We consider two mathematical problems that are connected and occur in the layer-wise production process of a workpiece using wire-arc additive manufacturing. As the first task, we consider the automatic construction of a honeycomb structure, given the boundary of a shape of interest. In doing this, we employ Lloyd’s algorithm in two different realizations. For computing the incorporated Voronoi tesselation we consider the use of a Delaunay triangulation or alternatively, the eikonal equation. We compare and modify these approaches with the aim of combining their respective advantages. Then in the second task, to find an optimal tool path guaranteeing minimal production time and high quality of the workpiece, a mixed-integer linear programming problem is derived. The model takes thermal conduction and radiation during the process into account and aims to minimize temperature gradients inside the material. Its solvability for standard mixed-integer solvers is demonstrated on several test-instances. The results are compared with manufactured workpieces.

我们考虑两个相关的数学问题，这些问题在使用线弧增材制造的工件的分层生产过程中会发生。作为第一项任务，我们考虑给定感兴趣形状的边界，自动构造蜂窝结构。为此，我们在两种不同的实现中采用了劳埃德算法。为了计算合并的Voronoi镶嵌，我们考虑使用Delaunay三角剖分或eikonal方程。我们比较和修改这些方法，以结合其各自的优势。然后，在第二个任务中，找到一个保证最小生产时间和高质量工件的最佳刀具路径，得出一个混合整数线性规划问题。该模型考虑了过程中的热传导和辐射，旨在最小化材料内部的温度梯度。在几个测试实例上证明了其对于标准混合整数求解器的可解性。将结果与制造的工件进行比较。