Combined Parameter and Shape Optimization of Electric Machines with Isogeometric Analysis

TL;DR

This work tackles the joint optimization of geometry parameters and shape for electric machines by leveraging Isogeometric Analysis (IGA) to provide smooth, CAD-consistent geometry updates. The authors develop a gradient-based PDE-constrained framework where shape derivatives are obtained analytically via an adjoint method and parameter derivatives through a semi-analytical chain rule, all within a discretize-then-optimize paradigm. The approach is demonstrated on a 2D magnetostatic PMSM, achieving substantial reductions in magnet material use and torque ripple compared to parameter-only or shape-only designs, and showing potential for significant design-time reductions. The combination of exact geometry representation, efficient sensitivity computation, and a modular optimization setup suggests broad applicability to multi-physics machine design and rapid exploration of high-dimensional design spaces.

Abstract

In structural optimization, both parameters and shape are relevant for the model performance. Yet, conventional optimization techniques usually consider either parameters or the shape separately. This work addresses this problem by proposing a simple yet powerful approach to combine parameter and shape optimization in a framework using Isogeometric Analysis (IGA). The optimization employs sensitivity analysis by determining the gradients of an objective function with respect to parameters and control points that represent the geometry. The gradients with respect to the control points are calculated in an analytical way using the adjoint method, which enables straightforward shape optimization by altering of these control points. Given that a change in a single geometry parameter corresponds to modifications in multiple control points, the chain rule is employed to obtain the gradient with respect to the parameters in an efficient semi-analytical way. The presented method is exemplarily applied to nonlinear 2D magnetostatic simulations featuring a permanent magnet synchronous motor and compared to designs, which were optimized using parameter and shape optimization separately. It is numerically shown that the permanent magnet mass can be reduced and the torque ripple can be eliminated almost completely by simultaneously adjusting rotor parameters and shape. The approach allows for novel designs to be created with the potential to reduce the optimization time substantially.

Figures (11)

• Figure 1: Types of structural optimization.
• Figure 2: Exemplary B-Spline basis functions in 1D. The knot vector indicates the support of the basis functions, i.e., a different basis function begins or ends for each entry.
• Figure 3: NURBS mapping of a two-dimensional surface from the parametric domain to the physical domain. The control points $\mathbf{\mathbf{C}}_k$ in red prescribe the mapping.
• Figure 4: PMSM geometry, rebuilt from JMAG. Geometry parameters, material definitions and boundary conditions are provided.
• Figure 5: Visualization of the numerical evaluation of control point changes depending on a parameter used in the evaluation of \ref{['eq:ChainRule']}.