# Digital Twin-Based Clamping Sequence Analysis and Optimization for Improved Geometric Quality
## 4 Results  
The proposed method for the clamping sequence analysis and optimization considering non-nominal geometries is established for the digital twin and applied to the two reference assemblies. The following are presented in this section, according to the assembly steps. * Positioning the non-nominal parts in a rigid fixture. * Locking the in-plane translation and rotations. * Applying the clamping points in a sequence. * Capturing the weld relative displacements after clamping. * Optimizing the damping sequence according to the objective above. * Performing spot welding with the optimal clamping sequence. * Optimizing the welding sequence with respect to the assembly displacements. * Releasing the assembly from the fixture and evaluating the assembly springback. To visualize the impact of the clamping and welding sequence optimization, the results are presented while both the optimal and unideal (worst registered sequence) sequences are taken into consideration. Figure 4 is the optimization plot of the clamping sequence with respect to the weld relative displacements. The reference assemblies are held in the fixture, and the stepwise optimization process, Section 2.2, is applied to the assemblies. Table 1 presents the summary of the optimization process. For Assembly I, the optimal clamping sequence of the eight included clamping points results in a weld relative displacement of 0.228 mm in the three weld points. An unideal clamping sequence results in 0.579 mm for this measure. The range of the differences in the weld relative displacements caused by the optimal and unideal clamping sequences is 1.5 times the optimal clamping sequence weld relative displacements, Figure 4a. This indicates the great impact of the clamping sequences on the follow-up welding step. After clamping, the assembly is welded with a sequence and then released from the fixture. The root mean square (RMS) of the nodal displacements of all the nodes in the assembly is considered for optimization of the assembly with respect to the welding sequences. Figure 5 visualizes the color plot of Assembly I after the welding and springback while it is welded with the welding sequence W1-W2-W3. This figure visualizes the impact of the clamping step on a predetermined welding sequence. In Figure 5a, the assembly is clamped with the optimal clamping sequence, Table 1. In Figure 5b, Assembly I is clamped with an unideal clamping sequence specified in Table 1.  
Similarly, for Assembly II, the clamping sequence is optimized. Six included clamping points are analyzed for the best clamping sequence, Figure 4b. The range of the differences in the weld relative displacements is 4.54 mm. The assembly is welded with the sequence W1-W2-W3-W4-W5-W6-W7-W8, and the color plot of the outcome of each clamping sequence is visualized in Figure 6. Figure 6a is Assembly II while it is clamped with the optimal clamping sequence, and Figure 6b is the color plot of Assembly II while it is clamped with an unideal clamping sequence, specified in Table 1, and welded with the sequence above. The welding sequence optimization is performed for the optimal and unideal clamping sequences. Figure 7a shows the welding sequence optimization steps on Assembly I with the optimal and unideal clamping sequences.The range of the displacements of this assembly after optimizing the welding sequences is 0.08 mm, which comprises 87% of the total optimal geometric deviation. The welding sequence optimization is also performed for Assembly II with the optimal and unideal clamping sequences. The optimization results are visualized in Figure 7b. The range of the differences in the displacements for this assembly with the optimal clamping and welding sequences and unideal clamping and welding sequences is 3.3 times the total optimal geometric deviation. Figure 4: Clamping sequence optimization of the two reference assemblies. Figure 5: Reference Assembly I welded with optimal and unideal clamping sequences. Figure 6: Reference Assembly II welded with optimal and unideal clamping sequences.