## 4 Demonstration, Application, and Discussion  
The developed algorithm is demonstrated on the elementary cell of a body-centric cubic lattice structure in three different versions (Figure 10). First, the elementary cell is constructed with constant strut radii and periodic node distribution (Figure 10). Then, the construction is repeated but with randomly shifted nodes (Figure 10) and with randomly varied strut radii (Figure 10). Figure 8: Construction stages for nodes with edge-free half-spaces. The subfigures (**a–c**) represent the sequence of action. Figure 9: Different stages of fallback in geometric continuity due to sharp angles and short struts. (**a**) \(G^{1}\)-continuity. (**b**) Partly fallback to \(G^{0}\)-continuity. (**c**) Complete fallback to \(G^{0}\)-continuity. In Figure 11, a lattice structure composed of eight elementary cells with different unit cell topologies is shown in two versions. First, only the topology is varied (Figure 11a). Then, strut radii and node positions are varied additionally (Figure 11b). It is seen in these examples that the proposed method is able to construct unit-cell-based lattice structures as well as such with irregular topologies. Furthermore, radii gradients between the ends of a single strut and between struts converging to the same node can be realised and parametrised exactly. As a result, the proposed method is particularly suitable for lightweight design constructions of irregular shapes and radius gradients such as lattice structures derived form topology optimisation results. Nevertheless, it is not always possible to construct a smooth node. Just as the lattice construction methods in [13; 14], the proposed general node construction step shows difficulties with sharp and wide angles between struts because the convex hull-based smooth transition design is not practicable. To overcome this issue, different special-case workarounds and a fallback mechanism in geometric continuity are integrated in the algorithm. Among the nodes shown in Figure 10, there is one node in the top-left corner of Figure 10b, to which the automatic fallback mechanism in geometric continuity is applied. All the other nodes are constructable with the general node construction step. Also in the lattice shown in Figure 11b, the fallback is used twice. However, due to the subdivision of the lattice construction task in node-wise steps, the nodes that are constructed using the fallback mechanism remain accessible for post-processing and their construction can be repeated with different parameters without affecting other parts of the lattice structure. Furthermore, another advantage of the node-wise subdivision of the construction task is the resulting parallelisability. Therefore, the speed can be increased by orders of magnitude, which is crucial to handle large-scale lattice structures and to integrate it in existing construction software. Requiring, besides basic geometric operations, only the three different patch construction methods described in Section 2 for the generation of surface patches, the developed algorithm can be integrated in any standard CAD software that supports the latter. The construction time of the 35-node lattice in Figure 11b was recorded. It took five minutes and eight seconds to construct the whole lattice structure even without using the parallelisation on a standard desktop PC. This corresponds to a construction time of 8.8 seconds, per node. Compared to the construction time of approximately one hour per node using the manual scheme in [17], the developed method in this paper is distinctly faster. The lattice construction algorithm was applied to construct lattice infills for an aircraft component [8]. Pictures of the corresponding additively manufactured component produced using selective laser melting (SLM) can be seen in Figure 12. The position of the lattice infills is chosen in areas where the topology should be strengthened without using solid material.