A solid-liquid coupling simulation method with surface tension effect based on the material point method

Abstract

The present invention discloses a solid-liquid coupling simulation method with surface tension effect based on the material point method, belonging to the fields of computer graphics and physics-based simulation. The system used in the simulation method includes solid model modeling, fluid model modeling, Level Set implicit surface construction, Marching Cubes display surface construction, surface particle resampling, and collision detection. The system simulates fluids and solids on the basis of the material point method, constructs Level Set to resample fluid surface particles, and adds surface tension to the fluid using the CSF model. Different background grids are used to simulate fluids and solids respectively to solve the inherent adhesion phenomenon during MPM solid-liquid coupling and achieve the slip contact effect during solid-liquid coupling. Finally, the coupling phenomenon of incompressible fluid with surface tension effect and nonlinear elastic solid is realized.

Description

Technical Field

[0001] The present invention relates to the fields of computer graphics and physics-based simulation, and in particular to a solid-liquid coupling simulation method with surface tension effect based on the material point method. Background Art

[0002] Surface tension is very important for free surface fluids, and how to more realistically simulate fluids with surface tension has also been a research hotspot in the field of graphics in recent years.

[0003] In the prior art, some extend the GEM method to handle incompressible fluids, and the influence of surface tension is added; some use surface tension as an explicit boundary condition in the projection step and use the Level Set surface to simulate the contact angle of water droplets. However, when the surface tension effect dominates the scene, explicit surface tension calculations can lead to system instability, such as simulating bubbles; some use a semi-implicit method based on the CSF model to calculate surface tension; some propose a semi-implicit surface tension calculation method based on Level Set, using the divergence of the velocity field of the fluid surface as the surface curvature of the fluid to simulate more realistic bubbles; some use a significantly different method, by solving a volume conservation equation based on the mean curvature instead of deriving the surface tension formula. Compared with grid-based calculation methods, Lagrangian surface tension calculations usually perform explicit geometric discretization; some use deformation operators to simulate droplets on the surface grid; some simulate complex bubble structures, using non-manifold Lagrangian triangular meshes and integrating surface tension information into the mesh vertices; some propose a fluid framework with only a surface and apply a general three-dimensional fluid solver to the surface grid. Lagrangian surface tension calculations can also be implicitly processed; some propose a hybrid framework that calculates implicit surface tension on a Lagrangian grid and solves for pressure on a background grid; some use the Lagrangian formula for incompressible fluids to handle specific surface tension in a fully implicit manner. Some propose a new three-way coupling method for simulating the interaction between solids and fluids driven by strong surface tension.

[0004] Commonly used solid-liquid coupling solvers usually use the Euler method to simulate fluids and the Lagrangian method to simulate solids, and their coupling usually uses the velocity of the solid as the boundary condition of the fluid.

[0005] In the prior art, some use a relevant contact surface as the interaction and solve for the fluid and solid separately on the contact surface; some use a similar method, but use the predicted solid position as the velocity limit for the fluid projection step. This weak coupling method usually has certain limitations in terms of accuracy and stability. These problems can be solved by using an overall strong coupling method, that is, putting the fluid and solid into the same linear system for solution; some have proposed the first fully implicit and stable solid-fluid bidirectional coupling solver, who integrated the fluid pressure projection equation and the velocity update equation of the deformable body into an asymmetric linear equation system; some have proposed a positive definite and symmetric equation system to solve the coupling of the fluid and rigid body by minimizing the kinetic energy; some have further modified the above positive definite and symmetric equation system using algebraic transformation; some have proposed a multigrid-based incompressible fluid and rigid body bidirectional coupling solver, which can solve the solid-liquid coupling system more efficiently; some have realized the coupling of the fluid and sub-grid solid; some have proposed a bidirectional coupling method for non-viscous non-linear elastic solids and incompressible fluids.

[0006] The MPM method has been introduced into the field of graphics. The MPM method can automatically support grid-based collision handling and bidirectional coupling interaction, but there is a no-slip boundary condition at the collision contact surface.

[0007] In the prior art, some have proposed an APIC transmission method that can better ensure the conservation of angular momentum of the system. Some have used MPM to realistically simulate hair; some have proposed using MPM to simulate viscoelastic fluids, foams, and sponges; some have used precise friction contact calculations to couple MPM with rigid bodies; some have extended the MPM method to be used for simulating sand; some have proposed a generalized interpolation material point method and performed adaptive subdivision on specific regions; some have used anisotropic drag in a separate velocity grid to achieve more flexible fluid-cloth interaction.

[0008] As can be seen from the above prior arts, in traditional surface tension effect simulation methods, the vast majority of research only simulates a single fluid with surface tension added, or only adds a non-slip boundary condition for solid-liquid coupling, while ignoring the surface tension effect during solid-liquid interaction. Summary of the Invention

[0009] The technical problem to be solved by the present invention is to provide a solid-liquid coupling simulation method with surface tension effect based on the material point method. On the basis of the material point method, the fluid is simulated. The fluid surface particles are resampled by constructing a Level Set, and the surface tension is added to the fluid using the CSF model. Different background grids are used to simulate the fluid and the solid respectively to solve the inherent adhesion phenomenon during MPM solid-liquid coupling, achieve the slip contact effect during solid-liquid coupling, and finally realize the coupling phenomenon of incompressible fluid with surface tension effect and nonlinear elastic solid.

[0010] To solve the above technical problem, the technical solution adopted by the present invention is:

[0011] A solid-liquid coupling simulation method with surface tension effect based on the material point method, comprising the following steps:

[0012] Step 1, create MPM fluid particles and solid particles, and perform initialization, and store physical quantities on the fluid particles and solid particles;

[0013] Step 2, construct an implicit Level Set surface of the fluid, construct a spherical Level Set surface for each fluid particle, generate a signed distance field, and then combine each spherical Level Set into an implicit surface of the entire fluid simulation region;

[0014] Step 3, use the signed distance field to calculate the gradient field and Laplacian operator field of the fluid region;

[0015] Step 4, convert the implicit surface into a Marching Cubes display surface, and use the zero isosurface of the signed distance field as the isosurface for constructing Marching Cubes;

[0016] Step 5, resample surface particles on the display surface. The generated surface particles only have position information, do not have physical quantities such as mass and velocity, and the surface particles are generated at the beginning of each time step and deleted at the end of each time step;

[0017] Step 6, perform grid marking;

[0018] Step 7, calculate the surface tension on the surface particles;

[0019] Step 8, map the surface tension on the surface particles to the fluid particles;

[0020] Step 9, map the particle information to the background grid. Here, the standard APIC mapping method is used to map the momentum and mass of the MPM particles to the background grid;

[0021] Step 10, add external forces on the background grid and solve for the new velocity;

[0022] Step 11, impulse collision handling, calculating the fluid velocity after collision using the impulse method to generate an impact effect, and adding an appropriate velocity decay to the fluid in the tangential direction of the solid-liquid contact surface;

[0023] Step 12, mapping the information on the background grid back to the MPM particles, using the standard APIC mapping method to transfer the particle velocity on the background grid back to the MPM particles, and updating the affine velocity of the MPM particles.

[0024] A further improvement of the technical solution of the present invention is that in Step 1, the physical quantities include position, velocity, deformation gradient, affine velocity, and mass.

[0025] A further improvement of the technical solution of the present invention is that in Step 2, an implicit Level Set surface is constructed:

[0026] A spherical Level Set surface is constructed for each fluid particle, and the expression of Level Set is:

[0027] f(x) = |x - c| - r (1)

[0028] In the formula, x represents the position of the particle, r represents the radius of the spherical Level Set, and c represents the position of the liquid particle.

[0029] A further improvement of the technical solution of the present invention is that in Step 3, the gradient field of the fluid region represents the normal vector field of the fluid, and the Laplacian operator field represents the curvature field of the fluid.

[0030] A further improvement of the technical solution of the present invention is that in Step 6, the fluid particle grid, fluid surface grid, solid particle grid, solid surface grid, air grid, and fluid-solid contact grid are respectively marked.

[0031] A further improvement of the technical solution of the present invention is that in Step 7, in order to calculate the surface tension more accurately, the surface tension is only calculated on the fluid free surface grid, and the fluid surface at the solid-liquid contact surface is not calculated for surface tension.

[0032] A further improvement of the technical solution of the present invention is that in Step 8, the surface tension calculated at the surface particles is first transferred to the fluid background grid through the B-spline interpolation function, and then transferred from the fluid background grid to the fluid surface particles.

[0033] A further improvement of the technical solution of the present invention lies in: in steps 7 and 8, for the calculation method of the fluid surface tension, the gradient field of the signed distance field is used as the normal vector field of the fluid, and the Laplacian operator field of the signed distance field is used as the curvature field. A bilinear interpolation function is used to calculate the normal vector and curvature of the resampled surface particle positions, and then the surface tension is calculated at the surface particles; after calculating the surface tension at the surface particles, the calculated surface tension is transmitted to the background grid through the bilinear interpolation function, and then to the surface fluid particles; use to represent the free surface of the fluid or the part of the fluid affected by the surface tension at time t.

[0034] A further improvement of the technical solution of the present invention lies in: in step 12, when dealing with the solid-liquid coupling, the treatment method of the solid and fluid velocities needs to ensure that the normal velocity of the fluid is consistent with the normal velocity of the solid to prevent the fluid from being bounced off; after the solid-liquid contact, the velocities are respectively:

[0035]

[0036] In the formula, v r is the relative velocity between the fluid and the solid, is the velocity of the fluid at the solid-liquid contact surface, and n i is the direction of the normal vector of the solid-liquid contact surface;

[0037]

[0038] In the formula, is the velocity of the solid at the solid-liquid contact surface.

[0039] Due to the adoption of the above technical solution, the technical progress achieved by the present invention is:

[0040] 1. The present invention proposes a method for coupling a fluid with surface tension effect and a solid under slip boundary conditions, which is different from the complete separation of the solid and the liquid in traditional solid-liquid coupling, and more realistically simulates the solid-liquid coupling.

[0041] 2. The present invention solves the adhesion problem existing at the solid-liquid contact surface inherent in traditional MPM, and more precisely solves the adhesion problem generated when the fluid and the solid are coupled.

[0042] 3. The present invention calculates the surface tension of the entire fluid surface in the traditional method, and more precisely processes the calculation method of the fluid surface tension at the solid-liquid contact surface through grid marking. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 is the flow chart of the simulation process in the embodiment of the present invention;

[0044] Figure 2 is the process diagram of surface particle resampling in the embodiment of the present invention;

[0045] Figure 3 It is the surface tension calculation process diagram in the embodiment of the present invention;

[0046] Figure 4 It is the surface tension grid marking diagram in the embodiment of the present invention;

[0047] Figure 5 It is the experimental result diagram in the embodiment of the present invention. Specific implementation manners

[0048] The embodiment of the present application provides a solid-liquid coupling simulation method with surface tension effect based on the Material Point Method (MPM), which solves the problem of the inherent solid-liquid contact surface adhesion in the prior art, more precisely solves the adhesion phenomenon problem generated when the fluid and the solid are coupled, and more precisely solves the fluid surface tension calculation problem under the slip boundary condition. By simulating the fluid and the solid on the basis of the Material Point Method, by constructing Level Set to resample the fluid surface particles, using the grid marking method to determine the free surface of the fluid, and using the Continuum Surface Force (CSF) model to add surface tension to the fluid to solve the fluid surface tension calculation problem under the slip boundary condition; using different background grids to simulate the fluid and the solid respectively to solve the inherent adhesion phenomenon during the MPM solid-liquid coupling and achieve the slip contact effect during the solid-liquid coupling; finally realizing the coupling phenomenon of the incompressible fluid with surface tension effect and the nonlinear elastic solid, so as to obtain a more realistic simulation and animation design effect.

[0049] It should be noted that the terms "first", "second", etc. in the description and claims of the present invention and the above-mentioned drawings are used to distinguish similar objects, and do not have to be used to describe a specific order or sequence. It should be understood that such data can be interchanged under appropriate circumstances so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "comprising" and "having" and any variations thereof are intended to cover non-exclusive inclusion. For example, a process, method, system, product or device comprising a series of steps or units does not have to be limited to those steps or units clearly listed, but may include other steps or units not clearly listed or inherent to these processes, methods, products or devices.

[0050] MPM is the abbreviation of Material Point Method, which is translated into Chinese as: Material Point Method;

[0051] CSF is the abbreviation of The Continuum Surface Force Model;

[0052] The following further describes the present invention in detail with reference to the drawings and embodiments:

[0053] The system of the present invention includes solid model modeling, fluid model modeling, Level Set implicit surface construction, Marching Cubes display surface construction, surface particle resampling, collision detection, etc.

[0054] (1) Solid model modeling voxelizes the obj model into a discrete point model to facilitate the construction of MPM solid particles and initialize their positions.

[0055] (2) Fluid model modeling is used to initialize MPM fluid particles.

[0056] (3) Level Set implicit surface construction is used to construct the implicit surface of the fluid. Since different Level Sets can be combined with each other, a spherical Level Set implicit surface is created around each fluid particle, and each spherical implicit surface is combined to form a signed distance field representing the entire simulation area, i.e., the fluid surface.

[0057] (4) Marching Cubes display surface construction is used to construct the display surface of the fluid. The display surface and the implicit surface can be converted into each other. The distance field generated using the implicit surface is converted into an explicit Marching Cubes surface for resampling surface particles.

[0058] (5) Surface particle resampling is used to generate massless surface particles, and the CSF model is used to calculate the surface tension of the fluid on the surface particles.

[0059] (6) Collision detection uses the impulse method to process the collision velocity to solve the inherent viscous problem of MPM.

[0060] As Figure 1 shown, the solid-liquid coupling simulation method with surface tension effect based on the material point method specifically includes:

[0061] 1. Construct an implicit Level Set surface:

[0062] A spherical Level Set surface is constructed for each fluid particle. The expression of Level Set is:

[0063] f(x) = |x - c| - r (1)

[0064] where x represents the position of the particle, r represents the radius of the spherical Level Set, and c represents the position of the liquid particle;

[0065] The signed distance field (SDF) of each fluid particle can be calculated using Equation (1). By taking the minimum value of different signed distance fields and merging them, the signed distance field of the entire fluid region is obtained.

[0066] 2. Resample the fluid surface particles:

[0067] First, use the zero isosurface of the signed distance field of the fluid region as the isosurface for generating the Marching Cubes to generate an explicit surface. Then, resample the generated explicit surface into surface particles. The positions of the generated particles are the positions of the fluid surface particles. The resampled surface particles are denoted by x q The particle resampling process is as Figure 2 shown.

[0068] 3. Calculate the surface tension:

[0069] The calculation of the surface tension is carried out on the surface particles. Use the CSF model to calculate the surface tension on the sampled surface particles. The expression of the CSF model is:

[0070]

[0071] In the formula, α is the surface tension coefficient of the fluid, κ is the average curvature of the fluid surface, n is the unit normal vector at the fluid surface, is the surface gradient.

[0072] The first part of Equation (2) acts in the normal direction and affects the change in the fluid surface area through the local curvature. The second part acts in the surface tangential direction, causing the fluid particles to flow from the position with low surface tension to the position with high surface tension. If the surface tension coefficient α is a constant, the tangential component is zero.

[0073] The gradient field of the signed distance field can be used as the normal vector field of the fluid, and the Laplacian operator field of the signed distance field can be used as the curvature field. Then, calculate the surface tension at the surface particles. After calculating the surface tension at the surface particles, transfer the calculated surface tension to the background grid through the interpolation function, and then transfer it to the surface fluid particles. The specific process is as Figure 3 shown.

[0074] More accurate calculations are carried out at the solid-fluid interface, and dynamic coupling between the solid and the fluid is supported. As Figure 3 shown, the left figure represents a solid-liquid coupling system with a free-slip boundary condition.

[0075] Use a Cartesian grid to store the signed distance field Then, the curvature field and the normal vector field of the simulation region can be calculated through The curvature field The normal vector field Using the bilinear interpolation function δ i (x) represents the bilinear interpolation function at grid node i. By associating the surface particles with the Cartesian background grid using the interpolation function δ i (x), the curvature and normal vector at the position of the surface particles can be calculated. Using to represent the fluid free surface or the part of the fluid affected by surface tension at time t, with the grid markings as shown in Figure 4 , the surface tension of the surface particles can be expressed as:

[0076]

[0077] where α is the fluid surface tension coefficient, represents the curvature at grid node i at time step n, represents the normal vector at grid node i at time step n, represents the position of surface particle q at time step n.

[0078] 4. MPM-based simulation method

[0079] 4.1 Constitutive model

[0080] In material mechanics, the constitutive model is a physical model that describes the constitutive relationship of materials. Since the MPM algorithm treats each particle as a part of the material, the energy of each part of the material is calculated using the energy density function, and the potential energy of the entire material can be obtained by integrating the potential energy carried by each part of the material.

[0081] The motion of the material can be represented as a mapping description. The initial region of the material is Ω 0 , and the spatial region of the material at time t is represented as Ω t . X ∈ Ω 0 represents the position of the material particle at the initial moment, and x ∈ Ω t represents the position of the material particle at time t. The expression of the deformation mapping can be represented as φ:(·,t):Ω 0 →Ω t , d = 2, 3.

[0082] Through the mapping function, the calculation formula for the deformation gradient F can be obtained as:

[0083]

[0084] The deformation gradient can be used to quantify the local deformation of the material. By obtaining J = det(F), the value of the determinant of the deformation gradient is obtained, and the value of J represents the volume change of a material point.

[0085] In the MPM, each particle represents a part of the material. The Cauchy stress is defined based on the energy density function of the hyperelastic body and is defined as follows:

[0086]

[0087] where P is the pressure and Ψ is the energy density function.

[0088] Calculated using the fixed - corotated constitutive model, the expression for the energy density function is:

[0089]

[0090]

[0091] where F is the deformation gradient, J is the value of the determinant of the deformation gradient, λ and μ are the lame parameters, representing the resistance of the material to deformation and volume change, and σ i is the eigenvalue of the deformation gradient F, obtained by performing SVD decomposition on the deformation gradient F.

[0092] 4.2 Interpolation Function

[0093] Denote the B - spline interpolation function at the grid node i by N i (x). The particles and the grid are associated through the interpolation function N i (x), which determines the intensity of the interaction between the particles and the grid. If the distance between the particle and the grid is closer, the weight is larger; conversely, the farther the distance, the smaller the weight.

[0094] The expression for the interpolation function N i (x) is:

[0095]

[0096] 4.3 Mapping Particle - Carried Information to the Background Grid

[0097] Use the standard APIC mapping method to transfer momentum and mass from the particles to the background grid.

[0098]

[0099]

[0100] where m p is the mass of particle p, is the position of particle p at time step n, is the mass stored at grid node i, is the velocity of particle p at time step n, is the velocity stored at grid node i at time step n, xi is the position of grid node i, is the affine velocity of particle p at time step n.

[0101] Here, the B-spline interpolation function is also used to transfer the surface tension calculated at the surface particles to the background grid.

[0102]

[0103] where is the surface tension at surface particle q at time step n.

[0104] 4.4 Grid Solving

[0105] The total potential energy of the MPM-simulated material can be expressed by the energy density function ψ. By integrating the energy of all MPM particles, it can be obtained as:

[0106]

[0107] where is the volume of the material, e is the total energy of the material, and the elastic force of the material is the negative gradient of the total potential energy of the material at its position. Therefore, all the forces on the material can be calculated in the background grid as:

[0108]

[0109] According to the momentum theorem, the change in momentum on the grid node is:

[0110]

[0111] 4.5 Mapping Background Grid Information to Particles

[0112] After updating the velocity of the grid, the velocity on the grid is also transferred back to the particles in the APIC manner, and the affine velocity of the particles is updated.

[0113]

[0114]

[0115] 5. Solid-Liquid Coupling

[0116] To solve the inherent adhesion problem of MPM during solid-liquid interaction, different grids are used to simulate the solid and the fluid respectively. Iterations are carried out between the solid and the fluid, and the impulse effect is used to handle the solid-liquid coupling contact surface to better handle the coupling between the fluid and the elastic body and solve the adhesion problem at the same time.

[0117] Using impulse to handle collisions can generate the impact effect of liquid on solid during collisions. To ensure that the fluid does not separate due to impulse calculation during collisions, the normal velocity of the fluid is made consistent with the normal velocity of the solid at the contact surface, rather than directly adding the impulse effect, which would cause the fluid to bounce off. First, calculate the velocity of the fluid relative to the solid:

[0118]

[0119] In the formula, is the velocity of the fluid at the solid-liquid contact surface, is the velocity of the solid at the solid-liquid contact surface.

[0120] Calculate the normal vector of the fluid surface using the gradient of the previously generated signed distance field, and calculate the tangential velocity of the fluid relative to the solid surface:

[0121]

[0122] The grid impulse calculation expression during solid-liquid coupling is:

[0123]

[0124] Use the Coulomb friction model to calculate the frictional force. The dynamic friction coefficient is denoted as μ. When applying the impulse effect, the maximum allowable change in the tangential velocity due to Coulomb friction is To prevent the fluid from separating from the contact surface due to impulse calculation, ensure that the normal velocity of the fluid is consistent with the normal velocity of the solid to prevent the fluid from bouncing off. Therefore, after solid-liquid contact, the velocities are respectively:

[0125]

[0126] In the formula, v r is the relative velocity of the fluid and the solid, is the velocity of the fluid at the solid-liquid contact surface, n i is the direction of the normal vector at the solid-liquid contact surface.

[0127]

[0128] In the formula, is the velocity of the solid at the solid-liquid contact surface.

[0129] The experimental results are as Figure 5 shown.

[0130] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A solid-liquid coupling simulation method with surface tension effect based on the material point method, characterized in that: It includes the following steps: Step 1, create MPM fluid particles and solid particles, and perform initialization, and store physical quantities on the fluid particles and solid particles; Step 2, construct an implicit Level Set surface of the fluid, construct a spherical Level Set surface for each fluid particle, generate a signed distance field, and then combine each spherical Level Set into an implicit surface of the entire fluid simulation region; Step 3, use the signed distance field to calculate the gradient field and Laplacian field of the fluid region; Step 4, convert the implicit surface into a Marching Cubes display surface, and use the zero isosurface of the signed distance field as the isosurface for constructing Marching Cubes; Step 5, resample surface particles on the display surface. The generated surface particles only have position information, do not have physical quantities such as mass and velocity, and the surface particles are generated at the beginning of each time step and deleted at the end of each time step; Step 6, perform mesh marking; Step 7, calculate the surface tension on the surface particles; Step 8, map the surface tension on the surface particles to the fluid particles; Step 9, map the particle information to the background grid. Here, the standard APIC mapping method is used to map the momentum and mass of the MPM particles to the background grid; Step 10, add an external force on the background grid and solve for the new velocity; Step 11, impulse collision processing, use the impulse method to calculate the fluid velocity after collision to produce an impact effect, and add an appropriate velocity decay to the fluid in the tangential direction of the solid-liquid contact surface; Step 12, map the information on the background grid back to the MPM particles. Use the standard APIC mapping method to transfer the particle velocity on the background grid back to the MPM particles and update the affine velocity of the MPM particles.

2. The solid-liquid coupling simulation method with surface tension effect based on the material point method according to claim 1, wherein: In Step 1, the physical quantities include position, velocity, deformation gradient, affine velocity, and mass.

3. A solid-liquid coupling simulation method with surface tension effect based on the material point method according to claim 1, characterized in that: In Step 2, construct the implicit Level Set surface: Construct a spherical Level Set surface for each fluid particle. The expression of Level Set is: f(x) = |x - c| - r In the formula, x represents the position where the particle is located, r represents the radius of the spherical Level Set, and c represents the position of the liquid particle.

4. A solid-liquid coupling simulation method with surface tension effect based on the material point method according to claim 1, characterized in that: In Step 3, the gradient field of the fluid region represents the normal vector field of the fluid, and the Laplacian field represents the curvature field of the fluid.

5. A solid-liquid coupling simulation method with surface tension effect based on the material point method according to claim 1, characterized in that: In Step 6, mark the fluid particle grid, fluid surface grid, solid particle grid, solid surface grid, air grid, and fluid-solid contact grid respectively.

6. A solid-liquid coupling simulation method with surface tension effect based on the material point method according to claim 1, characterized in that: In Step 7, in order to calculate the surface tension more accurately, it is only calculated on the fluid free surface grid, and the surface tension is not calculated on the fluid surface of the solid-liquid contact surface.

7. A solid-liquid coupling simulation method with surface tension effect based on the material point method according to claim 1, characterized in that: In Step 8, first transfer the surface tension calculated at the surface particles to the fluid background grid through the B-spline interpolation function, and then transfer it from the fluid background grid to the fluid surface particles.

8. A solid-liquid coupling simulation method with surface tension effect based on the material point method according to claim 1, characterized in that: In Steps 7 and 8, the specific calculation method of the fluid surface tension is: The gradient field of the signed distance field serves as the normal vector field of the fluid, and the Laplacian operator field of the signed distance field serves as the curvature field. The bilinear interpolation function is used to calculate the normal vector and curvature of the resampled surface particle positions, and then the surface tension is calculated at the surface particles; after calculating the surface tension at the surface particles, the calculated surface tension is transmitted to the background grid through the bilinear interpolation function and then to the surface fluid particles; Use represents the fluid free surface or the fluid part affected by surface tension at time t.

9. A solid-liquid coupling simulation method with surface tension effect based on the material point method according to claim 1, characterized in that: In step 12, when dealing with the solid-liquid coupling, the treatment methods of the solid and fluid velocities need to ensure that the normal velocity of the fluid is consistent with the normal velocity of the solid to prevent the fluid from being bounced off; after the solid-liquid contact, the velocities are respectively: Where, v r is the relative velocity of the fluid and the solid, is the velocity of the fluid at the solid-liquid contact surface, n i,f is the tangential velocity of the fluid relative to the solid surface, I i is the grid impulse during solid-liquid coupling; Wherein, is the solid velocity of the solid-liquid contact surface, and n i is the direction of the normal vector of the solid-liquid contact surface.

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