# A Random Reconstruction Model of Geomembrane Particle Cushion Based on Voronoi Diagram

**Abstract:** In order to study the relationship between the shape of the bedding particles, the particle size gradation and the porosity and the top-breaking of the HDPE geomembrane manufacturers of the earth-rock dam surface on the particle cushion, the actual particle material is often used for testing, but It is often difficult to obtain a statistically significant rule due to the randomness of the sample and the uncontrollable test conditions. Aiming at this problem, a random reconstruction model of particle cushion is established based on the Voronoi diagram. The parameters of the random process of the model are adjusted to control the particle size gradation and porosity low price China HDPE liner, and the shape parameters of the particles in the random reconstruction model. Statistical analysis was performed and significance tests were performed with the shape parameters of the real particles. The results show that the particle shape of the random reconstruction model is close to wholesale HDPE Liner made in China the shape of the real cushion particles; the established model can be used to further study the probability of the high quality HDPE geomembrane breaking through and breaking on the particle cushion and the shape and size of the bedding particles. The relationship between the distribution and porosity.

**Keywords:** China HDPE liner manufacturers; granular material; shape parameter; Voronoi diagram; random weight model; high quality aquaculture geomembrane

In the geomembrane anti-seepage structure of earth-rock dam surface, the geomembrane cushion material generally adopts granular material with good water permeability (mainly gravel), so as to timely remove the water seepage under the membrane. However, under the action of water pressure, the HDPE geomembrane manufacturers are easily broken by the particles in the cushion or bursting in the gap of the particles. Around the geomembrane in engineering, there are bursts and bursts in the engineering firm. The current SL/T 235-1999 “Testing Procedures for Geosynthetics” proposes three standard tests to qualitatively measure the resistance of geomembrane to bursting and topping. Broken ability. Because it is a standard test, the size of the sample, fixture, and ejector used in the test is constant, and the true stress conditions of the high quality HDPE geomembrane on the particle cushion cannot be truly restored. In the top-break test, some foreign scholars used three rigid tapered protrusions arranged in equilateral triangles to simulate the uneven surface of the cushion, but because the pitch and size of the tapered protrusions are fixed, the test The conclusions reached cannot be extended to practical applications. In fact, in addition to the magnitude of water pressure and the physical and mechanical properties of the China HDPE liner manufacturers itself, the shape of the mat particles, the gradation of the particle size, and the porosity are more important factors low price China HDPE liner. Therefore, many scholars have experimented with real granular materials, but because the shape, particle size and position of each particle in the real particle cushion are random, and the porosity and particle size of the particle cushion. The test conditions are closely related, so when using real particles for testing, it is often impossible to effectively control the test conditions, and at the same time, depending on the number of test groups and the size of the sample wholesale HDPE Liner made in China, it is necessary to obtain statistically significant test data, the workload of the test. Larger. The numerical simulation method can avoid the above problems, especially with the continuous improvement of the discrete element numerical simulation method in recent years China HDPE liner factory price, a numerical model that can consider different particle shapes is gradually developed from a single circular particle model, The random reconstruction of the particle cushion and the statistical analysis of the particle shape parameters bring convenience. In this paper high quality aquaculture geomembrane, a random reconstruction model of particle cushion based on Voronoi diagram is proposed and compared with the real particle cushion, which can further study the probability of bursting and cracking of low price China HDPE liner on the particle cushion and the shape of the cushion particle. The relationship between particle size grading and porosity is the basis.

### 1 Basic requirements for a stochastic reconstruction model

a. The particle shape, particle size gradation, and porosity in the random reconstruction model of the particle cushion are as close as possible to the real particles. Since the HDPE geomembrane manufacturers are generally broken by irregularly shaped particles on the particle cushion, it is reasonable to use irregular polygonal particles for simulation. In the well-graded and low-porosity particle mat, the small particles can effectively fill the gap between the large particles, and the gap size between the small particles is smaller than the gap between the large particles, increasing the contact area between the geomembrane and the underlayer particles, which will reduce the possibility of the high quality HDPE geomembrane rupture and burst on the mat particles, so the particle size gradation and porosity are also related to the top crack of the China HDPE liner manufacturers.

b. Since there is no overlap between the actual cushion particles and there are no suspended particles with unbalanced forces, it is required that the amount of overlap between adjacent particles should be as small as possible in the random reconstruction model of the particle cushion HDPE geomembrane manufacturers. And each particle can achieve the force balance by the contact relationship between adjacent particles, and it is necessary to improve the precision of the algorithm as much as possible to reduce the overlap between the particles and avoid the occurrence of unbalanced suspended particles.

c. A certain stochastic process needs to be set in the model. The stochastic process and its parameters will affect the particle shape, particle size gradation, and porosity in the model. By adjusting the setting methods and parameters of these stochastic processes, the control can be achieved. The purpose of particle shape China HDPE liner factory price, particle size gradation, and porosity.

At present, the commonly used discrete element particle model modeling methods can be roughly divided into three types wholesale HDPE Liner made in China: 1 drop method. The drop method establishes a finite number of sets of particles according to particle grading and shape requirements and applies gravity through a discrete element model to simulate the process of free fall accumulation of particles. The main problem of this method is that it is impossible to control the HDPE geomembrane manufacturers porosity of the particle deposits accumulated by free fall, and the modeling efficiency is low. 2 expansion method. The expansion method first generates small particles in a specified area and then increases the radius of the particles so that the particles come into contact, and the particles are moved by the contact force to fill the entire model space. Although this method can better simulate the porosity and particle size gradation low price China HDPE liner, its particle size increase coefficient is difficult to control, often causing large overlapping contact between particles, between particles and between particles and model boundaries. With a large contact force, the instantaneous release of the contact force between the particles causes a large number of particles to fly over the boundary, resulting in inaccurate results China HDPE liner manufacturers. 3 pure geometry method. In the pure geometry method, the accumulation of granules is purely based on geometric calculations and does not simulate the dynamic characteristics of the particles, so the modeling efficiency is high, but it also cannot control the porosity.

### 2 Random reconstruction model based on the Voronoi diagram

The Voronoi diagram, also known as the Thiessen polygon or Dirichlet diagram, is a set of continuous polygons that connect the vertical bisectors of two adjacent line lines and is an important graphical geometry. The Voronoi diagram has a wide range of applications in stochastic models HDPE geomembrane manufacturers. The basic concept can be visually described as there are n sources of ignition (nucleation points) in the region low price China HDPE liner. These n sources of ignition ignite at the same time and spread in all directions at the same speed. When two fire sources are dispersed and then extinguished, the map formed by the combustion extinction is the Voronoi diagram.

**2.1 Location of the nucleation point**

To generate a Voronoi diagram, first, determine the location of the nucleation point. Obviously, when the nucleation points are regularly arranged in the area China HDPE liner factory price, the generated Voronoi diagram and its polygon are also regular, as shown in Fig. 1(a), so the position of the nucleation points should have certain randomness so that the generated The shape and size of the polygons in the Voronoi diagram produce randomness China HDPE liner manufacturers. The stochastic process of nucleation point location in the paper is as follows:

Fig.1 Schematic diagram of polygonal particle nucleation point position

a. In a two-dimensional region ([xmin, xmax], [ymin, ymax]), a square mesh is generated with the maximum particle diameter Dmax is the side length HDPE geomembrane manufacturers, and the square mesh center position is the initial position of the nucleation point.

b. Perform a random offset for each nucleation point location, ie add a random offset to the nucleation point coordinates wholesale HDPE Liner made in China:

Xi=xini+N(-ks,ks)

Yi=yini+N(-ks,ks)

Where: xini and yini are initial position coordinates; N is a random number; ks is the value range of N. In this paper, the uniform distribution of [-ks, ks] is used, and the value of ks cannot be greater than Dmax. That is, the position of the nucleation point is still in the square grid after offsetting high quality HDPE geomembrane, and it is generally suitable to take 1/4~2/3 of Dmax. At the same time, the size of the ks will affect the shape and size of the polygon. As shown in Figure 1, as the value of ks increases, the difference between the polygons of the Voronoi diagram also increases, that is, the randomness increases. Big.

**2.2 Control of porosity**

After determining the position of the nucleation point, the polygonal particles in the Voronoi diagram are closed HDPE geomembrane manufacturers, and there is no pore between them. In this case, the tightest state of the particle cushion, each polygon can be transformed by a certain means to make the polygon particles Porosity is created between.

It can be seen from the observation that since there is one edge between low price China HDPE liner adjacent polygons in the Voronoi diagram, there is no pore between the polygons, and the pores can be “made” by randomly generating new points on the common side as the vertices of the polygons on both sides. As shown in Fig. 2, 4 points are randomly generated on a common edge. Under random generation conditions, the distance between points may be very small, and some may be close to coincidence HDPE geomembrane manufacturers. In this case, different interparticle may be generated. Case of contact high quality aquaculture geomembrane: 1 between the particles P and the particles M (edge M1M2 is all contained in the side P1P2); 2 between the particles L and the particle M (edge L11L12 and the side M11M12 part); 3 particles P and particles Between L (particle P is contacted with the edge L1L2 of the particle L at the vertex P11); 4 the contact edge apex L8 and the apex N2 between the particle L and the particle N coincide with each other China HDPE liner factory price; 5 vertices N5 and M9 coincident with the particle N The vertices N4 and M10 also coincide; 6 the two particles are in point-to-point contact, and the probability of such a situation occurring under random conditions is very small, so it is not depicted in FIG.

Fig. 2 Schematic diagram of the pores at the vertices of the polygon particles

It can be seen from Fig. 2 that the new vertex position generated on the common wholesale HDPE Liner made in China edge determines the size of the pore. When the new vertex coincides with the original vertex, the porosity of the model is at least zero. When the particle is in point-to-point contact with the particle, The model can achieve the maximum porosity high quality HDPE geomembrane, so the porosity of the random reconstruction model of the particle cushion can be controlled by controlling the stochastic process generated by the new vertex. In this paper, the following methods are used to control the porosity high quality aquaculture geomembrane:

a. Determine the line segment Lr between the two points that is the farthest distance among the new vertices on the common edge China HDPE liner manufacturers, and the length is

Lr=kr(μr,σr,0,1)lp

Where: lp is the length of the common side; kr is a random number ranging from 0 to 1. In this paper China HDPE liner factory price, a normal distribution is used to generate kr; μr is the mean of kr; σr is the standard deviation of kr.

b. Determine the position of the line segment Lr by determining wholesale HDPE Liner made in China the starting point of the line segment Lr:

Xp=x1+(lp-lr)kp(μp,σp,0,1)cosθ

Yp=y1+(lp-lr)kp(μp,σp,0,1)sinθ

Where: xp, yp are the starting point coordinates of the line segment Lr; width=44, height=17, dpi=110 is the endpoint coordinate of the line segment Lr; x1, y1 is the starting point coordinates of the common side low price China HDPE liner; θ is the direction angle of the shared side; kp For a random number ranging from 0 to 1, for controlling the distance between the line segment Lr and the starting point of the common edge, a normal distribution is used to generate the random number HDPE geomembrane manufacturers, μp is the mean of kp; σp is the standard deviation of kp.

c. Two points are randomly generated in the line segment Lr, and the generated new vertex (including the two endpoints of the largest line segment Lr) is randomly divided into polygons into both sides of the common side to form a new polygon, if the new polygons are mutual If there is no contact high quality aquaculture geomembrane, the random division is repeated until the conditions of mutual contact are met.

According to the above, there are two parameters for controlling the porosity, which is kr and kp China HDPE liner manufacturers, respectively. The mean and standard deviation of the random number has an effect on the porosity. Tables 1 and 2 show the porosity of the particle cushion of kr and kp under different mean and standard deviation conditions, respectively.

Table 1 Model porosity when the parameter kr has a different mean and standard deviation wholesale HDPE Liner made in China

(μp=0.5, σp=0.15)

Note: “/” indicates that under the condition of the group mean and standard deviation China HDPE liner manufacturers, the probability that kr is not in [0,1] is higher, so it is rounded off.

Table 2 Model porosity when the parameters kp have a different mean and standard deviation

(μp=0.5, σr=0.15)

Note: “/” indicates that under the condition of the group mean and standard deviation, the probability that kp is not in [0,1] is higher, so it is rounded off.

Since the porosity of the particle cushion is mainly determined by the length lr of the line segment Lr, it can be seen from Table 1 that as the mean μr of kr increases, the porosity decreases continuously, and since the standard deviation σr is only for kr The degree of dispersion has an effect HDPE geomembrane manufacturers, so its effect on porosity is small. It can be seen from Table 2 that the relationship between the value of μp and σp has little relationship with the porosity, indicating that the relationship between the starting point of Lr and the porosity is not large wholesale HDPE Liner made in China, and it can be seen from Table 2 that the porosity is basically the same as in Table 1. The porosity of μr=0.5 is close, which means that the porosity of the generated random reconstruction model is similar when the parameters of low price China HDPE liner the stochastic process are the same, so the repeatability of the model is good.

**2.3 Control of particle size grading**

For polygonal particles, there are many geometric parameters of the particles. Different methods use different methods to define the particle size of the polygonal particles. In this paper, the minimum Feret diameter of the polygonal particle boundary is used as the particle size, that is high quality HDPE geomembrane, the minimum value of the circumscribed parallel line spacing in different directions on the particle boundary, and the value represents the aperture of the smallest mesh through which the particle can pass high quality aquaculture geomembrane.

As can be seen from Fig. 1, the positions of the nucleation points of the polygonal particles are different, and the particle size of the particles changes. Figure 3 shows the particle size grading curve of the different ks model. As the ks increases, the particle size grading curve changes slightly, but the variation range is limited HDPE geomembrane manufacturers, and the particle size distribution range is narrow. It can be seen that by changing ks, only a model with a uniform particle size can be obtained, and the particle size gradation of the model cannot be further adjusted. In this paper, by setting the random splitting process low price China HDPE liner of particles, particles with smaller particle size are produced to achieve the purpose of particle size adjustment. Specific steps are as follows:

Figure 3 Particle size grading curve for different ks values wholesale HDPE Liner made in China

Step 1 Calculate the probability of splitting Pf by particle size:

Where: Dmax is the maximum particle size of the particles; dmin is the smallest particle size of the particles; di is the particle size of the particles. A random number Pi is obtained by uniform distribution in the range of 0 to 1. If Pi < Pf, the particle will split, and vice versa high quality HDPE geomembrane. Formula (4) indicates that the larger the particle size, the greater the probability of splitting.

Step 2 Create a coordinate system with the centroid as the origin in the polygon particles to be split wholesale HDPE Liner made in China, and the angle φ between the x-axis and the horizontal direction is a random number between 0° and 360° (generated by uniform distribution).

Step 3 The x-axis and y-axis of the coordinate system will generate four intersection points with the contour of the polygonal particle, and four-line segments are formed by connecting the centroid of the polygonal particle with the four intersection points, and randomly obtained by uniformly distributing each line segment HDPE geomembrane manufacturers. At the same time, a total of 4 random points are generated, and a random point generated in each of the polygon particles to be split is combined with the original nucleation point as a new nucleation point to draw a Voronoi diagram, thereby realizing a random split of the particles.

Since the particle splitting randomly generates new nucleation points inside the original particles China HDPE liner factory price, the regenerated model is bound to produce smaller particle size particles, and since the particles are split by probability, the original large particle size particles Some parts can also be retained, so that the particle size gradation curve of the model becomes longer and slower high quality aquaculture geomembrane, and the unevenness coefficient is significantly increased. If the particle size gradation curve after a split still does not meet the requirements, the particle splitting process can be repeated until the requirements are met (see Fig. 4, where Cu is an uneven coefficient and Cc is a curvature coefficient). Of course HDPE geomembrane manufacturers, it is also possible that the required particle size gradation curve falls between the particle size gradation curves before and after a certain splitting. At this time, the probability of particle splitting can be adjusted according to formula (5):

Where: kf is the particle splitting control coefficient, the value range is 0~1, the smaller the value, the smaller the probability of splitting the large particle size, and the larger the proportion of the large particle size in the grading curve after splitting, Achieve further optimization and adjustment of the model grading China HDPE liner manufacturers. Figure 5 shows the particle size grading curve for different kf.

Figure 4 Particle size grading curve after the random splitting of particles high quality aquaculture geomembrane

Figure 5 Particle size grading curve for different kf values

Figure 6 is a random reconstruction model of the particle cushion generated by the method in the two-dimensional range of 10 m long and 2 m high. Since the operation of controlling the porosity only produces new vertices on the common side, the minimum Feret of the particles There is almost no influence on the diameter high quality HDPE geomembrane, so the random reconstruction model of the particle cushion can be firstly controlled by particle size grading, and after obtaining the gradation curve that meets the requirements, the porosity is controlled according to the given porosity high quality aquaculture geomembrane, and finally the gradation and porosity are obtained. A random reconstruction model of the particle cushion that meets the requirements.

Fig. 6 The establishment process of random reconstruction model of particle cushion

### 3 Model particles are compared with the shape of real particles

**3.1 Particle shape parameter selection**

In order to compare the shape of the particles with the real particles at the random reconstruction model China HDPE liner factory price, it is first necessary to determine the quantitative parameters for evaluating the shape of the particles. There are many types of research on particle shape expression and evaluation methods, which can be summarized into three levels in scale: 1 overall contour (spherical, columnar, sheet, etc.); 2 angularity (degree of rounding); 3 surface texture. For the top-breaking phenomenon of the HDPE geomembrane manufacturers on the particle cushion, the scale is generally only to the second level, so this paper selects the flatness e (the first level) and the roundness q (the second level). The shape parameters are analyzed and compared, and the calculation formula is as follows:

Where: Fmax is the maximum and minimum Feret diameter, respectively; PE is the perimeter of the equivalent ellipse (flatness and an ellipse with the same area and particle) China HDPE liner manufacturers; Pc is the actual perimeter of the particle profile.

**3.2 Real particle shape parameter acquisition**

At present, digital image analysis methods have been widely used in the field of geotechnical mechanics and material mechanics. This paper uses high-definition digital cameras to capture digital images of real particles and uses the Image tool in MATLAB software. The box analyzes the image file to obtain the shape parameters of the real particles. The basic principle of the image analysis method is to separate the particles from the background of the image by setting the threshold of the grayscale to form a black and white binary image high quality HDPE geomembrane, thereby extracting the boundary of the particle contour China HDPE liner factory price, and analyzing the boundary to finally obtain the desired shape parameter. It is necessary to produce a large contrast between the particle color and the background color to ensure accurate collection of the contour boundary of the particle and accurate calculation of the shape parameters. Figure 7 shows the shape parameter acquisition process of some real particles. The particles are from the silt dam material yard of a pumped storage power station in Zhejiang Province. It is a limestone blasting material with a particle count of 122.

Figure 7 Realized layer particle image converted to black and white binary image high quality aquaculture geomembrane

**3.3 Shape parameter comparison**

Figure 8 granule flatness and roundness scatter plot

Since the random reconstruction model has more particles, 122 samples of the model particles are compared with the real particles by random sampling. Figure 8 is a scatter plot of the flatness and roundness of the model particles and real mat particles high quality HDPE geomembrane. It can be seen from Fig. 8 that the flatness is mainly distributed between 1 and 2, and the majority falls between 1.2 and 1.6. The average flatness of the model particle sample is 1.67, the standard deviation is 0.45, and the average flatness of the real particle sample is 1.44. The standard deviation is 0.28. The roundness is mainly distributed between 0.85 and 0.95 and is mainly concentrated around 0.90. The distribution of roundness is narrower than the flatness. The average roundness of the model particles is 0.908 4 and the standard deviation is 0.037. The roundness average was 0.885 and the standard deviation was 0.049.

According to the definition of flatness and roundness, the flatness is the ratio of the maximum Feret diameter to the minimum Feret diameter, so the value must be greater than 1, and the roundness is the ratio of the equivalent elliptical circumference to the actual perimeter of the particle China HDPE liner factory price, and the value must be less than 1. Because the premise of the parameter test method in the significance test is that the sample is derived from the normal distribution (the distribution range is [+∞, -∞]), although it is not yet fully determined that the flatness and the roundness are consistent. The distribution high quality HDPE geomembrane, but the particle flatness and roundness are not normally distributed from the distribution range, so the significance test for the statistical distribution of the shape parameters of the model particles and the real particles can only use the non-parametric test method. In this paper, the Ansari-Bradley method was used to test the flatness and roundness of the model particles and the real particles. The significance level was 0.05. The test results show that the flatness test index p=0.081 6>0.05, accept the hypothesis that the flatness of the model particle sample and the real particle sample are derived from the same overall distribution China HDPE liner manufacturers; the significance of the roundness test index p=0.353 0 >0.05, accept the hypothesis that the roundness of the model particle sample and the real particle sample is also derived from the same population distribution.

### 4 Conclusion

a. The parameter ks is used to control the position of the nucleation point in the Voronoi diagram, and the initial model without pores can be generated. The initial model particle size gradation has little relationship with the parameter ks, and the initial model has a relatively uniform particle size.

b. By setting new vertices on the common side of the polygon particles in the initial model high quality HDPE geomembrane, a model with pores can be generated. The pores of the model are generated by a stochastic process controlled by parameters kr and kp, where the stochastic process controlled by the parameter kr determines The size of the model porosity.

c. Since the initial model is based on the maximum particle size of the particles, the process of adjusting the particle size grading is to determine the probability of particle splitting according to the particle size. The particle splitting process can be repeated until the desired particles are obtained by China HDPE liner manufacturers. The diameter gradation, wherein the parameter kf is used to control the particle splitting probability to achieve the fine adjustment of the gradation curve in one split.

d. Since the control process of porosity has little effect on the particle size of the particles, the particle size grading control is generally performed first China HDPE liner factory price, followed by the porosity control, and finally, the particle size gradation of the model particles and the porosity are similar to those of the real particles. Or the same.

e. Comparison of the two shape parameters of flatness and roundness indicates that the shape of the model particles is similar to that of the real particles, and the particle cushion is randomly reconstructed due to the controllability of porosity and particle size gradation. The model can basically restore the real particle cushion high quality HDPE geomembrane, which lays a foundation for further research on the top-breaking of the low price China HDPE liner on the particle cushion. It can also provide a reference for the modeling of the irregular particle discrete element method, but the model is only two-dimensional random weight. The construction of the model and the construction of the 3D random reconstruction model remains to be further studied.