Study on the Contact Parameter Calibration of the Maize Kernel Polyhedral Discrete Element Model

Abstract

During maize production and transportation, maize kernels frequently interact with mechanical components. To accurately simulate the interaction process between maize and mechanical components, it is essential to establish a reliable maize kernel model and input precise contact parameters. This study established polyhedral discrete element models of different maize kernels and calibrated the contact parameters between maize kernels and steel plates using the inclined plane method. The coefficients of restitution, static friction, and dynamic friction between maize and steel sheets were measured to be 0.5, 0.545, and 0.213, respectively. Subsequently, the contact parameters between maize kernels were determined through steepest climb tests and central composite design response surface tests. Then, the above parameters were optimized using Design-Expert software. The coefficients of restitution, static friction, and dynamic friction between maize kernels were measured to be 0.318, 0.182, and 0.232, respectively. Finally, the optimized parameters were validated using the angle of repose experiment, which found that the relative error between the experiment and the simulation was only 1.24%. The results indicated that the obtained contact parameters were accurate and reliable.

1. Introduction

Maize predominantly exists as a bulk material. It is currently one of the most produced grains in China, containing substantial amounts of starch and various glucose components [1]. As a result, it is extensively used as a raw material in both the food and industrial sectors [2]. According to the National Bureau of Statistics of China, the total national maize production in 2023 reached 288.842 million tons, with a yield of 6532.1 kg per hectare. These statistics clearly indicate that maize has substantial market potential. With the increasing rate of agricultural mechanization, maize kernels interact with the mechanical components of agricultural equipment at various stages of production. For example, machines are involved in sowing, field management, harvesting, and post-harvest processing. Therefore, accurately calibrating the micro-contact parameters of maize kernels is highly significant for understanding the interaction between maize and mechanical components and for optimizing and improving agricultural machinery.

In recent years, simulation technology has been widely applied in agriculture [3]. In particular, the discrete element method (DEM) [4,5] has significant advantages in simulating discontinuous media. In order to improve the efficiency of maize sowing and achieve precision sowing, Wang [6] et al. and Liu [7] et al. designed pneumatic seed dischargers and finger-clamp precision seed dischargers, respectively, to optimize the sowing process and enhance seed discharge performance. Once maize matures, it must be harvested and threshed promptly for storage. The efficiency of threshing operations is directly linked to the economic benefits of the crop. Therefore, to improve the efficiency of maize threshing, Cui et al. [8] established a discrete element model of maize cobs and simulated threshing experiments. The study provided a basis for the design of maize threshing machinery. After the maize harvest, another challenge emerges. Maize stubble significantly impacts the subsequent operation of agricultural equipment. Zhang et al. [9] as well as Adajar [10] et al. developed and calibrated discrete element models for various crop stubble–soil systems through a series of experiments. Their research facilitated the subsequent study of the interaction between machinery and soil. The above research highlights the pervasive influence of the DEM throughout the entire crop production process, underscoring its indispensable role in equipment research and development.

Most particles in nature are irregularly shaped, and particle models established using the DEM can be divided into two categories based on different modeling approaches. The first is the Multi-Sphere Method (MSM) [11], which fits the contours of the particles by stacking multiple spherical particles. The basic unit of this method is the spherical particle. Nonetheless, while this method is relatively straightforward, the resulting model contours are coarser in the aggregate. The second method is the polyhedral non-spherical particle method, which shapes the particle contours by connecting the heads and tails of multiple planes. It is mentioned that this method creates cohesive particle shapes, leading to a comparatively smoother model surface. Regardless of the particle modeling method, the influence of micro-contact parameters on force and displacement during the interaction between maize kernels and mechanical components is crucial. Obtaining accurate contact parameters between maize kernels and other materials is essential for research related to maize granules.

On the one hand, there is a multitude of contact parameters utilized in numerical simulations. On the other hand, the calibration process is complex, leading to a substantial workload. Coetzee et al. [12] summarized the two main methods for correcting contact parameters: the direct measuring approach and the bulk calibration approach. However, due to the complex shapes of real kernels in nature and other simplifications and assumptions made in DEM (such as contact models, kernel shapes, and surface textures), the direct measuring approach is difficult for determining contact parameters accurately [13]. In summary, the bulk calibration approach is a more reliable approach. Numerous scholars have conducted extensive research on parameter calibration. Yan’s research [14] can help reduce the workload and complexity of calibrating discrete element models. Michael [15] proposed a systematic calibration approach for discrete element models. Song et al. [16] proposed a novel parameter calibration method that employs an optimized Latin hypercube to construct a variable matrix and determine the combination of data sampling and parameters to be calibrated. The sample points extracted from the optimized Latin hypercube are fitted using a Radial Basis Function Neural Network (RBFNN). Then, a global search on the approximate model is performed using the Multi-Island Genetic Algorithm (MIGA) to obtain the desired solution set. Finally, the solution set obtained from the optimized search is validated through the silo unloading method. Wet sand and gravel particles, representing polyhedral particles, were modeled, sieved, and stacked using the JKR contact model by Zhou et al. [17]. Chen et al. [18] conducted size analyses on horse-tooth-shaped, spherical-cone-shaped, and spherical maize kernels. He used the MSM to establish models by filling them with varying numbers of spherical particles. A preliminary verification was conducted to assess the feasibility and validity of the modeling approach for maize seed particle assemblies and individual maize seeds. Wang et al. [19] studied three types of maize kernels, including round particles, wedged particles, and quadrate maize kernels. Furthermore, they respectively determined the functional relationship between the restitution coefficient of maize kernels and the angles of impact and rebound through physical experiments. Li et al. [20] developed discrete element models and bending models for different parts of the corn cob, determining the contact parameters through bending experiments. Coetzee [21,22] calibrated non-spherical and spherical maize using different schemes, demonstrating that a special draw-down test is sufficient to calibrate both the static and dynamic friction coefficients. In summary, the utilization of the DEM for parameter calibration in this study is thoroughly documented.

This study gathered a specified quantity of Jinsui series maize samples, then measured and statistically analyzed the characteristic size data of maize kernels. Based on these measurements, discrete element models were established for three types of maize kernels. Meanwhile, the contact parameters between maize and steel plates, as well as between different maize kernels, were determined through a series of experiments. Finally, the optimized contact parameters were verified using the angle of repose experiments [23,24,25]. This study provides a crucial reference for the rational design and optimization of the operational parameters of equipment structures associated with maize kernel quality control, agricultural material transportation, seeding, and crushing.

2. Maize Granule Modeling

2.1. Materials

The Jinsui series maize variety, extensively cultivated in Gansu Province of China, was selected as the experimental subject. For subsequent analysis, the average moisture content of the maize seeds was determined to be 36.54 ± 0.04% using the drying method. After weighing, the thousand-kernel weight was documented as 351 g.

The geometric shape of maize kernels is complex, and various scholars have classified them into different types [8,26,27,28]. A total of 1000 maize kernels were selected for shape feature analysis. The maize kernels of this variety can be divided into five types according to their shapes: horse-tooth shape (HTS, trapezoidal in front and back, with thickness much smaller than width), flat shape (FS, flat in cross-section, with thickness less than width), spherical-cone shape (SCS, circular in cross-section, trapezoidal in front and back), nearly spherical shape (NS, with uniform diameter), and irregular shape (IS, with obvious defects or peculiar shapes). The numbers for each type are 416 (41.6%), 263 (26.3%), 195 (19.5%), 87 (8.7%), and 39 (3.9%), respectively, as presented in Figure 1. Among them, the combined number of horse-tooth-shaped, flat-shaped, and spherical-cone-shaped kernels accounted for 87.4% of the total sample size.

2.2. Measurement of Characteristic Particle Size of Maize Kernels

To accurately model the morphology of HTS, FS, and SCS kernels, their characteristic dimensions are delineated independently. HTS kernels constitute the largest proportion of the sample and have characterized complex morphology. Therefore, the morphology of HTS maize is defined by five characteristic variables: upper width (W_s1), lower width (W_s2), height (H_s), upper thickness (T_s1), and lower thickness (T_s2), as illustrated in Figure 2a. The cross-section of HS kernels approximates a circular shape, and their thickness is relatively uniform. Consequently, HS kernels can be characterized by two variables: the diameter (D) and the thickness (T_f) of the outer circle, as illustrated in Figure 2b. The morphology of SCS kernels resembles that of HTS kernels, but the range of variation in width and thickness is minimal. Therefore, the characteristic dimensions of SCS kernels are defined solely by width (W_c), height (H_c), and thickness (T_c), as shown in Figure 2c.

Next, we further investigated the relationship between the dimensions of various shape features and provided a dimensional basis for the accurate modeling of kernels. In this section, a total of 150 HTS, 150 FS, and 150 SCS maize kernels were randomly selected from a pool of 1000 samples to measure their characteristic dimensions. The results of their measurements are presented in the following figure (

Table 1. Relationship between the characteristic sizes of three types of maize kernels.

Type	Feature Size	Irrelevance	Mean Square	R2	Pearson’s r	F	Prob > F
HTS	(WS2)	WS2 = (1.9158 − 0.0951 WS1) WS1	0.79	0.37	−0.6139	89.51	6.65 × 10−17 (**)
	(Hs)	H = (3.3856 − 0.2230 WS1) WS1	4.74	0.66	−0.8164	295.9	4.03 × 10−37 (**)
	(TS1)	Ts1 = (1.3687 − 0.1038 WS1) WS1	0.94	0.57	−0.7557	196.9	5.48 × 10−29 (**)
	(TS2)	Ts2 = (1.5175 − 0.1168 WS1) WS1	1.19	0.50	−0.7124	152.5	1.57 × 10−24 (**)
FS	(D)	D = (3.5233 − 0.3388 T) Tf	4.92	0.76	−0.8699	460.2	2.85 × 10−47 (**)
SCS	(Hc)	Hc = (2.5683 − 0.1634 Wc) Wc	1.35	0.60	−0.7789	228.3	8.45 × 10−32 (**)
	(Tc)	Tc = (1.6281 − 0.1105 Wc) Wc	0.61	0.41	−0.6420	103.8	8.45 × 10−19 (**)

Note: R² and Pearson’s r both indicate the degree of correlation between the two parameters, ** indicate data significance.

).

Figure 3 reveals that the characteristic dimensions of the HTS, FS, and SCS kernels all follow a normal distribution, allowing for the determination of the size ranges for each shape. Specifically, the range of variation for the upper width (W_s1) of HTS kernels is 4.16–8.90 mm, for the lower width (W_s2) is 5.13–10.89 mm, for the height (H_s) is 10.46–13.99 mm, for the upper thickness (T_s1) is 3.46–6.02 mm, and for the lower thickness (T_s2) is 3.65–7.95 mm. The range of variation for the diameter (D) of FS kernels is 8.13–11.05 mm, and their thickness (T_f) varies from 3.80–7.05 mm. The range of variation for the width (W_c) of SCS kernels is 5.82–9.57 mm, for the height (H_c) is 7.9–11.80 mm, and for the thickness (T_c) is 4.79–8.68 mm.

In addition, a series of characteristic dimensions was calculated to investigate potential relationships between the measured feature dimensions and enhance the accuracy of kernel modeling. Linear fitting was conducted on the feature dimensions of the three kernel shapes. The results are presented in

Table 2. Modeling dimensions of three maize types.

Kernel Shape	HTS Kernel					FS Kernel		SCS Kernel
Feature Size	Ws1	Ws2	Hs	Ts1	Ts2	D	Tf	Wc	Hc	Tc
Modeling dimensions	7.22	8.82	12.16	4.41	4.80	9.05	5.00	8.05	10.03	5.91

. Therefore, in modeling maize kernels, the upper width, diameter, and width can be selected as the primary dimensions for HTS, FS, and SCS kernels, respectively.

Other characteristic dimensions can be considered secondary and can be calculated sequentially. The calculated kernel population size and size distribution closely approximate the actual measurements. The aforementioned methods for determining feature dimensions are typically employed when sample sizes are limited. However, in this study, a total of 1000 maize kernel samples were measured. Therefore, the average dimensions of the shape features for each kernel were used as the initial modeling size and applied to the discrete element modeling of maize kernels (Figure 4). The reference dimensions for modeling the three kernel shapes are presented in Table 2. Based on the shape parameters of the maize kernels in Table 2, polyhedral maize kernel models were established, as illustrated in the following figure.

2.3. Discrete Element Method Control Equations

Cundall and Strack first proposed the use of the DEM to describe the mechanical behavior of granular materials in 1979 [29]. This method has proven to be an effective tool for studying the constitutive relationships of particles. The DEM is based on the soft sphere model. This approach provides substantial advantages in accurately forecasting the mechanical behavior of particles. The basic principle of DEM is to regard each particle or block as a unit, calculate the contact forces based on interactions between particles at every timestep, and then iteratively compute the motion using Newton’s laws of motion to predict the behavior of the object.

The force and displacement changes of a single particle in the simulation are highly correlated with the contact changes at each timestep. Consequently, the detection of particle contact is a critical component of this procedure. Up to now, contact retrieval between spherical particles has been relatively simple. However, due to the complex shapes of polyhedral particles, they require more complex and time-consuming contact retrieval. Polyhedral contact types are diverse and complex, such as point-to-point, edge-to-edge, edge-to-face, and face-to-face. To optimize processing time, the contact detection operation is performed in two independent stages. In the first stage, a rough search determines the list of particles closest to each particle in the simulation. And in the next stage, the distance between adjacent particles is precisely calculated. Finally, all relevant geometric information for each pair of adjacent particles or adjacent particle boundaries is calculated, updating to the latest position of the particles in the simulation. The force and displacement changes of a single particle follow Newton’s second law, with the motion control expressions as follows:

$$m_i{\displaystyle \frac{d{\nu }_i}{\mathrm{dt}}}=\sum (F_{ij}^n+F_{ij}^t)+G_i$$

(1)

$$I_i{\displaystyle \frac{\mathrm{d}{\omega }_i}{\mathrm{dt}}}=\sum (R_i\times F_{ij}^{\mathrm{t}}-{\tau }_{ij}^{\mathrm{r}})$$

(2)

where m is the particle mass; i is the particle number; ν_i is the particle translation velocity, (m/s); ω_i is the particle angular velocity, (rad/s); G_i is the gravity of the particle, (N); I_i is the moment of inertia of the particles, (kg·m²); $F_{ij}^n$ is the contact normal force of the particles, (N); $F_{ij}^t$ is the contact tangential force of the particles, (N); R_i is the distance between the center of the particle and the force at the contact point, (mm); ${\tau }_{ij}^r$ is the torque due to rolling friction, (N·m).

3. Contact Parameter Calibration

The contact parameters include the micro-parameters of the interactions between maize and steel, as well as between maize kernels, mainly comprising the restitution coefficient, static friction coefficient, and dynamic friction coefficient [30,31,32]. Due to the large spatial volume of steel in the interaction between maize and steel, it is easier to design measurement experiments. Thus, the micro-contact parameters between maize and steel can be readily obtained through measurement.

However, the interaction between maize kernels has only a small contact area and a brief process. That makes it challenging to experimentally determine the contact parameters. Therefore, the micro-contact parameters between maize kernels are generally obtained using parameter calibration methods. In this section, the restitution coefficient between maize and steel is obtained from reference [33], while the static and dynamic friction coefficients are directly measured. The micro-contact parameters between maize kernels were calibrated and studied through a combination of angle of repose experiments and simulations. First, the steepest climb test was used to obtain the variation range of each parameter. Furthermore, the central composite design response surface experiment was used to establish a mathematical model between each factor and the angle of repose, and to analyze the interaction effect. In the end, the parameters were optimized to determine the optimal parameter combination for maize kernels.

3.1. Calibration of Contact Parameters between Maize and Steel

3.1.1. Determination of the Coefficient of Restitution

The restitution coefficient between maize kernels is defined as the ratio of the velocity after collision to the velocity before collision. In this study, the kernels in the DEM are modeled as soft spheres, where energy dissipation occurs during collision, resulting in a non-elastic collision. Considering the extensive research conducted by both domestic and international scholars on the restitution coefficient of maize kernels [19] and the challenge in obtaining accurate data through experiments, this study utilizes literature sources to determine the restitution coefficient for maize kernels, establishing that the restitution coefficient between maize kernels and steel is 0.5.

3.1.2. Determination of the Static Friction Coefficient

This study employed the inclined plane method to determine the static friction coefficient between maize kernels and steel plates. The measurement platform consists of two steel plates, a high-speed camera, and a height controller, as illustrated in Figure 5.

First, place the maize on horizontal steel plate 1, fix steel plate 2, and adjust the height controller to rotate steel plate 1 slowly and evenly. When the maize begins to slide along the steel plate, stop the rotation immediately and record the tilt angle of the steel plate at that moment. Repeat the experiment 10 times, and the average tilt angle measured is 28.6°. The tangent of the tilt angle is the static friction coefficient; thus, the static friction coefficient between maize and the steel plate is 0.545:

$$\mu ={\displaystyle \frac{F_f}{F_N}}={\displaystyle \frac{m\mathrm{g}\sin \alpha }{m\mathrm{g}\cos \alpha }}=\tan \alpha$$

(3)

where F_f is the maximum static friction force between the contact surfaces, N; F_N is the normal force of the steel plate on the maize kernels, N; m is the mass of the maize kernels, g; α is the angle between steel plate 1 and the horizontal plane, (°).

3.1.3. Determination of the Dynamic Friction Coefficient

Similar to the static friction test method, the inclined plane method is used to measure the dynamic friction coefficient between maize kernels and a steel plate. The test bench setup is shown in Figure 6. When the maize kernels pass over a horizontal steel plate without bouncing and move smoothly for a certain distance, it indicates that the inclined plane angle and displacement meet the test requirements. Preliminary experiments revealed that the inclined plane angle that meets the requirements is 40° and the inclined plane length is 220 mm. Ten maize kernels were selected randomly and were released from rest at the highest point of the slope. The maize kernels rolled down the slope and slid a distance on the horizontal steel plate before stopping, and the sliding distance was measured. The average sliding distance of 10 repeated experiments was 95 mm.

The inclined rolling test was simulated using EDEM 2024 software. The restitution coefficient between maize and the steel plate was 0.5, and the static friction coefficient was 0.545. During the simulation, only one maize kernel was generated. In this case, the contact parameters between maize kernels were set to zero. The range of the dynamic friction coefficient between maize and the steel plate for the experiment was 0.01 to 0.3. When the dynamic friction coefficient was between 0.18 and 0.24, the sliding distance was between 85 and 112 mm (Figure 7). Therefore, the dynamic friction coefficient range was set between 0.18 and 0.24, and repeated tests were conducted. The test results are shown in the table below. Each experiment was repeated 10 times, and the average value was taken. The experimental scheme and results are shown in the

Table 3. Design scheme and results of dynamic friction coefficient simulation experiment.

Test	Dynamic Friction Coefficient	Horizontal Displacement (mm)
1	0.18	110.3
2	0.19	107.8
3	0.20	103.2
4	0.21	97.9
5	0.22	93.9
6	0.23	89.7
7	0.24	85.9

below.

Using Origin’s fitting function to perform polynomial fitting on the above data, the resulting fitting equation is as follows:

$$y=-226.19x^2-328.93x+177.52$$

(4)

where y is the sliding distance (mm), and x is the coefficient of dynamic friction. The goodness of fit is 0.995, indicating a very high degree of fit for the curve. By substituting the measured average sliding distance of 97 mm into the above equation, the dynamic friction coefficient is found to be 0.213. Five simulation experiments were conducted with the dynamic friction coefficient set at 0.213. The simulated average sliding distance was 95.6 mm, with a relative error of only 1.44% compared to the actual value. The results indicated that the measured dynamic friction parameters are accurate and reliable.

3.2. Calibration of Contact Parameters between Mazie Kernels

3.2.1. Angle of Repose Experiment

The angle of repose experiment is among the most widely employed experimental methods for calibrating material contact parameters. The method facilitated the establishment of a stable angle of repose between the material stacking slope and the horizontal plane. The formation of the angle of repose results from the combined effects of material shape, microscopic contact parameters, and other factors. This study employed the cylindrical lifting method for the angle of repose experiment. The cylindrical container was made of organic glass, and the lifting device was a texture analyzer. The lifting speed in the experiment was 1 mm/s. This is illustrated in Figure 8.

Manual measurement of the angle of repose often results in significant errors and it is challenging to obtain accurate data [34,35]. Therefore, this study employs image processing technology [36] to measure the angle of repose. First, take a frontal photo of the stable pile, remove redundant backgrounds, and convert it into a grayscale image to facilitate subsequent edge recognition and extraction. After binary processing, clear contour curves of the pile can be extracted. Finally, coordinates are established, and after curve fitting, the slope of the contour boundary is obtained. The calculated stacking angle of the maize kernels is 27.35°. The process of measuring the stacking angle is illustrated in Figure 9.

3.2.2. Steepest Climb Test

Referring to previous literature [37,38,39], it can be preliminarily determined that the static friction coefficient between maize kernels ranges from 0.05 to 0.35, the dynamic friction coefficient ranges from 0.15 to 0.45, and the restitution coefficient ranges from 0.2 to 0.5. During the calibration process of each parameter, the static friction coefficient, dynamic friction coefficient, and restitution coefficient will be used as experimental factors, while the angle of repose will be used as the experimental indicator. With the aim of improving the accuracy of subsequent response surface analysis, this study employed the steepest ascent test to determine the central values of each variable factor.

According to

Table 4. Steepest climb test plan and results.

Test	Experiment Factors			Results
	A	B	C	AoR°	Relative Error/%
1	0.05	0.15	0.20	22.7	17.00
2	0.10	0.20	0.25	31.2	14.08
3	0.15	0.25	0.30	29.6	8.23
4	0.20	0.30	0.35	27.9	2.01
5	0.25	0.35	0.40	26.1	4.57
6	0.30	0.40	0.45	24.6	10.05
7	0.35	0.45	0.50	22.9	16.27

, the relative error between the simulated angle of repose (AoR) and the actual value first decreases and then increases. If the relative error is minimized near Experiment No. 4, the optimal parameters are located near the fourth set of experimental parameters. Therefore, a central composite design response surface experiment is conducted with the fourth set of experimental parameters as the central value and the third and fifth sets of experimental data as the horizontal values. The AoR is used as the response value to determine the optimal parameters, where the static friction coefficient, dynamic friction coefficient, and restitution coefficient of maize are denoted by A, B, and C, respectively.

3.2.3. Center Composite Design Response Surface Experiment

This paper employed the central composite design, a widely utilized method in response surface methodology. It was widely used to systematically integrate various experimental factors and develop a comprehensive experimental plan. The method effectively estimates both first-order and second-order interactions between experimental factors and response variables, ensuring that the experimental data comprehensively encompasses the entire sample space. It efficiently determines the optimal combination of operational parameters by developing a mathematical model that links experimental factors with response variables.

Table 5. Factor encoding in the angle of repose simulation experiments.

Code	Experiment Factors
	A	B	C
−1.682	0.082	0.182	0.232
−1	0.15	0.25	0.30
0	0.20	0.30	0.35
1	0.25	0.35	0.40
1.682	0.318	0.418	0.468

presents the levels and codes of the experimental factors. The experimental plan and results are detailed in

Table 6. Center composite design response surface experiment plan and results.

Test	Experiment Factors			AoR D/°	Relative Error E/%
	A	B	C
1	0	0	1.682	28.58	4.50
2	−1	−1	−1	29.03	6.14
3	−1	−1	1	28.21	3.14
4	1	−1	−1	27.57	0.80
5	0	0	0	27.89	1.97
6	0	0	0	28.04	2.52
7	0	0	−1.682	28.97	5.92
8	0	0	0	28.11	2.78
9	1	1	−1	29.26	6.98
10	0	−1.682	0	27.64	1.06
11	0	0	0	27.75	1.46
12	0	1.682	0	29.29	7.09
13	1.682	0	0	27.73	1.39
14	1	−1	1	27.65	1.10
15	0	0	0	27.81	1.68
16	−1.682	0	0	28.81	5.34
17	−1	1	−1	29.21	6.80
18	1	1	1	29.3	7.13
19	0	0	0	27.89	1.97
20	−1	1	1	29.11	6.44

, where the static friction coefficient, dynamic friction coefficient, and restitution coefficient of maize are denoted by A, B, and C, respectively.

Using Design-Expert software for the experimental design, the regression equation for each factor’s effect on the static angle of repose D was obtained through quadratic fitting of the experimental data:

$$\begin{array}{c}D=27.91-0.2633A-0.5268B-0.1066C+0.2825AB+0.13AC+0.1526A^2\\ +0.2216B^2+0.3312C^2\end{array}$$

(5)

The results of the significance test and analysis of variance (ANOVA) for the regression equation are presented in

Table 7. Regression equation analysis of variance.

Source	Sum of Squares	Degree of Freedom	Mean Square	F	p
Model	7.98	9	0.8864	33.6	<0.0001 **
A	0.947	1	0.947	35.9	0.0001 **
B	3.79	1	3.79	143.67	<0.0001 **
C	0.1552	1	0.1552	5.88	0.0357 *
AB	0.6385	1	0.6385	24.2	0.0006 **
AC	0.1352	1	0.1352	5.12	0.0471 *
BC	0.0578	1	0.0578	2.19	0.1696
A2	0.3357	1	0.3357	12.72	0.0051 **
B2	0.7075	1	0.7075	26.81	0.0004 **
C2	1.58	1	1.58	59.91	<0.0001 **
Residual	0.2638	10	0.0264
Lack of Fit	0.1707	5	0.0341	1.83	0.2612
Pure Error	0.0931	5	0.0186
Cor Total	8.24	19

Note: * indicates significant (p ≤ 0.05), ** indicates extremely significant impact (p ≤ 0.01).

.

As indicated in Table 7, the p-value of the model is below 0.0001, reflecting an exceptionally significant fit to the data. The non-significant mismatch term suggests that no additional factors are influencing the response variables. This concordance with empirical observations indicates that the model accurately represents the relationships among the variables A, B, C, and D. The R² value of 0.96 demonstrates that the predicted angle of repose from the regression equation closely aligns with the actual measurements, thereby validating its effectiveness in forecasting test outcomes within the optimization experiment.

3.2.4. Interaction Analysis of Experiment Factors

The response surfaces of the impact of the dynamic friction coefficient, static friction coefficient, and the restitution coefficient on the static angle of repose were obtained by analyzing the data using Design-Expert 13 software.

In Figure 10a, it can be seen that when the static friction coefficient is certain, the angle of repose increases with the increase in the dynamic friction coefficient. In Figure 10b, the collision recovery coefficient is a constant value. The larger the static friction coefficient is, the larger the angle of repose is. According to the regression equation, the factors influencing the stacking angle, ranked in descending order of their impact, are the restitution coefficient, static friction coefficient, and dynamic friction coefficient.

4. Experimental Verification

4.1. Optimal Parameter Solution

Using the optimization module in Design-Expert, the parameters are optimized by minimizing the relative error between the simulated angle of repose and the actual angle of repose. The constraints and objectives are as follows:

$$\left\{\begin{array}{l}\mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}0.082\le A\le 0.318\\ 0.182\le B\le 0.418\\ 0.232\le C\le 0.468\end{array}\right.\\ \min E(A,B,C)\end{array}\right.$$

(6)

Following the optimization calculations, the software determined the optimized values for the static friction coefficient, dynamic friction coefficient, and restitution coefficient to be 0.318, 0.182, and 0.232, respectively.

4.2. Comparison and Verification of Angle of Repose Experiments

To validate the accuracy of the optimized parameters, this study utilized the angle of repose (AoR) experiment and subsequently applied these parameters in a simulation. The remaining parameters for the simulation experiment were sourced from references [40,41]. The simulated AoR value was then compared with the experimentally obtained AoR. The experimentally obtained AoR was 27.35°, while the simulated AoR was 27.67°. The relative error was calculated to be 1.24%, demonstrating the reliability of the optimized contact parameters between maize kernels (Figure 11).

5. Discussion

With the significant increase in China’s agricultural mechanization rate, machinery related to the corn industry is developing rapidly. Therefore, it is important to accurately calibrate the relevant contact parameters of maize in order to elucidate the interaction mechanism between maize and machinery components. This is the significance of this study. At the same time, there are some limitations in this study. Merely obtaining the parameters related to corn kernels is not enough, we also observed the problem of the clogging of silos in the industry of making seed corn, which we will work on in the future.

6. Conclusions

This study focused on the locally cultivated Jinsui series maize variety. Polyhedral discrete element models were developed for three distinct types of maize kernels by accurately measuring their characteristic dimensions. Utilizing the inclined plane method in conjunction with simulation experiments, we calibrated two crucial contact parameters between maize kernels and steel surfaces. Employing a Central Composite Design approach, this research conducted AoR experiments and numerical simulations on maize kernels, effectively validating the optimization of three widely utilized microscopic contact parameters in discrete element simulations. The study draws the following conclusions:

(1) A comprehensive selection and classification of 1000 Jinsui series maize kernels was conducted based on their characteristic dimensions, resulting in horse-tooth-shaped, flat-shaped, spherical-cone-shaped, spheroid, and irregular-shaped kernels comprising 41.6%, 26.3%, 19.5%, 8.7%, and 3.9% of the sample, respectively.

(2) The inclined plane method experiment quantified the static friction coefficient between maize kernels and the steel plate as 0.545. By employing discrete element method (DEM) software to simulate the sliding behavior of maize kernels on an inclined plane, the dynamic friction coefficient was precisely determined to be 0.213.

(3) The steepest ascent experiment revealed that the static friction coefficient between maize kernels varies from 0.082 to 0.318, the rolling friction coefficient spans from 0.182 to 0.418, and the restitution coefficient ranges from 0.232 to 0.468. The calibrated parameters were precisely set at 0.318 for the static friction coefficient, 0.182 for the rolling friction coefficient, and 0.232 for the restitution coefficient, respectively.

(4) Verification experiments demonstrated that the actual angle of repose was 27.35°, whereas the simulated angle was 27.69°. The relative error between the simulation and the experiment was merely 1.24%, underscoring the accuracy and reliability of the optimized contact parameters for maize kernels.