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A Halbach array is a specialized arrangement of permanent magnets designed to generate a strong, uniform magnetic field in the designated region. This unique configuration has been widely utilized in various applications, including magnetic levitation (maglev) systems, electric motors, particle accelerators, and magnetic seals. The advantages of Halbach arrays include high efficiency, reduced weight, and precise directional control of the magnetic field. Halbach arrays are commonly categorized into two configurations: linear and cylindrical. A linear Halbach array produces a concentrated magnetic field on one face and is frequently employed in maglev trains and conveyor systems to ensure stable and efficient operation. In contrast, a cylindrical Halbach array consists of magnets arranged in a ring, generating a uniform magnetic field within the cylinder while suppressing the external field. This configuration is particularly advantageous in applications such as brushless electric motors and magnetic resonance imaging (MRI) systems. Traditionally, the design of electromagnetic systems incorporating Halbach arrays relied on engineers’ expertise and intuition due to the complexity of the permanent magnet configuration. However, advancements in numerical methods, particularly topology optimization, have introduced a systematic approach to optimizing the shape and distribution of permanent magnets within a given design domain. In the context of Halbach array design, topology optimization aims to maximize the total magnetic flux within a designated region while simultaneously determining the optimal material distribution to achieve a specified design objective. This approach enhances the performance and efficiency of Halbach arrays, providing a more precise and automated framework for their development. In this paper, we propose a Cardinal Basis Function (CBF)-based level-set method for designing a circular Halbach array capable of generating a uniform magnetic field within a designated region. The CBF-based level-set method offers significant computational advantages by reducing the computational cost and accelerating the convergence process. This approach enhances the efficiency of the optimization process, making it a promising technique for the systematic design of Halbach arrays. 

Abstract Synchronous reluctance motors (SynRMs) have gained considerable attention in the field of electric vehicles as they reduce the need for permanent magnets in the rotor, resulting in less material and manufacturing costs. However, their lower average torque and torque ripple vibrations have been identified as key issues that require resolution. In this study, we present a SynRM design framework employing the cardinal basis functions (CBF)-based parametric level set method. The SynRms design problem is recast as a variational problem constrained by Maxwell’s equations which describe the behavior of electric and magnetic fields in the SynRM. A continuum shape sensitivity analysis is carried out using the material derivative and adjoint method. A distance regularization energy function is employed to maintain the level set function as a signed distance function during the optimization. The parametric topology optimization problem is computationally solved using the Method of Moving Asymptotes (MMA). To demonstrate the effectiveness of our approach, we present a numerical example that compares the torque characteristics of the optimal design with those of a reference design. Preliminary results show that the optimized SynRM has a 30.30% increase in average torque, along with a slight increase in torque ripple, compared to the reference model. 

Generators are considered as the core application of electromagnetic machines, which require high-cost rare-earth-based permanent magnets. The development of generators is moving toward high efficiency and increased environmental friendliness. Minimizing the use of rare earth materials such as magnetic materials under the premise of machine performance emerges as a challenging task. Topology optimization has been promisingly applied to many application areas as a powerful generative design tool. It can identify the optimal distribution of magnetic material in the defined design space. This paper employs the level-set-based topology optimization method to design the permanent magnet for generators. The machine under study is a simplified 2D outer rotor direct-drive wind power generator. The dynamic and static models of this generator are studied, and the magnetostatic system is adopted to conduct the topology optimization. The optimization goals in this study mainly focused on two aspects, namely the maximization of the system magnetic energy and the generation of a target magnetic field in the region of the air gap. The continuum shape sensitivity analysis is derived by using the material time derivative, the Lagrange multiplier method, and the adjoint variable method. Two numerical examples are investigated, and the effectiveness of the proposed design framework is validated by comparing the performance of the original design against the optimized design.