## 3 Method
### Box Morphing Approach  
The Box Morphing approach is also referred to as free-form-deformation (FFD) and was first introduced by [26]. Other literature uses the term control block morphing [3]; in this paper, we name the method _Box Morphing_. Other than the Direct Morphing approach, Box Morphing does not act directly on the FE nodes [3]. To deform the original geometry, first a box structure must be created. In case the structure only consists out of parallelepipeds, the used approach is the FFD [26]. In the case that more complex volumes, such as e.g., prismatic ones, are used, the approach is called extended-free-form-deformation (EFFD) [27]. Since the nodes defining these volumes are the ones driving the deformation, we refer to them as control nodes. For parallelepipedic volumes, the control points are the nodes of the box. For more complex structures, additional control points might be added on each edge to change their curvature. The free and fixed nodes can generally be all FE nodes. Fixed nodes are always those outside the deformed morphing box, unless a box is displaced as whole. Then, the included nodes are free nodes. Additionally to that, relevant FE nodes might be defined as fixed nodes so their location stays unchanged. If none of those are defined, the free nodes are all FE nodes included in the deformed morphing box. The FE nodes of the part are shifted through connecting the movement of the control nodes to the FE nodes. In the Box Morphing approach, a lattice of morphing boxes is constructed. Then, the FE nodes are linked to the surrounding box by calculating their local coordinates \(s\), \(t\) and \(u\). By moving the control nodes, the FE nodes also move. [26]  
The local coordinates of an FE node \(\mathbf{x}\) can be calculated through  
\[\mathbf{x}=\sum_{s=0}^{v}\sum_{t=0}^{w}\sum_{u=0}^{r}\mathbf{c}_{stu}B_{s}B_{t}B_{u} \quad. \tag{8}\]  
Hereby, \(B_{s}\), \(B_{t}\) and \(B_{u}\) are arbitrary basis functions that only depend on the local coordinate shown by their index. The control nodes \(\mathbf{c}_{stu}\) are also numbered consecutively in each local coordinate so that the whole equation describes a linear combination of the control nodes and basis functions [28]. As a basis function, [26] originally used a Bernstein polynomial but referred to other functions as a B-spline [29] being suitable. The whole process is considered a mapping from \(\mathbb{R}^{3}\) to \(\mathbb{R}^{3}\)[26]. By not only considering the location of the control points before morphing, but also after morphing \(\mathbf{c}^{\prime}_{ijk}\), the displacement of each FE node  
\[\Delta\mathbf{x}=\sum_{s=0}^{v}\sum_{t=0}^{w}\sum_{u=0}^{r}(\mathbf{c}^{\prime}_{stu}- \mathbf{c}_{stu})B_{s}B_{t}B_{u} \tag{9}\]  
is calculated [28]. Since the Box Morphing approach is considered Indirect Morphing, the used control points are not situated on the FE mesh itself, but are defined in the box structure [3]. In the morphing structure created for this paper, all planes of morphing boxes that are relevant for the deformation of the component are oriented parallel to the planes of the global coordinate system. Due to that, shifting one plane in the direction of its normal vector only results in a displacement of the geometry in the same direction, which makes it easy to track the movement. Since this paper evaluates metamodels achieved through different morphing methods, the morphing process needs to be automated. This is achieved by a tool chain combining the morphing environment in the FE preprocessor _ANSA_ by _BETA CAE_[30] and _Python_ scripts. The morphing itself is also automated in a _Python_ script that only needs the displacement at each kinematic hard point of the component as input values. The morphing process is fragmented into the four steps, presented in Figure 4.  
The first two steps represent the translation of the tracked hard point. The process is partitioned for two reasons. First, the planes being moved are the ones representing the outer layer of the geometry.