In this paper, an adaptive degrees-of-freedom finite element method is extended to three-dimensional (3-D) nonlinear magnetic field analysis. The error distribution of the discrete solution changes along with the iterative process, and mesh coarsening or other operations with the same effect is needed to keep the scale of the problem small. In this proposed adaptive method, dispensable degrees of freedom (DoFs) are eliminated from the unknown list by constraining them with supplementary interpolation functions, which are formulated with master DoFs. Compared with mesh coarsening, the administration of DoFs and geometric data are no longer required, while the topology of the mesh is maintained. To extend to 3-D problems, a novel constraint, which produces accurate coefficients for an alterable number of master DoFs, is presented. The constraint is integrated into the element algebraic equation, followed by a conventional assembly. Other techniques for the adaptive algorithm are also included in this method. Several numerical examples are tested to showcase the effectiveness of this method in 3-D problems.

本文将自适应自由度有限元方法扩展到三维（3-D）非线性磁场分析。离散解决方案的误差分布会随着迭代过程而变化，并且需要进行网格粗化或其他具有相同效果的操作才能使问题的规模变小。在此提出的自适应方法中，通过用辅助插值函数（由主DoF公式化）来约束可分配的自由度（DoF），从未知列表中消除可分配的自由度。与网格粗化相比，在保持网格拓扑的同时，不再需要对DoF和几何数据进行管理。为了扩展到3-D问题，提出了一种新颖的约束条件，该约束条件可为可更改数量的主DoF生成精确的系数。将约束条件整合到元素代数方程式中，然后进行常规装配。该方法中还包括用于自适应算法的其他技术。测试了几个数值示例，以展示此方法在3-D问题中的有效性。