Greedy permanent magnet optimization

TL;DR

This paper introduces greedy algorithms for optimizing permanent magnet placement, formulating the problem as quadratic programs and demonstrating superior performance in designing magnets for stellarator plasmas.

Contribution

It formulates the binary permanent magnet problem as quadratic programs and proposes simple greedy algorithms that outperform existing methods in speed and flexibility.

Findings

Algorithms produce sparse, grid-aligned, binary solutions.

Proposed method outperforms state-of-the-art algorithms.

Demonstrated effectiveness in stellarator plasma magnet design.

Abstract

A number of scientific fields rely on placing permanent magnets in order to produce a desired magnetic field. We have shown in recent work that the placement process can be formulated as sparse regression. However, binary, grid-aligned solutions are desired for realistic engineering designs. We now show that the binary permanent magnet problem can be formulated as a quadratic program with quadratic equality constraints (QPQC), the binary, grid-aligned problem is equivalent to the quadratic knapsack problem with multiple knapsack constraints (MdQKP), and the single-orientation-only problem is equivalent to the unconstrained quadratic binary problem (BQP). We then provide a set of simple greedy algorithms for solving variants of permanent magnet optimization, and demonstrate their capabilities by designing magnets for stellarator plasmas. The algorithms can a-priori produce sparse, grid-aligned, binary solutions. Despite its simple design and greedy nature, we provide an algorithm that outperforms the state-of-the-art algorithms while being substantially faster, more flexible, and easier-to-use.