A Simulation Method for Layered Filling of Grain Piles Based on the Discrete Element Method

Abstract

The Discrete Element Method (DEM) has been widely employed to investigate the behavior of particle systems at a macroscopic scale. However, effectively simulating the gradual filling of bulk cereal grains within silos using the DEM remains a formidable challenge due to time constraints. Thus, there is a critical need to develop a simplified model capable of substantially reducing the computational time required for simulating cereal grain accumulation. This study introduces a Layered Filling Method (LFM) designed to expedite the computational process for cereal grain piles within silos. By utilizing particle kinetic energy as a specific criterion, this model identifies particles as stable situations when their kinetic energy drops below a designated threshold. Throughout the filling process, lower particles that were judged to satisfy the condition of stability are isolated, forming sub-heaps that are exempt from persistent detection. The whole particle heap is subsequently segregated into multiple sub-piles and a main pile till the process’s culmination, and these divisions are merged back together. In order to validate the model’s feasibility and accuracy, a comparative analysis was performed on the characteristics of the porosity and airflow patterns of grain piles generated using the LFM and the progressive filling method (PFM), respectively. The research results indicate that there is a marginally higher porosity value in the grain pile simulated by the LFM in comparison to the PFM. However, the average relative error remains below 5.00%. Both the LFM and PFM exhibit a similar spiral upward trend in the simulated airflow paths. Notably, the LFM demonstrates a substantial reduction in the time required to construct grain piles.

1. Introduction

The DEM serves as a potent tool for investigating particle behavior and extracting insights from grain piles. The DEM has been widely used for investigating pore structure within grain piles in agriculture, analyzing grain drying processes, studying particle flow from silos, and optimizing associated equipment [1,2,3]. However, during the process of grain accumulation in a silo, the sheer quantity of particles can reach tens of millions or even billions. Due to the computational cost of directly simulating the process of filling a silo with grain particles by the DEM becoming so computationally expensive, it is difficult or practically impossible to complete large-scale simulations within a reasonable period of time. The computational expense associated with directly simulating the grain storage filling process using the DEM becomes prohibitively high.

To reduce computational costs, researchers have put forth an array of methods. One approach is to accelerate simulations. With the advancements in high-performance computing based on Graphics Processing Units (GPU) [4,5,6,7,8] and improvements in parallel performance [9,10], an increasing number of scientific and industrial fields have adopted GPU acceleration solutions in recent years [11,12]. Najafi-Sani and Mansourpour introduced a GPU-based simulation approach for simulating systems with wide size distributions and multiple dispersed particles, which has been successfully applied to simulate grain unloading in granaries [12].

Another approach to reduce computation costs is to simplify the DEM, which involves modifying the geometric and physical properties of the simulated particles. Some researchers have employed the technique of modifying particle sizes to decrease the overall number of particles that are required for computation [13,14,15], thus enabling longer time steps. Washino explored the effectiveness of the Scale-Up Particle (SUP) model in reducing computational requirements [16]. The number of particles to be considered is reduced by scaling them up proportionally when the occupied space is much larger than that occupied by the particles under investigation. Additionally, other researchers have simplified the model by altering particle properties rather than the sizes or shapes [17,18,19]. He achieved accelerated simulation speed by reducing the stiffness of particles in additive manufacturing simulations [20]. For some simulations, another way to reduce the computation cost is to simplify the generation process of the particles under study. González-Montellano analyzed the differences between the PFM and the global filling method. The PFM that simulates the actual particle storage process is closer to reality than the mass filling method [21].

While the aforementioned methods have been adopted to reduce computational costs, the simulation cost of a huge grain pile in a silo still exceeds the acceptable time range. Consequently, reliance on empirical formulas for porosity distribution in ventilation simulations has become prevalent [22,23]. This method is simple and reliable to implement, but at the same time, it causes obvious deviations from the actual situation. If reducing the calculation cost of the DEM is available, it is possible to apply almost the real porosity distribution in the ventilation simulation of large granaries. To address the challenge of reducing computational burden in simulating grain particle accumulation within silos, this study introduces the Layered Filling Method (LFM). The overall thinking of this method is that the particles can be considered stable when the kinetic energy is below a certain threshold. As the filling procedure advances, particles meeting the stability criteria are segregated into sub-piles when the piles reach a designated height; this can obviate further involvement in computations and effectively diminish computational expenses. To substantiate the feasibility and accuracy of this method, differences in porosity distribution and airflow tortuosity obtained from various approaches were analyzed, and air path characteristics were also discussed.

2. Materials and Methods

2.1. DEM Formulation

The DEM approach uses Newton’s second law to solve particle motion states and forces, which was first described by Cundall and Strack [24]. The motion of individual particles is determined using Equations (1) and (2).

$$m_i\frac{\mathrm{d}{v}_i}{\mathrm{d}t}=\sum _j\left(F_n^{ij}+F_t^{ij}\right)+m_{i}g$$

(1)

$$I_i\frac{\mathrm{d}{\omega }_i}{\mathrm{d}t}=\sum _j\left(R_i+F_t^{ij}\right)+{\tau }_{rij}$$

(2)

where $m_i$, $v_i$, $I_i$, ${\omega }_i$, and $R_i$ are the mass, linear velocity, moment of inertia, angular velocity, and particle radius i, respectively, $\mathrm{k}\mathrm{g}$, $\mathrm{m}/\mathrm{s}$, $\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^2$, and $\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$. $F_n^{ij}$, $F_t^{ij}$, and ${\tau }_{rij}$ are the normal and tangential forces and the torque between the particles $i$ and $j$, respectively, $\mathrm{N}$, $\mathrm{N},$ and $\mathrm{N}\cdot \mathrm{m}$.

Hertzian contact theory has been widely applied in DEM simulations involving various agricultural grains. In this study, the non-linear Hertz–Mindlin contact model [2] was chosen. The expressions for normal contact force $F_n$ and damping normal force $F_n^d$ acting between two contacting spheres are expressed as follows:

$$F_n=\frac{4}{3}\sqrt{R^{*}}{\delta }_n^{3/2}$$

(3)

$$F_n^d=-2\sqrt{\frac{5}{6}}\beta \sqrt{S_n{m}^{*}}{v}^{{\overrightarrow{rel}}_n}$$

(4)

where parameter definitions are shown in

Table 1. Fundamental equations.

Parameters	Equations
Equivalent contact radius	$\frac{1}{R^{*}}=\frac{1}{R_i^{*}}+\frac{1}{R_j^{*}}$
Equivalent Young’s modulus	$\frac{1}{E^{*}}=\frac{1-v_i^2}{E_i}+\frac{1-v_j^2}{e_j}$
Equivalent mass	$\frac{1}{m^{*}}=\frac{1}{m_i}+\frac{1}{m_j}$
Normal stiffness	$S_n=2E^{*}\sqrt{R^{*}{\delta }_n}$
Damping coefficient	$\beta =\frac{lne}{\sqrt{l{n}^{2}e+{\pi }^2}}$

. For more details, please refer to reference [25].

2.2. Layered Filling Method

In DEM simulations, a significant amount of computational time is dedicated to particle collision detection [26]. The objective of this algorithm is to reduce collision detection and optimize computation time. Drawing upon the Janssen effect [27,28,29], when a grain pile reaches a certain height, the influence of further filling at that height on the lower part of the pile can be disregarded. As a result, the need for contact detection during the calculation process can be reduced. The removed section of the pile is referred to as a sub-pile, while the remaining section is referred to as the main pile. The entire Implementation process is shown in Figure 1.

The typical DEM simulation loop can be summarized as follows: collision detection, solving interactions, updating, and miscellaneous tasks [30]. The LFM, as depicted in Figure 2, operates between the updating and miscellaneous tasks. The specific steps of the LFM are outlined below:

• At intervals, the LFM traverses all particle properties and labels particles that meet kinetic energy conditions. It marks particles that meet specified criteria (such as Kinetic energy);
• Once all particles within a certain range surrounding a labeled particle have been labeled, this particle is considered to be stable;
• The highest-positioned particle is located, and particles below this position are separated to form a sub-pile. The base of the pile consists of a static particle surface formed by the upper layer of particles from the former sub-pile;
• After the piling process, the sub-piles are merged with the main pile and gravity settling is allowed until the imbalance forces meet the predetermined requirements.

Kinetic energy can reflect the motion state of particles. In this study, kinetic energy is selected as the criterion for determining particle stability. Kinetic energy can reflect the motion state of particles to some extent. Subsequently, the particles within a specific range surrounding the labeled particles need to be examined. To accomplish this, a simplified search algorithm proposed by Murotani is utilized [31], which leverages GPU acceleration for faster computations. The following illustrates how the algorithm divides the computational domain into multiple buckets with defined widths based on Equation (5):

$$l_b=c_b\times r_{max}$$

(5)

where $c_b$ is a coefficient of bucket width determined by particle movement and $r_{max}$ is the radius of the maximum particle.

Figure 3 illustrates the process of determining the state of particles. The solid wireframes represent particles allocated to different block areas. When searching for neighboring particles, the search group is limited to particles within nearby bins, thereby enhancing search efficiency. For example, to determine whether the red particle in Figure 3a is in a stable state, it is necessary to determine whether the particle itself and the particles around it that can affect its state are also in a stable or quasi-stable state. The involved block areas are the block areas where the particle is located, as shown in Figure 3a, and the other 8 block areas surrounding it. In order to minimize the cost of calculation required, the specific search scope is limited to a square area with the judged particle as the center, which spans one block length both in the positive and negative directions, respectively, in the horizontal and vertical directions, and the area is marked by the dashed line box as shown in Figure 3a. If the judged particle is located at the boundary of the total calculation zone, the search area will be adjusted according to the position of the particle based on the shape of the blue particle area in Figure 3a, just as shown in Figure 3b. When all particles within a specific range around the marked particles meet the stability conditions, the marked particles can be considered in a stable state.

Figure 4a shows the specific process of step 3, where the blue color represents particles that have reached the stability condition. In Figure 4a, for the red particle P indicated by the arrow, there are no particles within a certain range (as indicated by the dashed circle) that meet the criteria. This particle is considered to be in a false stable state and will be removed from the marking in the subsequent calculations.

In order to form a relatively flat particle surface, a stable particle surface is established when a certain number of stable particles reach a specific height. In the subsequent processing, the particles within this plane that do not meet the stability condition will be regarded as stable. As depicted in yellow in Figure 4a, all particles comprising this plane will be deemed to satisfy the stability condition in the subsequent processing.

Process 2 in Figure 4a illustrates the process of segregating stable particles from the particle pile. The blue particles that meet the stability condition are separated to form a sub-pile, with the upper layer of particles from this sub-pile serving as the supporting surface for the main pile. To prevent smaller particles within the main pile from permeating through the supporting surface due to pressure, it may be necessary to increase the thickness of the particle surface, as the green particles show in Figure 4b. This necessitates a minimum height ($H_{min}$) requirement for the sub-pile, ensuring that the sub-pile has a sufficient height to construct an appropriate particle surface.

In step 4, Figure 5 illustrates the process of merging three sub-piles into a single pile. The most important thing in the separation process is to identify whether the combinations of stabilizer particles, which are called sub-piles, are separated, and the topmost particle plane of a sub-pile combination forms the established stable foundation for the particle pile that needs to be continued to accumulate. As illustrated in Figure 5, the particle planes circled by the same dotted box in Sub-pile 1 and Sub-pile 2 are actually the same plane. Similarly, this duplication of particle planes is observed between Sub-pile 2 and Sub-pile 3. Consequently, when merging the individual sub-particle piles, it becomes imperative to eliminate duplicate particles. At the same time, due to the method of layered successive stacking, the effect of compression caused by dead weight and other sub-piles has not been fully considered in the result of the merged particle pile, so gravity deposition is required until the ratio of the unbalanced force of the particles is reduced to below 0.01 [32].

2.3. Materials and Parameters

There are two primary approaches for determining parameter values: the direct measurement method and the parameter calibration method [33,34]. Due to the challenges associated with direct measurement, researchers often rely on the parameter calibration method to obtain the necessary material properties for DEM simulations. This process involves iteratively adjusting the attribute values used in the simulation to match the observed macroscopic behavior and achieve similar material properties.

In this study, a small, galvanized steel silo with a flat shape was chosen for analysis. The soil inside the silo has a vertical height (H) of 1000 mm and a diameter (D) of 100 mm, as depicted in Figure 6. To simplify the model, a single spherical particle is employed to represent soybeans, and the material properties of both the soybeans and the silo are presented in

Table 2. DEM simulation parameters reported by Yan [36].

Parameters	Symbol	Material
		Soybean Seed	Galvanized Steel
Density, $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$	$\rho$	1213	7850
Poisson ratio	$\nu$	0.4	0.3
Modulus of elasticity, Pa	$E$	6.1 × 108	7.90 × 1011
Coefficient of friction	$mu$	0.2	0.247
Coefficient of restitution	$e$	0.562	0.715
Translational normal damping coefficient, kg/s	${\eta }_{t,n}$	0.185	2.66
Translational tangential damping coefficient, kg/s	${\eta }_{t,t}$	0.06	0.88

. It is worth noting that the particle diameter distribution range is from 5.5 mm to 7.5 mm, as reported by Yuey [35].

In this study, four cases with distinct kinetic energy values are employed as evaluation metrics for particle stability, and they are compared with the case of the PFM (D1). In order to simplify the analysis, the impact of different kinetic energy values on the surrounding particles is not taken into account. The search range is confined to ten times the diameter of the largest particle, and the grain pile is examined at intervals of 0.75 s. The desired stacking height is set at 900 mm. The parameters for each scenario are outlined in

Table 3. Parameters for LFM cases.

Parameter	CASE
	C1	C2	C3	C4	D1
Kinetic energy	3.39 × 10−9	1.356 × 10−10	3.39 × 10−11	1.356 × 10−12	Reference condition
Search range	0.075	0.075	0.075	0.075	Reference condition
$H_s$	0.1	0.1	0.1	0.1	Reference condition
Cb	5	5	5	5	Reference condition

.

3. Results and Discussion

Figure 7 depicts the comprehensive process of the LFM. As the stacking process proceeds, a stable particle pile is continuously separated to form three sub-piles (Figure 7a–c) and the main pile, where the black areas represent the fixed particle planes, which are composed of the same particles at nearly the same height. It can be seen in Figure 7b–d that there are bottom particle planes to support the accumulation body. As shown in Figure 5, this layer of particles is the top particle plane of the lower sub-pile and the bottom particle plane of the upper sub-pile concurrently, and it is necessary to ensure that those particles are not double-counted when merging all sub-piles into an entire pile. Once the main pile is fully constructed (Figure 7d), the three sub-piles are combined with the main pile and then undergo gravity settling together, leading to a tightly stacked particle pile, as seen in Figure 7e.

3.1. Porosity Distribution and Tortuosity

The distribution of porosity is of significant importance for studying the ventilation system of grain piles. Typically, in the design of ventilation schemes, the porosity follows a specific curve. However, in actual grain piles, the porosity may exhibit local variations due to different stacking methods. Therefore, it is necessary to simulate grain piles using the DEM to obtain an approximate distribution of porosity.

Figure 8 illustrates the porosity distribution and error distribution of grain piles under different conditions. Due to the relatively large aspect ratio of the silo, the frictional forces on the silo walls result in minimal variation in porosity along the height direction. Simultaneously, the presence of dead corners at the bottom of the silo leads to an increased porosity in the lower part. In comparison to the PFM, the grain pile simulated using the LFM model exhibits slightly higher porosity. As the stability conditions of the particles decrease, the porosity gradually approaches that of the direct method, maintaining a consistent overall trend.

Based on Figure 8b, it is apparent that C2 exhibits the highest error at a height of 800 mm, reaching a maximum value of 1.14%. The average relative errors for the four cases are 3.05%, 4.56%, 2.93%, and 3.12%, respectively, all of which fall below 5.00%. Hence, it can be concluded that the influence of the LFM on porosity can be deemed negligible.

Tortuosity is an extensively employed key parameter to predict the transport characteristics of porous media and represents the airflow resistance within grain piles. To compare the variations in tortuosity under different conditions, three points were randomly selected at the bottom of the grain pile ((0, 0, 0), (0.015 m, 0, 0), and (0, −0.028 m, 0)) as the air entry points (AEPs). It is important to note that the AEP should be sufficiently distant from the vertical wall to minimize the impact of wall effects on the results.

As depicted in Figure 8, the most significant variation in porosity primarily occurs below the 800 mm mark. Hence,

Table 4. Path length and tortuosity.

AEP Location (x, y, z) (m)	Global Path Length (mm)					Average Local Path Length (mm)					Tortuosity
	D1	C1	C2	C3	C4	D1	C1	C2	C3	C4	D1	C1	C2	C3	C4
AEP1 (0, 0, 0)	1361.38	1428.71	1462.18	1409.93	1357.13	2.66	2.66	2.64	2.67	2.65	1.70	1.79	1.83	1.76	1.70
AEP2 (0.015, 0, 0)	1386.46	1464.11	1424.81	1375.67	1408.91	2.55	2.65	2.63	2.68	2.65	1.73	1.83	1.78	1.72	1.76
AEP3 (0, −0.028, 0)	1391.31	1364.13	1418.17	1361.45	1438.67	2.62	2.61	2.64	2.62	2.64	1.74	1.71	1.77	1.70	1.80

provides the length and tortuosity of the widest airflow path below this level. It is apparent that different AEPs yield distinct airflow paths. Among the three airflow paths observed under various operational conditions, the majority exhibit widths ranging from 2 mm to 3 mm, consistent with the findings of Yue [35]. The longest path measures 1462.18 mm, while the shortest path measures 1357.13 mm.

Based on the relative error variations of tortuosity, as shown in

Table 5. Relative error.

AEP	Global Path Length				Average Local Path Length				Tortuosity
	C1	C2	C3	C4	C1	C2	C3	C4	C1	C2	C3	C4
AEP1	4.95%	7.40%	3.57%	−0.31%	−0.30%	−1.09%	0.23%	−0.71%	4.95%	7.40%	3.57%	−0.31%
AEP2	5.60%	2.77%	−0.78%	3.06%	3.72%	3.17%	5.05%	3.84%	5.60%	2.77%	−0.78%	1.62%
AEP3	−1.95%	1.93%	−2.15%	5.63%	−0.08%	0.80%	0.11%	0.92%	−1.95%	1.93%	−2.15%	3.40%

, it is evident that tortuosity is influenced not only by the stability parameters but also by the position of the air inlet. The closer the air inlet is to the central position of the grain pile, the greater the impact of stability parameter accuracy on the tortuosity. In the case of AEP 2, C3 outperforms the others. Overall, the test conditions demonstrate smaller errors compared to D1.

3.2. Air Path Characteristics

To explore the factors influencing the disparity in tortuosity, this section specifically examines the C4 condition. Figure 9 showcases the simulated airflow path of AEP 1 in C4 using particles, where red denotes C4 and black represents D1. In Figure 9a, it is apparent that the airflow paths of AEP 1 in C4 and D1 exhibit overlapping trends at the inlet. However, as the paths extend, they gradually diverge and become nearly parallel to each other. It can also be observed that the tortuous characteristics of the two airflow paths formed by the particles constituting are roughly similar.

It is important to highlight that along the Z-axis, the two airflow paths progressively diverge and take on parallel trajectories until they converge at the outlet (Figure 9b). As depicted in Figure 9c, the simulated airflow paths display a spiral upward motion, with an approximate relative difference of 90° between the two paths. This phenomenon can be attributed to the static particle plane generated during the filling process by the LFM model, which hinders the transmission of force in the X–Y plane further downward through the particle plane. Consequently, this leads to differing twisting states between the two airflow paths.

3.3. Method Evaluation of LFM

In performance testing, the microcomputer with the main frequency of its processor is 3.1 GHz was utilized as the hardware platform for conducting the tests. The comparison of filling speed and total computation time under different conditions is presented in Figure 10. From the figure, it is apparent that the LFM computational model significantly reduces the time required for generating a grain pile in comparison to the PFM. For instance, under the D1 condition, the computation time for simulating flow every 0.75 s amounts to 80 min. However, this time is reduced to 70 min, 69 min, 68 min, and 25 min under the C1, C2, C3, and C4 conditions, respectively.

In Figure 10a, the dashed lines depict the turning points that indicate the separation points of the grain pile. The number of separations for the C1, C2, C3, and C4 conditions is 5, 4, 4, and 2, respectively. As depicted in Figure 10b, the total computation time is reduced by 55.73%, 51.17%, 47.8%, and 29.7% for the corresponding conditions. The proportion of gravity deposition time under each condition is 38.56%, 22.84%, 11.37%, and 5.31%, respectively. Thus, with enhanced accuracy in conditions and a decrease in separation frequency, the combined gravity deposition time diminishes while the total computation time increases. In Figure 10, the calculation time is mainly related to the number of separations. The more the number of separations, the shorter the total stacking time. Separation accuracy controls the number of separations, and the two are directly proportional.

4. Conclusions

The objective of this study was to minimize the formation time of grain piles by developing a layered filling model based on the Discrete Element Method. The kinetic energy of particles was utilized as the assessment element to verify the rationality of the model. Compared with the PFM, the LFM reduces the calculation cost of stabilized particles and reduces the calculation time. The main conclusions of this study are as follows:

(1)

In comparison to the direct deposition method, the LFM generated grain piles with slightly higher porosity. However, as the particle kinetic energy decreased, the porosity gradually approached that of the direct deposition method, with average relative errors below 5.00%. Hence, the internal distribution of internal porosity in grain piles generated using the LFM model and the direct deposition method, respectively, exhibited similarities.

(2)

Different air entrance points resulted in distinct flow paths, and both the LFM and the PFM demonstrated a spiral-like upward trend in the simulated airflow paths. Under specific precision conditions, the errors in the tortuosity between the airflow paths obtained using the LFM and the direct deposition method, respectively, were observed to be small.

(3)

Compared to the PFM, the LFM significantly reduced the time required to formulate grain piles. For instance, under the D1 condition, the generation time for the grain pile was 80 min, while under the C1–C4 conditions, it was reduced to 70 min, 69 min, 68 min, and 25 min, respectively.