This paper presents an optimization method for the design of the layout of an autonomous underwater vehicles (AUV) fleet to minimize the drag force. The layout of the AUV fleet is defined by two nondimensional parameters. Firstly, threedimensional computational fluid dynamics (CFD) simulations are performed on the fleets with different layout parameters and detailed information on the hydrodynamic forces and flow structures around the AUVs is obtained. Then, based on the CFD data, a backpropagation neural network (BPNN) method is used to describe the relationship between the layout parameters and the drag of the fleet. Finally, a genetic algorithm (GA) is chosen to obtain the optimal layout parameters which correspond to the minimum drag. The optimization results show that
Autonomous underwater vehicles (AUVs) are a kind of selfsailing, selfexecuting underwater robots, which play an important role in expanding people’s knowledge of the ocean. AUVs are capable of a wide range of applications, such as pipeline inspection [
When the AUVs travel in a fleet, the layout of the fleet becomes a key problem. Observations of animal motions such as birds in flocks suggest that some energy benefit may be obtained by certain fleet shapes [
(a) Migrating geese and (b) possible layout of an AUV fleet.
The drag of an individual AUV can be reduced by optimizing the shape of the hull, propeller, and surface control [
CFD simulations can predict the hydrodynamic performance of the AUV with high accuracy, however, at the cost of time. Combined optimization method based on the CFD, BackPropagation Neural Network (BPNN), and Genetic Algorithm (GA) can efficiently save the simulation time to obtain the optimal target. BPNN and GA are used at some of the nonlinear data to find the optimal value [
Inspired by the above research, this paper introduces an optimization procedure of an AUV fleet, which contains two AUVs, to minimize its drag. The layout of the fleet is defined by two nondimensional parameters. Threedimensional CFD simulations are performed to find the drag of each case. Then BPNN and GA optimization method is used to find the optimal layout which has the minimum drag.
The AUV considered in this paper has a torpedolike axisymmetric shape, which is characterized with a length of
Dimensional parameters of the AUV.








0.20  0.04  1.30  0.20  0.70  0.40  30 
Shape of the AUV.
As shown in Figure
Parameter definition for the AUV fleet.
A series of fleet layouts can be described by changing the two design parameters,
This study aims optimize the drag of the AUV fleet. The drag is normalized in the following way:
The drag coefficients and the flow structures for each simulation case are obtained by the CFD approaches separately. A group of transient representations of the threedimensional flow are carried out to improve the efficiency of the optimization process. Further, the results of this numerical model will be compared with existing experimental data for the model validation, which will be discussed in Section
The prediction of the flow around the AUVs is based on the incompressible NavierStokes equations [
The shear stress transport (SST)
The computational domain is a finite space used to simulate the flow around the AUV. In order to minimize the effect of block caused by the AUV, a rectangular domain with a dimension of
Computational domain and boundary conditions.
Steady and uniform velocity inlet boundary is set at the left surface of the domain. The magnitude of the inflow velocity is 6 m/s and a moderate turbulence intensity of 5% is chosen. Pressure outlet boundary is chosen for the right surface of the domain, which has the same turbulence intensity with the inlet boundary. Smooth wall conditions are imposed at the fourside surfaces, where the shear effects are neglected to minimize the influences of the walls. Standard wall conditions are applied to the surfaces of both AUVs.
In order to obtain more accurate results with smaller number of grids, the computational domain is meshed with structured hexahedral grids, as shown in Figure
Details of the computation mesh: (a) overview; (b) grids around the vehicle; and (c) prism layer grid elements.
A mesh verification study is carried out to determine the proper density of mesh. The verification simulations are performed on a single AUV with different meshes. Three meshes, with approximately 3.76 million (fine mesh), 2.80 million (mediate mesh), and 1.67 million (coarse mesh) elements, respectively, are generated for the grid resolution verification. The drag coefficients obtained by the three meshes are listed in Table
Results of the mesh verification study.
Mesh  Grid number 


Coarse  1670000  0.0786 
Mediate  2800000  0.0774 
Fine  3760000  0.0771 
A validation study is performed for the validation of the proposed numerical model. Simulations are performed according to an experiment of twin parallel bare prolate spheroids with transverse separations [
Wind tunnel test of the twin spheroids [
Results of the validation study: (a) coefficients of drag and (b) coefficients of side force.
This paper establishes an optimization method based on the combination of CFD method, BPNN and GA. The drag coefficients and the flow structures for 53 different layouts are obtained by the CFD approaches separately. The results are then used to build the agent model between
The artificial neural network is widely accepted as an alternative to providing solutions to complex and ambiguous problems [
The topology of the BPNN.
The BPNN is working based on the following principle:
(7) Return to step
It is noted that all the iterative processes of the BP neural network have a mean square error (MSE). In addition, the average accuracy of the prediction
In this paper, the GA tool in MATLAB is used for the optimization, which runs in the following principle:
Initialize the population, calculate the fitness value, and find the best chromosome from the population.
Iterative optimization:
Select: first, the solution to the problem is encoded by using the floatingpoint encoding. This function selects the chromosomes in each generation population for subsequent crossover and mutation. The method used is the roulette selection method.
Crossover: this function is a random selection of two chromosomes, according to determine the crossover probability to determine whether the cross, and the cross position is also random.
Mutation: this function performs the mutation operation. The mutation chromosomes and mutated positions are randomly selected. Finally, it will check the feasibility of chromosomes; otherwise, it will be recompiled.
Result analysis.
The optimal solution can be found after several generations. The establishment of BPNN approximate model and the numerical optimization of the genetic algorithm are shown in Figure
Neural network and genetic algorithm flowchart.
In order to more intuitively show the influences of the fleet layout on the drags of the two AUVs, the coefficients of drags are expressed in a nondimensional drag ratio by dividing
The drag ratios for the 53 different fleet layouts are shown in Figure
Drag ratios for different AUV fleet layouts.
The Parallel Region locates at
The contours of pressure and velocity around the two parallel AUVs are shown in Figure
Contours of (a) velocity and (b) pressure around two parallel AUVs at different latitudinal offsets.
The Tandem Region is where the follower AUV locates just behind the leader AUV (
The contours of pressure and velocity around the two tandem AUVs are shown in Figure
Contours of (a) velocity and (b) pressure around two parallel AUVs at different latitudinal offsets.
The Pull Region is where the drag of the leader AUV is increased and that of the follower AUV is reduced. The Pull Region mainly locates between
The contours of pressure and velocity around the two AUVs in the Pull Region are shown in Figure
Contours of (a) velocity and (b) pressure around two AUVs in the Pull Region.
The Push Region is where the drag of the leader AUV is reduced and that of the follower AUV is augmented. The Push Region mainly locates between
The contours of pressure and velocity around the two AUVs in the Push Region are shown in Figure
Contours of (a) velocity and (b) pressure around two AUVs in the Pull Region.
The predicted drag ratios for the AUV fleet obtained by CFD are given in Table
CFD data for the training samples.
Sample 






(1)  0.00  0.00  2.0000  2.0000  4.0000 
(2)  0.00  0.25  2.0000  2.0000  4.0000 
(3)  0.00  0.50  2.0000  2.0000  4.0000 
(4)  0.00  0.75  2.0000  2.0000  4.0000 
(5)  0.00  1.00  2.0000  2.0000  4.0000 
(6)  0.00  1.25  0.7910  1.0972  1.8882 
(7)  0.00  1.50  0.9392  0.9748  1.9141 
(8)  0.00  1.75  0.9757  0.9602  1.9359 
(9)  0.00  2.00  0.9885  0.9629  1.9514 
(10)  0.10  0.00  2.0000  2.0000  4.0000 
(11)  0.10  0.25  2.0000  2.0000  4.0000 
(12)  0.10  0.50  2.0000  2.0000  4.0000 
(13)  0.10  0.75  2.0000  2.0000  4.0000 
(14)  0.10  1.00  2.0000  2.0000  4.0000 
(15)  0.10  1.25  0.8245  1.1725  1.9970 
(16)  0.10  1.50  0.9432  1.0604  2.0036 
(17)  0.10  1.75  0.9765  1.0296  2.0061 
(18)  0.10  2.00  0.9887  1.0188  2.0075 
(19)  0.20  0.00  1.0552  1.0552  2.1104 
(20)  0.20  0.25  1.7136  0.3598  2.0733 
(21)  0.20  0.50  1.5153  0.5364  2.0517 
(22)  0.20  0.75  1.4027  0.5841  1.9868 
(23)  0.20  1.00  0.7591  1.2507  2.0098 
(24)  0.20  1.25  0.8671  1.1107  1.9778 
(25)  0.20  1.50  0.9450  1.0398  1.9848 
(26)  0.20  1.75  0.9728  1.0150  1.9879 
(27)  0.20  2.00  0.9841  1.0050  1.9891 
(28)  0.30  0.00  1.0247  1.0247  2.0494 
(29)  0.30  0.25  1.4041  0.6326  2.0367 
(30)  0.30  0.50  1.3595  0.6702  2.0297 
(31)  0.30  0.75  1.1989  0.7884  1.9873 
(32)  0.30  1.00  0.9257  1.0920  2.0177 
(33)  0.30  1.25  0.9135  1.0671  1.9806 
(34)  0.30  1.50  0.9557  1.0297  1.9853 
(35)  0.30  1.75  0.9761  1.0118  1.9879 
(36)  0.30  2.00  0.9852  1.0037  1.9889 
(37)  0.40  0.00  1.0146  1.0146  2.0292 
(38)  0.40  0.25  1.2386  0.7859  2.0245 
(39)  0.40  0.50  1.2489  0.7770  2.0259 
(40)  0.40  0.75  1.1302  0.8617  1.9920 
(41)  0.40  1.00  0.9947  1.0283  2.0229 
(42)  0.40  1.25  0.9487  1.0354  1.9841 
(43)  0.40  1.50  0.9669  1.0200  1.9869 
(44)  0.40  1.75  0.9800  1.0086  1.9886 
(45)  0.40  2.00  0.9871  1.0025  1.9896 
(46)  0.50  0.00  1.0094  1.0095  2.0189 
(47)  0.50  0.25  1.1488  0.8699  2.0187 
(48)  0.50  0.50  1.1753  0.8506  2.0259 
(49)  0.50  0.75  1.0950  0.9000  1.9951 
(50)  0.50  1.00  1.0243  1.0032  2.0275 
(51)  0.50  1.25  0.9723  1.0146  1.9868 
(52)  0.50  1.50  0.9769  1.0118  1.9887 
(53)  0.50  1.75  0.9843  1.0057  1.9900 
(54)  0.50  2.00  0.9891  1.0016  1.9907 
(55)  0.60  0.00  1.0062  1.0065  2.0127 
(56)  0.60  0.25  1.0963  0.9170  2.0132 
(57)  0.60  0.50  1.1263  0.8991  2.0253 
(58)  0.60  0.75  1.0724  0.9245  1.9969 
(59)  0.60  1.00  1.0327  0.9907  2.0235 
(60)  0.60  1.25  0.9873  1.0018  1.9891 
(61)  0.60  1.50  0.9850  1.0052  1.9903 
(62)  0.60  1.75  0.9883  1.0029  1.9912 
(63)  0.60  2.00  0.9911  1.0000  1.9911 
When the AUV fleet executes tasks underwater, a minimum drag is expected so that the fleet will work for a longer time and a wider range. Therefore, the layout of the AUV fleet should be optimized to obtain a minimum drag.
Using the data in Table
Comparisons of the drag ratio of the AUV fleet by BPNN and CFD.
A CFD simulation of the AUV fleet for the optimal parameters is carried out for the comparisons between the two methods. The optimal results are shown in Table
Comparisons of the optimal drag ratio of the AUV fleet between CFD and BPNN.


BPNN  CFD  Relative error (%) 

0.00  1.21  1.8949  1.8825  0.66% 
To further investigate the flow structures of the optimal fleet layout, Figure
Contours of pressure around the AUV: (a) single AUV and (b) optimal AUV fleet.
Pressure distributions along the longitudinal direction of the AUV: (a) single AUV and (b) optimal AUV fleet.
Although the optimal fleet layout analyzed in Section
Using the data in Table
Comparisons of the drag ratio of the follower AUV by BPNN and CFD.
Based on the network after the training process, the optimal output drag ratio of the follower AUV is predicted using the BPNN model. Typically, a CFD simulation for the optimal parameters is carried out for the comparisons between the two methods. The optimal results are shown in Table
Comparisons of the optimal drag ratio of the follower AUV between CFD and BPNN.


BPNN  CFD  Relative error (%) 

0.22  0.29  0.3481  0.3417  1.87% 
Figure
Contours of pressure around the AUV: (a) single AUV and (b) optimal AUV fleet.
Pressure distributions along the longitudinal direction of the AUV: (a) single AUV and (b) optimal AUV fleet.
The drag of the AUV fleet significantly influences the operating time and range of underwater vehicles. In this study, an optimization method is proposed to find the optimal layout of the AUV fleet which has the minimum drag. A combined method of CFD simulation, BPNN, and GA method is utilized for the optimization of the AUV fleet. Important conclusions of this study include the following:
Autonomous underwater vehicles
Backpropagation neural network
Computational fluid dynamics
Genetic algorithm
Maximum diameter of the AUV
Length of the AUV
Maximum crosssectional area of the (m^{2})
Velocity of the AUV
Density of seawater
Normalized position of the follower AUV
Position of the follower AUV
Drag coefficient of the leader AUV
Drag coefficient of the follower AUV
Drag coefficient of the fleet
Drag ratio of the leader AUV
Drag ratio of the follower AUV
Drag ratio of the fleet
Turbulence kinetic energy
Specific rate of dissipation.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was supported by the National Science Foundation of China (Grants nos. 51179159 and 61572404).