Processing method for simulating boundary force of wave impact structure by smoothed particle hydrodynamics

Abstract

The invention provides a smoothed particle fluid dynamics simulation wave impact structure boundary force processing method, which comprises the following steps: S1, initializing the speed, density and position of particles by establishing a numerical model; s2, dividing fluid particles and boundary particles; s3, calculating and correcting the density and speed variation of the particles according to the hydrodynamic equation and the boundary rejection effect; s4, calculating the pressure and the speed of boundary particles through a Schelerd interpolation method, and reflecting the interaction between the fluid and the solid; s5, based on the pressure and the position of the boundary particles, calculating the pressure intensity, obtaining the stress and the torque of the structure, and updating the degree of freedom; and S6, updating the particle state and entering the solution of the next time step. According to the method, when the influence of the boundary force is determined, the influence of the relative speed of fluid particles and wall surface particles is considered, meanwhile, the maximum speed in a flow field is approximately replaced by the artificial sound velocity, and the purposes of accurately simulating the interaction process of the ocean wave-ocean engineering structure and ensuring the stability of numerical simulation calculation are achieved.

Description

Technical Field

[0001] The present invention belongs to the field of computational fluid dynamics, and in particular relates to a method for processing boundary forces of a structure using smoothed particle hydrodynamics (Smoothed Particle Hydrodynamics, SPH) to simulate waves impacting the structure. Background Art

[0002] In marine engineering, numerical simulation is essential for understanding and predicting the interaction between waves and structures. As an advanced meshless numerical simulation method, the SPH method can effectively handle complex free surface flows, large deformation problems, and fluid-solid coupling phenomena. Compared with traditional grid-based methods, SPH provides higher flexibility and adaptability, enabling it to more accurately capture the details of the interaction between waves and structures. Especially when simulating the impact of waves on important structures such as oil and gas platforms and offshore wind power facilities, SPH shows its irreplaceable importance. However, when simulating waves impacting marine structures, the SPH method has the problem that fluid particles can easily penetrate the boundary, which can lead to a decrease in simulation accuracy and even calculation collapse. This problem poses a challenge to the accuracy and reliability of SPH simulations. Therefore, developing effective boundary processing techniques to solve this problem is crucial to enhancing the application value of the SPH method in marine engineering.

[0003] Some researchers have explored solutions to the problem that fluid particles easily penetrate solid boundaries. Monaghan et al. proposed a solid boundary processing method. When fluid particles approach the boundary, the boundary exerts a repulsive force on the fluid to prevent the fluid particles from penetrating the solid boundary. However, once the fluid particles are too close to the boundary, or the particle distribution near the boundary is extremely uneven, the boundary particles will exert a huge repulsive force on the fluid particles during the calculation process. This huge repulsive force often leads to calculation collapse, resulting in poor stability. In order for SPH to more accurately simulate the complex interaction between waves and marine structures, a better boundary force processing method that can simulate SPH simulation of waves impacting structures is needed, thereby providing important theoretical support for the design and optimization of marine engineering. Summary of the invention

[0004] In response to the problems in the prior art that "the SPH method has fluid particles penetrating the boundary when simulating waves impacting marine structures" and "the existing solid boundary processing methods have poor stability", the present application proposes an SPH boundary force processing method for simulating waves impacting structures. When determining the influence of the boundary force, the influence of the relative velocity between fluid particles and wall particles is considered, and the maximum velocity in the flow field is approximately replaced by an artificial sound velocity, so as to achieve the purpose of accurately simulating the interaction process between ocean waves and marine engineering structures and ensuring the stability of numerical simulation calculations.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is a method for processing boundary forces of wave impact structures by using smooth particle fluid dynamics simulation, which specifically includes the following steps:

[0006] Step S1: Based on the scale of the simulated sea area, the water body and the marine engineering structure in the ocean waves are discretized using smooth particle hydrodynamic particles, a numerical model is established, and the physical initial quantities of the particles are determined, including the velocity, density and position of the particles;

[0007] Step S2: In the numerical model, the particles are divided into fluid particles and boundary particles, and the density change of the fluid particles is calculated based on the distance kernel function between the fluid particles and the particles in their neighborhood and the velocity gradient of the fluid particles according to the smoothed particle hydrodynamics method;

[0008] Step S3: According to the fluid dynamics equation, the velocity change of the fluid particles is calculated by combining the gravity, viscosity and pressure acting on the fluid particles, and the boundary repulsion effect caused by the boundary particles is applied to correct the velocity change of the fluid particles caused by the boundary force generated by the boundary repulsion effect;

[0009] Step S4: using the Shepard interpolation method, based on the distance kernel function between the fluid particles and the boundary particles, the pressure and velocity of the boundary particles are calculated so that the boundary particles reflect the interaction between the fluid and the solid structure;

[0010] Step S5: For a non-fixed marine engineering structure with a degree of freedom, the pressure of the boundary particles calculated in step S4 is combined with the density and position of the boundary particles initialized in step 1, and the pressure of the boundary particles is further calculated using the fluid dynamics state equation, and the total force on the structure is obtained by numerically integrating the pressure of each boundary particle on the surface of the structure, or the torque on the structure is obtained by numerically integrating the moment of the pressure of each boundary particle relative to the rotation axis or center of mass of the structure, and then the displacement of each degree of freedom at the current moment and its first-order and second-order derivatives are calculated;

[0011] Step S6: According to step S2 and step S3, the density change and velocity change of the particles are obtained, and the density, velocity and position state of all fluid particles are updated. At the same time, according to the force and torque of the non-fixed marine engineering structure calculated in step S5, the velocity and position state of the boundary particles are updated, and then the overall degree of freedom of the structure is updated, and then the solution of the next time step is entered.

[0012] Furthermore, in step S2, the fluid particles represent fluids, including air, water, and oil, and the boundary particles represent specific boundaries, including coasts, marine engineering structures, ships, missiles, and wave tank boundaries.

[0013] Further, step S2 includes the following steps:

[0014] Step S201: When particle i is a fluid particle and there is a boundary particle j in the support domain, the density change of particle i is calculated according to the discrete continuity equation of the smooth particle fluid dynamics format:

[0015]

[0016] Among them, i is the density of particle i at the current time step, particle j is a particle in the support domain of particle i, and u i ,u j are the velocity vectors of particles i and j, respectively, is the kernel function gradient, m j ,ρ j are the mass and density of particle j, respectively, and t represents time;

[0017] Step S202: using the Navier-Stokes equation discretized by the smoothed particle hydrodynamics method to calculate the velocity change of particle i, that is, the velocity change of particle i is calculated according to the following formula:

[0018]

[0019] in,

[0020] π ij =(u j -u i )·(x j -x i ) / |r ij | 2

[0021] In the above formula, ρ i is the density of particle i at the current time step, particle j is a particle in the support domain of particle i, and u i ,u j are the velocity vectors of particles i and j, respectively, is the kernel function gradient, m j ,ρ j are the mass and density of particle j, t represents time, g is the gravitational acceleration vector, c is the speed of sound, ρ 0i is the reference density of particle i, p i ,p j is the pressure of particles i and j, W ij is the kernel function value determined by the distance between particle j and particle i, α is the artificial viscosity coefficient, and x i ,x j is the position vector of particle i and particle j, r ij=x i -x j , h is the smooth length of the particle.

[0022] Furthermore, step S3 includes the following steps:

[0023] Step S301: When particle i is a fluid particle and there is a boundary particle j in the support domain, the boundary particle j produces a boundary repulsion effect on the fluid particle i, and the acceleration dimension f is calculated. ij , represents the velocity change of particle i caused by the boundary force generated by the repulsive effect, and the calculation formula is as follows:

[0024]

[0025] Where c is the artificial sound speed used in the calculation, n j is the normal vector of the boundary particle j, that is, the normal vector of the position of particle j determined by the shape of the solid boundary, dim is the dimension of calculation, dim=2 in two-dimensional simulation, dim=3 in three-dimensional simulation;

[0026] Step S302: Compare the speed change obtained in step S2 with the calculated f ij Add them together, correct the velocity change, and calculate the final velocity change of particle i in the current time step:

[0027]

[0028] In the calculation of the boundary repulsion in step S301, its physical meaning is: (u i -u j )·n j ≥0, that is, the velocity difference between fluid particle i and boundary particle j (u i -u j ) and the boundary normal vector n j At an acute angle (or perpendicular to each other), it means that the particle is moving away from the boundary (or parallel to the boundary), so it is impossible for the particle to penetrate the boundary and no treatment is required. Similarly, when (u i -u j )·n j <0, particle penetration problems need to be avoided.

[0029] Furthermore, in step S3, the boundary repulsion calculation formula uses the artificial sound velocity c instead of the maximum velocity in the flow field to obtain a boundary repulsion size that matches the flow intensity. When the flow is intense, the maximum flow field velocity and the artificial sound velocity are larger, and the calculated boundary repulsion force is also larger, which can better prevent fluid particles from penetrating the boundary. When the flow is relatively gentle, the maximum flow field velocity and the artificial sound velocity are smaller, and the boundary repulsion force obtained by the calculation formula is also smaller, which can minimize the calculation error caused by the numerical method and make the simulation more in line with physical reality.

[0030] Furthermore, in step S4, when particle i is a boundary particle, Shepard interpolation is used to solve the pressure and velocity of particle i:

[0031]

[0032] Where f represents the fluid particles, p f ,m f ,ρ f are the pressure, mass and density of the fluid particles, W if is the kernel function value determined by the distance between the boundary particle i and the fluid particle f, u f is the velocity vector of the fluid particle f.

[0033] The beneficial effects of the present application are as follows: in response to the problems existing in the prior art that "the SPH method has the problem that fluid particles easily penetrate the boundary when simulating waves impacting marine structures" and "the existing solid boundary processing methods are not stable", it is proposed to consider the influence of the relative velocity between fluid particles and wall particles when determining the influence of the boundary repulsion size, so that the boundary's repulsion of fluid particles close to the boundary is more in line with physical reality. At the same time, the method described in the present application approximates the maximum velocity in the flow field with an artificial sound velocity, and there is no need to find the maximum velocity in the flow field after each calculation step. While ensuring that the size of the repulsive force can meet the size required by the problem, it also saves calculation requirements. Furthermore, the present application can achieve the purpose of accurately processing the fluid-solid interface when simulating ocean waves impacting structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 It is a schematic diagram of the standing wave simulation of the present invention.

[0035] Figure 2 This is a curve diagram of total kinetic energy versus time obtained by simulating standing wave flow with a dissipation coefficient of β=0.3432 using the SPH method in the present invention when there is and does not have a boundary force model.

[0036] Figure 3 This is a curve diagram of total kinetic energy changing with time obtained by simulating standing wave flow with dissipation coefficient β=0.1648 using the SPH method in the present invention when there is or is not a boundary force model.

[0037] Figure 4 It is a schematic diagram of the numerical wave tank of the present invention.

[0038] Figure 5 The SPH simulation of the regular wave obtained by the present invention at t = 1.3 (g / d) 1 / 2 Schematic diagram of particle distribution when , where the left side shows no boundary force applied, and the right side shows the boundary force applied.

[0039] Figure 6 It is a schematic diagram of the interaction of an embodiment of a wave-bottom hinged pendulum wave energy device of the present invention.

[0040] Figure 7 Schematic diagram showing the comparison between the experimental photo (left) and the SPH method simulation prediction result (right) of the bottom-hinged pendulum wave energy device embodiment of the present invention within one wave cycle. DETAILED DESCRIPTION

[0041] The present invention is further described and illustrated below in conjunction with specific embodiments. The embodiments are merely exemplary of the present disclosure and do not define the scope of limitation. The technical features of each embodiment of the present invention may be combined accordingly without conflicting with each other.

[0042] Example 1: The following uses a standing wave numerical water tank as an example to illustrate the technical solution of the present invention. The simulation includes the following steps:

[0043] Step S1: Initially, establish Figure 1 The numerical model shown in the figure is a water tank with an open top. The length of the water tank is L, the height of the walls on both sides of the water tank is 1.2L, and the water depth is d = L. At the initial time t = 0, the water surface is parallel to the bottom of the water tank, and the initial density is calculated by the formula calculate;

[0044] Step S2: If a particle i belongs to a fluid particle, its density increment is calculated according to the SPH format discretized continuity equation:

[0045]

[0046] The velocity increment of particle i is calculated using the Navier-Stokes equation discretized by the SPH method, that is, the velocity increment of particle i is calculated as follows:

[0047]

[0048] Step S3: If a particle i belongs to a fluid particle and there is a boundary particle j in the support domain, the boundary particle j should have a certain boundary repulsion f to the fluid particle i. ij :

[0049]

[0050] Calculate the boundary repulsion f ij Finally, the velocity change obtained in step 3) should be added with boundary repulsion, that is,

[0051]

[0052] Step S4: If a particle i is a boundary particle, use Shepard interpolation to solve the pressure and velocity of particle i;

[0053] Step S5: This embodiment does not involve non-fixed marine engineering structures, and there is no need to calculate the total force or torque on the structure;

[0054] Step S6: for all fluid particles, after obtaining the density change and velocity change of the particles according to steps S2 and S3, the density and velocity of the particles are updated and the solution of the next time step is entered.

[0055] The total kinetic energy of the standing wave is E k The diachronic evolution can be expressed analytically by Indicates that, where β = 4νk 2 is the damping coefficient, is kinematic viscosity. Figure 2 and Figure 3 The total kinetic energy obtained by simulating standing wave flow (boundary force and unbounded force models) using the SPH method under different damping coefficients is shown as a function of time. As shown in the figure, the total kinetic energy calculated by SPH under both the boundary force model and the unbounded force model is consistent with the theoretical results. This shows that in areas with low particle motion intensity, the boundary force treatment method proposed in this paper does not affect the calculation accuracy. However, the subsequent numerical results in Example 2 show that for wave problems, if there is no boundary force model in the calculation process, the liquid particles can penetrate the solid surface.

[0056] Example 2: The following uses a wave numerical water tank as an example to illustrate the technical solution of the present invention. The simulation includes the following steps:

[0057] Step S1: Initially, establish Figure 4 The numerical model shown in the figure. The water depth of the water tank is d, the length is L (L = 40d), the leftmost side of the water tank is a wave-making plate, and a push plate is used for numerical wave making. The simulated regular wave parameters are: wave height H equals 0.23d, wavelength λ equals 7.4d.

[0058] Step S2: If a fluid particle i is a free surface particle, its density increment is calculated according to the SPH format discretized continuity equation:

[0059]

[0060] The velocity increment of particle i is calculated using the Navier-Stokes equation discretized by the SPH method, that is, the velocity increment of particle i is calculated as follows:

[0061]

[0062] Step S3: If a particle i belongs to a fluid particle and there is a boundary particle j in the support domain, the boundary particle j should have a certain boundary repulsion f to the fluid particle i. ij :

[0063]

[0064] Calculate the boundary repulsion f ij Finally, the velocity change obtained in step 3) should be added with boundary repulsion, that is,

[0065]

[0066] Step S4: If a particle i is a boundary particle, use Shepard interpolation to solve the pressure and velocity of particle i;

[0067] Step S5: This embodiment does not involve non-fixed marine engineering structures, and there is no need to calculate the total force or torque on the structure;

[0068] Step S6: For all fluid particles, after obtaining the density change and velocity change of the particles according to steps S2 and S3, the density and velocity of the particles are updated according to the time advancement format used for calculation (such as the prediction-correction time format) and the solution of the next time step is entered.

[0069] Figure 5 It is shown that in the simulation, t = 1.3 (d / g) 1 / 2 Particle distribution near the wave-making plate obtained by SPH when . The results in the left figure are in the process of calculation, and the solid has no repulsive force on the liquid, while the results in the right figure apply repulsive force during the simulation. As shown in the figure, if the boundary force processing method described in this application is not used during the calculation, the liquid particles will penetrate into the solid wall. This non-physical penetration phenomenon not only destroys the conservation characteristics, but also causes flow field fluctuations, and finally leads to the collapse of the calculation. Therefore, the repulsive force model is very important to prevent liquid particles from penetrating into the solid wall, especially when simulating high-speed relative motion problems between liquid and solid.

[0070] Example 3: The following takes the bottom swing wave energy device (OWSC) as an example to illustrate the technical solution of the present invention. The simulation includes the following steps:

[0071] Step S1: At the initial moment, according to the specific size of the bottom hinged pendulum wave energy device, the corresponding fluid particles and boundary particles are arranged, such as Figure 6As shown: The device consists of a box with a width of 0.646m, a height of 0.310m, and a thickness of 0.0875m. There is a semi-cylinder at the bottom of the box. The model OWSC is installed on a hinge 0.1m from the bottom. The center of mass is located 12.2m away from the wave maker and 0.132m above the hinge. The mass is 4.27kg, and the moment of inertia of the hinge is 0:1147kg·m 2 ,like Figure 6 As shown, in the example simulation, the wave height H = 0.1m, the wave period T = 1.9s, the water depth d = 0.305m, the initial particle distance is 0.01m, the total number of particles is 55,834, and the calculation starts after the initialization is completed;

[0072] Step S2: If a particle i belongs to a fluid particle, its density increment is calculated by discretizing the continuity equation in SPH format, and the velocity increment of particle i is calculated by using the Navier-Stokes equation discretized by SPH method;

[0073] Step S3: If a particle i belongs to a fluid particle and there is a boundary particle j in the support domain, the boundary particle j should have a certain boundary repulsion f to the fluid particle i. ij :

[0074]

[0075] This example is a two-dimensional simulation, dim = 2;

[0076] Calculate the boundary repulsion f ij Finally, the velocity change obtained in step 3) should be added with boundary repulsion, that is,

[0077]

[0078] Step S4: If a particle i is a boundary particle, use Shepard interpolation to solve the pressure and velocity of particle i;

[0079] Step S5: Since the marine engineering structure simulated in this example is an articulated structure with one rotational degree of freedom, the torque on the structure should be calculated; the pressure on the surface of the structure calculated in step S4 is accumulated to obtain the torque on the structure by adding the moment of the pressure to the axis of rotation (or center of mass) of the structure, and the angular velocity and rotation angle of the structure are further calculated in combination with the time step;

[0080] Step S6: For all fluid particles, after obtaining the density change and velocity change of the particles according to steps S2 and S3, the density and velocity of the particles are updated according to the time advancement format used for calculation, and the solution of the next time step is entered.

[0081] Figure 7The experimental photos (left) and the SPH simulation prediction results (right) of the bottom-hinged pendulum wave energy device within a wave cycle are compared. At the beginning, the water pushed the baffle and flowed through the top (Frame A, B). Then, the free surface on the left side of the pendulum changed dramatically, even including certain breakage and cavitation (Frame C, D, E). When the pendulum rotated to the maximum seaward direction (Frame F, G, H), breakage and slamming occurred. However, no matter at which moment in a cycle, the SPH predicted structure movement can be consistent with the experiment. It can be seen that even in the design of wave breaking, strong impact and other problems where pressure is extremely easy to oscillate and particles are extremely easy to penetrate, the boundary force processing method proposed in this application for simulating wave impact on structures can still prevent fluid particles from penetrating the boundary and ensure good numerical accuracy.

[0082] The above-mentioned embodiments only express several implementation methods of the present invention, and the description is relatively specific and detailed, but it cannot be understood as limiting the scope of the present invention. For ordinary technicians in this field, several modifications and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A method for processing boundary forces of wave impact structures using smooth particle fluid dynamics simulation, characterized in that: The steps include: Step S1: Based on the scale of the simulated sea area, the water body and the marine engineering structure in the ocean waves are discretized using smooth particle hydrodynamic particles, a numerical model is established, and the physical initial quantities of the particles are determined, including the velocity, density and position of the particles; Step S2: In the numerical model, the particles are divided into fluid particles and boundary particles, and the density change of the fluid particles is calculated based on the distance kernel function between the fluid particles and the particles in their neighborhood and the velocity gradient of the fluid particles according to the smoothed particle hydrodynamics method; Step S3: According to the fluid dynamics equation, the velocity change of the fluid particles is calculated by combining the gravity, viscosity and pressure acting on the fluid particles, and the boundary repulsion effect caused by the boundary particles is applied to correct the velocity change of the fluid particles caused by the boundary force generated by the boundary repulsion effect; Step S4: using the Shepard interpolation method, based on the distance kernel function between the fluid particles and the boundary particles, the pressure and velocity of the boundary particles are calculated so that the boundary particles reflect the interaction between the fluid and the solid structure; Step S5: For a non-fixed marine engineering structure with a degree of freedom, the pressure of the boundary particles calculated in step S4 is combined with the density and position of the boundary particles initialized in step 1, and the pressure of the boundary particles is further calculated using the fluid dynamics state equation, and the total force on the structure is obtained by numerically integrating the pressure of each boundary particle on the surface of the structure, or the torque on the structure is obtained by numerically integrating the moment of the pressure of each boundary particle relative to the rotation axis or center of mass of the structure, and then the displacement of each degree of freedom at the current moment and its first-order and second-order derivatives are calculated; Step S6: According to step S2 and step S3, the density change and velocity change of the particles are obtained, and the density, velocity and position state of all fluid particles are updated. At the same time, according to the force and torque of the non-fixed marine engineering structure calculated in step S5, the velocity and position state of the boundary particles are updated, and then the overall degree of freedom of the structure is updated, and then the solution of the next time step is entered.

2. The method for processing boundary forces of smooth particle fluid dynamics simulation of wave impact structures according to claim 1, characterized in that: In step S2, the fluid particles represent fluids, including air, water, and oil, and the boundary particles represent specific boundaries, including coasts, marine engineering structures, ships, missiles, and wave tank boundaries.

3. The method for processing boundary forces of smooth particle fluid dynamics simulation of wave impact structures according to claim 1, characterized in that: Step S2 includes the following steps: Step S201: When particle i is a fluid particle and there is a boundary particle j in the support domain, the density change of particle i is calculated according to the discrete continuity equation of the smooth particle fluid dynamics format: Among them, i is the density of particle i at the current time step, particle j is a particle in the support domain of particle i, and u i ,u j are the velocity vectors of particles i and j, respectively, is the kernel function gradient, m j ,ρ j are the mass and density of particle j, respectively, and t represents time; Step S202: using the Navier-Stokes equation discretized by the smoothed particle hydrodynamics method to calculate the velocity change of particle i, that is, the velocity change of particle i is calculated according to the following formula: in, π ij =(u j -u i )·(x j -x i ) / |r ij | 2 In the above formula, ρ i is the density of particle i at the current time step, particle j is a particle in the support domain of particle i, and u i ,u j are the velocity vectors of particles i and j, respectively, is the kernel function gradient, m j ,ρ j are the mass and density of particle j, t represents time, g is the gravitational acceleration vector, c is the speed of sound, ρ 0i is the reference density of particle i, p i ,p j is the pressure of particles i and j, W ij is the kernel function value determined by the distance between particle j and particle i, α is the artificial viscosity coefficient, and x i ,x j is the position vector of particle i and particle j, r ij =x i -x j , h is the smooth length of the particle.

4. The method for processing boundary forces of smooth particle fluid dynamics simulation of wave impact structures according to claim 1, characterized in that: Step S3 includes the following steps: Step S301: When particle i is a fluid particle and there is a boundary particle j in the support domain, the boundary particle j produces a boundary repulsion effect on the fluid particle i, and the acceleration dimension f is calculated. ij , represents the velocity change of particle i caused by the boundary force generated by the repulsive effect, and the calculation formula is as follows: Where c is the artificial sound speed used in the calculation, n j is the normal vector of the boundary particle j, that is, the normal vector of the position of particle j determined by the shape of the solid boundary, dim is the dimension of calculation, dim=2 in two-dimensional simulation, dim=3 in three-dimensional simulation; Step S302: Compare the speed change obtained in step S2 with the calculated f ij Add them together, correct the velocity change, and calculate the final velocity change of particle i in the current time step:

5. The method for processing boundary forces of smooth particle fluid dynamics simulation of wave impact structures according to claim 4, characterized in that: In step S3, the boundary repulsion calculation formula uses the artificial sound velocity c instead of the maximum velocity in the flow field. When the flow is violent, the maximum flow field velocity and the artificial sound velocity are larger, and the calculated boundary repulsion force is also larger. When the flow is relatively gentle, the maximum flow field velocity and the artificial sound velocity are smaller, and the boundary repulsion force obtained by the calculation formula is also smaller.

6. The method for processing boundary forces of smooth particle fluid dynamics simulation of wave impact structures according to claim 1, characterized in that: In step S4, when particle i is a boundary particle, Shepard interpolation is used to solve the pressure and velocity of particle i: Where f represents the fluid particles, p f ,m f ,ρ f are the pressure, mass and density of the fluid particles, W if is the kernel function value determined by the distance between the boundary particle i and the fluid particle f, u f is the velocity vector of the fluid particle f.

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