This paper proposes a topology optimization formulation for the three-dimensional (3-D) design of segmented permanent magnet (PM) arrays. Specifically, the proposed formulation aims to find an optimal 3-D structural topology of PM segments and their discrete magnetization directions. To achieve this, a design variable is defined in the vector form of a Cartesian coordinate system. The magnitude of the design variable vector determines a PM density, and the directional cosines of the vector determine PM magnetization directions. To acquire a segmented PM design with discrete magnetization directions, a PM strength penalization scheme is proposed. In this scheme, a PM strength is controlled using a minimum distance function between a magnetization direction and target discrete directions. Here, the minimum distance function is approximated using the p-norm for sensitivity calculation. To validate the effectiveness of the proposed formulation, three design examples are provided. In the first example, a two-dimensional (2-D) dipole Halbach cylinder is designed to confirm that the proposed formulation can be applied for a 2-D design problem. The second example aims to design 3-D dipole PM arrays with a cuboid cavity. In the third example, a 3-D PM arrays are designed for maximizing the magnetic force acting on a soft ferromagnetic material.

本文为分段式永磁体（PM）阵列的三维（3-D）设计提出了一种拓扑优化公式。具体而言，提出的公式旨在找到PM段及其离散磁化方向的最佳3-D结构拓扑。为此，以笛卡尔坐标系的矢量形式定义一个设计变量。设计变量矢量的大小确定了PM密度，矢量的方向余弦确定了PM磁化方向。为了获得具有离散磁化方向的分段PM设计，提出了PM强度惩罚方案。在该方案中，使用磁化方向和目标离散方向之间的最小距离函数来控制PM强度。这里，最小距离函数使用p范数进行灵敏度计算。为了验证所提出配方的有效性，提供了三个设计实例。在第一个示例中，设计了二维（2-D）偶极哈尔巴赫圆柱体，以确认所提出的公式可以用于二维设计问题。第二个示例旨在设计具有长方体腔的3-D偶极子PM阵列。在第三个示例中，设计了3-D PM阵列以使作用在软铁磁材料上的磁力最大化。第二个示例旨在设计具有长方体腔的3-D偶极子PM阵列。在第三个示例中，设计了3-D PM阵列以使作用在软铁磁材料上的磁力最大化。第二个示例旨在设计具有长方体腔的3-D偶极子PM阵列。在第三个示例中，设计了3-D PM阵列以使作用在软铁磁材料上的磁力最大化。