A Double-Stage Surrogate-Based Shape Optimization Strategy for Blended-Wing-Body Underwater Gliders

In this paper, a Double-stage Surrogate-based Shape Optimization (DSSO) strategy for Blended-Wing-Body Underwater Gliders (BWBUGs) is proposed to reduce the computational cost. In this strategy, a double-stage surrogate model is developed to replace the high-dimensional objective in shape optimization. Specifically, several First-stage Surrogate Models (FSMs) are built for the sectional airfoils, and the second-stage surrogate model is constructed with respect to the outputs of FSMs. Besides, a Multi-start Space Reduction surrogate-based global optimization method is applied to search for the optimum. In order to validate the efficiency of the proposed method, DSSO is first compared with an ordinary One-stage Surrogate-based Optimization strategy by using the same optimization method. Then, the other three popular surrogate-based optimization methods and three heuristic algorithms are utilized to make comparisons. Results indicate that the lift-to-drag ratio of the BWBUG is improved by 9.35% with DSSO, which outperforms the comparison methods. Besides, DSSO reduces more than 50% of the time that other methods used when obtaining the same level of results. Furthermore, some considerations of the proposed strategy are further discussed and some characteristics of DSSO are identified.


Introduction
Underwater glider (UG) is a new kind of Autonomous Underwater Vehicle (AUV) which was first presented by Stommel (1989). Traditional AUVs always appear in a torpedo shape with a power propulsion system (Alam et al., 2014), while the UGs glide through the water by controlling their buoyancy and converting the lift on wings into propulsive force (Bachmayer et al., 2004). Since UGs have many merits over the traditional AUVs such as long-range, extended-duration and low costs, some typical gliders have already been successfully utilized in oceanographic sensing and data collection (Graver, 2005), such as Slocum (Webb et al., 2001), Spray (Sherman et al., 2001), Seaglider (Eriksen et al., 2001), and Deepglider (Osse and Eriksen, 2007). The shapes of the above mentioned UGs are mainly composed of revolving body, wings and control surfaces.
Lift-to-drag ratio (LDR) is one of the important indexes of UGs since a higher LDR can make UGs glide with a smaller gliding angle and the voyage can be extended in a gliding cycle. To this end, designing a UG with a higher LDR is of vital significance. Although the drag of revolving bodies is reduced in some studies (Stevenson et al., 2009;Ma et al., 2006), the LDRs of these torpedo shape UGs are still not high enough. To further improve the hydrodynamic performance of UGs, some blended-wing-body (BWB) shapes were applied to UGs (Hildebrand et al., 2009;ONR, 2006). The advantages of BWB shapes lie in higher maximum LDR and lower wetted area to volume ratio so that the UGs can glide farther in a gliding cycle.
In this study, a kind of BWBUG is optimized to gain a higher LDR. The LDRs of UGs are usually calculated with CFD-based simulations and a general optimization process requires calling the expensive simulation many times, which further increases the computation cost. In order to tackle this problem, a Surrogate-based Optimization (SBO) method is adopted here to alleviate the computational burden. SBO is widely used in modern engineering practice to reduce computational cost (Sacher et al., 2018;Wang et al., 2018). The representative methods for surrogate model (SM) construction include Polynomial Response Surface (PRS) method (Box and Draper, 1987), Support Vector Regression (SVR) (Smola and Schölkopf, 2004), kriging (Sacks et al., 1989), Radial Basis Function (RBF) (Mullur and Messac, 2005), etc. Among these methods, kriging is one of the most popular methods because it offers better approximations for nonlinear multidimensional problems and can provide the mean square error estimation at the to-be-predict locations.
SBO is also widely used for shape optimization of UGs. For example, Gu et al. (2009) compared several experimental design types and surrogate modeling techniques in terms of their capability to generate accurate approximations for the shape optimization of UGs, and the optimal plane shape of the UG was found based on the SM. Sun et al. (2015Sun et al. ( , 2017 applied the kriging-based Efficient Global Optimization (EGO) method to optimize the shape of the BWBUG. Results demonstrate that the lift-to-drag ratio or the maximum gliding range of the BWBUG is increased significantly. Wang et al. (2017) used the Gaussian kernel function algorithm to establish the SM of the flying-wing structure underwater glider, and the hydrodynamic performance of the UG is improved by the PSO algorithm. The above-mentioned shape optimization methods for UGs can reduce the computational cost compared with the direct simulationdriven optimization process. However, if one evaluation of the objective function is computationally expensive and the SBO process involves a large number of function evaluations, the whole optimization process might be still timeconsuming.
To cope with this deficiency, researchers have developed some alternative methods or simplified strategies to further reduce the computational cost in the engineering design. For example, multi-fidelity modeling techniques (Kuya et al., 2011;Leifsson et al., 2016) can be adopted to reduce the expensive evaluations if the objective function can be evaluated with different fidelities. Besides, based on the characteristics of a specific design problem, the optimization process can be simplified in some unique ways to alleviate the computational burden (Guo et al., 2013;Zhang et al., 2016). In this work, a novel Double-stage Surrogatebased Shape Optimization (DSSO) strategy is proposed based on the characteristics of the BWBUG shape to reduce the computational cost.
Different from the ordinary One-stage Surrogate-based Optimization (OSBO) methods (Dong et al., 2018(Dong et al., , 2019 which utilize an SM to approximate the output with respect to the original design variables, the DSSO strategy uses a Double-stage Surrogate Model (DSM) to optimize the BW-BUG. The First-stage Surrogate Models (FSMs) are constructed for the 2-D sectional airfoils and the Second-stage Surrogate Model (SSM) is constructed for the 3-D shape of the BWBUG by regarding the outputs of FSMs as the inputs. Then, a high dimensional SM is decomposed into several lower-dimensional SMs and the number of samples which are used to construct the SMs can be reduced. To fur-ther reduce the computational cost, an efficient Multi-start Space Reduction (MSSR) surrogate-based global optimization method (Dong et al., 2016) is used here to search for the optimum. The main contributions of this work include: (1) A DSSO strategy is newly proposed to reduce the computational cost for shape design optimization of BW-BUG.
(2) The proposed DSSO strategy is verified to be more efficient than the OSBO strategy in the test problem, which provides an alternative method for shape optimization of BWBUGs.
(3) A comparative study is made between DSSO and other six popular optimization methods, and the efficiency of the proposed strategy is validated.
(4) Some queries of the proposed DSSO strategy are further discussed and some characteristics of the DSSO strategy are illustrated.
The remainder of this paper is organized as follows. Section 2 describes the details of the shape optimization problem of the BWBUG. Section 3 illustrates numerical methods adopted in this paper. Then the proposed DSSO strategy is described in Section 4 and the MSSR method is also introduced in this section. In Section 5, the shape of the BWBUG is optimized by DSSO and OSBO strategies, respectively. Then, the other six popular optimization methods are used to make a comparative study. Besides, some queries are further discussed in Section 6 to get a better understanding of the DSSO strategy. In the end, the merits and deficiencies of this work are summarized.

Optimization problem description
Generally speaking, the shape of the BWBUGs is determined by the shapes of planform and cross-sections. For the sake of simplicity, the planform of the BWBUG shown in Fig. 1 is fixed in this work. Then, this BWBUG shape is only determined by the shapes of cross-sections. In order to reduce the complexity of the shape optimization problem, a BWBUG formed with three airfoil sections is used here as an example. The values of the planform parameters are listed in Table 1, where D 1 and D 2 are the offsets of Sections B and C along the X direction, respectively. L 1 is the distance between Sections A and B, and L 2 is the distance between Sections B and C. Besides, C 1 , C 2 and C 3 are the chord lengths of Sections A, B, and C. It should be noted that all the endpoints of the three sectional airfoils are in the XZ plane, so the Y coordinates of these endpoints are 0.
The design variables of the shape need to be defined first in this problem. As the shape of the BWBUG is mainly determined by three sectional airfoils, the airfoils need to be parameterized and the variables of the BWBUG are composed with these parameters. The Class-Shape function Transformation (CST) method is adopted in this paper to parameterize the airfoils, and the main formulas of the CST method are listed as Eqs. (1) ∆z te where C(x/c) is the class function, S(x/c) is the shape function, c is the chord length of the airfoil and is the trailing edge thickness. The exponents N 1 and N 2 define the type of geometry, and then an airfoil is generally represented by N 1 =1/2 and N 2 =1. W i denote the weights of the Bernstein polynomial and the coefficients K i,n can be calculated with Eq. (4).
where n is the order of the Bernstein polynomial and a fourorder Bernstein polynomial is chosen as the shape function in this work. Besides, a modified CST method is adopted in which the airfoil is parameterized based on a baseline airfoil. Then the upper part of an airfoil can be expressed as Eq. (5) and y 0 (x) refers to the baseline airfoil.
It can be seen from Eq. (5) that W i denotes the only parameters that need to be determined when representing an airfoil curve. As illustrated in Fig. 2, a symmetrical airfoil curve can be represented with five parameters and totally fifteen variables can describe the BWBUG.
In this work, the goal is to maximize the LDR of the BWBUG. Generally speaking, a thinner shape possesses a higher LDR but has a smaller inner space. However, it is essential for BWBUGs to have sufficient inner space which could hold more fuel and other essential equipment. In the process of parameterization, the standard NACA0022, NACA0016 and NACA0010 airfoils are selected as the baseline airfoil of Sections A, B and C, respectively. In order to make sure that the optimized shape possesses enough volume, the maximum relative thicknesses (MRTs) of the sectional airfoils are required to be larger than that of the baseline airfoils. Then the optimization problem can be expressed in Eq. (6).
where X represents the design variables of the problem, B L and B U are the lower and upper boundaries of the design space, T A0 , T B0 and T C0 denote the MRTs of the three baseline airfoils and are set as 0.22, 0.16 and 0.10, respectively. T A (X), T B (X) and T C (X) refer to the calculated MRTs of the three sectional airfoils in the optimization process, separately.

Numerical method
In this work, the LDRs of airfoils and UGs need to be calculated based on computational fluid dynamics (CFD). The commercial software ANSYS is adopted here to perform these calculations. In order to reduce the manual workload of the settings, the replay script in ANSYS ICEM and journal file in ANSYS Fluent are combined to build an automatic calculation procedure. The details of the simulation method in this paper are described below.

Domain and mesh settings
The structured mesh of the geometry is created by AN-SYS ICEM. The shape of the computational domain is a box-topology and the BWBUG is put in the center of the domain at the origin of the coordinates. The size of the domain is set to [−6C 1 , 9C 1 ] m×[−5C 1 , 5C 1 ] m×[0, 6C 1 ] m, where C 1 is the reference length of the computational model. In this work, the O-block grids are used to capture the viscous boundary layer and 674625 grids are generated totally, in which 100980 grids are generated in the O-block.   LI Cheng-shan et al. China Ocean Eng., 2020, Vol. 34, No. 3, P. 400-410 403 3.2 CFD solver ω The fluid material is water-liquid which is regarded as incompressible fluid with a density of 998.2 kg/m 3 , and the dynamic viscosity is 1.003×10 −3 Pa/s. The k− shear stress transport (SST) turbulence model is adopted to solve the Reynolds-averaged Navier−Stokes control equations. The primary purpose of this work is to propose a new shape optimization strategy for BWBUGs, which can reduce the computational cost. So, we mainly focus on optimizing a specific steady cruising process with a constant angle of attack and velocity, without considering the buoyancy control system and gravity longitudinal position control system. The magnitude of the inlet velocity and the angle of attack are set to constant 1 m/s and 6°, respectively. Besides, the pressure at the outlet is set to be 0 Pa. The convergence criterion is that the root-mean-square residual is smaller than 10 −5 for each equation or the total number of iterations reaches 900. Test results show that each simulation of the BWBUG costs about 24 minutes while one simulation of the airfoil costs about 2 minutes with a Quad-core processor (Intel Core i7-2600 CPU, 3.40 GHz).

Double-stage surrogate-based shape optimization strategy
Although SBO has dramatically reduced the amount of computational cost, the whole optimization process might be still time-consuming. Meanwhile, the number of function evaluations involved in the optimization increases with the increase of dimensions. Based on the characteristics of the BWBUG, the authors notice that SM of the objective function could be built in a different way and the dimensions of the SM can be reduced. As described before, the shape of the BWBUG is only determined by the shape of the sectional airfoils. Therefore, it is reasonable to assume that the LDR of the BWBUG can be represented with the LDRs of sectional airfoils. Based on this assumption, a DSSO strategy is proposed and the core idea of the DSSO is that the optimization is performed based on a novel DSM. DSM is different from the ordinary surrogate modeling process, which decomposes the design variables into several groups and the corresponding FSMs are built respectively. Then, the SSM is constructed with respect to the responses of FSMs.
The DSM owns some advantages over the ordinary SMs. Firstly, the structures of the DSMs are simpler and the modeling process is more efficient than that of ordinary SMs. Secondly, the dimensionality of the DSM is reduced and the number of samples that are needed to construct an accurate SM is also reduced. In this paper, the FSMs are built based on the LDRs of the sectional airfoil and the SSM is constructed based on the LDRs of the BWBUG with respect to the responses of the FSMs. By this means, the high dimensional model is decomposed into several low dimensional SMs. The proposed strategy is summarized as Al-gorithm 1 and the flowchart is shown in Fig. 5.
Algorithm 1: Procedures of DSSO (1) Define the sizes of the planform and the initial shape of the BWBUG.
(2) Parameterize the BWBUG with CST method. The parameters of the BWBUG are composed of three groups of design variables and each group of design variables can describe a sectional airfoil.
(3) Determine the scopes of the design variables according to the design requirements.
(4) Select N 1 samples for the sectional airfoils, respectively, and these samples constitute the N 1 samples of the BWBUG.
(5) Calculate the LDRs of the airfoils and BWBUGs with CFD-based simulations.
(6) Construct kriging SMs of LDRs for the sectional airfoils with calculated samples, and the SMs are denoted as y a , y b and y c .
(7) Verify the accuracies of the initial FSMs with crossvalidation technique. If the accuracies are not satisfied, select extra N 2 samples for the sectional airfoils, respectively and use them to refine the FSMs. Once the FSMs are accurate enough, go on to the next step.
(8) By regarding the LDRs of the three sectional airfoils as the inputs, the SSM for the BWBUG is constructed with calculated samples, which is denoted as y.
(9) Optimize the LDRs of the BWBUG with MSSR and the optimum is denoted as X opt .
(10) If X opt meets the design requirements, the optimization stops. Otherwise, add the candidate points selected in the optimization process to the current sample sets and switch to Step (5).
In order to validate the accuracy of the DSM, it is compared with the ordinary one-stage kriging model (OKM) in predicting the LDRs of the BWBUG. The sample size is set to 20 and the two types of SMs are built, respectively. Leave-one-out cross validation (LOOCV) method is adopted here to calculate the cross-validation variance and the relative error. The comparison results shown in Table 2 indicate that the DSM shows better accuracy than OKM with the same number of samples.
In order to find the global optimum efficiently, a surrogate-based optimization (SBO) method abbreviated as MSSR is used here to get the optima. MSSR defines three design spaces to perform the optimization, including Global Space, Medium Space and Local Space. Then, a multi-start SQP algorithm is adopted to find the local optima in these design spaces alternately and the start points are generated by Latin hypercube sampling (LHS) to keep the start points uniform. Afterwards, a few better local optimal locations are chosen as potential points and used to update the SM. If the multi-start SQP converges to the same point or no suitable points are found, the estimated mean square error (MSE) of kriging is maximized to explore unknown space. MSSR is validated to have better performance than some other SBO methods in dealing with computational-expensive problems.

Scopes of the design variables
During the optimization, the three sectional airfoils are confined with other two standard airfoils, respectively. As shown in Fig. 6, Airfoil A is confined between the NACA0016 and NACA0028, Airfoil B is confined between the NACA0010 and NACA0022, while Airfoil C is confined between the NACA0008 and NACA0012. Therefore, the boundaries of the design variables can be obtained by least square fitting and the scopes are listed in Table 3.

Sample sizes selection for FSMs
In the DSSO strategy, the SSM is constructed based on the outputs of the FSMs. If the outputs of the FSMs possess large errors, these errors will be transferred to the SSM and lead to incorrect responses. It could make it difficult for the optimizer to find the global optimum of the problem. Therefore, the accuracy of the FSM seems to be very important during the optimization and the initial FSMs need to be assured accurate enough. Therefore, it is essential to investigate how many samples can build accurate FSMs in this problem. Then, FSMs with 20, 50, 100 and 200 samples are constructed, respectively. The LOOCV method is also adopted to measure the accuracy of the FSMs and the relative errors are shown in Fig. 7. If the relative error is smaller than 5%, it is regarded that the SM is accurate enough. Therefore, the appropriate sampling sizes for FSMs are 50, 100, and 20 respectively. Since the SSM needs to be constructed with LDRs of the airfoils and the corresponding LDRs of BWBUGs, only 20 samples are used to construct the initial SSM.

Shape optimization with DSSO strategy
Based on the sample sets obtained, the DSSO strategy is then applied to optimize the BWBUG. Because of the limited computation resource, the optimization stops when the number of 3-D simulations exceeds 200. Fig. 8 gives the iteration process of the optimization and the result is compared with the initial shape in Table 4. The initial shape of the BWBUG is formed with NACA0024, NACA0018 and  NACA0012, respectively, which is within the design space. The calculated LDR of the initial shape is 17.2561 and results indicate that the proposed DSSO strategy has improved the LDR of the BWBUG by 9.35%. In order to verify the efficiency of the DSSO strategy, the BWBUG is also optimized with the OSBO strategy. The kriging model is constructed based on the LDRs of the BW-BUG by regarding the 15 design variables as the inputs, directly. Except for the construction of SMs, the other settings in the OSBO strategy are identical to that in DSSO, and the OSBO results are also shown in Fig. 8 and Table 4. With the help of the additional airfoil calculations and the use of DSMs, the DSSO strategy can obtain a better result with the same number of 3-D simulations. Besides, the number of iterations (NIT) involved in DSSO is smaller than that in the OSBO strategy.
It seems that the computational cost involved in DSSO is larger than that in OSBO strategy because it uses the   same number of 3-D simulations and some extra 2-D simulations. This phenomenon is mainly caused by the inappropriate stop criterion. If the number of 3-D simulations is smaller than 200, the optimization will continue. But as a matter of fact, the DSSO strategy is capable of reducing much computational cost when obtaining the same level of optimization result as OSBO strategy. As shown in Fig. 8, the pink horizontal dashed line represents the optimal value obtained by the OSBO strategy while the pink vertical dotted line denotes the number of iterations at which DSSO can get the same level of optimal value. The result shows that DSSO obtained the same level of optimal value at the 6th iteration, while OSBO used 86 iterations to get this optimum. In this sense, DSSO is more efficient than the OSBO strategy.
In the view of computational expense, DSSO also shows a great advantage over the OSBO strategy. Specifically, OSBO uses 168 3-D simulations to acquire the optimum, while DSSO only requires 38 3-D and 224 2-D simulations to obtain the same level of optimal value. As indicated in Table 4, the proposed DSSO strategy only requires 33.74% of the CPU time involved in OSBO when attaining the same level of optimal value.

Comparative study
As discussed above, the proposed DSSO strategy outperforms the OSBO strategy when MSSR is directly adopted. To compare the proposed DSSO strategy in a more extensive range, the other three state-of-art SBO methods, including the hybrid and adaptive meta-model-based (HAM) global optimization algorithm (Gu et al., 2012), Mixed Integer Surrogate Optimization (MISO) method (Müller, 2016) and a RBF-based algorithm using the Candidate point sampling strategy (CAND) (Müller, 2012), are also utilized to perform the shape optimization. Furthermore, three heuristic optimization algorithms, including Genetic Algorithm (GA) (Whitley, 1994), Differential Evolution (DE) al-gorithm (Storn and Price, 1997) and Grey Wolf optimizer (GWO) (Mirjalili et al., 2014), are also used to make a comparison. These methods are widely used to solve computationally expensive or black-box engineering design problems.
For the sake of fairness, the initial samples in the comparisons are the same as that in the DSSO strategy used above, and the initial samples are treated as the initial populations in three heuristic optimization algorithms. All the optimizations stop when the number of 3-D simulations exceeds 200 and the test results are listed in Table 5. Since heuristic methods use less iteration than SBO methods in the optimizations, the iteration processes of SBO and heuristic methods are illustrated in Fig. 9a and Fig. 9b, respectively. As Table 5 shows, when the numbers of 3-D simulations exceed the stop criterion, DSSO gets a higher LDR than other methods but requires some extra 2-D simulations. However, it is more reasonable to compare the CPU-time when obtaining the same level of results. As shown in Figs. 9a and 9b, the optimal result of MSSR is still selected as the reference value and denoted by a horizontal pink dotted line. Results indicate that only DSSO, MSSR, HAM, and CAND can achieve this goal within 200 times simulations. Specifically, DSSO uses less CUP-time than the comparison method except for HAM, when obtaining the same level of result. Though HAM uses less CPU-time than DSSO, the optimum of HAM remains unchanged as the optimization goes on, while DSSO can achieve a better result. The comparison results indicate that SBO methods are more efficient than heuristic methods in dealing with this kind of problems, and the proposed DSSO strategy outperforms these SBO methods by obtaining a better result or using much smaller computational cost to obtain the same level of optimum.

Further discussion
As demonstrated above that the proposed DSSO strategy is applicable and the result is better than that of the comparison methods. However, some issues need to be further discussed. In the proposed strategy, the FSMs are updated iteratively along with the SSM because the FSMs are designed to provide accurate outputs for SSM. However, if the initial FSM is relatively accurate as described in Section 5, is it necessary to update the FSMs? If not, the computational cost caused by airfoil calculations can be reduced. Besides, it is suggested in Section 4 that the initial FSMs need to be relatively accurate when performing the DSSO strategy, but how the accuracy of the initial FSMs affects the optimization is not investigated. To investigate these problems, the BW-BUG is optimized with DSSO strategy in different test cases which are described below. Case 1: The samples sizes for the initial FSMs are 50, 100 and 20, respectively. The FSMs and SSM are updated iteratively in the optimization process.
Case 2: The samples sizes for the FSMs are 50, 100 and 20, respectively. Only the SSM is updated iteratively in the optimization process.
Case 3: All the initial FSMs are constructed with 20 samples; Case 4: All the initial FSMs are constructed with 50 samples; Case 5: All the initial FSMs are constructed with 100 samples; Case 6: All the initial FSMs are constructed with 200 samples.
Results of Cases 1 and 2 are compared in Fig. 10a and Table 6, which indicate that updating the FSMs can obtain a slightly better result, but bring some extra computational cost. If the computational resources are limited, it is unnecessary to update the FSMs iteratively in the DSSO strategy when the FSMs are already relatively accurate. The FSMs are updated iteratively from Cases 3 to 6, and the test results are compared in Fig. 10b and Table 6. Results show that Case 5 performs slightly better than other cases. The results of Cases 6 and 5 are better than those of Case 3 and Case 4. In this sense, the FSMs with more initial samples give better results. However, Case 4 owns much more initial samples but performs even worse than Case 3. The optimal results of Cases 3 to 6 are close to each other, which indicates that increasing the accuracy of the initial FSMs has a small impact on the optimal result and it is not appropriate to improve the optimum by selecting too many samples for the initial FSMs.
The above test results show that both the initial sample sizes and whether to update the FSMs have little impact on the optimization results. The main reasons for this phenomenon are summarized as follows. First, the FSMs are  only 5-dimensional problems and the design scopes are relatively small in this paper. Then, the accuracies of the FSMs with 20 samples are passable. Therefore, whether to update the FSMs during the optimization has small effect on the results. For the same reason, more initial samples can improve the accuracy of the FSMs, but the improvement is limited, then the impact on the optimal results is also small. Besides, the characteristics of DSM also cause this phenomenon. The accuracy of DSM is not determined by the accuracy of FSMs, directly. So, even the FSMs are not very accurate, the optimization can obtain a good result if the DSM is relatively accurate.

Conclusions
In this paper, a novel shape optimization strategy is proposed based on the characteristics of the BWBUG shapes. The commonly used one-stage SMs are replaced with DSMs in the surrogate-based shape optimization. By this means, the original high-dimension SM is decomposed into several low-dimension SMs and the complexity of the SM is reduced. Besides, a previously proposed SBO method called MSSR is combined to perform the optimization. In order to validate the feasibility of the proposed DSSO strategy, a BWBUG shape is optimized with DSSO and the results indicate that DSSO can get a better result when using the same number of 3-D simulations. Furthermore, the other six popular methods, including three SBO methods and three heuristic methods, are used to make an extensive comparative study. Results also state that DSSO requires less computational cost than these comparison methods when obtaining the same level of result. To further investigate the characteristics of the DSSO strategy, six cases with different initial samples or model updating strategies are tested.
Based on the present research, some conclusions can be drawn as follows: (1) The proposed DSSO strategy is an applicable and more efficient strategy for shape design optimization of BWBUGs, which can provide an alternative way of optimizing the BWBUG shape. (2) Test results indicate that iteratively updating the FSMs in the optimization can slightly benefit the result, but bring more computational burden. Therefore, it is unnecessary to update the FSMs iteratively if the available computing resource is limited.
(3) Utilizing more samples to construct initial FSMs can not necessarily bring a better result, so it is not suggested to improve the result by using too many samples for initial FSMs.
There are also some deficiencies in this work. For ex-ample, the DSSO strategy is only validated through a relatively simple BWBUG shape, but some more complex BW-BUGs are not covered in this work. Furthermore, the application range of the proposed DSSO strategy is limited and how to extend this DSM based strategy to more problems is not discussed. Therefore, the above mentioned defects will be investigated in our future work.