Abstract
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.
Highlights
Additive manufacturing (AM) processes evolved in the past decades into a notable alternative to classical material-removing production techniques
In a first step, we consider the automatic construction of a honeycomb structure through centroidal Voronoi tesselations (CVTs), given the boundary shape of a structure
We compare two methods for finding VTs, namely the geometric or graph based approach utilizing a Delaunay triangulation (DT), and an approach based on partial differential equation (PDE) utilizing the fast marching (FM) method
Summary
Additive manufacturing (AM) processes evolved in the past decades into a notable alternative to classical material-removing production techniques. Given the shape of a workpiece, the problems of (i) finding a good inner structure in terms of functionality and stability and (ii) the best path for printing the desired layer arise. Often workpieces have to absorb impacts and tackle additional external force constraints, i.e., a method for structural optimization should allow for the consideration of the related, additional constraints For this reason, in a first step, we consider the automatic construction of a honeycomb structure through centroidal Voronoi tesselations (CVTs), given the boundary shape of a structure. By combining the path generation and one of the approaches for temperature calculation into a single model we obtain mixed-integer linear problems (MILPs), that are investigated on several test instances We compare their results in a qualitative way to the temperature distribution of real processed workpieces, built using the path computed by the models. We discuss a number of directions for future work
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