*5.5. Optimization Results*

Direct optimization and optimization using the K–L transform–based dimensionality reduction reconstruction method were performed to compare their optimization performances and efficiencies. The results are discussed below.

#### (1) Results of direct optimization

Using the hydrodynamic performance optimization platform, the *Y*-coordinates of the 10 variable points were defined as the design variables, and the hull surface was modified by varying the coordinates using the RBF method. The WMR was calculated using SHIPFLOW. A particle swarm optimization algorithm was used, with both the number of particles and iterations set as 50. The hull surface was directly optimized.

The resulting optimal hull form had a WMR coefficient of *Cw* = 4.90 × <sup>10</sup>−<sup>4</sup> and a WMR of 3.328 N, whereas the original shape hull had a WMR coefficient of *Cw* = 6.83 × 10−<sup>4</sup> and a WMR of 4.647 N. Thus, direct optimization decreased the WMR coefficient and WMR by 28.25% and 28.38%, respectively. The optimized hull form had a larger volume of displacement and smaller LCB and wetted surface area than the original hull form. Figure 15 shows the optimization convergence curve. The objective function is observed to converge after approximately 3000 iterations. Tables 6 and 7 summarize the optimization

results and the variations in the hydrostatic parameters, respectively. Figures 16 and 17 compare the line plans of the optimized and original hull forms. The shape of the bulbous bow of the optimized hull form is not markedly different from that of the original hull form. The optimized hull form is concave near the bow part of the design waterline, making the design waterline appear slenderer and changing the geometric shape under the design waterline from V- to U-shaped compared with the original hull form.

**Figure 15.** Optimization convergence curve of direct optimization.

**Figure 16.** Comparison of body plan views of direct optimized and original hull forms.

**Figure 17.** Comparison of profile views of direct optimized and original hull forms.


**Table 6.** Optimization results of direct optimization.

**Table 7.** Comparison of hydrostatic parameters of direct optimization.


#### (2) Results of dimensionality reduction optimization

The uniform experimental design method was used to sample the 10 design variables (*Y*1–*Y*10) described above to obtain 200 samples. The hull form was varied using the RBF method to generate 200 sample hull forms, which were screened to retain only those shapes that satisfied the predefined constraints of the volume of displacement and LCB. The offset information of the sample hull forms that satisfied the constraints was used to generate the offset matrix. A total of 113 sample hull forms satisfied the constraint conditions. The 3D model of DTMB 5415 was divided into three surfaces. Each surface was discretized into a 40 × 30 point cloud, generating a total of 3600 points. Each point was divided into three dimensions of *x*, *y*, and *z*. Thus, the offset matrix for a sample hull form had a size of 113 × 10,800.

The dimensionality was reduced using the K−L transform procedure, with the threshold of *β* set as 95%. A total of 406,800 eigenvalues were obtained and arranged in a descending order. Except for the first 113 eigenvalues, all eigenvalues were approximately zero. The sum of the first 6 eigenvalues was 2.95 (Table 8), and the sum of the first 113 eigenvalues was 3.065. The first 10 eigenvalues were significant in magnitude (Table 9). The ratio of the sum of the first 6 eigenvalues to the sum of the total eigenvalues was 96.24%, which was larger than the preset threshold of *β* (95%); this can be seen from Figure 18. Figure 19 shows the distribution of the first 6 eigenvalues. Thus, the eigenvectors corresponding to the first 6 eigenvalues after dimensionality reduction constituted the transformation matrix, which was projected to obtain new design variables *P*1–*P*6.

The offsets of the reconstructed hull forms were obtained using Equation (35). The original hull form was optimized using the new design variables, *P*1–*P*6. The WMR coefficient was calculated using SHIPFLOW. A particle swarm optimization algorithm was used, with both the number of particles and iterations set as 50.



**Table 9.** Ten largest eigenvalues.


**Figure 18.** Contribution rates of eigenvalues.

**Figure 19.** Distribution of eigenvalues.

The optimized hull form had a WMR coefficient of *Cw* = 4.98 × <sup>10</sup>−<sup>4</sup> and a WMR of 3.394 N, representing decrease of 27.07% and 26.96% compared with those of the original hull form, respectively. The optimized hull form had a larger volume of displacement, larger wetted surface area, and slightly smaller LCB than the original hull form. Figure 20 shows the convergence curve for the WMR optimization. The objective function is observed to converge after approximately 1000 iterations. Table 10 shows the optimization results, and Table 11 compares the hydrostatic parameters of the original and optimized hull forms. Figures 21 and 22 compare the original and optimized hull forms. The bulbous bow shape of the optimized hull form does not vary markedly from that of the original hull form. The optimized hull form is concave near the design waterline at the bow, making the design waterline appear slenderer and changing the geometric shape under the design waterline from V- to U-shaped compared with the original hull form.

**Figure 20.** Optimization convergence curve of dimensionality reduction optimization.

**Figure 21.** Comparison of body plan views of dimensionality reduction optimized and original hull forms.

**Figure 22.** Comparison of profile views of dimensionality reduction optimized and original hull forms.




**Table 11.** Comparison of hydrostatic parameters of dimensionality reduction optimization.

### (3) Comparison of direct and dimensionality reduction optimization results.

Figures 23 and 24 compare the line plans of the hull forms obtained using the two different optimization methods. The bulbous bow shape obtained from the dimensionality reduction optimization is slightly slenderer than that obtained from the direct optimization. The hull form is concave in the design waterline near the bow, making the design waterline appear slender. Moreover, it is convex in that near the midship, making the design waterline appear fuller and changing the geometric shape under the design waterline from V- to Ushaped. Figures 25–27 compare the wave disturbances of the hull forms obtained using the two optimization methods and the original hull form. The original hull form has markedly more scattered waves than the hull forms obtained by both the direct and dimensionality reduction optimizations. However, the hull form obtained by the dimensionality reduction optimization has fewer scattered waves than that by the direct optimization, as indicated by the red ellipses. Figure 28 compares the transverse cross sections of the longitudinal waves at *y*/*lpp* = 0.12. Owing to the optimized bow shape, the hull forms obtained by the direct and dimensionality reduction optimizations have low-magnitude wave disturbances around the hull, which are favorable, and markedly low waves at the bow and midship, which decrease the WMR. Tables 12 and 13 compare the results and time consumptions and the hydrostatic parameters of the hull forms obtained by the two optimizations, respectively. The hull forms obtained by the two optimizations have larger volumes of displacement and LCBs closer to the stern than the original hull form. The optimal hull forms obtained by the two optimizations and the original hull form do not have significantly different wetted surface areas. The hull form obtained by the dimensionality reduction optimization has a larger wetted surface area, a larger volume of displacement, and an LCB closer to the stern than that obtained by the direct optimization. Compared with the WMR coefficient of the original hull form (*Cw* = 6.833 × <sup>10</sup>−4), those of the hull forms obtained by the direct and dimensionality reduction optimizations decrease by 28.25% and 27.07%, respectively. Compared with the WMR of the original hull form, those of the hull forms obtained by the direct and dimensionality reduction optimizations decrease by 28.38% and 26.96%, respectively. Thus, the results of the direct and dimensionality reduction optimizations do not differ significantly.

**Figure 23.** Comparison of body plan views of hull forms obtained by direct and dimensionality reduction optimizations.

**Figure 24.** Comparison of profile views of hull forms obtained by direct and dimensionality reduction optimizations.

**Figure 25.** Comparison of wave disturbances of original hull form and hull form obtained by direct optimization.

**Figure 26.** Comparison of wave disturbances of original hull form and hull form obtained by dimensionality reduction optimization.

**Figure 27.** Comparison of wave disturbances of hull forms obtained by direct and dimensionality reduction optimizations.

**Figure 28.** Comparison of transverse cross sections of longitudinal waves at *y*/*lpp* = 0.12.


**Table 12.** Comparison of optimization results.

**Table 13.** Comparison of hydrostatic parameters of three hull forms.


Both optimizations were performed using a computer equipped with a 6-core 12 thread i7-8700 CPU, 16 G memory, and 64-bit operating system. The direct optimization converged after approximately 3000 iterations, consuming 15 h. The dimensionality reduction optimization reduced the number of design variables from 10 to 6 and converged after 1000 iterations, consuming 5 h, representing a 75% time saving. This suggests that the K–L transform–based hull form reconstruction method effectively reduces the number of design variables, considerably reduces the time consumption of optimization, and improves the efficiency of optimization while yielding results that differ insignificantly from those by the direct optimization.

#### **6. Conclusions**

This study derived the equations for a K–L transform–based hull form reconstruction method and optimized the DTMB 5415 hull form by dimensionality reduction and direct optimizations. The following can be concluded from the results. The dimensionality reduction reduced the number of design variables, markedly reduced the number of iterations at optimization convergence, and significantly reduced the computation time compared with the direct optimization. The results yielded by the dimensionality reduction and direct optimizations differed insignificantly. This suggests that the K–L transform– based dimensionality reduction method reduces the time consumption and improves the optimization efficiency while retaining the variability of the design space. This proves the feasibility of the developed hull form optimization method combining the dimensionality reduction method and SHIPMDO-WUT.

During the study, several aspects could not be investigated thoroughly due to time constraints, and these are summarized as follows:


physical information of a new hull form can be directly obtained by dimensionality reduction reconstruction. This is a major direction for future research.

**Author Contributions:** Conceptualization: C.W., H.C. and Z.L.; methodology: C.W., H.C. and Z.L.; software: B.F., C.Z. and X.C.; validation: C.W. and H.C.; formal analysis: C.W., H.C. and Z.L.; data curation: C.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (grant numbers 51979211, 52271327, 52271330), Key Research and Development Plan of Hubei Province (2021BID008), and 111 Project (BP0820028).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available in this article (tables and figures).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**


#### **References**


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**Hui Li 1, Yan Feng 1,\*, Muk Chen Ong 2, Xin Zhao <sup>1</sup> and Li Zhou <sup>3</sup>**


**Abstract:** Selecting an optimal bow configuration is critical to the preliminary design of polar ships. This paper proposes an approach to determine the optimal bow of polar ships based on present numerical simulation and available published experimental studies. Unlike conventional methods, the present approach integrates both ice resistance and calm-water resistance with the navigating time. A numerical simulation method of an icebreaking vessel going straight ahead in level ice is developed using SPH (smoothed particle hydrodynamics) numerical technique of LS-DYNA. The present numerical results for the ice resistance in level ice are in satisfactory agreement with the available published experimental data. The bow configurations with superior icebreaking capability are obtained by analyzing the sensitivities due to the buttock angle *γ*, the frame angle *β* and the waterline angle *α*. The calm-water resistance is calculated using FVM (finite volume method). Finally, an overall resistance index devised from the ship resistance in ice/water weighted by their corresponding weighted navigation time is proposed. The present approach can be used for evaluating the integrated resistance performance of the polar ships operating in both a water route and ice route.

**Keywords:** bow configuration; level ice; ice resistance; sensitivity analysis; integrated evaluation
