A hybrid of CFD and PSO optimization design method of the integrated slipper/swashplate structure in seawater hydraulic axial piston pump

An integrated slipper/swashplate structure is developed for seawater hydraulic axial piston pump. Considering the overturning moment of the integrated slipper, the transient hydrodynamic lubrication model of the integrated slipper is established, and the body-fitted grid and five-point finite difference method is employed to accurately solve the model. An optimization design method based on a hybrid of CFD (Computational Fluid Dynamics) and PSO (Particle Swarm Optimization) for the integrated slipper is proposed for the interface structure with the low leakage and power loss. The optimal sealing belt of the integrated slipper is obtained, and the leakage and power loss are reduced by 27.3% and 16.7% under the same gap size, respectively. The effects of working condition on the film-forming characteristics and power consumption characteristics of the lubricating film are analyzed based on the optimized integrated slipper. It is found that the leakage and total power loss increase significantly with the increasement of working pressure and shaft speed, and decrease with the increasement of temperature. The excellent lubrication characteristics and strong anti-overturning ability of the integrated slipper/swashplate structure can improve the performance of the seawater axial piston pump, and its optimization design method proposed in this study can provide a new reference for engineers.


Introduction
Seawater hydraulic power system which directly uses natural seawater instead of traditional mineral oil as the working medium has the advantages of environmental compatibility, higher security and lower operating cost, which makes water hydraulic system more and more attractive, especially in buoyancy regulation system (Yang & Pan, 2015;Yin et al., 2018). Figure 1 shows the schematic diagram of the buoyancy regulation system. In this system, seawater hydraulic axial piston pump (SHAPP) is employed to directly extract or inject seawater from the buoyancy regulating tank. The system is controlled by an electric directional valve to realize the floating, diving and suspension of underwater equipment. Obviously, the performance of SHAPP will directly affect the overall characteristics of seawater hydraulic power system and even the entire underwater equipment (Xu et al., 2013).
The configuration of a traditional SHAPP is illustrated in Figure 2. The rotating kit mainly includes a cylinder which is connected to the drive shaft and piston/slipper assembly. A slipper is coupled to the piston through a pair of ball and socket joint to form a ball joint friction CONTACT Songlin Nie niesonglin@bjut.edu.cn; Fanglong Yin yfl@bjut.edu.cn pair. When the shaft rotates, the piston/slipper assembly slides on the swashplate, and forces each piston to make a periodic linear reciprocating motion inside the cylinder. Generally, the main lubrication interfaces of SHAPP would adopt the 'soft-to-hard' material combination of corrosion-resistant metals and engineering materials, such as PEEK/17-4PH (Huang et al., 2020;Nie et al., 2019). However, those underwater equipment would be subject to the increased working pressure and rotational speed in some specific occasions, which will lead to the abnormal work of SHAPP. The high-working pressure significantly would be inclined to cause the deformation of softer material and increase the lateral force on the piston, bend the piston and accelerate its wear, especially in poorly lubricated seawater (Nie et al., 2021;Yin et al., 2021a,b;Zhu et al., 2020); the high shaft speed would make the slipper generate considerable centrifugal force due to its own structure causing severe sliding wear (Bergada et al., 2010;Suo et al., 2021).
To improve the anti-overturning and anti-wear ability of slippers, several researchers have tried several improvements. Through numerical simulations and experiments, Xu et al. (2012) found that the adequate slope on the inner edge helped to improve the carrying ability and reduce the leakage of slipper. Ye et al. (2020) and Tang et al. (2021) recognized that reasonable texture was helpful to improve the load-carrying capacity of slipper/swashplate interface. Furthermore, other researchers have tried to find some new materials to reduce the slippers wear. Zhai et al. (2022) and Wang et al. (2016) found through tribological experiments that a combination of cermet/cermet exhibits excellent potential in the development of seawater hydraulic components. Other scholars have conducted several research on the anti-tilting of the piston/cylinder interface in the macroscopic and microscopic aspects of the structure. Kumar and Bergada (2013) found that the use of grooves on the piston surface increase the stiffness of film in piston/cylinder interface and restoring torque of the piston. Chao et al. (2022) proposed a novel integrated slipper retainer mechanism for high-speed axial piston pumps, which can avoid direct contact between the slippers and swash plate and thereby eliminate the slipper wear under severe operating conditions.
The above efforts have made great contributions to reducing the wear slippers and improving the reliability of slipper/swashplate and piston/cylinder interface. However, the slippers cannot avoid the overturning and surface wear caused by centrifugal force due to its inherent structure. Under high pressure conditions, the lateral force exerted by the piston cannot be eliminated, which hinders the application of hard materials in the piston/cylinder interface for the deep sea environment. Therefore, a new integrated slipper structure with synchronous drive pin is proposed, which can not only reduce the surface wear of the integrated slipper under high shaft speed but also greatly reduce the lateral force on the piston under high-working pressure. Figure 3 shows the schematic diagram of an improved SHAPP incorporating an integrated slipper interface. The integrated slipper is connected with multiple pistons through the connecting rod with double ball joint and driven by the drive pin. The high-pressure water flows into the kidney shaped chamber at the bottom of the integrated slipper through three orifices of the piston, connecting rod and integrated slipper, and leaks to the case through the sealing belt of the support plate, constructing hydrostatic pressure across the lubricating interface. Meanwhile, the relative rotation of the integrated slipper and swashplate produces additional hydrodynamic pressure. The combined hydrostatic and hydrodynamic pressures are balanced against other external forces and moments acting on the integrated slipper.  Comparing Figure 3 with Figure 2, it can be found that the improved integrated slipper is significantly different from the traditional slipper in structure, force balance and driving mode. First, the mass center of the integrated slipper can be set to coincide with the mass center of the shaft ball twisting. The overturning moment of the integrated slipper is completely provided by the extremely small viscous friction of the seawater film. This design can eliminate the overturning problem caused by centrifugal force to a large extent and reduce the wear of the integrated slipper. The lateral force of the piston exerted by the integrated slipper is borne by the shaft ball twisting, which could greatly improve the lubrication state of the piston/cylinder interface, due to the use of double ball-end connecting rods. Second, the pressure inside piston chamber is no longer borne by the multiple slippers but by the integrated slipper. This strategy can lessen the risk of sudden pump failure due to a single slipper failure. Finally, the integrated slipper rotates synchronously with the cylinder through the synchronous driving pin, which can avoid the direct contact between the connecting rod and the cylinder, thereby to reduce the wear of the piston and the connecting rod. Therefore, the integrated slipper interface not only can reduce the slipper wear, improve the lubrication condition of the piston/cylinder interface, but also can extend the service life of the piston pump through 'hard to hard' material combination, even under the condition of suspended sand in the ocean environment.
The purpose of this paper is to develop an integrated slipper/swashplate structure in seawater hydraulic axial piston pump. Considering the overturning moment of the integrated slipper, the transient hydrodynamic lubrication model of the integrated slipper is established, and the body-fitted grid and five point finite difference method is employed to accurately solve the model. The advantage of this mesh generation method is that the generated mesh is continuous and smooth, good orthogonality and completely matches the complex shape of the real geometry. An optimization design method based on a hybrid of CFD and PSO is developed for the integrated slipper with the lowest leakage and power loss. The water film lubrication characteristics of the integrated slipper are analyzed under considering the inertial force and centrifugal force. The influences of working pressure, shaft rotational speed, water film thickness, temperature on the integrated slipper are investigated.

Dynamics model of the integrated slipper
When the cylinder rotates, the connecting rod, pistons and integrated slipper assembly rotate synchronously around the drive shaft under the action of the drive pin, and reciprocate along the axis of the cylinder. The integrated slipper rotates close to the surface of swashplate under the action of spindle ball and chamber pressure. The motion trajectory of rotating assembly is illustrated in Figure 4. Assuming that the upper dead center is the initial position, a global coordinate system o-xyz is established to analyze the kinematics and dynamics of the integrated slipper interface. To analyze the trajectory of the integrated slipper, a new coordinate system o-x 1 y 1 z 1 is obtained by rotating the coordinate system o-xyz counterclockwise around the oz axis, of which the y 1 -axis passes through the center of the integrated slipper ball sockets.
The trajectory of the integrated slipper is an ellipse in coordinate system o-xyz. By convention, the initial shaft rotational position angle, ϕ = 0, is defined to be outer dead center at the positive y-axis. The angle α between the connecting rod and the piston axis is a function of the shaft rotational position angle, when ϕ = 0, α = α 0 . Therefore, when the cylinder rotates at any angle ϕ, the coordinates of the center of ball joint A and ball joint B in the coordinate system o-xyz are: where R s is the radius of the ball sockets distribution on the integrated slipper, R p is the radius of the cylinder distribution and γ is the swashplate angle. The piston position displacement s p , axial velocity v p and axial acceleration a p in the global coordinate system o-xyz can be defined as: where L is connecting rod length and ω is the cylinder block rotation velocity. The values of dα/dt, d 2 α/dt 2 , sinα and cosα in Equations (4) and (5) can be obtained from the given swashplate angle γ and shaft rotational position angle ϕ through Equation (6): Considering the change of the angle between the connecting rod and the piston is very small over one revolution, it can be assumed that the connecting rod is not inclined. When the pump is working, a series of forces acting on the integrated slipper are illustrated in Figure 5. The pressure inside the displacement chamber pushes the piston with an equivalent force F dc , which switches between suction and delivery stroke varying greatly with time. The inertial force F a and the centrifugal force F ω act on the mass center of the piston and connecting rod. The viscous friction force F f acts on the piston surface and F tθ acts on the integrated slipper. The spring force F k acts on integrated slipper. The reaction force, with an equal but opposite magnitude of fluid film force F sp , can be decomposed into a force component parallel (F spx ) and perpendicular (F spy ) to the shaft axis. The clamping force on the integrated slipper includes the displacement pressure force F dc , the spring force F k , the axial inertia force and the component force of the centrifugal force of the connecting rod in the x 1 direction F ωcx1 . The displacement pressure force F dc can be expressed as: where d p is the piston outer diameter, p pi is the pressure inside the i-th piston chamber, and p case is the case drain pressure. The spring force F kx 1 can be expressed as: where k is the center spring stiffness and x is the spring precompression length. The overall inertia force of the piston connecting rod assembly acting on integrated slipper is where m c and m p are the masses of the connecting rod and piston respectively. The component of centrifugal force of connecting rod in x 1 direction is: Finally, the instantaneous value of F fx can be obtained as: Since the pressure inside each displacement chamber is different, the clamping force acting on the integrated slipper is also different at each instant, and two torques about the y 1 -axis and z 1 -axis are generated.
When the integrated slipper slides at a high-speed relative to the swashplate, a wedge-shaped fluid film will be formed between the sealing belt on the bottom of the integrated slipper. The pressure field of fluid film is composed of integrated slipper pocket pressure and sealing belt pressure.
where p is the water film pressure field at the integrated slipper sealing land, p ipocket is i-th integrated slipper pocket pressure, A pocket is integrated slipper pocket area and is the integral region. Because the pressure field on the bottom of the integrated slipper is not radially symmetric, the fluid moments about y 1 and z 1 axis of the integrated slipper local coordinate system are: The viscous shear stress is integrated over the lubrication area to calculate the F t : The magnitude of those moments acting on the integrated slipper in local coordinate system y 1 -axis and z 1 -axis can be calculated: where l sp is the distance from the integrated slipper bottom to the mass center. The translation and rotation of the three degrees of freedom can be obtained by satisfying the defined forces and moments above-mentioned.

Integrated slipper pocket pressure model
The integrated slipper pocket is connected to the displacement chamber through three small orifices (fixed damping), and is almost sealed by the integrated slipper sealing land (variable damping) at the outer radius. The high-pressure fluid inside the displacement chamber first flows into the integrated slipper pocket through three small orifices, and then enters the case through the integrated slipper sealing land. In this process, high pressure fluid produces a static pressure supporting force on the integrated slipper to balance part of the pressing force on the integrated slipper (Schenk & Ivantysynova, 2015). To compare with the lubrication performance of traditional slipper interface, the fluid compressibility is ignored, and the pressure inside the integrated slipper pocket is approximately as constant throughout the volume. In this way, the fluid 'pressure build-up equation' in the integrated slipper pocket can be approximately obtained by the expression of pressure-flow characteristics.
As shown in Figure 6, for three fixed damping, the pressure-flow characteristics can be expressed as follows: where p ir is the pressure at the right end of the i-th orifice, p il is the pressure at the left end of the i-th orifice, d i is the diameter of the i-th orifice and l i is length of the i-th orifice.
In the case of neglecting the compressibility of fluid, the flow through fixed damping and variable damping is equal according to the incompressible continuity equation. Therefore, the integrated slipper pocket pressure model can be expressed as follows: where p dc is the pressure inside piston chamber, p sp is the pressure inside integrated slipper pocket. The variable damping pressure-flow characteristics of the integrated slipper sealing belt is expressed as Figure 6. Integrated slipper pocket-control volume. (Brennen, 2011): For annular hydrostatic supports, the leakage coefficient k q is: According to the principle of flow continuity, it can be known that Q F = Q V . Combining Equations (20) and (24), we can get: Finally, fixed damping and variable damping constitute the negative feedback dynamic regulation system of the integrated slipper interface about pressure and flow. This integrated slipper/swashplate structure can make the liquid film have a certain stiffness like the traditional slipper, and can be adjusted dynamically with the external load under a good lubricating state.

Water film thickness equation
The distribution of water film thickness field of integrated slipper is similar to that of the cylinder/valveplate interface. Without considering the deformation of the integrated slipper (as shown in Figure 7), with respect to the cylindrical reference system o-rθz, the water film thickness h(r, θ, t) can be presented as: The normal squeeze motion of the integrated slipper is defined by deriving with respect to time the fluid film thickness defined by Equation (24), as indicated in Equation (25):

Reynolds equation
The water film pressure distribution can be obtained through the Reynolds equation. In this research the spatial gradients of velocity is considered, but the gradient of pressure along the thickness direction is ignored. The two-dimensional Reynolds equation is expressed by Equation (26) in Cartesian coordinates (Gropper et al., 2016). where: According to the dynamic model and hydrodynamic model of the integrated slipper, the configuration of the integrated slipper can be optimized with the minimum energy consumption. The traditional CFD tool for solving the N-S equations can accurately describe the flow by considering the inertial effect and building a threedimensional model, but it also has the disadvantage of time-consuming (Chen et al., 2019). In the process of solving the lubrication model of the slipper/swashplate, the structured grid defined in the polar coordinate system suitable for the finite volume method (FVM) is widely used (Tang et al., 2017). However, due to the complex geometry of the new integrated slipper and the constantly changing computational domain, the orthogonal grid defined in polar coordinate system could only be approximated. If the sealing land area is poorly approximated, the size error will have a great influence on the prediction of film thickness. Although the non-orthogonal grid could almost accurately describe the complex clearance geometry, the calculation of gradient will become a problem. The finite volume method introduces the socalled secondary gradient or non-orthogonal correction, which would force a linear problem to become nonlinear, resulting in a slow overall solution process (Jasak et al., 2015;Murthy & Mathur, 2002). Therefore, in this research a body-fitted grid technology is employed to solve the above problem accurately and efficiently, which can not only satisfy unstructured requirements, but also accurately describe complex geometric shapes.
The Reynolds equation in Curvilinear coordinate system (ς , η) can be obtained by using the derive method of parametric implicit function.
where the coefficients of coordinate transformation are:

Energy dissipation characteristics
The energy dissipation, including volumetric and frictional power loss, is the mechanical energy accumulated in form of pressure. The viscous friction power loss can be obtained by Equation (30) considering the fluid as Newtonian. As well as the volumetric power loss, the leakage flow varies during one shaft revolution due to the different conditions encountered by the integrated slipper. In the numerical domain shown in Figure 8, the leakage from high pressure groove is described by Equations (31) and (32).

Body-fitted grid generation
The grid is generated by Thompson-Thames-Martin (TTM) method (Lin et al., 2009) in the model. Figure 8 represents the interface physical domain, which can be equivalent to the computational domain by using the body-fitted grid technique. Nodes 1, 3, 4 and 2 are the initial points, while nodes 1 ', 3', 4 'and 2' are the termination points. The dotted line 5-6, 7-8, 9-10-11-12, 13-14, 15-16, 17-18, 19-20 and 21-22 are the common boundary between the two computational domains, as shown in Figure 8. First, it is assumed that the coordinate variables ζ and η satisfy the Laplace equation in their rectangular coordinates.
Then, the equations equivalent to Equations (31) and (32) are as follows in (ζ , η) coordinates: where a = x 2 η + y 2 η , b = x ξ x η + y ξ y η , c = x 2 ξ + y 2 ξ . The boundary conditions for solving Equations (35) and (36) are defined as the x and y coordinates on the water film boundary with the values of ζ and η defined. Solving (x, y) of each (ζ , η) grid point, the body-fitted grid is automatically generated. Elements are organized in planar layers, composed of η rings of size ζ . Each direction is identified by a counter i and j for the radial and circumferential directions respectively. The grid spacing is 360 in ζ direction and 60 in η direction. Therefore, the whole computing domain has 360×60 grids, that is, 21,600 grids. It is evident that the body-fitted grids perfectly match the complex shapes of the real geometries.

Discretization of the Reynolds equation
The finite difference method is used to solve the water film pressure distribution. In the calculation grid, the pressure value of each node is used to form the difference quotient of each order, which approximately replaces the derivative of Reynolds equation. The Reynolds equation is simplified into a set of algebraic equations, and the pressure value on each node is solved. The pressure field in the water film can be approximately expressed by a set of discrete pressure values. After solving the pressure field, the corresponding numerical integration can be used to obtain the lubrication characteristic parameters such as the bearing capacity of the water film, the viscous friction force, the flow rate and the total power loss. In the numerical domain shown in Figure 8, the five-point difference scheme is applied, and the difference Reynolds equation equivalent to Equation (28) is as follows: where, The discrete equation of pressure field at the interface is obtained by re-arranging Equation (37).
For an element inside the grid, as shown in Figure 8, the definition of algebraic coefficient is listed in Equation (49): The Successive-Over-Relaxation (SOR) method is applied to accelerate the solution process of the water film pressure field. In this example, the over-relaxation coefficient ω = 1.7.
The accuracy requirements for the iterative solution of the water film pressure field are: where the allowable relative error is taken as δ = 1×10 −3 . The Dirichlet boundary condition are used in the solution of water film pressure field:

Particle swarm optimization (PSO) method
For the improved SHAPP, it seems to be auspicious to use the pressure inside the displacement chamber to lubricate the integrated slipper at the beginning, however, it would decrease the overall efficiency of the pump. Therefore, it is essential to improve the pump efficiency as possible through reasonable structural design on the premise of ensuring sufficient lubrication of the integrated slipper. In this study, the standard PSO methodology is employed to optimize the sealing land size of the integrated slipper, which is a typical multi-dimensional nonlinear discrete optimization problem. Particle Swarm Optimization (PSO) method can effectively search the optimal solution through the cooperation and information sharing among particles, especially for the complex and nonlinear problems that are tough to solve through traditional search methods (Li et al., 2008). Generally, the update method of particle position and velocity in standard PSO methodology is where x i is the particle current positon, v i is the velocity of particle, c 1 = c 2 = 1.494 are the learning factors, rand() is a random number between (0,1) and ω = 0.729 is the inertia factor. As mentioned above, the diameter of the ball socket distribution circle R s has been given. Here we assume that the width of kidney groove is W g , the width of internal land is W i , the width of external land is W o and integrated slipper pocket range angle is θ g . Their relationship is as follows: Therefore, the variable matrix is defined by Therefore, the sum of volumetric and frictional power loss can be taken as the objective function of this optimization.

Optimization procedure of integrated slipper
Based on MATLAB software, the complete solution of the integrated slipper design method based on PSO methodology is described in Figure 9. First of all, the maximum number of iterations N = 100 is set, and the speed and position combining with the local search schedule are randomly initialized. PSO with different number of particles (swarm size) has reasonably similar performance, and swarm's population size is 100. According to different particles, the energy consumption characteristics of integrated slipper interface can be calculated. The particles are iterated and updated towards the direction of minimum energy consumption. When a convergence is reached, the optimal structure of integrated slipper interface is obtained.

Parameter setting and grid independence checking
According to the actual structural configuration, the main structure sizes of the integrated slipper is required to be within a certain range. Here 5 mm ≤ W g ≤ 15, The structural parameters of the improved SHAPP are listed as Table 1. For the model built in this study, the orthogonal body-fitted grid system (ζ , η) is selected. The grid independence test about lubrication characteristics was conducted on two different grid sizes (360×60) and (720×120). Figure 10 presents the sampling pressure distribution under p p = 14 MPa, n = 1500 r/min and T = 25°C. For the case of fine grid density (720×120), good simulation results can be obtained. It can be seen from the Figure 10 that, the difference between the fine grid (720×120) and the medium grid (360×60) is less than 1%, and the results are independent of the grid size. Therefore, considering  the calculation accuracy and the required resources, the medium grid is adopted.

Results analysis and discussion
Figure 11(a) shows the convergence graph of structural optimization based on PSO. It can be known that in the 11th generation (about 2.97 h), the optimal structural parameters are obtained. The structural parameters of the integrated slipper optimized by using medium mesh (360×60) are illustrated in Figure 11(b).

Water film lubrication performance comparison
The structure of integrated slipper has natural advantages in anti-overturning and reducing the lateral force of pistons, but the characteristics of lubricating water film in power loss need to be verified. Therefore, it is necessary to compare the power losses of traditional slipper and integrated slipper. The volumetric power loss P leak and frictional power loss P friction of the slipper/swashplate interface can be calculated (Canbulut et al., 2009): where h G is the fluid film thickness, p G is the slipper pocket pressure, r outG is the outer pocket radius, r inG is the inner pocket radius, l o is the slipper orifice length and d o is the slipper orifice diameter.
It is feasible to directly design the integrated slipper pair using the size of the traditional slipper seal belt, but it is not necessarily optimal. It can be seen from Figure 12(a) that when the gap size is less than 4 μm,  the leakage of the integrated slipper and the traditional slipper is equivalent. With the increasement of the gap size, the leakage of the traditional slipper is larger than that of the integrated slipper, and this trend is becoming more severe. It can be seen from Figure 12(b) that, the total power loss (including P leak and P friction ) of the traditional slipper increases with the increasement of the gap size, while the total power loss of the integrated slipper shows a little low. The total power loss of the integrated slipper first decreases and then increases, and obtains a minimum value under the gap size of 5 μm. The phenomenon may be attributed to that the frictional power loss dominates the total power loss when the gap size is less than 5 μm, while the leakage power loss dominates when the gap size is greater than 5 μm. Meanwhile, it can be found that the total power loss of the integrated slipper is always smaller than that of the traditional slipper when the gap size is greater than 5 μm. In addition, the performance of the optimized integrated slipper is obviously better than that before optimization. Theoretically, it is proved that the leakage loss and power loss of the designed slipper pair structure are superior to the traditional slipper structure. Figure 13 show the relative rate of leakage and total power loss of the integrated slipper and the traditional slipper under different gap size. It can be found from Figure 13(a) that the relative change rate of leakage is all negative, indicating that the leakage of integrated slipper is less than traditional slipper. It can be seen from Figure 13(a) that the leakage of the integrated slipper is smaller than that of the slipper/swashplate interface whether it is optimized or not, and this property does not change with the gap size. When the gap size is 1 μm, the total power loss of the traditional slipper is obviously better than that of the integrated slipper, and when the gap size is more than 5 μm, the total power loss of the integrated slipper becomes more excellent as shown in Figure 13(b). It is worth noting that through the optimized design method in this study, when the water film gap size is 5 μm, the leakage of the optimized integrated slipper can be reduced by 47.8% and 27.3% compared with the traditional slipper and the un-optimized the integrated slipper, respectively. Moreover, the total power loss of the optimized integrated slipper is reduced by 28.4% and 16.7% respectively in comparison with the traditional slipper and the un-optimized integrated slipper. This further verifies the advantages of the structural performance of the designed integrated slipper pair, and lays a theoretical foundation for the optimal design of SHAPP.

Characteristics of water film lubrication
When analyzing the water film pressure field, several specific points are selected according to the working conditions of the pistons. The piston at the outer dead center is numbered as No.1, and the others are labeled in turn, as illustrated in Table 2.
It can be seen from Table 2 that the different cylinder rotation angles represent different working states of the piston. When the cylinder rotation angle is 0°, 5°, 15°T and 25°respectively, the four working states of the piston are presented, thus the study of the piston at these positions is more representative for the analysis of water film pressure field on the bottom surface of the integrated slipper sealing land, as shown in Figure 14. The shape of the pressure field exhibits a wedge-shaped constriction within the fluid film area, and the constriction amplitude of the pressure field under the adjacent highpressure kidney grooves becomes smaller. It can be seen from Figure 14 that the pressure in the kidney groove changes with the pressure change inside the piston cavity. Obviously, the dynamic pressure effect of film pressure field is the strongest when the shaft angle is 0°, especially in the sealing area between kidney grooves No. 2 and No. 3. It can be predicted that when the integrated slipper passes shaft angle of 0°, the water film thickness will have a large adjustment. At this time, the piston No. 1 is located at the initial point of the low to high pressure region, while the piston No. 6 is located in the high to low pressure region. With the drive shaft turns 25°, the piston No.1 turns to the low to high pressure region, the piston No.6 completes the pressure relief process, and the piston No.5 enters the high to low pressure region. It is not difficult to find that the pressure rise and drop processes of the piston will be repeated every 40°. The simulated water film characteristics of the integrated slipper interface are shown in Figure 15. It can be seen from Figure 15(a) and (b) that, the three-point thickness value of the water film has obvious periodicity, a large period with 1/f p and a small period with 1/f sp . The small periods are caused by the rotating pistons, so the frequency f sp is related to the number of pistons N p in one revolution. The large periods are due to the combined force of inertial force and centrifugal force acting on integrated slipper, which show periodic changes every 360°. There is such a large-amplitude oscillation in the water film thickness, but it has little effect on the water film lubrication characteristics. The average thickness of water film is 5.9 μm, the average leakage is 1.57 L/min (volumetric efficiency of the pump at the displacement of 86 mL/r is 98.78%) and the average power loss is 309.4 W, which is acceptable.
The central film thickness of the above thickness field is taken to calculate the power loss of slipper/swashplate for a comparison with integrated slipper, as shown in Figure 15(c) and (d). The leakage and power loss are consistent with the variation of water film thickness field. Moreover, it should be noted that the leakage and power loss of the integrated slipper are reduced by 19.7% and 35.4% respectively in comparison with the traditional slipper under the same inclination. The reduction of leakage and power loss will greatly improve the efficiency of SHAPP. However, more changes in performance compared with traditional structures need to be verified by future experiments.

Transient analysis under variable operating conditions
The effects of medium physical properties and working conditions on the performance of water film are studied in this section. Table 3 lists the changing working conditions as input parameters. The water film lubrication characteristics of the integrated slipper were analyzed in this study. Figure 16 show the variation curves of water film lubrication characteristic parameters in three shaft revolutions and illustrate the influence of temperature on average water film thickness, water film pressure, leakage and total power loss. As shown in Figure 16(a), there is a similar trend of the average water film thickness between different temperatures, and the average water film thickness decreases with the increase of the temperature. With the increase of temperature, the viscosity of water film decreases, the leakage increases, and the carrying capacity of water film decreases. This can be explained by referring to Figure 16(c), which shows the relationship between the water film temperature and the leakage. This just confirms the negative feedback characteristic of the integrated slipper. As the leakage increases with the increase of temperature, the total power loss of water film undoubtedly increases, as shown in Figure 16(d).

Effects of water film temperature
When the temperature increases from 20°C to 80°C, the average water film thickness decreases from 5.9 μm to 3.6 μm, the average leakage decreases from 1.57 L/min to 1.18 L/min, the average total power loss decreases from 309.39 W to 220.7 W, and the volumetric efficiency increases from 98.78% to 99.85%. It is worth noting that the increase of temperature does not bring great change to the water film pressure field. The pressure results at different temperatures are very close with a maximum difference of 0.1%, as shown in Figure 16(b). Figure 17 show the variation curves of water film lubrication characteristic parameters in three shaft revolutions and illustrate the influence of working pressure on average water film thickness, water film pressure, leakage and total power loss. Figure 17(a) shows that the water film thickness increases abnormally with the increase of working pressure, and the oscillation amplitude of water film thickness increases. This may be due to the fact that the integrated slipper has a large water film area. Under higher load pressure, the dynamic pressure effect is more significant, causing a large oscillation of water film thickness, and increasing leakage and power loss, as shown in Figure 17(c) and (d). When the working pressure increases from 10 MPa to 16 MPa, the average water film thickness increases from 3.44 μm to 6.76 μm, the average leakage increases from 0.24 L/min to 3.13 L/min, the average total power loss increases from 153.17 W to 618.78 W, and the volumetric efficiency decreases from 99.8% to 97.57%. Obviously, the pump working pressure strongly influences the pressure results, but the trend of each line is similar as shown in Figure 17(b). Figure 18 show the variation curves of water film lubrication characteristic parameters in three shaft revolutions and illustrates the influence of shaft rotational speed on average water film thickness, water film pressure, leakage and total power loss. As shown in Figure 18(a), (c) and (d), the average thickness of water film, leakage and total power loss all show oscillatory trends and increase with the increase of shaft speed. When the shaft speed increases from 500 r/min to 2000 r/min, the average water film thickness increases from 4.36 μm to 6.55 μm, the average leakage rises from 0.51 L/min to 2.21 L/min, and the average total power loss rises from 99.7 W to 443.92 W. This shows that the stability of water film decreases with the increasement of shaft speed. High shaft speed will bring violent oscillation, which is unfavorable to the lubrication of the integrated slipper interface. Fortunately, the period of time when the water film thickness is large is short. Therefore, from the whole revolution, the impact of high-speed conditions on pump performance is acceptable. For different shaft speeds, the water film pressure fields are also very close, with a difference of only about 0.3% as illustrated in Figure 18(b).

Sensitivity analysis of water film
To express the lubrication characteristics of water film under different working conditions more clearly, the average value of water film lubrication characteristic parameters in Section 4.3 is taken to illustrate. Figure 19 shows the relationship between the lubrication performance of the integrated slipper and the operating conditions. The two operating parameters of temperature, working pressure and rotational speed were kept unchanged, and one of them was changed. The influence of variable operating parameters on the lubrication characteristics of integrated slipper was analyzed. For example, when the three reference parameters, namely, the temperature is 20°C, the working pressure is 10 MPa, and the speed is 500 r/min. The results show that the leakage is 1.57, 0.24 and 0.51 L/min, and the total power loss is 309.39, 153.17 and 99.7 W, respectively. Intuitively, with the increasement of temperature, the leakage and total power loss decrease, but the amplitude is not large. Fortunately, the excellent cooling conditions of the deep sea make up for these deficiencies. On the contrary, the increasement of temperature has a positive effect on the performance of the integrated slipper. However, with the increase of working pressure and shaft speed, both leakage and total power loss would increase significantly. Obviously, as the working pressure increasing, the leakage increases sharply, and the increase in total power loss is not as large as the leakage. This shows that the increase of working pressure increases the thickness of water film and reduces the frictional power loss. Therefore, it can be concluded that the working pressure seriously affects the volumetric efficiency of the piston pump. The effect of increasing shaft speed on the leakage and power loss is similar and almost linear. That is, with every 500 r/min increase in shaft speed, the leakage and total power loss would increase by 0.65 L/min and 132 W, respectively.

Conclusions
The purpose of this paper is to develop an integrated slipper structure in seawater hydraulic axial piston pump through PSO intelligent methodology, and to analyze its dynamic lubrication characteristics. Through the analysis above, we can get the following valuable conclusions: (1) An integrated slipper/swashplate structure in SHAPP is proposed, which can effectively weaken or even eliminate the lateral force on pistons and has superior performance of low power loss and high volumetric efficiency.
(2) An optimization design method based on a hybrid of CFD and PSO for the integrated slipper is proposed, and the leakage and power loss of the optimized   slipper is improved by 27.3% and 16.7% compared with that before optimization.
(3) The influence of multiple working conditions on the performance of the integrated slipper is investigated. The leakage and total power loss increase significantly with the increasement of working pressure and shaft speed, and decrease with the increasement of temperature.
(4) For the integrated slipper, the larger interface area enhances the hydrodynamic pressure effect under high load and shaft speed conditions. As the temperature increases, the viscosity of the water film inside the integrated slipper interface would decrease, and the thickness and load-carrying capacity of the water film decreases, thus the leakage and power loss of the water film decrease. Fortunately, the excellent cooling conditions of the deep sea make up for these deficiencies.
Furthermore, due to the limitation of time and experimental equipment, the experimental investigation of integrated sliding pair is not complete. In the future, a monitoring system for the thickness of the water film of the key friction pair of the seawater piston pump will be built to verify the correctness of the dynamic pressure lubrication model of the integrated sliding pair.

Disclosure statement
No potential conflict of interest was reported by the author(s).