# Multiscale Discrete Element Modeling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- Starting from macro-element deformation, we calculate the DE deformation tensor:$${D}_{e}=\left({r}_{1}-{r}_{0}\left|{r}_{2}-{r}_{0}\right|{r}_{3}-{r}_{0}\right)\cdot {\left({r}_{1}^{*}-{r}_{0}^{*}|{r}_{2}^{*}-{r}_{0}^{*}|{r}_{3}^{*}-{r}_{0}^{*}\right)}^{-1},$$
- Assuming that the atomic sample associated with the element undergoes the same deformation as the element, we get the strain tensor for AS:$${D}_{a}={D}_{e}$$
- Having the deformation tensor, we can calculate new atom positions in the global coordinate system:$${P}_{g}={D}_{a}\cdot Q\cdot B\cdot P,$$
- Atomic dynamics is simulated solving the system of ordinary differential Equations (1)–(2) with numerical method (3). The system can be written in a matrix form:$$\{\begin{array}{c}\dot{V}=F{M}^{-1}=-\mathbf{\nabla}E\left({P}_{\mathrm{g}}\right){M}^{-1}\\ {\dot{P}}_{\mathrm{g}}=V\end{array}$$
- After the steady state is reached in the process of atomic dynamics simulation, we calculate the stress tensor of AS:$$\begin{array}{c}{S}_{a}=\frac{1}{\left|{D}_{a}\cdot Q\cdot B\right|}\left(-(V-\overline{V})M{(V-\overline{V})}^{\mathrm{T}}+\frac{1}{2}{\displaystyle {\displaystyle \sum}_{i,j\ne i}}{r}_{ij}{f}_{ij}^{\mathrm{T}}\right)\\ \overline{V}=\frac{V\cdot {1}^{N\times 1}\cdot {1}^{1\times N}}{N}\end{array}$$Here, $\left|{D}_{a}\cdot Q\cdot B\right|$ is the AS volume and $\overline{V}$ is the average atom velocity.
- Assuming that the stress tensors of AS and DE are equal, we get the stress tensor for the element$${S}_{\mathrm{e}}={S}_{\mathrm{a}}$$
- Having the stress tensor ${S}_{\mathrm{e}}$, we can calculate the forces acting on every vertex of the DE. Consider one face of the DE. The force ${f}_{\mathrm{lmk}}$ acting on the k-th vertex (k = 1,2,3) of the m-th face of element l is calculated using the force acting on the face ${f}_{\mathrm{lm}}$:$${f}_{\mathrm{lmk}}={f}_{\mathrm{lm}}\frac{{a}_{lmk}}{{a}_{lm}}\phantom{\rule{0ex}{0ex}}{f}_{\mathrm{lm}}={S}_{\mathrm{e}}\cdot {n}_{\mathrm{lm}}\phantom{\rule{0ex}{0ex}}{n}_{\mathrm{lm}}=\frac{1}{2}\left[({r}_{{\mathrm{lm}}_{1}}-{r}_{{\mathrm{lm}}_{0}})\times ({r}_{{\mathrm{lm}}_{2}}-{r}_{{\mathrm{lm}}_{0}})\right],$$
- After forces acting on vertexes and vertex masses are determined, we calculate new vertex positions using the system of differential Equations (1)–(2) for vertex dynamics and numerical method (3).
- Getting new vertex positions, we turn to step 1, and the computational process continues.

## 3. Results and Discussion

_{ij}is the distance between atoms i and j and θ

_{ijk}is the bond angle between the ij and ik bonds. This potential has 14 parameters, but R and D are always fixed before fitting so that atoms interact only with the nearest neighbors. Additionally, m is generally fixed and equal to 3, so effectively, there are only 11 parameters that can be used in the fitting.

_{c}), bulk modulus (B), elastic constants (C′, C

_{11}, C

_{12}, C

_{44}

^{0}, C

_{44}), and unit cell size (a). Here, C

_{44}

^{0}is the theoretical value obtained for C

_{44}in the absence of internal displacements. All target properties can be extracted from ab initio calculations; we do not dive into it but use some already known values. Table 1 contains the values used [25]. The unit cell size for a given parameter set is not directly computed, but instead, the loss function contains a derivative of the cell energy with respect to the cell size, which is required to be zero at the actual cell size so that the energy is minimized for the unit cell of appropriate size. All elastic properties are considered equally important so that the chosen loss function is a weighted root mean square error:

_{44}

^{0}, the deformation matrix and resulting evaluation are as follows:

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**The modeled object as a set of discrete macro-elements (

**a**) and the atomic sample associated with the element (

**b**).

**Figure 3.**Comparison of the stress obtained from the model with that obtained from the linear elasticity theory.

**Figure 4.**Comparison of the volume change obtained from the model with that obtained from the linear elasticity theory.

**Figure 5.**Pressure wave propagation in a silicon plate. Velocity distribution at different times: (

**a**) 5 ns; (

**b**) 12.5 ns.

**Figure 6.**Pressure wave propagation in a silicon rod. Velocity distribution at different times: (

**a**) 5 ns; (

**b**) 10 ns.

**Figure 7.**Shear wave propagation in a silicon rod. Velocity distribution at different times: (

**a**) 10 ns; (

**b**) 15 ns.

Experimental (Ab Initio) Values | Calculated Properties | |
---|---|---|

E_{c} [eV] | −4.63 | −4.63 |

B [GPa] | 99 | 97 |

C’ [GPa] | 51 | 51 |

C_{11} [GPa] | 167 | 166 |

C_{12} [GPa] | 65 | 63 |

C_{44}^{0} [GPa] | 106 | 113 |

C_{44} [GPa] | 81 | 78 |

Parameter | Value |
---|---|

A | 3821.34 |

B | 113.17 |

λ_{1} | 3.36252 |

λ_{2} | 1.27279 |

λ_{3} | 1.19417 |

β | 0.132272 |

n | 4.16334 |

γ | 5.71477 |

c | 9.69902 |

d | 2.35646 |

cos(θ_{0}) | −0.40882 |

R | 2.85 |

D | 0.15 |

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**MDPI and ACS Style**

Zhuravlev, A.A.; Abgaryan, K.K.; Reviznikov, D.L.
Multiscale Discrete Element Modeling. *Symmetry* **2021**, *13*, 219.
https://doi.org/10.3390/sym13020219

**AMA Style**

Zhuravlev AA, Abgaryan KK, Reviznikov DL.
Multiscale Discrete Element Modeling. *Symmetry*. 2021; 13(2):219.
https://doi.org/10.3390/sym13020219

**Chicago/Turabian Style**

Zhuravlev, Andrew A., Karine K. Abgaryan, and Dmitry L. Reviznikov.
2021. "Multiscale Discrete Element Modeling" *Symmetry* 13, no. 2: 219.
https://doi.org/10.3390/sym13020219