# Thermal Strain-Based Simplified Prediction of Thermal Deformation Caused by Flame Bending
## 4 Application of Method
### Results Comparison  
To compare results, an accurate comparison between the results predicted by the SBD method and the design features is required. Therefore, in order to converge data on two different coordinate systems, we applied iterative close point (ICP) algorithms [23], which are commonly used mainly for feature registration, which is a method of convergence using unit quaternion presented by [PERSON] [24]. The ICP algorithm was commonly used for registration problems with two coordinate systems located differently. The 3D plot results of the experiment and the prediction by SDB method using shell elements are shown in Figure 11. The computational time was about 90 s, and the convergence results for comparing the experimental and predicted results are shown in Figures 11 and 12. The reference point was set to four corners of the curved plate, and the tolerance was set \(\pm\)3 mm. The maximum difference was 9.98 mm and the area within the tolerance range was 5.47% of the whole plate, with a root mean square (RMS) error of 8.89 mm.  
In addition, the results of the comparison by different data are described as shown in Figure 11. After matching the position of the four corners of the plate as reference points, the deformations were analyzed. The comparison results of changing the reference points are shown in Figure 12. In addition, the information of numerical results are summarized in Table 2. The case with the least root mean square error (RMS) is shown in the calculation results, and the largest ratio of area within the tolerance range is Case 5 in Figure 12d with 7.09%.  
Figure 11: Coordination reference shape and target shape (Case 1).  
and irregular, then the distance between the surrounding points and the heating lines, which has less spacing than the breadth of the inherent strain region, was considered to assume the proper inherent strain distribution. The results of thermal deformation in the SDB method based on thermal strain considering phase transformation showed good agreements within the error of 7%. As a result, it has been verified that the SDB method based on thermal strain can predict reasonable deformed shape, which can replace the three-dimensional thermo-elasto-plastic analysis.  
In the actual flame bending (or line heating) during ship construction, more complex heating lines are imposed on the plate. Thus, the SDB method based on thermal strain considering the phase transformation can accurately and rapidly predict the thermal deformation of thick steel plate. However, in this work, we conducted a limited case study. Therefore, various case studies will be conducted on concave, saddle twist-type shapes to improve the accuracy of the prediction method.  
Conceptualization, S.-Y.H., J.-H.L. and J.H.; methodology, J.-H.L. and J.H.; validation, K.-G.P. and J.-H.L.; formal analysis, J.H. and K.-G.P.; investigation, S.-Y.H., J.H. and J.-H.L.; data curation, S.-Y.H. and [PERSON]; writing--original draft preparation, [PERSON], [PERSON] and [PERSON]; writing--review and editing, [PERSON] and [PERSON]; visualization, [PERSON] and [PERSON]; supervision, [PERSON]; project administration, [PERSON]; funding acquisition, [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government (MOTIE) (P0001968, HRD Program for Industrial Innovation).  
Not applicable.  
This work was partly supported by the 'IoT and AI based development of Digital Twin for Block Assembly Process (20006978)' Program of the Korean Ministry of Trade, Industry and Energy, Republic of Korea and Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government (MOTIE) (P0001968, HRD Program for Industrial Innovation).  
The authors declare no conflict of interest.