Loading and servo control method suitable for complex cross-section columnar sample

Abstract

The application discloses a loading and servo control method suitable for a complex cross-section columnar sample, and is based on a particle flow numerical calculation method, wherein a cross-section polygonal contour of a model of an arbitrary shape columnar sample is firstly determined, and an outer boundary polygon is extended into a polyhedron and internal particles are generated according to a model height requirement. Then, from the bottom, particles with corresponding radii are generated on the outer boundary in sequence, and after a complete boundary film particle is generated, the generation is started on the next layer, and then the same generation method is adopted to generate layer by layer until the top of the column is reached. After that, the overall outer boundary film particles are set with a lower modulus and a higher bonding strength, a servo force is started to be applied, and the size and direction of the force are updated at a certain frequency until the servo is finished. The application generates boundary particles according to each boundary in layers, and is more in line with the shape characteristics of the sample in actual engineering; and the soft servo method can not limit the lateral free deformation of the sample.

Description

Technical Field

[0001] This invention belongs to the technical field of numerical simulation calculation of deformation and failure processes of rock and soil using particle flow, and specifically relates to a loading and servo control method suitable for columnar specimens with complex cross-sections. Background Technology

[0002] The particle discrete element method (DEM) is a popular discontinuous numerical simulation method in geotechnical engineering. This method can reflect the microscopic properties of soil and rock masses and the deformation and failure mechanisms of soil and rock media under complex stress paths, thereby revealing macroscopic physical and mechanical phenomena.

[0003] When using the discrete element method (DEM) to study problems, it is often necessary to calibrate the micromechanical parameters using indoor triaxial tests, which is an important guarantee for the reliability of numerical simulation results. Currently, commonly used triaxial test specimens are standard specimens with a height / diameter ratio of 2.0, or cuboid specimens with a height / width ratio of 2.0. A servo mechanism is used to ensure uniform particle and contact distribution within the specimen. The principle is that a rigid wall is set at the outer boundary of the column, and the wall is moved along the normal direction of the outer boundary for servoing. This operation is not necessarily applicable to specimens with special requirements for the specimen profile (such as columnar jointed rock masses with irregular cross-sections). The number of outer boundaries is often large and varies in length. Using traditional rigid servo methods is often labor-intensive, and it is difficult to control the stress state of the irregular particle system within the interior, resulting in poor parameter applicability and making it difficult to obtain ideal results in numerical simulation.

[0004] In this context, if a loading and servo control method suitable for cylindrical specimens with complex cross-sections can be established, it can solve the problem of lateral loading on the one hand, and the problem of stress state control of the internal particle system on the other hand. This will enable the specimen to meet the design requirements in terms of loading deformation and failure, and the resulting micromechanical parameters can better approximate actual engineering practice, and the numerical simulation results will be more reasonable. Summary of the Invention

[0005] The purpose of this invention is to address the aforementioned problems and shortcomings by providing a loading and servo control method suitable for cylindrical specimens with complex cross-sections. By retrieving the boundary shape of the cylindrical specimen, boundary particles are generated layer by layer according to each boundary, which better matches the shape characteristics of specimens in actual engineering. A soft servo method is adopted, which applies servo to the specimen model through the flexible wall formed by the boundary particles, without restricting the lateral free deformation of the specimen.

[0006] To achieve the above objectives, the technical solution adopted is:

[0007] A loading and servo control method suitable for cylindrical specimens with complex cross-sections, comprising:

[0008] a. Determine the outer boundary contour shape of the model's cross-section. Based on the model's height, extrude the outer boundary contour into a columnar polyhedron and generate internal particles.

[0009] b. Based on the set particle radius, starting from the selected outer boundary contour, membrane particles are generated sequentially for each edge to form the contour membrane particles of the layer. Then, along the height direction, contour membrane particles of other layers are generated layer by layer until the entire columnar polyhedron forms the outer boundary membrane particles.

[0010] c. Set modulus parameters and bonding strength parameters for the outer boundary membrane particles. When servo force is applied, the corresponding modulus parameters ensure that the outer boundary membrane particles do not hinder the lateral deformation of the model, and the corresponding bonding strength parameters ensure that the outer boundary membrane particles do not break.

[0011] d. Apply servo force, update the magnitude and direction of the servo force at the set frequency, and perform model calculation and analysis until the internal stress state meets the design requirements.

[0012] According to the loading and servo control method applicable to columnar specimens with complex cross-sections of the present invention, preferably, in step b, the number of particles n on each side and the center position of each particle on each side are determined based on the coordinates of each endpoint of the geometric boundary of the selected layer of the columnar polyhedron and the radius r of the target particles, thus forming the contour particles of the selected layer.

[0013] Based on the coordinates and radius of the contour particles in the selected layer, the generation positions of particles in adjacent layers are determined. Then, based on the particle radius and the positional relationship between particles, the center positions of particles on each edge of other layers are generated in the same way. The particle height on the corresponding edge of two adjacent layers changes by 2r layer by layer.

[0014] According to the loading and servo control method for cylindrical specimens with complex cross-sections of the present invention, preferably, the coordinates of the two endpoints of one side are set as (x1, y1, z1) and (x2, y2, z1), and the number of particles n on this side is:

[0015]

[0016] The coordinates of the center position of the kth particle on this edge are:

[0017]

[0018] Where 1≤k≤n.

[0019] According to the loading and servo control method applicable to columnar specimens with complex cross-sections of the present invention, preferably, when the length of one side of the geometric boundary of the selected layer is not a multiple of the particle radius, the number of particles n is rounded down, and the particle radius is revalued according to the length of the side.

[0020] According to the loading and servo control method for cylindrical specimens with complex cross-sections of the present invention, preferably, the maximum particle radius r is set. max and minimum particle radius r min Under the control of the generated columnar polyhedral body outline, randomly distributed soil and rock particles are generated according to porosity.

[0021] According to the loading and servo control method applicable to cylindrical specimens with complex cross-sections of the present invention, preferably, the ray method is used to traverse all particles. If the particle ray intersects the contour surface at only one point, it indicates that the particle is inside the polyhedron and is retained; otherwise, the particle is deleted.

[0022] According to the loading and servo control method of the present invention applicable to columnar specimens with complex cross-sections, preferably, servo forces are applied to the particles on each side respectively. When a servo force is applied to a certain particle, the normal direction is determined according to the positional relationship between the particle and the particles above, below, left and right. Then, the servo force in the normal direction is superimposed to obtain the final applied servo force.

[0023] The steps to obtain the magnitude of the servo force finally applied to the target particle are as follows: First, find the direction vectors of the target particle and its corresponding particles in the up, down, left, and right directions. Based on the servo stress σ applied to the target particle and its directional relationship with the particles above, below, left, and right, the magnitudes F1, F2, F3, and F4 of the servo force of the target particle in the corresponding normal direction are calculated respectively.

[0024]

[0025]

[0026]

[0027]

[0028] The magnitude of the servo force applied to the target particle can be obtained by superposition. By traversing each particle, the servo force applied to each particle is obtained, and the magnitude and direction of the servo force are continuously updated during the iteration process. Each time step calculation is called an iteration step. Through iteration step calculation, it is determined whether the particle system meets the servo requirements. If the servo requirements are met, the process exits.

[0029] The beneficial effects achieved by adopting the above technical solution are:

[0030] This application, based on determining the cross-sectional contour shape of the model, extrudes the outer boundary contour into an outer boundary columnar polyhedron. Starting from the bottom layer, boundary particles are generated sequentially for each edge to form the outer boundary particles. Then, the outer boundary contour is generated layer by layer upwards until the top layer. After generating the outer boundary particles of the top layer, a low modulus is set for the outer boundary membrane particles of the overall columnar specimen to avoid affecting lateral deformation, and a high bonding strength is set to prevent damage to the outer boundary contour during servoing. Finally, servoing is initiated by applying a servo force until the model meets the servoing requirements and the servoing is completed. The above process steps, by retrieving the boundary shape of the columnar specimen and generating boundary particles layer by layer according to each boundary, better reflect the shape characteristics of specimens in actual engineering. The soft servoing method applies servoing to the specimen model through the flexible wall formed by the boundary particles, without restricting the lateral free deformation of the specimen.

[0031] This application attempts to find an equilibrium solution within a very small time step by continuously updating contact, displacement, velocity, and force information during the iterative solution process. If the model is not in equilibrium at the end of the iteration, another round of iteration will be performed. With each iteration, the solution obtained will be closer to the equilibrium state; finally, an equilibrium solution is obtained through multiple iterations. After calculating a certain number of time steps, it is determined whether the particle system meets the servo requirements. This is determined by the tolerance of the servo force or the unbalanced force of the particle system. Attached Figure Description

[0032] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings of the embodiments of the present invention will be briefly described below. The drawings are merely illustrative of some embodiments of the present invention and are not intended to limit the scope of the present invention to all embodiments.

[0033] Figure 1 This is a schematic diagram of the polygonal boundary contour in an embodiment of the present invention.

[0034] Figure 2 This is a schematic diagram of the outer boundary contour of the columnar polyhedron in an embodiment of the present invention.

[0035] Figure 3 This is a schematic diagram of boundary particle generation in an embodiment of the present invention.

[0036] Figure 4 This is a schematic diagram of the generation of layered particles at the bottom in an embodiment of the present invention.

[0037] Figure 5 This is a schematic diagram of the particle coordinate dimensions of different layers in an embodiment of the present invention.

[0038] Figure 6 This is a schematic diagram of the generation of outer boundary membrane particles in an embodiment of the present invention.

[0039] Figure 7 This is a schematic diagram illustrating the application of particle servo force in an embodiment of the present invention.

[0040] Figure 8 This is a schematic diagram showing the completion of loading of the soil-rock mixture sample model in an embodiment of the present invention.

[0041] Figure 9 This is a schematic diagram showing the completion of loading of the rock sample model in an embodiment of the present invention.

[0042] Figure 10 This is the stress-strain curve of the soil-rock mixture sample in the embodiment of the present invention.

[0043] Figure 11 This is the stress-strain curve of the rock sample in the embodiment of the present invention. Detailed Implementation

[0044] The exemplary solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Unless otherwise defined, the technical or scientific terms used in this invention should have the ordinary meaning understood by one of ordinary skill in the art.

[0045] It should be noted that terms such as "up," "down," "left," and "right," which indicate orientation or positional relationship, are only used to express relative positional relationship. They are used for the convenience of describing the present invention and do not mean that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation. When the absolute position of the object being described changes, the relative positional relationship may also change accordingly.

[0046] The working principle of this invention is as follows: Based on the numerical calculation method of granular flow, firstly, the cross-sectional polygonal outline of an arbitrary-shaped cylindrical sample model is determined. According to the model height requirements, the outer boundary polygon is extended into a cylindrical polyhedron, and internal particles are generated. Then, starting from the bottom, particles of corresponding radii are generated sequentially on the outer boundary. After generating a complete layer of boundary film particles, generation begins on the next layer, and so on, using the same generation method until the top of the column is reached. Afterward, a low modulus and a high bonding strength are set for the overall outer boundary film particles, and a servo force is applied, updating the magnitude and direction of the force at a certain frequency until the servo operation ends.

[0047] The technical solution for implementing the loading and servo control method for cylindrical specimens with complex cross-sections, as described in this application, specifically includes the following steps:

[0048] Based on the determined cross-sectional contour shape of the model, the outer boundary contour is extruded into an outer boundary columnar polyhedron. Starting from the bottom layer, boundary particles are generated sequentially for each edge to form the outer boundary particles. Then, the outer boundary contour is generated layer by layer upwards along the height until the top layer. After the generation of the top layer outer boundary particles is completed, a low modulus is set for the outer boundary membrane particles of the overall columnar sample to avoid affecting lateral deformation, and a high bonding strength is set to prevent damage to the outer boundary contour during servoing. Finally, servoing is initiated by applying a servo force until the model meets the servoing requirements and the servoing process is complete.

[0049] This invention is achieved through the following steps:

[0050] Step 1: Draw a cross-sectional polygon according to the outer boundary contour of the 3D cylindrical model. The polygon consists of M vertices and M corresponding edges.

[0051] Step 2: Set the radius of the boundary particles and the height of the model to generate the vertices of the top-level outer boundary contour, and extrude the outer boundary polygon into a cylindrical polyhedron.

[0052] Step 3: Based on the vertex coordinates and particle radius of each layer in Step 2, generate servo particles on each edge. Repeat this process for each subsequent layer until the top layer is reached.

[0053] Step 4: For boundary contact, to ensure that the servo film particles are not damaged and can deform freely laterally during the servo process, a lower modulus and a higher bond strength are set. Servo force is applied to the boundary particles according to the confining pressure setting;

[0054] Step 5: Update the magnitude and direction of the servo force at a certain frequency, and perform model calculation and analysis until the internal stress state meets the design requirements.

[0055] To implement the loading and servo control method for cylindrical specimens with complex cross-sections as described in this invention, in step 1, the cross-sectional boundary polygon of the model is drawn, and the outline of the polygon boundary is determined. The number of sides of the polygon can be arbitrary, but based on engineering practice experience, the polygon should be a convex polygon, such as... Figure 1 The figure shows a hexagonal cross-section.

[0056] To sort the vertices and edges of the polygon boundary, first determine the starting vertex and number it 1, then number each vertex in a counter-clockwise direction until all vertices have been passed. Finally, determine the number of each edge according to the vertex numbering order.

[0057] To implement the loading and servo control method for cylindrical specimens with complex cross-sections as described in this invention, in step 2, the maximum particle radius r of the generated particles is set. max Minimum particle radius r min Parameters such as model height h. Based on the endpoint coordinates (x, y) of each side of the geometric boundary.m ,y m ,z m Given the model height h, determine the vertex positions (x, y) of the top-level outer boundary. m ,y m ,z m +h), extrude the outer boundary polygon into a cylindrical polyhedral body outline, such as Figure 2 As shown, under the control of the generated columnar polyhedral external contour, randomly distributed soil and rock particles are generated according to porosity.

[0058] The ray traversal method is used to examine all particles. If a particle's ray intersects the contour surface at only one point, it indicates that the particle is inside the polyhedron and is retained. Otherwise, the particle is deleted. This ensures that all particles are located within the defined columnar sample model.

[0059] To implement the loading and servo control method for cylindrical specimens with complex cross-sections as described in this invention, in step 3, the number of particles n on each side and the center positions of other particles on the same side are determined based on the endpoint coordinates of each side of the geometric boundary and the radius r of the target particle. Since each side may be tilted relative to the coordinate axis, the formula for the center position of the membrane particle is calculated as follows:

[0060] The coordinates of the endpoints of the corresponding edges are (x1, y1, z1) and (x2, y2, z1);

[0061] The number of membrane particles, n, is:

[0062]

[0063] The center position of the k-th particle along this edge:

[0064]

[0065] Where 1≤k≤n, and so on, layer by layer upwards until the top layer is generated.

[0066] Since the length of each side is not a perfect multiple of the particle radius, the calculated number of particles on each side will not be an exact integer. Therefore, an approximate rounding method is used to determine the particle count. This can be done by directly removing the decimal part or by rounding the decimal part to the nearest whole number, ensuring that the particle radii on each side are approximately equal. This forms the outer boundary contour particle layer, as shown below. Figure 3 As shown.

[0067] Based on the coordinates and radius of the particles generated in the first layer, determine the generation positions of the particles in the second layer. For example, if the center position of the first particle in the first layer is (x1, y1, z1), then... Figure 4As shown, based on the positional relationship between the particles, the center position of the first particle in the second layer is (x1, y1, z1+2r). The center positions of other particles on the same edge are generated by analogy based on the particle radius. The height of the remaining membrane particles increases by 2r layer by layer.

[0068] To implement the loading and servo control method for cylindrical specimens with complex cross-sections as described in this invention, in step 4, contact setting parameters are set for the divided groups. A lower modulus ensures lateral free deformation of the model during servoing, while a higher bonding strength ensures that the servo membrane will not be damaged during servoing. The servo force is applied separately to the particles on each edge. When a servo force is applied to a certain membrane particle, the normal direction can be determined based on the positional relationship between the target particle and the particles above, below, to the left, and to the right. Then, the servo force in the normal direction is superimposed to obtain the final applied servo force. First, the direction vectors of the particle and the corresponding membrane particles above, below, to the left, and to the right are found. like Figure 7 As shown, the servo force of the target particle in the corresponding normal direction is calculated based on the servo stress σ applied to the target particle and its directional relationship with the particles above, below, left, and right. The servo force applied to the target membrane particle can be obtained by superimposing the servo stress σ. The servo force applied to each membrane particle is obtained by iterating through each membrane particle, and the magnitude and direction of the servo force are continuously updated during the iteration process.

[0069] in,

[0070]

[0071]

[0072]

[0073]

[0074] To implement the loading and servo control method for cylindrical specimens with complex cross-sections as described in this invention, in step 5, the particle system is calculated through iterative steps (each time step is called an iterative step) to determine whether the servo requirements are met. If the servo requirements are met, the process exits.

[0075] The solution iteration step is an attempt to find an equilibrium solution by continuously updating information such as contact, displacement, velocity, and force in very small time steps. If the model is not in equilibrium at the end of the iteration, another round of iteration will be performed.

[0076] With each iteration, the solution obtained will be closer to the equilibrium state; sometimes many iterations are required to obtain an equilibrium solution. After calculating a certain time step, it is determined whether the particle system meets the servo requirements. This is determined by the tolerance of the servo force or the unbalanced force of the particle system.

[0077] This application also provides an embodiment to further illustrate and explain the solution and effects of this application:

[0078] A regular hexagonal prism rock sample and a soil-rock mixture sample were constructed, each with a side length of 1m and a height of 4m. During the construction of the numerical model of this sample, a servo system was applied to the model to ensure uniform particle and contact distribution. The present invention was used to implement soft servo control of this prism sample, and the steps are as follows:

[0079] (1) In the engineering drawing software AutoCAD, draw the cross-sectional shape diagram of the prism specimen, i.e., the outer boundary contour diagram. Based on the outer boundary contour diagram of the specimen, determine the number and location of the vertices and edges of the cross section, and number the vertices and edges in sequence, such as... Figure 1 As shown, the cross-section of the hexagonal prism is a regular hexagon. The vertices are numbered in counterclockwise order, and the order of the sides is determined by the vertex numbers.

[0080] (2) Define some basic design parameters for generating hexagonal prisms in the PFC3D granular flow numerical calculation platform. Use these parameters to generate the walls that construct the hexagonal prisms. Based on the outer boundary vertex numbers and coordinates obtained in step 1 and the model height information, use the coordinates obtained in step 1 as the coordinates of the bottom vertex of the model. Add the model height to obtain the coordinates of the top vertex of the model. Connect them sequentially to form a closed columnar model outline, such as... Figure 2 As shown. With a minimum radius rlo = 0.02m, a maximum radius rhi = 0.04m, and an initial particle porosity pro of 0.34, 56012 soil and rock particles were randomly generated. After a certain number of time steps (10000 steps in this case), the initial equilibrium state was reached.

[0081] (3) Generate outer boundary membrane particles based on the positional relationship of each edge of the prism. Starting from the bottom edge, first generate membrane particles at the first vertex. Then, based on the inclination relationship between this edge and the coordinate axis and the setting of the membrane particle radius, generate a row of membrane particles on this edge. Calculations show that 16 particles can be generated on each edge (excluding the two vertices). Continue in this manner to generate the first layer of membrane particles, such as... Figure 3 As shown, the same method is used to generate membrane particles for each edge of the remaining layers. The height of the center position of the membrane particles in each layer increases by 2r (approximately 0.06m in this example), as shown. Figure 4-6 As shown.

[0082] (4) Apply a servo stress of 1MPa as required. In order to ensure the effect of soft servo action, set a higher bonding strength and a lower modulus. Traverse each membrane particle and apply four corresponding normal servo forces to each membrane particle according to the positional relationship between each membrane particle and the membrane particles above, below, left and right. By superimposing, obtain the total servo force applied to each membrane particle.

[0083] (5) Because the modulus of the membrane particles is relatively small during the servo process, the membrane particles will not hinder the lateral deformation of the model. The normal direction of each membrane particle and the plane containing the particles above, below, left, and right is constantly changing. Therefore, it is necessary to update the magnitude and direction of the servo force of the membrane particles every certain time step. Determine whether the particle system meets the servo requirements and whether it satisfies the equilibrium condition. If it does, exit; otherwise, continue calculating until the particle system is in equilibrium. Set the tolerance of the servo force to 0.05 and the limit of the unbalanced force to 1e-5MPa. When the error between the applied servo force and the target stress is less than 0.05 or the unbalanced force is less than 1e-5MPa, the servo is considered to meet the requirements, and the servo ends. The servo result is as follows: Figure 6 As shown.

[0084] (6) After the servo operation ends, by adjusting the parameters of the particles inside the two samples, the two samples are set as a soil-rock mixture and a rock sample. By loading the two samples respectively, a soil-rock mixture sample can be obtained. Figure 8 ) and rock samples ( Figure 9 The deformation and failure model of the specimen, and the stress-strain curves of the two specimens, as shown in the figure. Figure 10 and Figure 11 As shown, where Figure 10 The stress-strain curves of the soil-rock mixture sample are shown. Figure 11 This is the stress-strain curve of the rock sample.

[0085] like Figure 10 and Figure 11 As shown, the stress-strain curve of the soil-rock mixture obtained by this soft servo method does not exhibit the phenomenon of post-peak softening causing discrepancies with the soil deformation curve, resulting in better performance.

[0086] The preferred embodiments for implementing the present invention have been described in detail above. However, it should be understood that these embodiments are merely illustrative and not intended to limit the scope, application, or construction of the invention in any way. The scope of protection of the present invention is defined by the appended claims and their equivalents. Those skilled in the art can make numerous modifications to the foregoing embodiments under the teachings of this invention, and all such modifications fall within the scope of protection of this invention.

Claims

1. A loading and servo control method suitable for cylindrical specimens with complex cross-sections, characterized in that, include: a. Determine the outer boundary contour shape of the model's cross-section. Based on the model's height, extrude the outer boundary contour into a columnar polyhedron and generate internal particles. b. Based on the set particle radius, starting from the selected outer boundary contour, membrane particles are generated sequentially for each edge to form the contour membrane particles of the layer. Then, along the height direction, contour membrane particles of other layers are generated layer by layer until the entire columnar polyhedron forms the outer boundary membrane particles. c. Set modulus parameters and bonding strength parameters for the outer boundary membrane particles. When servo force is applied, the corresponding modulus parameters ensure that the outer boundary membrane particles do not hinder the lateral deformation of the model, and the corresponding bonding strength parameters ensure that the outer boundary membrane particles do not break. d. Apply servo force, update the magnitude and direction of the servo force at the set frequency, and perform model calculation and analysis until the internal stress state meets the design requirements.

2. The loading and servo control method for cylindrical specimens with complex cross-sections according to claim 1, characterized in that, In step b, based on the coordinates of each endpoint of the geometric boundary of the selected layer of the columnar polyhedron and the target particle radius r, the number of particles n on each side and the center position of each particle on each side are determined to form the contour particles of the selected layer. Based on the coordinates and radius of the contour particles in the selected layer, the generation positions of particles in adjacent layers are determined. Then, based on the particle radius and the positional relationship between particles, the center positions of particles on each edge of other layers are generated in the same way. The particle height on the corresponding edge of two adjacent layers changes by 2r layer by layer.

3. The loading and servo control method for cylindrical specimens with complex cross-sections according to claim 2, characterized in that, Let the coordinates of the two endpoints of one of the edges be (x1, y1, z1) and (x2, y2, z1), and the number of particles n along this edge be: The coordinates of the center position of the kth particle on this edge are: Where 1≤k≤n.

4. The loading and servo control method for cylindrical specimens with complex cross-sections according to claim 3, characterized in that, When the length of one of the edges in the geometric boundary of the selected layer is not a multiple of the particle radius, the number of particles n is rounded down, and the particle radius is recalculated based on the length of that edge.

5. The loading and servo control method for cylindrical specimens with complex cross-sections according to any one of claims 1-4, characterized in that, Set the maximum particle radius r max and minimum particle radius r min Under the control of the generated columnar polyhedral body outline, randomly distributed soil and rock particles are generated according to porosity.

6. The loading and servo control method for cylindrical specimens with complex cross-sections according to claim 5, characterized in that, The ray traversal method is used to traverse all particles. If the particle ray intersects the contour surface at only one point, it indicates that the particle is inside the polyhedron and is retained; otherwise, the particle is deleted.

7. The loading and servo control method for cylindrical specimens with complex cross-sections according to claim 1, characterized in that, Servo forces are applied to the particles on each edge. When a servo force is applied to a particle, the normal direction is determined based on the positional relationship between the particle and the particles above, below, left, and right. The servo force in the normal direction is then superimposed to obtain the final applied servo force. The steps to obtain the magnitude of the servo force finally applied to the target particle are as follows: First, find the direction vectors of the target particle and its corresponding particles in the up, down, left, and right directions. Based on the servo stress σ applied to the target particle and its directional relationship with the particles above, below, left, and right, the magnitudes F1, F2, F3, and F4 of the servo force of the target particle in the corresponding normal direction are calculated respectively. The magnitude of the servo force applied to the target particle can be obtained by superposition. By traversing each particle, the servo force applied to each particle is obtained, and the magnitude and direction of the servo force are continuously updated during the iteration process. Each time step calculation is called an iteration step. Through iteration step calculation, it is determined whether the particle system meets the servo requirements. If the servo requirements are met, the process exits.

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