## 5 Case Studies
### Validation by Simulation  
The actual machining process is simulated by a finite element simulator to obtain a large amount of part deformation force data for the ensuing neural network training and testing. The simulation was carried out in commercial finite element software ABAQUS. The thin-walled design part has a deformation-prone 3D geometry as shown in Figure 4b. The residual stress data are shown in Figure 5. The lower surface edge of the part is restrained to simulate the positioning and clamping in the finite element environment. The deformation force at any clamping point is obtained by applying the fixed constraints, and the calculated node force of the deformation monitoring point is equivalent to the deformation force. The deformation force dataset generated from the simulation is then fed to the training model. Gaussian noise with a variance of 50 N is applied during the training process. If \(\hat{F}_{b}\leq 0\), it is considered that the direction of deformation within the machining step is the same as that of the support element arrangement under the confidence level \(\alpha\). In this case, a smaller clamping force can be used according to our past experience. However, if \(\hat{F}_{b}>0\), it is considered that the deformation direction in the machining step is opposite to the support element arrangement at the confidence level \(\alpha\). The deformation force verification data are generated the same way as the training data. For the verification data set, if \(\hat{F}_{b}\geq F_{b}\), the clamping force can be monitored. The prediction probability model of deformation force is compared with the traditional approach of using the mean value of deformation force in the training set as the basis of applying the pre-support force. The monitoring success rate of clamping force in each machining step is shown in Figure 6. If the support force is applied according to the average value of the deformation force, the clamping force monitoring is effective at around 50%. If pre-support is applied only according to the end-to-end model (\(\alpha=50\%\)), the results may be too poor at certain machining steps. The simulation results show that the application of pre-support forces according to a probabilistic model can effectively improve the success rate of clamping force monitoring. Next, the machining process is simulated with a finite element solver to verify the effectiveness of the proposed method. The new residual stress data are generated by adding noise to the simulated residual stress data. When using the finite element solver, the effect of auxiliary support is simulated by exerting a displacement. Different from actual machining, the applied displacement is calculated based on the deformation force distribution, the confidence degree (\(\alpha=84.1\%\)), and the equivalent flexibility. As previously stated, the node force of the deformation monitoring point is equivalent to the clamping force. In addition, the real deformation force is obtained by exerting the fixed constraints. The equivalent  
Figure 5: Residual stress distribution. flexibility is obtained by simulation too, as shown in Figure 7. The variance of the measured values is set empirically, i.e., 5 in this work. Figure 8 shows the discrepancy between the exerted pre-support forces and the calculated deformation forces, which confirms that the proposed model learns the distribution of deformation force data well, including the deformation direction. The error in the estimation of the deformation force is 10.93%, which suggests that the pre-support force calculated based on the estimation of deformation force can be used with sufficient confidence. Basically, when the in-process workpiece deforms inward, we exert a small support force (20 N) based on the past experiences; on the other hand, when the in-process workpiece deforms outward, we will exert a support force based on the distribution of deformation force, the confidence degree (\(\alpha=84.1\%\)), and the equivalent flexibility.