## 5 Modeling Complex Manufacturing by Using Case Based Reasoning  
Manufacturing processes, such as a forming process, sometimes exhibit complex manufacturing interactions that cannot be formalized directly by analytic relationships in CSP via constraints. In this case, external surrogate models (e.g., finite element models) must be used to model the process interactions. Since the use of process simulations is usually associated with a high computational time, the following section presents the structure of a case-based inference logic using the example of the joining zone expression of an extruded tailored forming shaft. This inference logic allows external surrogate models  
Figure 5: Constraint network of a tailored forming process chain (data from [11]). to be incorporated into the CSP in order to restrict the domains for individual variables in the CSP to process-conformant value assignments in advance. The joining zone in the case of a multimaterial component is the area where the different materials of the component are joined together. In the case of the extruded tailored-forming shaft with serial material arrangement, the inner part of the joining zone is made of steel and the outer part of aluminum (see Figure 6). The geometry of the joining zone of a tailored forming shaft is significantly influenced by the impact extrusion process. The main factors influencing this are the geometry of the previously joined surface \(\alpha\) (e.g., flat or pointed), the degree of forming \(\eta\), the shoulder angle of the extrusion die \(\beta\), and the process parameters \(p_{i}\) (e.g., temperature or back pressure) [48]. In order to obtain the knowledge about the forming behavior of the joining zone, parameter studies are carried out in advance with the aid of a finite element simulation of the impact extrusion. The main influencing factors are varied by means of a full-factorial experimental design and the expression of the joining zone is analyzed after the impact extrusion simulation. In order to formalize the results of the parameter study and make them available by computer, the geometry of the joining zone must be described mathematically. For this purpose, [PERSON] et al. [49] derived a 2D functional profile f(x), which gives the surface of the joining zone by a complete rotation and covers a wide range of expression shapes. \[f(x)=R\cdot\sqrt{1-\frac{x(\frac{V}{\pi\cdot R^{2}L-V})}{\gamma L}};\forall L >\frac{V}{\pi\cdot R^{2}}, \tag{1}\]  
This function is given by three physical parameters: Length \(L\), Radius \(R\), and Volume \(V\). Here, the starting point is at \((0,R)\), where R is the radius of the shaft at the given interface, and the end point is at \((L,0)\), where \(L\) is the length of the joining zone. The volume \(V\) corresponds to the volume enclosed in the joining zone (see Figure 6, right). With this mathematical modeling, the simulated joining zones can now be described by the parameter length \(L\), radius \(R\), and volume \(V\). These parameters can be identified by curve fitting, in which a curve is placed through the finite element nodes of the resulting joining zone. The Python library SciPy was used for this (see Figure 6). These parameters are then merged with the input variable that led to this joining zone expression in a tabular case base. The case-based reasoning logic works in three steps (see Figure 7):  
1. At the beginning, the joining zones are read out from the component to be examined. Since this is already modeled parametrically in the CAD model via the function (1), the parameter length \(L\), radius \(R\), and volume \(V\) can be read directly from the parameter set. Figure 6: Joining zone curve fitting from a finite element simulation of a impact extrusion process. 2. This parameter set is compared with the joining zones from the case base to identify similarcases (see Figure 8). For this purpose, the joining zone profiles of the investigated component and the simulated case are discretized by 100 points.