Computational fluid dynamics-based hull form optimization using approximation method

Figures & data

Table 1. Parameters.

Method	N	c1	c2	ω			Δd	θ	η
PSO	40	2	2	0.5	–	–	–	–	–
IPSO	40	2	2	–	0.8	0.5	1e^−4	0.2	0.1

Table 2. Optimization results of different algorithms.

	D	Minimum value	PSO	IPSO	IPSO (Li, 2012)
f1(x) (Rosenbrock)	10	0	5.3067	0	4.855
f2(x) (Schaffer)	10	0	0.009716	0	0.0097
f3(x) (Rastrigrin)	10	0	16.9250	0	3.669
f4(x) (Griewank)	10	0	1.0277	0	0.0694

Figure 1. Convergence history of the different algorithms (a) f₁(x), (b) f₂(x), (c) f₃(x), (d) f₄(x).

Figure 2. Geometry deformation using two parameters.

Figure 3. Grid division of computational domain and hull (a) Mesh of the computational domain, (b) Computational mesh on the ship geometries.

Figure 4. Comparison of samples (a) Random latin hypercube design, (b) Optimal latin hypercube design (above: Present study, below: Wang & Lu, 2015).

Figure 5. Space distributions of samples (a) DTMB5512, (b) WIGLEYIII.

Table 3. Experiment samples.

	DTMB5512					WIGLEY III
No.	a11	a12	a21	a22	CT1*10^−3	b11	b12	b21	b22	CT2*10^−3
1	−0.0201	−0.02442	0.0623	−0.0479	4.341	−0.0369	−0.5494	−0.061	−0.00275	5.536
2	−0.2533	−0.03809	0.0573	−0.0468	4.373	−0.3791	−0.5237	−0.6747	0.04178	5.206
3	−0.1065	−0.09362	−0.1148	0.0884	4.343	−0.0932	−0.7357	−0.0161	0.00361	5.577
4	−0.201	−0.02528	−0.0143	−0.0214	4.342	−0.3068	−0.2731	−0.2924	0.03976	5.369
5	−0.3075	−0.08337	−0.0721	−0.1115	4.472	−0.249	−0.0867	−0.7229	0.04294	5.297
6	−0.2432	−0.01503	−0.145	−0.019	4.341	−0.008	−0.3373	−0.1221	0.036	5.426
7	−0.3176	−0.0808	−0.0759	−0.0329	4.387	−0.3936	−0.3502	−0.1896	0.02993	5.311
8	−0.0422	−0.00392	−0.1136	0.0769	4.352	−0.1044	−0.5783	−0.1542	0.01287	5.502
9	−0.0683	−0.10131	−0.0683	−0.0075	4.497	−0.2506	−0.1382	−0.4305	−0.00969	5.402
10	−0.203	−0.13975	0.0824	0.0561	4.383	−0.3438	−0.045	−0.3181	0.04612	5.536
…	…	…	…	…	…	…	…	…	…	…
…	…	…	…	…	…	…	…	…	…	…
197	−0.2915	−0.12779	−0.057	−0.0098	4.493	−0.355	−0.4016	−0.7454	0.02212	5.211
198	−0.3437	−0.06286	0.0673	0.0792	4.361	−0.3213	−0.0257	−0.2024	0.01402	5.291
199	−0.3276	−0.07739	0.0523	0.0098	4.362	−0.1141	−0.3181	−0.6008	0.05913	5.358
200	−0.2854	−0.10387	−0.0043	0.033	4.451	−0.0273	−0.498	−0.3566	0.00448	5.502

Figure 6. The framework of ElmanNN (above: Present study, below: Wang et al., 2011).

Figure 7. Prediction results of C_T1 for DTMB5512ship (a) The prediction of the C_T1using ElmanNN, (b) The prediction of the C_T1using IPSO-ElmanNN.

Figure 8. Prediction results of C_T2 for WIGLEYIIIship (a) The prediction of the C_T2using ElmanNN, (b) The prediction of the C_T2using IPSO-ElmanNN.

Table 4. Result of the total resistance coefficient prediction based on the different algorithms.

	Average Error (%) for 200 sampling hull forms
Training algorithms	DTMB5512	WIGLEY III
ElmanNN	8.89*10^−5	1.41*10^−4
IPSO-ElmanNN	3.2*10^−5	4.7*10^−5

Figure 9. Flow chart of the optimization loop.

Figure 10. Modified region of hull forms.

Table 5. Major characteristics of two ships.

Hull forms	DTMB5512	WIGLEY III
Design speed Fr	0.28	0.3
Length between perpendiculars Lpp(m)	3.048	3
Draft d (m)	0.132	0.1875
Breath B (m)	0.409	0.3
Displacement Δ (kg)	86.4	78
Block Coefficient CB	0.507	0.4622

Table 6. Control parameters of two ships.

Hull forms	Design variables	No.	Moving directions	Constraints
DTMB5512	a11	1	X	−0.4 ≤ a11 ≤ 0
	a12	1	Z	−0.15 ≤ a12 ≤ 0.02
	a21	2	Y	−0.15 ≤ a21 ≤ 0.1
	a22	3	Y	−0.13 ≤ a22 ≤ 0.1
WIGLEY III	b11	4	X	−0.4 ≤ b11 ≤ 0
	b12	5	Y	−0.8 ≤ b12 ≤ 0
	b21	6	Y	−0.8 ≤ b21 ≤ 0
	b22	4	Z	−0.012 ≤ b22 ≤ 0.06

Figure 11. Control positions of two ships.

Figure 12. The total resistance coefficients for a range of Fr numbers (a) DTMB5512, (b) WIGLEYIII.

Table 7. Resistance results of the optimized hulls.

Method	Parent hulls	Optimized hulls	Fr	CTorg/CTopt
IPSO-ElmanNN + IPSO	DTMB5512	Optimized hull-A	0.28	1.0591
	WIGLEY III	Optimized hull-B	0.3	1.05468

Figure 13. Comparison of hull lines (a) DTMB5512, (b) WIGLEYIII.

Figure 14. Comparison of wave profile at y/L_pp = 0.082 (a) DTMB5512, (b) WIGLEYIII.

Figure 15. Comparison of the wave patterns around the vessels (a) DTMB5512, (b) WIGLEYIII.

Figure 16. Comparison of the static pressure on the ship surfaces (a) DTMB5512, (b) WIGLEYIII.

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