## 3 Method
### Direct Morphing Approach  
Each Direct Morphing process consists of three subprocesses, shown in Figure 3. First, we define the nodes for the three different node types, mentioned in Section 2.2. Control nodes are points in  
Figure 2: Workflow from initial finite element (FE) model to optimization. space, that impose a given displacement, free nodes are nodes of the FE mesh that are free to move and fixed nodes are points in space that stay at their locations during the morphing process. Next, we apply the actual morphing algorithm onto the set of free nodes and receive their new coordinates. This is done separately for the three directions \(x_{1}\), \(x_{2}\) and \(x_{3}\). This results in the preliminary morphed mesh. Mesh quality criteria indicate where node relaxation or remeshing is necessary in a last step, resulting in the final morphed mesh. For the morphing step, we use the tool chain we proved working in an earlier publication [21] in a slightly modified version. Moreover, [10] suggest using linear morphing and hypothesize natural neighbor as a method, possibly resulting in better element quality by using control nodes in the natural neighborhood [10] (p. 297). In our application, we are confronted with control nodes close to each other, so we needed a method that interpolates independently from the distance to control nodes. We decided to use a combined approach, as [22] suggests as an extension for his approach [22]. Our morphing approach thus consists of two separate interpolation algorithms. First, we generate a three dimensional regular grid of nodes that encases the complete component. We perform a natural neighbor interpolation from the control and fixed nodes onto this grid. Second, we interpolate from this grid onto the actual free FE nodes. In the following, we describe the morphing algorithm. We use bold symbols to label vectors and underlined symbols to label matrices. Our examples are situated in a global Cartesian coordinate system with the directions \(x_{1}\), \(x_{2}\) and \(x_{3}\). The last index in vector components specifies the direction. For the first process step, we define three matrices for sets of control nodes, free nodes and fixed nodes. As our application demands the investigation of design coordinate changes, we want to define displacements at a non-uniform set of points in space, not only on nodes of the FE mesh. For example, the location coordinate of a bushing lies in its center. As the bushing is pressed into a hole of the component, there is no FE node present at this location. The \(k\) control nodes  
\[\underline{x}_{c}=[\boldsymbol{x}_{c,1},\boldsymbol{x}_{c,2},\ldots,\boldsymbol {x}_{c,k}]=\left[ \begin{pmatrix}x_{c,1,1}\\ x_{c,1,2}\\ x_{c,1,3}\end{pmatrix} , \begin{pmatrix}x_{c,2,1}\\ x_{c,2,2}\\ x_{c,2,3}\end{pmatrix} ,\ldots, \begin{pmatrix}x_{c,k,1}\\ x_{c,k,2}\\ x_{c,k,3}\end{pmatrix} \right] \tag{1}\]  
are points in space, defined by their coordinates, independent from the FE mesh. Each control node receives an imposed displacement in the form of a displacement vector, originating from the corresponding control node. This yields a second matrix for the control nodes  
\[\underline{x}_{c}=[\boldsymbol{v}_{c,1},\boldsymbol{v}_{c,2},\ldots,\boldsymbol {v}_{c,k}] \tag{2}\]  
containing all the imposed displacements. The \(m\) fixed nodes  
\[\underline{x}_{f}=\left[\boldsymbol{x}_{f,1},\boldsymbol{x}_{f,2},\ldots, \boldsymbol{x}_{f,m}\right] \tag{3}\]  
Figure 3: Detailed Direct Morphing process from Figure 2.  
are points in space, defined by their coordinates, independent from the FE mesh, too.