A discretization method suitable for complex computational domain of lattice discrete particle model

Abstract

A kind of discretization method suitable for complex computational domain of lattice discrete particle model, including 1, concrete component three-dimensional entity modeling;2, round spherical concrete aggregate particle generation;3, on the surface and inside three-dimensional entity randomly throw aggregate particle;4, aggregate inclusion generation.The present application introduces three-dimensional entity modeling before aggregate particle generation and throwing, so that LDPM pre-processing method is more complete, in entity modeling stage, various shapes of entity model can be generated by the establishment, transformation and assembly of parts, so that the pre-processing method can process various shapes of concrete test piece, so that it is more flexible and versatility;The uniformity of aggregate particle size distribution and spatial distribution is improved, so that aggregate particle model is close to real concrete material;Delaunay tetrahedron and each face thereon are marked, so that the pre-processing method has the ability to process complex geometric feature concave entity with notch, with hollow and the like.

Description

Technical Field

[0001] This invention belongs to the field of concrete simulation technology, specifically relating to a discretization method suitable for complex computational domains of lattice discrete particle models. Background Technology

[0002] Concrete, as a typical heterogeneous material, is characterized by multiple length scales, such as the atomic scale, nanoscale, microscale, mesoscale, and macroscale. When considering the heterogeneity of concrete to understand how phases affect its macroscopic behavior, fracture mechanics, and how to improve its performance, modeling concrete at the mesoscale is considered the most practical and effective method.

[0003] Based on the requirement of continuity, mesoscale models are mainly divided into continuous mesoscale models and discrete mesoscale models. Continuous medium mesoscale models represent the mesostructure of concrete as a continuous medium and use the finite element method to model the interfaces between coarse aggregate blocks, mortar matrix, and inclusion matrix. This type of model requires complex constitutive relations to represent the behavior of each component and needs to select an appropriate mesh size to avoid the influence of mesh sensitivity on the simulation results. Even for very small specimens, an excessively small mesh size can lead to a computational system with a huge number of degrees of freedom. In contrast, discrete mesoscale modeling, as an alternative to the finite element method, avoids the continuum assumption and uses different elements (such as spheres, lattice beams, trusses, etc.) to represent the mesostructure of concrete.

[0004] In recent decades, lattice and discrete particle-based microstructure models of concrete have made significant progress and have been widely used in simulating fracture and damage associated with multiphysics fields or processes in concrete. Casatis developed the Lattice Discrete Particle Model (LDPM) by combining the Constrained Shear Lattice (CSL) model and the Discrete Particle Model (DPM). Because LDPM introduces more interparticle contact surfaces into the microstructure and provides a quantitative description of volume effects, it improves stress resolution, crack or damage distribution, modeling, and prediction capabilities, particularly by enhancing the simulation of compaction under cyclic loading and confining pressure.

[0005] After more than a decade of development, LDPM has made significant progress in the microscopic simulation of the failure behavior of heterogeneous brittle materials such as concrete, reinforced concrete, fiber-reinforced concrete, and masonry, as well as in the microscopic simulation of other physicochemical processes coupled with cracking in brittle materials (such as alkali-aggregate reaction, fluid diffusion or flow, creep, etc.). However, LDPM has not yet received sufficient attention or widespread application in materials or structures such as concrete. Besides the high computational demands of LDPM, another crucial limiting factor is the lack of geometric preprocessing techniques capable of reconstructing the internal microstructure of concrete that meets the requirements of LDPM.

[0006] Unlike other preprocessing methods for concrete microstructure models, LDPM requires the simultaneous addition of random spherical aggregates and the placement of potential cracks in a specific manner to form so-called aggregate inclusions. Aggregate inclusions are not "units" in the traditional sense, but rather polyhedra formed by many triangles surrounding aggregate particles. The shape of the polyhedron depends on the number and position of adjacent aggregates and can be convex or concave. Generating this aggregate inclusion model requires both the addition of random aggregates and the formation of inclusion units around the aggregates, presenting challenges in both geometric model representation and algorithm development, thus limiting the widespread application of LDPM.

[0007] Currently, most geometric models related to LDPM and its extended applications are implemented in the Structural Response Modeling and Analysis (MARS) computational environment, which is a specialized computational software that is difficult to access on the market, and its LDPM preprocessing capabilities are still unknown.

[0008] Equally important, advancements in computational technology have reduced the time and memory costs required to solve relevant field equations, thus computational cost is not necessarily the primary concern for small-scale analytical problems. Conversely, demanding tasks such as model building, result interpretation, and potential model modifications can become bottlenecks in the simulation process. With the increasing demand for LDPM simulations, developing LDPM pretreatment methods applicable to concrete geometries is particularly urgent and important.

[0009] In concrete specimens or structures with various complex geometries in the computational domain, concave solids with notches or holes are more difficult to discretize using LDPM than convex solids. Summary of the Invention

[0010] The purpose of this invention is to overcome the shortcomings of the prior art and provide a discretization method suitable for complex computational domains of lattice discrete particle models. By leveraging the data structures and algorithms for entities, meshes, and point sets in the computational geometry algorithm library, the discretization from the 3D entity model to the LDPM random aggregate model is achieved through modules for 3D entity modeling, aggregate particle generation, aggregate particle placement, and aggregate inclusion generation, in order to change the situation where LDPM geometric preprocessing is difficult to implement.

[0011] The technical problem solved by this invention is achieved through the following technical solution:

[0012] A discretization method applicable to complex computational domains of lattice discrete particle models, characterized in that: the method comprises the following steps:

[0013] S1. Three-dimensional solid modeling of concrete components;

[0014] S2, Generation of spherical concrete aggregate particles;

[0015] S3. Randomly place aggregate particles on the surface and inside of the three-dimensional solid;

[0016] S4, Aggregate inclusion formation.

[0017] Moreover, S1 specifically refers to:

[0018] 1a. Part Creation: Generate a Polyhedron_3 object provided by CGAL using a custom format 3D solid surface mesh file. Multiple data files can be read at once to create multiple parts.

[0019] 1b. Part Transformation and Assembly: Convert one or more Polyhedron_3 objects into Nef_polyhedron_3 objects, and perform translation, rotation or scaling transformations as needed. Then perform Boolean operations to generate a three-dimensional solid model corresponding to the concrete component, i.e., the LDPM computational domain.

[0020] 1c. Composite body processing: In order to put aggregate particles and generate aggregate inclusions, prepare the Linear Cell Complex object and Surface_Mesh object of the three-dimensional solid model in advance, and calculate the three-dimensional solid volume V through the SM object.

[0021] Moreover, S2 specifically refers to:

[0022] 2a. Calculate the volume fraction of aggregate in the concrete specimen: Using the concrete mix design parameters (cement content c, water-cement ratio w / c, and the volume fraction of trapped or entrained air v)... air The density of water is ρ w The density of cement is ρ c Substituting into the following formula, the volume fraction v of aggregate in the concrete specimen can be calculated. a :

[0023]

[0024] 2b. Calculate the volume fraction of the simulated aggregate: Assume the maximum and minimum particle sizes of the simulated aggregate are d and d, respectively. max and d min Based on the Fuller particle size distribution curve, the volume fraction v of the simulated aggregate can be calculated using the following formula. a0 :

[0025] v a0 =[1-F(d) min )]v a =[1-(d min / d max ) 12 ]v a

[0026] 2c. Generating simulated aggregates by sampling through inverse transformation of the cumulative distribution function: A random number P is generated in the range [0,1] using a random number generator. i Substituting this into the Stroeven particle size distribution function in the particle size range [d min ,d max The inverse transform of the cumulative distribution function on the [] yields the particle size d of the random aggregate. i :

[0027]

[0028] 2d. Continuous sampling until the sampling stops: Calculate the volume V of the simulated aggregate in the concrete specimen. a =v a0 V, repeat step 2c until the sum of the volumes of all collected aggregate particles exactly exceeds V. a ,Right now:

[0029]

[0030] Moreover, S3 specifically refers to:

[0031] 3a. Place the smallest aggregate particles onto the surface of the three-dimensional solid in the order of edge first, then particle last:

[0032] First, iterate through each edge of the 3D solid LCC object, ensuring a spacing of at least 1.1d. min The requirement is to place aggregate particles of the smallest diameter on each side;

[0033] Then, iterate through each face of the LCC object, rotating each face to horizontal to generate its two-dimensional rectangular bounding box. Using the Monte Carlo method, randomly place the smallest aggregate particle within the rectangular bounding box. If an aggregate particle falls within the LCC face and the distance between it and the aggregate particles already placed on that face is not less than 1.1d, then the minimum aggregate particle size is considered. min If the particle is considered to have been successfully placed, then all successfully placed aggregate particles are rotated back to their original positions on the LCC surface.

[0034] 3b. Traverse all aggregate particles placed on the surface of the three-dimensional entity, check the particle spacing, and remove aggregate particles whose particle spacing does not meet the requirements while retaining the geometric features of the three-dimensional entity.

[0035] 3c. Aggregate particles are placed inside the 3D solid in descending order of particle size: Similar to surface placement, the Monte Carlo method is used for particle placement. First, a bounding box of the 3D solid is generated. Random positions are selected within the bounding box. If a position falls inside the 3D solid, then the particle size is d. iThe algorithm searches for aggregate particles and simultaneously identifies the nearest aggregate particle. Let the particle size of the found aggregate particle be d. j Then, determine whether the spacing between the two aggregate particles satisfies the condition of not less than d. i / 2+d j / 2+0.2d min If the requirements are met, the feeding is considered successful. This process is repeated until all aggregate particles generated in S2 are successfully fed.

[0036] Moreover, S4 specifically refers to:

[0037] 4a. Use the Triangulation package in CGAL to perform Delaunay triangulation on the point set generated by aggregate delivery;

[0038] 4b. Mark each Delaunay tetrahedron and its face relative to the 3D solid. Use the numbers 1 and -1 to mark the Delaunay tetrahedron inside and outside the solid, respectively. Use the numbers 2, -1, 0 and 1 to mark the internal face, external face, external boundary face and internal boundary face, respectively.

[0039] 4c. Traverse the edges, faces, and volumes of the Delaunay tetrahedron respectively, and calculate the coordinates of the midpoints (net center), face centers (net face centers), and volume centers (net volume centers) of the edges after deducting the aggregate particles.

[0040] 4d: For each aggregate particle, traverse all Delaunay tetrahedrons associated with it. Within each tetrahedron, traverse the three edges passing through the center of the aggregate particle sphere. On a tetrahedron, each edge has two associated tetrahedral faces. Traverse these two associated faces and, based on the face's identifier, perform the following processing:

[0041] If the numeric identifier of the associated face is 2, it means that the face is an internal face. The net center of the edge where the aggregate particle is located, the net body center of the current tetrahedron, and the net face center of the associated face are connected in sequence to form a wrapping face of the current aggregate particle's wrapping body.

[0042] If the numeric identifier of the associated surface is 0 or 1, it means that the surface is a boundary surface. The net center of the edge where the aggregate particle is located, the net center of the associated surface, and the center of the current aggregate particle sphere are connected in sequence to form an outer surface of the current aggregate particle's encapsulation.

[0043] If the numeric identifier of the associated face is -1, it means that the face is an external face, and no processing is performed on it; it can be skipped.

[0044] The advantages and beneficial effects of this invention are as follows:

[0045] 1. This invention introduces three-dimensional solid modeling before aggregate particle generation and placement, making the LDPM preprocessing method more complete. In the solid modeling stage, various shapes of solid models can be generated by creating, transforming and assembling parts, making the preprocessing method able to handle concrete specimens of various shapes, thus making it more flexible and versatile.

[0046] 2. This invention improves the uniformity of aggregate particle size distribution and spatial distribution by randomly generating and randomly placing aggregate particle components into concrete aggregate particle models, making the aggregate particle models closer to real concrete materials.

[0047] 3. Based on the Delaunay triangulation of the aggregate particle model, the present invention marks the Delaunay tetrahedron and its faces, enabling the preprocessing method to handle concave solids with complex geometric features such as notches and voids. Attached Figure Description

[0048] Figure 1 This is a flowchart of the present invention.

[0049] Figure 2 A 100mm×100mm×350mm prism diagram generated for an embodiment of the present invention;

[0050] Figure 3 A cylindrical drawing with an R50mm×200mm diameter generated for an embodiment of the present invention;

[0051] Figure 4 A 100mm×200mm×150mm block diagram generated for an embodiment of the present invention;

[0052] Figure 5 The figure shows the assembly A1 formed by combining two cylinders C1 and C2 and one block B1 according to an embodiment of the present invention.

[0053] Figure 6 This is a diagram of a dog-bone-shaped solid D formed by subtracting prism P and two blocks B1 and B2 in an embodiment of the present invention.

[0054] Figure 7 The corresponding LCC object diagram is obtained by transforming the dog bone-like entity D in this embodiment of the invention;

[0055] Figure 8 This is a dot diagram of aggregate particles placed on the surface of the calculation domain of the dog bone specimen in an embodiment of the present invention;

[0056] Figure 9 This is a point diagram of aggregate particles placed on the entire computational domain of the dog bone specimen in an embodiment of the present invention;

[0057] Figure 10This is a Delaunay triangulation diagram of a dog bone specimen based on aggregate particle points, according to an embodiment of the present invention.

[0058] Figure 11 This is an internal unit diagram of the Delaunay tetrahedron of the dog bone specimen according to an embodiment of the present invention;

[0059] Figure 12 a is a net center diagram of the edge of the Delaunay tetrahedron according to an embodiment of the present invention; 12b is a net centroid diagram of the face of the Delaunay tetrahedron according to an embodiment of the present invention; 12c is a net centroid diagram of the Delaunay tetrahedron according to an embodiment of the present invention.

[0060] Figure 13 a is a diagram of the aggregate particle inclusion surface in the Delaunay tetrahedron of the present invention; 13b is a model diagram of the aggregate particle inclusion obtained by discretizing the computational domain of the dog bone specimen of the present invention; 13c is a model diagram of the aggregate particle inclusion in the lower half of the dog bone specimen of the present invention. Detailed Implementation

[0061] The present invention will be further described in detail below through specific embodiments. The following embodiments are merely descriptive and not limiting, and should not be used to limit the scope of protection of the present invention.

[0062] like Figure 1 As shown, a discretization method suitable for complex computational domains of lattice discrete particle models is presented. Its innovation lies in the following steps:

[0063] Step 1: Create a 3D solid model of the dog bone concrete component

[0064] Step 1a: Create the individual parts of the combined dog bone 3D solid model. The dog bone specimen solid model is composed of 7 basic parts: 1 prism, 4 cylinders, and 2 blocks. Figure 2 The figure shows a prism with a length, width, and height of 100mm, 100mm, and 350mm, respectively, denoted as P. Figure 3 The image shows a cylinder with a radius of 50 mm and a length of 200 mm, denoted as C. The block is shown below. Figure 4 As shown, its length, width, and height are 200mm, 100mm, and 150mm respectively, and it is denoted as B.

[0065] Step 1b: The first step in creating the dog bone geometry is to combine two cylinders C1 and C2 and one block B1 through rotation, translation, and merging operations to form the following... Figure 5 The composite A1 is shown. The remaining two cylinders C3 and C4, and one block B2 are similarly combined to form composite A2. The second step is to move A1 and A2 to the left and right sides of prism P respectively, and form a dog bone-shaped solid, denoted as D, through Boolean difference operations.

[0066] Step 1c: First, convert the Nef_Polyhedron_3 object of the dog bone entity into a Polyhedron_3 object, and then further convert it into an LCC object. Figure 7 The surface mesh object (as shown) was calculated to have a volume of 0.0022m. 3 .

[0067] Step 2: Generation of spherical concrete aggregate particles.

[0068] Step 2a: Read the concrete material parameters listed in Table 1 from the data file and calculate the volume fraction of aggregate in the concrete specimen. The calculation result is 0.7248.

[0069] Table 1 Concrete Material Parameters

[0070]

[0071] Step 2b: The maximum and minimum particle sizes of the simulated aggregate are 8 mm and 4 mm, respectively. The volume fraction of the simulated aggregate is calculated to be 0.2123.

[0072] Step 2c: Perform inverse transform sampling on the cumulative distribution function of the Stroeven particle size distribution function to generate simulated aggregate. The generated aggregate particle size all falls within the interval [4mm, 8mm].

[0073] Step 2d: Continuously sample until the cumulative volume of all aggregate particles first exceeds 0.000467 m³. 3 (0.2123×0.0022m 3 Sampling was stopped. A total of 5520 aggregate particles were generated, with a maximum particle size of 7.9809 mm and a minimum particle size of 4.0002 mm.

[0074] Step 3: Randomly place aggregate particles on the surface and inside of the three-dimensional solid.

[0075] Step 3a: Enter the aggregate particle feeding module and evenly feed the smallest aggregate particles along each side of the LCC. After completion, using these particles as a reference, randomly feed the smallest aggregate particles on each surface of the LCC. In this example, a total of 5241 smallest aggregate particles were fed on the surface.

[0076] Step 3b: Perform a distance check while preserving geometric features, and remove particle points that do not meet the spacing requirements. In this example, a total of 349 particle points were removed because they did not meet the requirements. Figure 8 The image shows aggregate particles placed on the boundary of a dog bone specimen.

[0077] Step 3c: The particles are sequentially placed inside the dog bone body in descending order of size, ultimately forming a structure like... Figure 9The aggregate particle model shown has a total of 10,761 aggregate particles.

[0078] Step 4: Constructing the aggregate inclusion model

[0079] Step 4a: Perform Delaunay triangulation on the aggregate particles added in Step 3. After triangulation, the original solid model is replaced by spatial tetrahedral elements based on point sets. It should be noted that for computational domains formed by solids with notches or holes, these notches or holes will also be filled with tetrahedrals. The dog bone specimen belongs to a computational domain with notches, and its Delaunay triangulation is as follows: Figure 10 As shown, there are a total of 57,872 tetrahedral elements.

[0080] Step 4b: In order to remove the tetrahedral elements in the notch of the dog bone specimen, the tetrahedrons and their element faces are marked. The marked model contains 51,177 internal tetrahedrons, 97,212 internal faces, 6,183 external boundary faces and 4,101 internal boundary faces, and 6,695 external tetrahedrons and 11,488 external faces. Figure 11 What is being shown is the internal unit.

[0081] Step 4c: Traverse the edges, faces, and volumes of the Delaunay tetrahedron, and calculate the midpoints of the edges after deducting the range of aggregate particles. Figure 12 E in a 12 ), noodle center ( Figure 12 F4 in b) and body ( Figure 12 The T position coordinates in c.

[0082] Step 4d: Traverse the aggregate particles. For each aggregate particle, traverse its associated Delaunay tetrahedral elements. If the Delaunay tetrahedron is an internal element, find the three edges passing through the center of the aggregate particle. For each edge, sequentially query the two faces associated with that edge. If the current associated face is an internal face, sequentially connect the net center of the edge containing the aggregate particle, the net body center of the current tetrahedron, and the net face center of the associated face to form a wrapping face of the current aggregate particle's enclosure, such as... Figure 13 As shown in diagram a; if the current associated surface is an external or internal boundary surface, connect the net center of the edge where the aggregate particle is located, the net center of the associated surface, and the center of the current aggregate particle sphere in sequence to form an outer surface of the current aggregate particle inclusion. Finally, the computational domain of the dog bone specimen is discretized as shown in diagram a. Figure 13 The aggregate inclusion model shown in b has its lower half as follows: Figure 13 As shown in c.

[0083] Although embodiments and drawings of the present invention have been disclosed for illustrative purposes, those skilled in the art will understand that various substitutions, variations and modifications are possible without departing from the spirit and scope of the present invention and the appended claims. Therefore, the scope of the present invention is not limited to the contents disclosed in the embodiments and drawings.

Claims

1. A discretization method suitable for complex computational domain of lattice discrete particle model, characterized in that: The steps of the method are: S1, three-dimensional entity modeling of the concrete member; S2, generation of spherical concrete aggregate particles; S3, random placement of aggregate particles on the surface and inside the three-dimensional entity; S4, aggregate wrapping body generation; The S2 is specifically: 2a, calculate the volume fraction of aggregate in the concrete specimen: the concrete mix design parameters cement content c, water-cement ratio w / c and the volume fraction of captured or entrained air is v air , the density of water is p w , and the density of cement is p c Substitute the following formula to calculate the volume fraction of aggregate in the concrete specimen v a : 2b, calculate the volume fraction of the simulated aggregate: assuming that the maximum and minimum particle sizes of the simulated aggregate are d max and d min , according to the Fuller particle size curve, the volume fraction v a0 of the simulated aggregate can be obtained according to the following formula: v a0 = [1 - F(d min )]v a = [1 - (d min / d max ) 1 / 2 ]v a 2c. Generating simulated aggregate by sampling the inverse transform of the cumulative distribution function: use a random number generator to generate a random number P in the range [0, 1] i , into the inverse transform of the cumulative distribution function of the Stroeven particle size distribution function over the interval [d min , d max ] gives the particle size d i of the random aggregate: 2d, continue sampling until the stopping condition is met: calculate the volume V of the simulated aggregate of the concrete test piece a = v a0 V, repeat step 2c until the sum of the volumes of all the aggregate particles collected is just more than V a i.e.: The S4 is specifically: 4a, using the Triangulation package in CGAL to perform Delaunay triangulation on the point set generated by aggregate placement; 4b, according to the relative position relationship between the Delaunay tetrahedron and each face thereof and the three-dimensional entity, marking one by one using the numbers 1 and -1 to mark the internal and external Delaunay tetrahedrons respectively, and using the numbers 2, -1, 0 and 1 to mark the internal face, external face, external boundary face and internal boundary face respectively; 4c, respectively traversing the Delaunay tetrahedron edge, face and body, and sequentially calculating to obtain the net center, net face center and net body center position coordinates excluding the aggregate particle range; 4d: for each aggregate particle, traversing all the Delaunay tetrahedrons associated therewith, in each tetrahedron, sequentially traversing the three edges passing through the spherical center of the aggregate particle, in a tetrahedron, there are two tetrahedron faces associated with each edge, traversing the two associated faces, according to the identifier of the face, respectively processing as follows: If the numerical identifier of the associated face is 2, it means that the face is an internal face, sequentially connecting the net center of the aggregate particle, the net body center of the current tetrahedron, and the net face center of the associated face to form a wrapping face of the current aggregate particle wrapping body; If the numerical identifier of the associated face is 0 or 1, it means that the face is a boundary face, sequentially connecting the net center of the aggregate particle, the net face center of the associated face, and the spherical center of the current aggregate particle to form an external surface of the current aggregate particle wrapping body; If the numerical identifier of the associated face is -1, it means that the face is an external face, and no processing is performed thereon, which can be skipped.

2. The discretization method suitable for complex computational domain of lattice discrete particle model according to claim 1, characterized in that: The S1 is specifically: 1a, part creation: generating a Polyhedron_3 object provided by CGAL through a three-dimensional entity surface mesh file in a self-defined format, and reading multiple data files at a time to create multiple parts; 1b, part transformation and assembly: converting one or more Polyhedron_3 objects of the created parts into Nef_polyhedron_3 objects, and performing translation, rotation or scaling transformation as needed before performing Boolean operation to generate a three-dimensional entity model corresponding to the concrete member, that is, the LDPM calculation domain; 1c, combined body processing: preparing a Linear Cell Complex object and a Surface_Mesh object of the three-dimensional entity model in advance for aggregate particle placement and aggregate wrapping body generation, and calculating the three-dimensional entity volume V through the SM object.

3. The discretization method suitable for complex computational domain of lattice discrete particle model according to claim 1, characterized in that: The S3 is specifically: 3a, placing the aggregate particles with the smallest particle size on the surface of the three-dimensional entity in the order of edge first and then face; First, each edge of the three-dimensional solid LCC object is traversed, and the smallest particle size of aggregate particles is placed on each edge in accordance with the requirement that the pitch is not less than 1.1d min ; Then, traverse each face of the LCC object, rotate each face to horizontal, generate its two-dimensional rectangular bounding box, use Monte Carlo method to randomly throw the minimum particle size aggregate particles in the rectangular bounding box, if the aggregate particles fall in the LCC face and the distance between the aggregate particles and the aggregate particles already thrown on the face is not less than 1.1d min , consider that the particle throwing is successful, and finally rotate all the successfully thrown aggregate particles to the original position of the LCC face; 3b, traversing all the aggregate particles placed on the surface of the three-dimensional entity, checking the particle spacing, and removing the aggregate particles with spacing not meeting the requirements on the basis of preserving the geometric characteristics of the three-dimensional entity; 3c, put the aggregate particles in the three-dimensional entity according to the order from large to small: similar to the surface putting, the Monte Carlo method is used for the particle putting operation, firstly, the bounding box of the three-dimensional entity is generated, a random position is taken in the bounding box, if the position falls into the three-dimensional entity, the aggregate particle with the particle size d i is put, and the nearest aggregate particle is searched, and the particle size of the searched aggregate particle is d j ; then, it is judged whether the distance between the two aggregate particles satisfies the requirement of not less than d i / 2+d j / 2+0.2d min , if yes, it is considered that the putting is successful, and the iteration is repeated until all the aggregate particles generated in S2 are successfully put.

Images