变比例广义粗粒化方法的多颗粒场景验证

doi:10.11949/0438-1157.20250338

摘要/Abstract

摘要：

粗粒化模型被广泛用于加速离散元方法模拟。对于多分散颗粒系统，颗粒数量随颗粒粒径比的三次方增加。传统粗粒化模型由于使用相同的比例缩放各粒径颗粒，无法减少由于粒径比增大导致的巨大计算成本。为解决该问题，基于无量纲接触方程的一致性，提出了变比例广义粗粒化方法，通过引入“N对1接触”假设，构建了更符合物理实际的变比例接触模型，并通过调整等效杨氏模量减小几何误差。在变比例广义粗粒化方法的基础上，通过休止角和单轴压缩过程验证其在多颗粒场景下的适用性。结果显示，该方法在大颗粒粒径不变、小颗粒放大两倍时，可以减少计算时间约80%，相对误差均值低于2%。

关键词: 粗粒化模型, 离散元方法, 变比例, 休止角, 单轴压缩, 粒子, 粒度分布, 模拟

Abstract:

Coarse-grained models are widely used to accelerate discrete element method simulations. For polydisperse particle systems, the number of particles increases with the cube of the particle size ratio. Traditional coarse-grained models cannot reduce the huge computational cost caused by the increase in particle size ratio because they use the same scale to scale particles of different sizes. To solve this problem, a variable-scale generalized coarse-grained model is proposed based on the consistency of the dimensionless contact equation, includes a new model named the constant surface energy (CSE) model, alongside the existing constant relative overlap (CRO) and constant absolute overlap (CAO) models. The applicability of the variable-scale generalized coarse-grained model in multi-particle simulations is verified through the angle of repose and uniaxial compression process. The CRO and combined CSE-CRO models show better consistency with the original bed using uniform and variable scaling ratios, respectively. The results show that the CSE-CRO model can reduce the calculation time by about 80% when the small particle scaling ratio is 2, and the mean relative error is less than 2%.

Key words: coarse-grained model, discrete element method, variable scale, angle of repose, uniaxial compression, particle, particle size distribution, simulation

中图分类号:

TQ 015.9

方延玮, 柳冠青, 张易阳, 朱泽鹏, 方筑, 李水清. 变比例广义粗粒化方法的多颗粒场景验证[J]. 化工学报, 2025, 76(11): 5630-5644.

Yanwei FANG, Guanqing LIU, Yiyang ZHANG, Zepeng ZHU, Zhu FANG, Shuiqing LI. Validation of the generalized coarse-graining model in multi-particle simulations[J]. CIESC Journal, 2025, 76(11): 5630-5644.

图/表 18

表1 广义粗粒化模型的缩放准则

Table 1 Scaling criteria for the generalized coarse-grained model

广义粗粒化模型特例	$χ$	$l v$	$l ω$	$l γ$	$l E$	$K M$	$l μ S$	$l θ R$	$l μ T$	$l t J K R$
CSE	2	$1$	$l R - 1$	$1$	$l R - 2.5$	$l R - 1$	$1$	$1$	$l R - 0.5$	$l R 2$
CRO	1	$1$	$l R - 1$	$l R$	$1$	$1$	$1$	$1$	$1$	$l R$
CAO	0	$1$	$l R - 1$	$l R 2$	$l R 2.5$	$l R$	$1$	$1$	$l R 0.5$	$1$

表1 广义粗粒化模型的缩放准则

Table 1 Scaling criteria for the generalized coarse-grained model

广义粗粒化模型特例	$χ$	$l v$	$l ω$	$l γ$	$l E$	$K M$	$l μ S$	$l θ R$	$l μ T$	$l t J K R$
CSE	2	$1$	$l R - 1$	$1$	$l R - 2.5$	$l R - 1$	$1$	$1$	$l R - 0.5$	$l R 2$
CRO	1	$1$	$l R - 1$	$l R$	$1$	$1$	$1$	$1$	$1$	$l R$
CAO	0	$1$	$l R - 1$	$l R 2$	$l R 2.5$	$l R$	$1$	$1$	$l R 0.5$	$1$

图1 休止角模拟示意图（实心圆表示原始颗粒，颗粒颜色表示颗粒类型，实线箭头表示速度向量）

Fig.1 Schematic diagram of repose angle simulation (solid circles represent original particles, particle color represents particle type, and solid arrows represent velocity vectors)

表2 休止角过程的模拟参数

Table 2 Simulation parameters of the repose angle process

模拟参数	模拟参数取值
模拟参数	相同比例缩放	变比例缩放
原始颗粒直径 $d$ /mm	$d 2 = 5$
原始颗粒粒径比 $β p$	$β p = 1,2$	$β p = 5$
粗粒比 $l r$	$l r = 1,1.5,2$	$l r, 1 = 1,1.5,2, 2.5; l r, 2 = 1$
密度 $ρ$ /(kg/m³)	$ρ p = ρ w = 3000$
杨氏模量 $E$ /GPa	$E p = E w = 0.03$
泊松比 $σ$	$σ p = σ w = 0.3$
表面能 $γ$ /(mJ/m²)	$γ p - p = γ p - w = 50$
碰撞恢复系数 $e$	$e p - p = e p - w = 0.5$
滑动摩擦因数 $μ S$	$μ S, p - p = μ S, p - w = 0.4$
临界滚动角 $θ R$ /rad	$θ R, p - p = θ R, p - w = 0.1$	$θ R, p - p = θ R, p - w = 0.05,0.1,0.2$
扭转摩擦因数 $μ T$	$μ T, p - p = μ T, p - w = 0.4$
小颗粒的体积分数/%	$Φ 1 = 25$	$Φ 1 = 2,4, 8$

表2 休止角过程的模拟参数

Table 2 Simulation parameters of the repose angle process

模拟参数	模拟参数取值
模拟参数	相同比例缩放	变比例缩放
原始颗粒直径 $d$ /mm	$d 2 = 5$
原始颗粒粒径比 $β p$	$β p = 1,2$	$β p = 5$
粗粒比 $l r$	$l r = 1,1.5,2$	$l r, 1 = 1,1.5,2, 2.5; l r, 2 = 1$
密度 $ρ$ /(kg/m³)	$ρ p = ρ w = 3000$
杨氏模量 $E$ /GPa	$E p = E w = 0.03$
泊松比 $σ$	$σ p = σ w = 0.3$
表面能 $γ$ /(mJ/m²)	$γ p - p = γ p - w = 50$
碰撞恢复系数 $e$	$e p - p = e p - w = 0.5$
滑动摩擦因数 $μ S$	$μ S, p - p = μ S, p - w = 0.4$
临界滚动角 $θ R$ /rad	$θ R, p - p = θ R, p - w = 0.1$	$θ R, p - p = θ R, p - w = 0.05,0.1,0.2$
扭转摩擦因数 $μ T$	$μ T, p - p = μ T, p - w = 0.4$
小颗粒的体积分数/%	$Φ 1 = 25$	$Φ 1 = 2,4, 8$

图2 原始颗粒的颗粒堆（黄色虚线为圆筒壁面位置，颜色表示颗粒配位数）：(a) βp=1（单分散）；(b) βp=2

Fig.2 Particle piles of the original particles (the yellow dashed line is the position of the cylinder wall, and the color indicates the particle coordination number): (a) βp=1 (monodisperse); (b) βp=2

图3 相同粗粒比的单分散颗粒堆（βp=1，lr=2，颜色代表颗粒类型，红色为原始颗粒，蓝色为粗晶颗粒；黄色虚线为圆筒壁面位置）：(a)原始颗粒；(b)CSE模型；(c)CRO模型；(d)CAO模型

Fig.3 Monodisperse particle piles with the same coarse-grained ratio (βp=1，lr=2, color represents particle type, red is original particles, blue is coarse-grained particles; the yellow dotted line is the position of the cylinder wall): (a) original particles; (b) CSE model; (c) CRO model; (d) CAO model

图4 相同粗粒比的双分散颗粒堆（βp=2，lr=2，红色为原始颗粒，蓝色为粗晶颗粒，黄色虚线为圆筒壁面位置）：(a)原始颗粒；(b)CSE模型；(c)CRO模型；(d)CAO模型

Fig.4 Bidisperse particle piles with the same coarse-grained ratio (βp=2,lr=2, red is original particles, blue is coarse-grained particles, the yellow dotted line is the position of the cylinder wall): (a) original particles; (b) CSE model; (c) CRO model; (d) CAO model

图5 相对转向误差示意图

Fig.5 Schematic diagram of relative deflection error

图6 相同粗粒比的休止角及其相对误差

Fig.6 The angle of repose and its relative error for the same coarse-grained ratio

图7 不同粗粒比的双分散颗粒堆（βp=5，Φ1=8%，θR=0.1，lr,1=2.5，lr,2=1，红色为原始颗粒，蓝色为粗晶颗粒，黄色虚线为圆筒壁面位置）：(a)原始颗粒；(b)CSE-CRO模型；(c)CRO-CRO模型；(d)CAO-CRO模型

Fig.7 Bidisperse particle piles with different coarse-grained ratios (βp=5,Φ1=8%,θR=0.1,lr,1=2.5,lr,2=1, red is original particles, blue is coarse-grained particles; the yellow dotted line is the position of the cylinder wall): (a) original particles; (b) CSE-CRO model; (c) CRO-CRO model; (d) CAO-CRO model

图8 不同粗粒比的休止角及其相对误差

Fig.8 The repose angle and relative error of different coarse-grain ratios

图9 变比例广义粗粒化模型在模拟休止角过程中的性能

Fig.9 Performance of the variable-scale generalized coarse-grained model in simulating the angle of repose process

图10 单轴压缩模拟示意图（实心圆表示原始颗粒，颗粒颜色表示颗粒类型，实线箭头表示速度向量）

Fig.10 Schematic diagram of uniaxial compression simulation (solid circles represent original particles, particle color represents particle type, and solid arrows represent velocity vectors)

表3 单轴压缩过程的模拟参数

Table 3 Simulation parameters of the uniaxial compression process

模拟参数	模拟参数取值
模拟参数	相同比例缩放	变比例缩放
原始颗粒直径 $d$ /mm	$d 2 = 2$
原始颗粒粒径比 $β p$	$β p = 1,2, 3$	$β p = 10$
粗粒比 $l r$	$l r = 1,1.5,2$	$l r, 1 = 1,2, 4,6; l r, 2 = 1$
密度 $ρ$ /(kg/m³)	$ρ p = ρ w = 3000$
杨氏模量 $E$ /GPa	$E p = E w = 0.1$
泊松比 $σ$	$σ p = σ w = 0.3$
表面能 $γ$ /(mJ/m²)	$γ p - p = γ p - w = 50$	$γ p - p = γ p - w = 5,50,500$
碰撞恢复系数 $e$	$e p - p = e p - w = 0.5$
滑动摩擦系数 $μ S$	$μ S, p - p = μ S, p - w = 0.3$
临界滚动角 $θ R$ /rad	$θ R, p - p = θ R, p - w = 0.01$
扭转摩擦系数 $μ T$	$μ T, p - p = μ T, p - w = 0.3$
小颗粒的体积分数/%	$Φ 1 = 50$	$Φ 1 = 2,4, 8$

表3 单轴压缩过程的模拟参数

Table 3 Simulation parameters of the uniaxial compression process

模拟参数	模拟参数取值
模拟参数	相同比例缩放	变比例缩放
原始颗粒直径 $d$ /mm	$d 2 = 2$
原始颗粒粒径比 $β p$	$β p = 1,2, 3$	$β p = 10$
粗粒比 $l r$	$l r = 1,1.5,2$	$l r, 1 = 1,2, 4,6; l r, 2 = 1$
密度 $ρ$ /(kg/m³)	$ρ p = ρ w = 3000$
杨氏模量 $E$ /GPa	$E p = E w = 0.1$
泊松比 $σ$	$σ p = σ w = 0.3$
表面能 $γ$ /(mJ/m²)	$γ p - p = γ p - w = 50$	$γ p - p = γ p - w = 5,50,500$
碰撞恢复系数 $e$	$e p - p = e p - w = 0.5$
滑动摩擦系数 $μ S$	$μ S, p - p = μ S, p - w = 0.3$
临界滚动角 $θ R$ /rad	$θ R, p - p = θ R, p - w = 0.01$
扭转摩擦系数 $μ T$	$μ T, p - p = μ T, p - w = 0.3$
小颗粒的体积分数/%	$Φ 1 = 50$	$Φ 1 = 2,4, 8$

图11 原始颗粒床的单轴压缩过程:(a)应力-应变曲线;(b)体积分数-应力曲线

Fig.11 Uniaxial compression process of the original particle bed: (a) stress-strain curve; (b) volume fraction-stress curve

图12 相同粗粒比的单轴压缩过程

Fig.12 Uniaxial compression process with the same coarse-grain ratio

图13 相同粗粒比的单轴压缩特征参数及其相对误差

Fig.13 Uniaxial compression characteristic parameters and their relative errors for the same coarse-grained ratio

图14 不同粗粒比粗晶颗粒床的压缩过程

Fig.14 Compression process of coarse-grained particle beds with different coarse-grain ratios

图15 变比例广义粗粒化模型在模拟单轴压缩过程中的性能

Fig.15 Performance of the variable-scale generalized coarse-grained model in simulating uniaxial compression

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