# Integrated Tolerance and Fixture Layout Design for Compliant Sheet Metal Assemblies
## 4 Case Study and Results  
The method that is presented in Section 3 is applied to an industrial sample case to evaluate its performance. The sample case is a spot-welded assembly of three sheet metal parts. Figure 2 depicts this assembly and the defined requirements of its KPCs. Each part of the presented assembly can be produced through several production scenarios. Each scenario results in different geometrical quality and production costs. Figure 3 depicts the tolerance zone of the first part. This part can be produced in four strategies, each resulting in a \(t_{1j}\) in the tolerancing zone. In multi-station assemblies, positions of the common locating point among different stations are reused in each station to avoid variation propagations. Consequently, some locating positions of the parts are already defined by the fixture layout that is used in the next station to assemble this assembly to other parts. These positions are the datum positions of the measuring the KPCs in this assembly. In the presented sample case, all of the datum positions of the assembly are in the first part. Therefore, in the fixture layout of this part, the hole and slot location are not included in the design variables, because they should be in the same positions as the datum positions of the assembly. Figure 3 presents these positions though the hole and slot are not displayed in the model. Figure 2: The utilized sample case and its KPC tolerances. Figure 4 presents the tolerance zone of the second part. This part can also be produced in four production strategies. The tolerance zone of this part is the entire surface of the part. The third part has two tolerances. The first tolerance (\(t_{3/1}\)) is the surface profile that applies to the entire surface of the part. The second tolerance \(t_{3/2}\) is allocated to the positions of the holes. Different production strategies of this part result in four tolerance limits for the surface profile and three for the position tolerance. Figure 5 visualizes these tolerances. Table 3 denotes the value of each \(t_{ij}\). The associated cost of each tolerance is defined by Equation (12). The reason for having an inverse relation between the each \(t_{ij}\) and \(c_{ij}\) is tightening the production tolerances (i.e., higher geometrical qualities), commonly increasing the production cost [13]. Figure 4: Tolerances of the second part. Figure 5: Tolerances of the third part. Figure 3: Tolerances of the first part. \[c_{ij}=\frac{1}{t_{ij}} \tag{12}\]  
Based on the tolerance limits that are presented in Table 3, the optimal \(j\) of each part (\(j_{1}\) and \(j_{2}\) for the third part) should be defined by the optimization algorithms. Before conducting the optimization, the feasible areas of each part to be the location of a locator are defined. Thereafter, these areas are divided into smaller areas, and a representative point from each area is defined. The outcome of this procedure is 15, 6, and 14 representative nodes for the first, the second, and the third part, respectively. Figure 6 depicts the representative nodes in each part and the allocated number to each node. The variation simulation model of the sample case is generated in the RD&T program to determine the geometrical quality of each production strategy and fixture layout in the objective function of the optimization. In RD&T program (www.rdnt.se) (accessed on 1 October 2020) non-linear compliant simulations can be conducted while considering contacts, gravity, and welding sequence. Hence, this program is utilized in this study to determine the geometrical variation of assemblies in each function evaluation of the optimization. The deformations being elastic and linear, and that the heat effects of spot welds are negligible are the main assumptions in conducting variation simulations in this study [50]. Figure 6: The representative locating points for fixture layout design.