Multiply connected freeform surface features are widely encountered in industrial components, where toolpath generation often suffers from discontinuities, sharp turns, non-uniform scallop heights, and incomplete boundary coverage. This paper proposes a scalar-field variational optimization method for milling that produces continuous, boundary-conforming, and non-self-intersecting toolpaths with smoother transitions, more uniform spacing, and reduced redundant path length. A feasible singularity-free initial scalar field with boundary-conforming iso-level sets is first constructed via conformal slit mapping. The optimization is then reformulated as a topology-preserving mesh deformation process governed by boundary-synchronous updates, whereby the continuity, boundary-conformity, and non-self-intersection requirements of the toolpath are converted into mesh-shape constraints maintained throughout the iterative optimization. As a result, the proposed method achieves globally optimized path spacing and improved scallop-height uniformity while preserving trajectory smoothness. Milling experiments show that, compared with a state-of-the-art conformal slit mapping-based method, the proposed approach improves machining efficiency by 14.24%, enhances scallop-height uniformity by 5.70%, and reduces milling impact-induced vibrations by over 10%. The proposed strategy provides an effective solution for high-performance machining of complex multiply connected freeform components.

翻译：多连通自由曲面特征广泛存在于工业零部件中，其刀具路径生成常面临不连续性、急转弯、非均匀残留高度及边界覆盖不完整等问题。本文提出一种面向铣削加工的标量场变分优化方法，可生成连续、边界贴合且无自交的刀具路径，具有更平滑的过渡、更均匀的间距及更短的冗余路径长度。首先通过共形狭缝映射构建可行且无奇点的初始标量场，其等值线集合与边界贴合。进而将优化重构为受边界同步更新支配的拓扑保持网格变形过程，将刀具路径的连续性、边界贴合性及无自交性要求转换为迭代优化全程保持的网格形状约束。由此，本方法在保持轨迹平滑性的同时实现全局优化的路径间距与改善的残留高度均匀性。铣削实验表明，与现有最先进的共形狭缝映射方法相比，本方法加工效率提升14.24%，残留高度均匀性改善5.70%，铣削冲击振动降低超10%。所提策略为复杂多连通自由曲面构件的高性能加工提供了有效解决方案。