Kriging-based optimization design for a new style shell with black box constraints

Abstract

Complex engineering applications generally have the black box and computationally expensive characteristics. Surrogate-based optimization algorithms can effectively solve expensive black box optimization problems. This paper employs the kriging predictor to construct a surrogate model and uses an initial multistart optimization process to realize the global search on this kriging model. Based on a proposed trust region framework, a local search is carried out around the current promising solution. The whole optimization algorithm is implemented to solve a new style shell design of the autonomous underwater vehicle. Based on finite element analysis, buoyancy–weight ratio, maximum von Mises stress, and buckling critical load of the new style shell are calculated and stored as expensive sample values to construct the kriging model. Finally, the better design parameters of the new shell are obtained by this proposed optimization algorithm. In addition, compared with the traditional shell, the new shell shows the stronger stability and better buoyancy–weight ratio.

Keywords

Autonomous underwater vehicle shell design surrogate-based optimization kriging black box problem

Introduction

Autonomous underwater vehicles (AUVs) are unmanned, self-propelled robotic devices, which have an increasingly ubiquitous role in the scientific, commercial, and military fields in recent years.^1,2 AUVs were originally deployed for marine geosciences, which normally had a torpedo-shaped streamlined body, but some with more complex configurations for special assignments. With the economic, military divers increasing rapidly, these vehicles are employed as platforms to collect marine data on a large scale.³ Meanwhile, substantial scientific researches and engineering projects are concentrated on AUVs and higher design standards are proposed.⁴ In the future, the design of AUVs’ shell structure must gradually meet the requirement of large operational depth and be able to adapt the hostile environments of marine better.

The traditional design of AUVs’ shell structure is based on the empirical formulas of torpedo shell.⁵ The main torpedo shells are ring-stiffened cylindrical shells (RSCSs) which are widely used in various submersible vehicles. However, as the depth of the marine increases, RSCSs cannot provide the permissible stress and better stability. Although, advanced composite material can enhance the performance of AUVs’ shell, the cost of production will be added enormously. Furthermore, the expenses of the compressive strength testing are also quite expensive. Hence, in the conceptual design stage, changing AUVs’ shell structure form by computer-aided engineering and numerical analysis to improve its strength and stability turns into a significant means.⁶

In previous studies, various types of submersible vehicles’ hulls have been analyzed. Yuan et al.⁷ presented a novel swedged-stiffened pressure hull and investigated how the cone angle of the hull would affect the entire performance. Liang et al.⁸ researched on the geometric parameters, gravity–buoyancy ratio, high strength material of the spherical and elliptical shells using the finite element software. Li et al.⁹ did research on nonlinear stability of a manned submersible and gave the relationship of buckling critical pressure, thickness, and radius. Hsu et al.¹⁰ utilized the finite element program of Hibbitt and Karlsson to analyze the failure mode of the spherical shell with a circular hole. The static mechanical property of multiple intersecting spherical (MIS) deep-submerged pressure hulls was investigated theoretically and numerically by Wu et al.¹¹ Song et al.¹² did further research on MIS shells and analyzed the feasibility of the hulls employed in AUVs.

In this paper, according to the actual mission requirement of large operational depth, one kind of ring-stiffened MIS shell (RSMISS) appropriate for AUVs is provided. In the process of conceptual design, in order to offer the optimal consequence for the subsequent detailed design, surrogate-based optimization (SBO) is employed.

SBO^13–16 proves to be an efficient and feasible approach for simulation-based conceptual design. Generally, SBO includes three phases:^17–20 infill criterion, constructing surrogate models (SMs), and optimizing SMs. Infill criterion is a plan of sampling, which provides a set of suitable sample points to construct an accurate SM. In engineering design,^21,22 there exist several frequently used construction methods that are the response surface method (RSM),²³ the kriging interpolation method, and the radial basis function (RBF) method, respectively. RSM is the classical regression method and is widely used in engineering design, but it is difficult to deal with the multidimensional model. RBF²⁴ and Kriging²⁵ are most prominent and commonly used for interpolation. RBF is represented as a sum of basis functions each associated with an appropriate coefficient to emulate complicated design landscapes and Kriging treats the system response as a realization of a stochastic process to predict the nonlinear problem better. There are plenty of papers about SBO and many scholars show great interest in this subject used in complex engineering applications. In a review paper, Forrester and Keane²⁶ introduced an attractive SBO method. This approach utilized one local infill criterion called “minimizing the predictor” (MP) which minimizes the predictor and runs several true function evaluations within the neighboring region around the optimal value. After several iterations, the SM is sufficiently reconstructed near the exact local optimal solution and the agreeable solution is obtained. However, this method depends on the accuracy of the initial SM and cannot solve multimodal engineering problems. What’s more, if the optimal solution of the predictor is far away from the true optimal point, it will cost lots of expensive evaluations.

In this paper, according to the characteristics of RSMISSs, the kriging-based model was used and a SBO algorithm was proposed. Since RSMISSs have the more complex structure, there exist nonlinear relations between its geometric parameters and response outputs. Kriging can better predict the low-dimensional nonlinear model, thus this algorithm was developed based on the kriging model. Meanwhile, in order to possess the better parallelism, the algorithm involves a multistart optimization method²⁷ and a local infill criterion called “enhanced MP” (EMP). The presented EMP method explores the unknown region by a management framework of SMs, which can dynamically adjust the search direction and the length of sampling radius to decrease the computational cost. Ultimately, the proposed SBO algorithm was implemented to obtain the optimal design of the presented AUVs’ RSMISSs. Through comparative analysis, RSMISSs performed better than the traditional RSCSs on AUVs.

Proposed SBO algorithm

First of all, several sample points will be obtained by DOE^28,29 (in this paper, optimal Latin hypercube sampling (OLHS)³⁰ is employed) to construct the kriging-based SM. These points need to be evaluated using expensive simulation results. The expensive sample points and values are archived in a database where the subsequent resampled points can also be stored. After the initial SM is constructed, a multistart optimization method is used, which utilizes multiple sequential quadratic programming optimizers from different starting points to search the local optimal solutions of the SM. Simultaneously, these local optimal solutions will be evaluated and added to the previous database. One local optimal value of the SM is not always around the true local optimal value of the actual problem. When the optimization process goes on, the current local optimal position of the SM may change.²⁷ This process is repeated until several different local optimal solutions go close to some similar regions.

After the promising regions of the SM are confirmed, the local search begins to run. Among these local optimal values of the SM, the first to be considered is the minimum value (if it is a minimization problem). In order to obtain better design, an EMP method is proposed to fully explore the local region with fewer simulating calculations. In “Local infill criterion: EMP” section, a detailed introduction will be provided.

Kriging-based SM

As our previous discussion, kriging is an interpolation method to predict the value of a function at an untested point by computing a weighted sum of the known values of the function in the neighborhood of the point.³¹

Kriging defines the correlation model between two points x ¹ and x ² as follows

R (Θ, x^{1}, x^{2}) = Π_{j = 1}^{n} R_{j} (θ_{j}, x_{j}^{1} - x_{j}^{2})

(1)

Here, n is the dimension of sample points. If the Gaussian correlation function is employed, it is formulated as

R_{j} (θ_{j}, x_{j}^{1} - x_{j}^{2}) = exp (- θ_{j} | x_{j}^{1} - x_{j}^{2} |^{2})

(2)

Generally, the predictor and the estimated mean squared error (MSE) of kriging can be used to explore the design space in a SBO process. Based on the correlation model introduced above, the predictor and MSE are expressed as follows

\hat{f} (x) = \hat{μ} + r^{T} (x) R^{- 1} (y - 1 \hat{μ})

(3)

{\hat{s}}^{2} (x) = {\hat{σ}}^{2} [1 - r^{T} (x) R^{- 1} r (x) + \frac{(1 - 1^{T} R^{- 1} r (x))^{2}}{1^{T} R^{- 1} 1}]

(4)

Assume that there are M sample points $s^{(1)}, s^{(2)}, \dots, s^{(M)}$ and their true values are evaluated $y = [y (s^{(1)}), y (s^{(1)}), \dots, y (s^{(M)})]^{T}$ . In Equations (3) and (4), R is an $M \times M$ correlations matrix whose (i, j) entry is defined as $R (Θ, s^{(i)}, s^{(j)})$ and $r (x) = [R (Θ, x, s^{(1)}), R (Θ, x, s^{(2)}), \dots, R (Θ, x, s^{(M)})]^{T}$ . Where, parameters $\hat{μ}, {\hat{σ}}^{2}, Θ$ can be obtained by maximum likelihood estimation.

The MSE function reflects the uncertainty of the kriging predictor. The approximation accuracy depends on the distance between untested location and the given sample points. Intuitively, the kriging predictor can perform better if the untested location is closer to the sample points. Here, a graphical example on Himmelblau function is employed to demonstrate the prediction capacity of kriging. As Figure 1 shows the original function, and Figure 2 shows the kriging model constructed by 25 sample points. The whole shape in Figure 2 is similar to the true function in Figure 1.

Figure 1.

Original Himmelblau function.

Figure 2.

Predictive model by kriging.

Figure 3.

Proposed surrogate-based optimization algorithm.

Local infill criterion: EMP

This section introduces the EMP method, which includes a trust region framework to determine when to change the search region and direction.

For minimization problems, the local minimum solution of the SM is defined as the initial point x ₀ and meanwhile y( x ₀) is evaluated to renew the SM. Due to the better approximation performance of kriging in the vicinity of the known sample points, a search area is set to search the next point x ₁. The initial δ ₀ is given at user-defined value. The next point can be acquired by exploring the interval $x_{0} \pm {\underline{δ}}_{0}$ . The new SM will be rebuilt through ( x ₀, y( x ₀)) ( x ₁, y( x ₁)) and the initial samples. The parameter δ ₁ will also be updated as follows. Equation (5) is used to evaluate the performance of the prediction

r = \frac{y (x_{m - 1}) - y (x_{m})}{y (x_{m - 1}) - \hat{y} (x_{m})}

(5)

y( x _m _-1) is the evaluation at the prior point x _m ₋ ₁. Likewise, y ( x _m ) is the evaluation at the current point x _m . The response value of SM at the current point x _m is $\hat{y} (x_{m})$ . The new size of the search interval can be calculated through Equation (6)

δ_{m} = {{c_{1} | x_{m}^{1} - x_{m - 1}^{1} |, c_{1} | x_{m}^{2} - x_{m - 1}^{2} |, \dots c_{1} | x_{m}^{n} - x_{m - 1}^{n} |} if r < r_{1} {c_{2} | x_{m}^{1} - x_{m - 1}^{1} |, c_{2} | x_{m}^{2} - x_{m - 1}^{2} |, \dots c_{2} | x_{m}^{n} - x_{m - 1}^{n} |} if r > r_{2} {| x_{m}^{1} - x_{m - 1}^{1} |, | x_{m}^{2} - x_{m - 1}^{2} |, \dots | x_{m}^{n} - x_{m - 1}^{n} |} otherwise

(6)

The parameter n denotes the dimension of the design vector. The coefficients c₁ and c₂ determine how much the shrinkage or expansion of the new search region will be, and the parameters r₁ and r₂ determine when to shrink or expand. In summary, the search region will expand when prediction performs better, and otherwise the region will shrink. In addition, since design variables in most of engineering applications have different units, the distance between two locations is calculated on each dimension.

Although local search runs in a trust region, the right direction may still be lost. Since the search region expands, the approximation performance of the kriging-based SM will deteriorate. Meanwhile, it is difficult to utilize the gradient of the exact problem. Here a parameter ɛ is set as Equation (7)

ɛ = {0 if y (x_{m}) > y (x_{m - 1}) and y (x_{m - 1}) > y (x_{m - 2}) 1 otherwise

(7)

ɛ is defined as 0 or 1. (For minimization problems) When ɛ is 1, it means that the next point performs better than the prior one or the next point may have chance to change the direction into the right way and iterations go on along the direction based on the trust region framework. On the contrary, when ɛ is 0, it means the next two points lose the right direction and d + 1 points will be obtained by OLHS (d is the dimension of x ) in the vicinity of x _m ₋ ₂ to improve the approximation effect of the SM there. The size parameter δ of resampled area around x _m ₋ ₂ is $(c_{1} | x_{m}^{1} - x_{m - 2}^{1} |, c_{1} | x_{m}^{2} - x_{m - 2}^{2} |, \dots c_{1} | x_{m}^{n} - x_{m - 2}^{n} |)$ . Here, n denotes the dimension. If it is a constrained optimization problem, y( x ) should include a penalty factor. After the SM is updated, x _m ₋ ₂ will be regarded as x ₀ and δ ₀ will be $(| x_{m}^{1} - x_{m - 2}^{1} |, | x_{m}^{2} - x_{m - 2}^{2} |, \dots | x_{m}^{n} - x_{m - 2}^{n} |)$ . The algorithm will go on from the current x ₀ and repeat the above process until it satisfies the stopping criterion described below

{g (x_{m}) \leq 0 | f (x_{m}) - y_{goal} | \leq 0.5 % \cdot | y_{goal} |

(8)

Here, is the expected value of a true black box problem. The Equation $g (x_{m}) \leq 0$ denotes that the constraint needs to be satisfied.

RSMISS for AUVs

For a torpedo-shaped AUV, the security of its parallel middle section is always considered primarily. This is because the middle section has the largest diameter, which is more liable to failure. Besides, the inner space of the middle section has many important instruments to be protected. The traditional shell of the torpedo-shaped AUV is the RSCS that is commonly designed under the condition of about 500 m depth. As the depth of water increases, the mass will get larger and the shell will be more unstable. Therefore, a kind of new shell structure (RSMISS) was proposed. The MIS shell has proven to have high strength and high stability. However, no one tries to utilize it on AUVs. In order to keep more spacious internal space of AUVs, ring-shaped ribs proposed by Liang et al.³² are setup around the outer hull as shown in Figure 4. According to the cylindrical characteristic of torpedo-shaped AUVs, the RSMISS was provided and analyzed on AUVs.

Figure 4.

Ring-stiffened MIS shell of AUVs. AUV: autonomous underwater vehicle.

The structural style and design parameters of RSMISSs are shown in Figure 4. Among these parameters, R defined as a constant denotes the radius of the shell. B₁, B₂, h are the width and height of main and auxiliary ribs, respectively. Since auxiliary rib is responsible for the safety and airtightness of bolted connection between two spherical shells and auxiliary rib can be regarded as the extension of the main rib, the parameter B₂ will be defined equal to B₁ in the subsequent optimization process. The distance between the top of the rib and the horizontal axis is set equal to R. At last four design variables are confirmed as follows: the thickness of shell t, the radius of face blend r, the radian of shell α, and the width of main rib B₁.

In this paper, R is 162 mm, the depth of water is 1000 m, and four multiple intersecting spheres are analyzed. The mechanical properties of the aluminum alloy are shown in Table 1, which are used in the finite element analysis (FEA).³³ Buoyancy–weight ratio, maximum von Mises stress, and buckling critical load of the RSMISS are calculated as response values, respectively.

Table 1.

Mechanical properties of aluminum alloy.

Properties	Aluminum alloy
Young’s modulus	70 GPa
Density	2800 kg/m³
ultimate strength	450 MPa
Poisson’s ratio	0.3

SBO on AUVs’ RSMISS

As our previous discussion, there are four design variables and three response values in this optimization problem. The objective is maximizing the buoyancy–weight ratio. In the step of conceptual design, two security coefficients are proposed in constraints. The maximum von Mises stress needs to be smaller than 85% of the ultimate strength. Meanwhile, the buckling critical load should be more than 120% of the calculating pressure that is equal to 12 MPa in this paper. Generally, a reference solution near the constrained boundary should be provided which can contribute to set the design range. In this paper, the reference solution is (4.25, 5, 5, 1.003) and the design range is set around it. In order to meet the requirement of inner space and processing feasibility, the specific optimization expression is formulated as follows

MaxBG (t, B_{1}, r, α) st . 2.5 mm \leq t \leq 6 mm 3 mm \leq B_{1} \leq 7 mm 3 mm \leq r \leq 7 mm 0.6981 rad \leq α \leq 1.309 rad 0 \leq σ_{max} (t, B_{1}, r, α) \leq 0.85 σ_{s} 1.2 P_{j} \leq Pcr (t, B_{1}, r, α)

(9)

According to the SBO process shown in Figure 3, first 3d + 2 sample points were obtained by OLHS and evaluated by FEA. Three kriging models were constructed by these initial sample points, which can predict buoyancy–weight ratio, maximum von Mises stress, and buckling critical load of the RSMISS. Since the initial DOE just provided fewer sample points, the accuracy of the predictor was low and some unreal local optimal locations appeared. In this paper, the multistart optimization approach was employed to acquire these local optimal locations of kriging models and the predictor got updated. When two adjacent solutions were closer and the feasible solution appeared, the multistart optimization stopped. In this example, five iterations were carried out and a local area was confirmed, which is a neighboring region around $(t, B_{1}, r, α) = (3.62, 4.84, 6.86, 0.7887)$ .

Table 2 shows intermediate results about the multistart optimization process. From Table 2, a discovery can be obtained that the number and locations of local optimal solutions change as the iteration runs. Meanwhile, the overall accuracy of the SMs improves. After the fifth iteration, the feasible solution (3.62, 4.84, 6.86, 0.7887) was found that was closer to the solution (3.76, 4.40, 7, 0.8308) from the fourth iteration. According to the known samples, the four-variable nested plots of kriging models were provided by Figures 5 to 7 to express the overall tendency. In Figures 5 to 7, the parameters t and B₁ represent the horizontal and vertical axis, respectively. The variable r varies along the horizontal axis of each tile and α changes along the vertical axis of each tile. The color denotes the size of the predicted response values.

Figure 5.

Nested plot of the RSMISS’s buoyancy–weight ratio. RSMISS: ring-stiffened MIS shell.

Figure 6.

Nested plot of the RSMISS’s maximum von Mises stress. RSMISS: ring-stiffened MIS shell.

Table 2.

The local optimal solutions of the kriging-based predictor and response values solved by finite element analysis.

The local optimal solutions	Design variables				Objective	Constraints		Feasibility
The local optimal solutions	t (mm)	B₁ (mm)	r (mm)	a (rad)	BG	σ_max (MPa)	Pcr (MPa)	Yes/No
After the first multistart optimization	3.67	5.76	7.00	0.6981	4.248	340.40	56.429	Yes
	6.00	7.00	7.00	0.6981	2.906	237.23	124.056	Yes
	6.00	4.16	3.00	0.6981	3.242	364.19	89.311	Yes
	6.00	4.26	7.00	0.6981	3.156	262.25	95.072	Yes
	6.00	7.00	7.00	1.3090	3.745	318.50	13.476	No
After the second multistart optimization	2.50	5.41	7.00	0.7635	5.746	473.91	26.394	No
	2.50	6.12	7.00	0.7907	5.639	465.29	26.452	No
	6.00	4.23	7.00	1.3090	3.798	334.28	13.396	No
	2.50	6.37	6.61	0.7573	5.457	433.88	32.575	No
	5.34	4.92	7.00	1.3090	4.206	368.27	10.798	No
After the third multistart optimization	2.86	5.59	5.52	0.7285	5.212	420.38	39.646	No
	4.85	5.40	3.00	1.0313	4.283	477.43	11.295	No
	5.34	4.11	3.00	1.1784	4.227	487.96	9.008	No
After the fourth multistart optimization	3.76	4.40	7.00	0.8308	4.764	391.97	32.542	No
	2.50	6.71	4.95	0.8369	5.804	439.43	35.437	No
	5.62	7.00	3.35	1.1326	3.824	408.84	9.497	No
After the fifth multistart optimization	3.62	4.84	6.86	0.7887	4.708	374.85	44.608	Yes
	2.50	7.00	5.56	0.7286	5.241	426.82	34.762	No
	6.00	5.03	7.00	1.2573	3.745	343.92	12.225	No
	5.09	4.39	7.00	1.2147	4.320	380.18	9.551	No
Best solution	3.62	4.84	6.86	0.7887	4.708	374.85	44.587	Yes

Figure 7.

Nested plot of the RSMISS’s buckling critical load. RSMISS: ring-stiffened MIS shell.

In order to further improve the performance of the shell structure, EMP was employed for local search. Based on these obtained results in the preliminary design, designers need to provide an expected objective value y_goal as Equation (8) advises. In this paper, y_goal is set equal to 4.85. Simultaneously, the initial point and size parameter of the search region are defined as $x_{0} = (3.62, 4.84, 6.86, 0.7887)$ and ${\underline{δ}}_{0} = (0.14, 0.45, 0.15, 0.05)$ , respectively. In this example, the objective function y( x ) was modified by supplementing a penalty factor 1000 to implement Equation (7), when design point x does not meet the constraints.

The maximization problem can be transformed into a minimization one, when the objective function multiplies −1. On the basis of the optimization process proposed by Figure 3, the EMP method ran and the procedure’s results are shown in Table 3. In the process of EMP, after four iterations the right direction was lost. The second iteration got the objective value −4.791, the third iteration got −4.910 + 1000, and the fourth iteration got −4.886 + 1000. According to Equation (7), five sample points need to be added to update the kriging models. Eventually, the satisfactory solution was obtained with 10 evaluations by FEA. On the other hand, as Table 4 suggests, the MP method ran with 12 evaluations, while it did not get a better solution.

Table 3.

The local search process with the EMP infill criteria.

The EMP iteration procedures	Design variables				Objective	Constraints		Feasibility
The EMP iteration procedures	t (mm)	B₁ (mm)	r (mm)	a (rad)	BG	σ_max (MPa)	Pcr (MPa)	Yes/No
The first iteration	3.5643	5.0739	6.7113	0.8254	4.821	386.40	36.070	No
The second iteration	3.4821	4.7505	6.8366	0.7621	4.791	376.39	50.028	Yes
The third iteration	3.3588	4.3853	6.7970	0.7282	4.910	387.94	50.054	No
The fourth iteration	3.3943	4.7290	6.7763	0.7623	4.886	388.59	48.869	No
Confirm the right search direction by OLHS	3.4548	4.7655	6.8585	0.7620	4.813	391.52	49.817	No
	3.5093	4.7571	6.8187	0.7622	4.765	378.79	50.266	Yes
	3.4821	4.7402	6.7988	0.7620	4.795	387.93	49.961	No
	3.5365	4.7486	6.8785	0.7621	4.737	387.35	50.611	No
	3.4310	4.7350	6.8410	0.7622	4.844	395.19	49.380	No
The fifth iteration	3.4276	4.7318	6.8386	0.7622	4.848	382.37	49.417	Yes
Best solution	3.4276	4.7318	6.8386	0.7622	4.848	382.37	49.417	Yes

EMP: enhanced minimizing the predictor; OLHS: optimal Latin hypercube sampling.

Table 4.

The local search process with the MP infill criteria.

The MP iteration procedures	Design variables				Objective	Constraints		Feasibility
The MP iteration procedures	t (mm)	B₁ (mm)	r (mm)	a (rad)	BG	σ_max (MPa)	Pcr (MPa)	Yes/No
First OLHS to update the Kriging-based model	3.4821	5.2923	6.8556	0.8137	4.816	374.72	39.546	Yes
	3.5521	4.3923	6.9278	0.7637	4.802	399.24	48.408	No
	3.6921	5.0673	6.7834	0.7387	4.479	352.37	56.726	Yes
	3.6221	4.6173	6.7113	0.8387	4.893	394.18	31.355	No
	3.7621	4.8423	7.0000	0.7887	4.576	379.32	45.446	Yes
Optimal solution	3.3548	5.2895	6.7000	0.8118	4.945	399.93	39.344	No
Second OLHS to update the Kriging-based model	3.4248	5.5145	6.8500	0.7868	4.750	367.03	47.239	Yes
	3.4948	5.2895	6.7000	0.8618	4.950	398.08	29.642	No
	3.2848	5.7395	6.6250	0.8468	5.032	429.70	33.171	No
	3.3548	5.0645	6.5500	0.7618	4.860	393.82	49.028	No
	3.2148	4.8395	6.7750	0.8118	5.193	408.69	36.491	No
Optimal solution	3.3001	5.1456	6.8471	0.7630	4.884	384.25	48.1872	No
Best solution	3.4821	5.2923	6.8556	0.8137	4.816	374.72	39.546	Yes

MP: minimizing the predictor; OLHS: optimal Latin hypercube sampling.

The simulation results which involve the distribution of von Mises stress and the total deformation by FEA are displayed in Figures 8 to 15. Figures 8 and 9 show the simulation results of the initial reference design mentioned above which cannot meet the constraint conditions. Figures 10 and 11 show the best results obtained after the multistart optimization process. Figures 12 and 13 show the best results obtained after the local search by EMP. Figures 14 and 15 provide the corresponding consequences of the RSCS, which are the optimal response values under the same operating conditions as the RSMISS. For the sake of a better visuality, the figures of total deformation are magnified by a factor of 28. The obtained optimal design parameters of the RSCS are $(t, l, t_{2}, l_{2})$ = $(4.91, 126.85, 13.96, 16.20)$ and its buoyancy–weight ratio is 4.32. Here, t denotes the thickness of the shell, l means the frame spacing, t₂ is defined as the width of a rib, and l₂ represents the height of a rib. The detailed analysis can be found from the authors’ previous research.⁵ The results show that the buoyancy–weight ratio of the best RSMISS is 12.2% larger than that of the optimal RSCS. Meanwhile, the best RSMISS provides a 235% higher loading capacity than the RSCS does.

Figure 8.

Distribution of von Mises stress at the RSMISS’s reference solution. RSMISS: ring-stiffened MIS shell.

Figure 9.

Total deformation at the RSMISS’s reference solution. RSMISS: ring-stiffened MIS shell.

Figure 10.

Distribution of von Mises stress at the RSMISS’s best solution obtained after the multistart optimization process. RSMISS: ring-stiffened MIS shell.

Figure 11.

Total deformation at the RSMISS’s best solution obtained after the multistart optimization process. RSMISS: ring-stiffened MIS shell.

Figure 12.

Distribution of von Mises stress at the RSMISS’s best solution obtained after the local search. RSMISS: ring-stiffened MIS shell.

Figure 13.

Total deformation at the RSMISS’s best solution obtained after the local search. RSMISS: ring-stiffened MIS shell.

Figure 14.

Distribution of von Mises stress at the RSCS’s optimal solution. RSCS: ring-stiffened cylindrical shell.

Figure 15.

Total deformation at the RSCS’s optimal solution. RSCS: ring-stiffened cylindrical shell.

As Table 3 shows, the initial kriging model was optimized for the first time and the best solution $(3.67, 5.76, 7, 0.6981)$ was acquired. The obtained solution is feasible and its buoyancy–weight ratio is 4.25. After the multistart optimization process implemented, the best performance is improved from 4.25 to 4.71 and the rate of increase is 10.82%. In addition, Table 3 indicates that the local infill criterion (EMP) improves the performance from 4.71 to 4.848, the growth rate is 2.9%. Simultaneously, the comparison of Tables 3 and 4 suggests that the EMP method can find the agreeable solution more quickly than MP. In conclusion, the proposed SBO process effectively improves the performance of RSMISSs at the concept design stage and the RSMISS proves to have a better property than the RSC under the condition of large operational depth.

Conclusion

According to the torpedo-shaped AUVs’ characteristics, a kind of RSMISS is proposed in this paper. In order to explore the black box model, a SBO algorithm is provided to acquire the optimal design. Based on FEA, the SBO process finds the satisfactory solution with fewer simulating evaluations. In addition, compared with the traditional RSCS, the RSMISS performs better and may be more suitable for AUVs in the future. The main conclusion can be addressed as follows:

The RSMISS is proposed for AUVs. Compared with the traditional RSCS, it can provide larger buoyancy–weight ratio and higher loading capacity (better stability).

The proposed SBO algorithm involves the global and local exploration on the kriging model. The multistart optimization approach is utilized to do a global search until a local promising region is obtained. The presented EMP method is employed for local search in this promising region.

An EMP method is presented. The EMP method can sequentially supplement samples and change the size of the trust region in the process of local search. The results show that EMP can find the satisfactory solution with fewer simulating evaluations than MP.

Footnotes

Acknowledgment

The author is grateful to the editor and the anonymous referees for their insightful and constructive comments and suggestions, which have been very helpful for improving this paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (Grant No. 51375389) and the National High Technology Research.

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