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Tracing Pareto-optimal points for multi-objective shape optimization applied to electric machines

Alessio Cesarano, Peter Gangl

TL;DR

The paper tackles multi-objective PDE-constrained shape optimization for electric machines by tracing the Pareto front between the torque-related objective $\mathcal{J}_1(\Omega)$ and rotor volume $\mathcal{J}_2(\Omega)$ using a homotopy continuation $\mathcal{H}(\Omega,t)$ of first- and second-order shape derivatives. A predictor–corrector scheme based on shape derivatives and a shape-Newton update guides the design along the front, with interior deformations filtered and boundary normal deformations enforced. The approach is demonstrated on a synchronous reluctance rotor with free-form geometry, yielding a set of Pareto-optimal designs and showing efficient exploration of non-convex front regions, aided by a multi-resolution refinement strategy. This method offers practical benefits for industrial rotor design by enabling controlled spacing of Pareto points and reducing the computational burden compared to multi-start gradient methods.

Abstract

In the context of the optimization of rotating electric machines, many different objective functions are of interest and considering this during the optimization is of crucial importance. While evolutionary algorithms can provide a Pareto front straightforwardly and are widely used in this context, derivative-based optimization algorithms can be computationally more efficient. In this case, a Pareto front can be obtained by performing several optimization runs with different weights. In this work, we focus on a free-form shape optimization approach allowing for arbitrary motor geometries. In particular, we propose a way to efficiently obtain Pareto-optimal points by moving along to the Pareto front exploiting a homotopy method based on second order shape derivatives.

Tracing Pareto-optimal points for multi-objective shape optimization applied to electric machines

TL;DR

The paper tackles multi-objective PDE-constrained shape optimization for electric machines by tracing the Pareto front between the torque-related objective and rotor volume using a homotopy continuation of first- and second-order shape derivatives. A predictor–corrector scheme based on shape derivatives and a shape-Newton update guides the design along the front, with interior deformations filtered and boundary normal deformations enforced. The approach is demonstrated on a synchronous reluctance rotor with free-form geometry, yielding a set of Pareto-optimal designs and showing efficient exploration of non-convex front regions, aided by a multi-resolution refinement strategy. This method offers practical benefits for industrial rotor design by enabling controlled spacing of Pareto points and reducing the computational burden compared to multi-start gradient methods.

Abstract

In the context of the optimization of rotating electric machines, many different objective functions are of interest and considering this during the optimization is of crucial importance. While evolutionary algorithms can provide a Pareto front straightforwardly and are widely used in this context, derivative-based optimization algorithms can be computationally more efficient. In this case, a Pareto front can be obtained by performing several optimization runs with different weights. In this work, we focus on a free-form shape optimization approach allowing for arbitrary motor geometries. In particular, we propose a way to efficiently obtain Pareto-optimal points by moving along to the Pareto front exploiting a homotopy method based on second order shape derivatives.
Paper Structure (8 sections, 12 equations, 3 figures)

This paper contains 8 sections, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem description
  3. Homotopy methods
  4. Shape Newton
  5. Application to model problem
  6. Pareto tracing for shape optimization
  7. Numerical results
  8. Conclusions

Figures (3)

  • Figure 1: Left: Initial design. Ferromagnetic material is in red, air is in blue and copper coils are in yellow. Right: Set of Pareto-optimal points traced starting by $A$ in which the negative torque is minimized (homotopy parameter t=0), towards points $B$ (t=0.999779) and $C$ (t=0.999864). Points $A_r$, $B_r$ and $C_r$ are obtained respectively from points $A$, $B$ and $C$ with one uniform refinement of the mesh and following gradient descent optimization.
  • Figure 2: Optimized designs corresponding to Pareto-optimal points $A$ (t=0), $B$ (t=0.999779) and $C$ (t=0.999864), from left to right.
  • Figure 3: Optimized designs corresponding to Pareto-optimal points $A_r$, $B_r$ and $C_r$, from left to right.

Theorems & Definitions (1)

  • Remark 5.1