increase in power density and a recent trend to high speed machines, thermal aspects play a more and more important role. In this work, we perform multi-material topology optimization of an electric machine, where the cost function depends on both electromagnetic fields and the temperature distribution generated by electromagnetic losses. We provide the topological derivative for this coupled multi-physics problem consisting of the magnetoquasistatic approximation to Maxwell's equations and the stationary heat equation. We use it within a multi-material level set algorithm in order to maximize the machine's average torque for a fixed volume of permanent-magnet material, while keeping the temperature below a prescribed value. Finally, in order to ensure mechanical stability, we additionally enforce a bound on mechanical stresses. Numerical results for the optimization of a permanent magnet synchronous machine are presented, showing a significantly improved performance compared to the reference design while meeting temperature and stress constraints.

Paper Structure

This paper contains 25 sections, 1 theorem, 97 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model problem
  3. Multi-material level set based topology optimization by the topological derivative
  4. Topological perturbations
  5. Multi-material level set description
  6. Level set algorithm
  7. Volume constraint
  8. Topological derivative for torque maximization
  9. Topological derivative for torque maximization
  10. Evaluation of the topological derivative
  11. Topological derivative for coupled electromagnetic-thermal problem
  12. Magnetoquasistatic
  13. Topological derivative for magnetoquasistatics
  14. Modeling eddy-current losses
  15. Temperature constraint
  16. ...and 10 more sections

Key Result

Lemma 1

Let $\psi_{\Omega}$ be a vector-valued level set function describing the configuration $\Omega$. If for a positive constant $c>0$ and all $z\in\Omega$, then $\Omega$ is locally optimal with respect to topological perturbations.

Figures (5)

  • Figure 1: 2d cross section of the reference machine (left), one pole of the reference machine $D_\mathrm{all}$ with radial boundaries $\Gamma_1, \Gamma_2$ consisting of rotor, stator and air gap (right). Stator consists of iron $D_S$ (red) with outer boundary $\Gamma_S$ and copper coils $D_{A^+},D_{B^-},D_{C^+}$ (yellow). The air gap $D_\mathrm{AG}$ (blue) is split by circle $\Gamma$ used for harmonic mortaring. The rotor consists of shaft $D_\mathrm{SH}$ (blue) with boundary $\Gamma_\mathrm{SH}$, design domain $D$ (greyed out) and iron ring $D_\mathrm{RI}$ (red) with outer boundary $\Gamma_R$. Design domain consists of iron $\Omega_f$ (red), air $\Omega_a$ (blue) and permanent magnets $\Omega_{m_1},\Omega_{m_2}$ (light blue, light green).
  • Figure 2: Initial design $\Omega_\mathrm{ini}$ (left), average loss density (mid), Von Mises stress distribution (right). Average torque $\overline{\mathcal{T}_\mathrm{ed}}(\Omega_\mathrm{ini})=653\mathrm{\,Nm}$, maximal temperature $\vartheta_\mathrm{max}(\Omega_\mathrm{ini})=72^\circ\mathrm{C}$, maximal Von Mises stress $s_\mathrm{max}(\Omega_\mathrm{ini})=340\,\mathrm{MPa}$.
  • Figure 3: Final design $\Omega_\mathcal{T}$ of maximizing average torque subject to magnetostatics \ref{['eq:CostTorque']} (left), average loss density (mid), Von Mises stress distribution (right). Average torque $\overline{\mathcal{T}_\mathrm{ed}}(\Omega_\mathcal{T})=858\,\mathrm{\,Nm}$, maximal temperature $\vartheta_\mathrm{max}(\Omega_\mathcal{T})=198^\circ\mathrm{C}$, maximal Von Mises stress $s_\mathrm{max}(\Omega_\mathcal{T})=772\,\mathrm{MPa}$.
  • Figure 4: Final design $\Omega_{\mathcal{T}_\mathrm{ed},\mathcal{C}_t}$ of maximizing average torque subject to magnetoquasistatics \ref{['eq:CosttorqueED']} with temperature constraint \ref{['eq:Ctemp']} (left), average loss density (mid), Von Mises stress distribution (right). Average torque $\overline{\mathcal{T}_\mathrm{ed}}(\Omega_{\mathcal{T}_\mathrm{ed},\mathcal{C}_t})=841\mathrm{\,Nm}$, maximal temperature $\vartheta_\mathrm{max}(\Omega_{\mathcal{T}_\mathrm{ed},\mathcal{C}_t})=92^\circ\mathrm{C}$, maximal Von Mises stress $s_\mathrm{max}(\Omega_{\mathcal{T}_\mathrm{ed},\mathcal{C}_t})=792\,\mathrm{MPa}$.
  • Figure 5: Final design $\Omega_{\mathcal{T}_\mathrm{ed},\mathcal{C}_t,\mathcal{C}_\mathrm{VM}}$ of maximizing average torque subject to magnetoquasistatics \ref{['eq:CosttorqueED']} with temperature constraint \ref{['eq:Ctemp']} and stress constraint \ref{['eq:consVM']} (left), average loss density (mid), Von Mises stress distribution (right). Average torque $\overline{\mathcal{T}_\mathrm{ed}}(\Omega_{\mathcal{T}_\mathrm{ed},\mathcal{C}_t,\mathcal{C}_\mathrm{VM}})=825\mathrm{\,Nm}$, maximal temperature $\vartheta_\mathrm{max}(\Omega_{\mathcal{T}_\mathrm{ed},\mathcal{C}_t,\mathcal{C}_\mathrm{VM}})=91^\circ\mathrm{C}$, maximal Von Mises stress $s_\mathrm{max}(\Omega_{\mathcal{T}_\mathrm{ed},\mathcal{C}_t,\mathcal{C}_\mathrm{VM}})=464\,\mathrm{MPa}$.

Theorems & Definitions (26)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Example 1
  • Lemma 1
  • proof
  • Remark 3
  • Remark 4
  • ...and 16 more