Multi-Material Topology and Magnetization Co-Optimization of Circular Halbach Arrays Via a Cardinal Basis Function-Based Level Set Method

A Halbach array is a specialized arrangement of permanent magnets that generates a strong, uniform magnetic field in a designated region and suppresses it elsewhere. This configuration has been widely applied in magnetic levitation systems, electric motors, particle accelerators, and magnetic resonance imaging devices due to its high efficiency, reduced weight, and precise directional control. Linear Halbach arrays concentrate the field on one side, making them suitable for applications such as maglev trains and conveyor systems. Cylindrical Halbach arrays, with magnets arranged in a circular configuration, produce a uniform internal field while minimizing the external field, which is advantageous in brushless motors and imaging systems. The traditional design of Halbach arrays has relied heavily on engineering intuition because of the complexity of magnet placement and orientation. With advances in numerical methods, topology optimization now provides a systematic approach to determining both material distribution and magnetization directions to maximize magnetic flux in the target region. In this article, we propose a cardinal basis function (CBF)-based level set method for the design of circular Halbach arrays capable of generating an area-averaged magnetic field. The finite difference method is employed to optimize magnetization directions simultaneously with geometry, providing additional design flexibility. The CBF-based level set method reduces computational cost and accelerates convergence, improving the overall efficiency of the optimization. Furthermore, multi-material topology optimization is incorporated, enabling the addition of permanent magnets with distinct magnetization directions to achieve greater control over magnetic flux distribution.