# Introducing Metamodel-Based Global Calibration of Material-Specific Simulation Parameters for Discrete Element Method

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## Abstract

**:**

## 1. Introduction

- During initial sampling, a large number of parameter sets will be generated at once. Depending on the available computational resources, these parameter sets can be evaluated in parallel. Most metamodel-based optimization algorithms generate only one new parameter set per iteration, so the total time needed for calibration strongly depends on the number of necessary iterations during optimization phase.
- Each calibration sequence starts with an initial sampling. This is necessary to generate a minimum amount of initial information needed for the optimization algorithms, e.g., for training of initial metamodels. Even with continuous improvement of optimization algorithms as well as metamodel types, the minimal computational effort per calibration sequence is limited to ${N}_{init}$ simulations. Depending on the available computational resources as well as the simulation model, the time required for a calibration sequence can still take several hours to days.
- Simulation results as well as metamodels generated in one calibration sequence cannot be reused for other calibration sequences with new bulk materials. This has several reasons. First, most metamodels do not predict simulated material responses but objective function results $z$. The objective function already contains the material-specific real material responses, which were previously determined in the course of experimental investigations. Another reason, which prevents the transferability of metamodels and simulation results to other bulk solids, is the fact that they are produced with discrete element models containing specific values for particle density, particle shape and particle size distribution. Thus, most metamodels are material-specific.
- The lack of reusability of metamodels as well as simulation results for other calibration sequences means that they are usually deleted immediately after the calibration sequence is completed. The only used output of a calibration sequence is the identified parameter set. If the output per calibration sequence is put in relation to the required time and computational effort, classical calibration has a very poor efficiency.
- Classical calibration aims at identifying a parameter set which leads to the desired material responses. Thus, it provides a punctual assignment between the input and output variables of the DEM simulation. The underlying effective relationships are not examined more closely and remain hidden. It is therefore a non-learning approach.

## 2. Basic Idea of Metamodel-Based Global Calibration

## 3. Global Metamodeling

#### 3.1. Material Domain

#### 3.2. Contact Model

#### 3.3. Parametric Particle Size Distribution

#### 3.4. Definition of Parameter Space

#### 3.5. Calibration Experiment and DEM-Modeling

#### 3.6. Sampling Strategy

#### 3.7. Metamodel

#### 3.7.1. Choice of a Metamodel

#### 3.7.2. Genetic Programming

## 4. Results

#### 4.1. Global Metamodelling

#### 4.2. Parameter Identification and Validation

^{−11}. If the parameters from Table 3, Table 4 and Table 5 are entered into Equations (6) and (7), the estimated response values are $\widehat{\phi}=35.530$ and $\widehat{\rho}=1478.78$.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Granular materials considered to quantify the boundaries of the investigated material domain: (

**a**) coke; (

**b**) stone chippings 2-5 mm; (

**c**) limestone; (

**d**) gravel 2/8 mm; (

**e**) gravel 16/32 mm; (

**f**) dry sand; (

**g**) woodchips; (

**h**) dry corn.

**Figure 4.**Comparison how good different analytical distributions functions fit the real particle size distribution of the example materials.

**Figure 5.**Structure of the shear box test: (

**left**) initial state before opening the flap; (

**right**) final state.

**Figure 6.**Alternative sequential sampling design strategies (initial samples in black dots, sequentially added samples in red squares): (

**a**) space-filling sampling; (

**b**) adaptive sampling [23].

**Figure 7.**Benchmark of different sequential space-filling methods for 2-dimensional space. Blue bars show the mean value of 10 different designs. Black whiskers show the minimum and maximum values. (

**a**) minimum intersite distance; (

**b**) cover measure.

**Figure 8.**Benchmark of different sequential space-filling methods for 10-dimensional space. Blue bars show the mean value of 10 different designs. Black whiskers show the minimum and maximum values. (

**a**) minimum intersite distance; (

**b**) cover measure.

**Figure 17.**Validation of the identified set of parameters. (

**a**) Simulated bulk material; (

**b**) real bulk material.

Metric | Mean Over All Materials | ||
---|---|---|---|

GGS | RRSB | LOGN | |

Coefficient of determination $({R}^{2})$ | 0.8155 | 0.9925 | 0.9935 |

Mean squared error $(MSE)$ | 0.0308 | 0.0012 | 0.0010 |

**Table 2.**Parameter ranges used for data generation. The ranges define the parameter space in which the metamodel is valid.

Parameter | Symbol | Unit | Range |
---|---|---|---|

Young’s modulus of particles | ${E}_{p}$ | $\mathrm{Pa}$ | 5 × 10^{6} … 5 × 10^{8} |

Young’s modulus of walls | ${E}_{w}$ | $\mathrm{Pa}$ | 5 × 10^{6} … 5 × 10^{8} |

Poisson ratio for particles | ${\nu}_{p}$ | - | 0.1 … 0.5 |

Coefficient of restitution particle-particle | ${e}_{pp}$ | - | 0.05 … 0.95 |

Coefficient of restitution particle-wall | ${e}_{pw}$ | - | 0.05 … 0.95 |

Coefficient of friction particle-particle | ${\mu}_{pp}$ | - | 0.0 … 1.0 |

Coefficient of friction particle-wall | ${\mu}_{pw}$ | - | 0.0 … 1.0 |

Coefficient of rolling friction particle-particle | ${\mu}_{r,pp}$ | - | 0.0 … 1.0 |

Coefficient of rolling friction particle-wall | ${\mu}_{r,pw}$ | - | 0.0 … 1.0 |

Particle density | ${\rho}_{p}$ | $\mathrm{kg}/{\mathrm{m}}^{3}$ | 250 … 3000 |

Median diameter of PSD | ${d}_{50}$ | $\mathrm{mm}$ | 0.40 … 25.6 |

Standard Deviation of PSD | ${\sigma}_{ln}$ | $\mathrm{mm}$ | 0.07 … 1.17 |

Parameter/Response Value | Symbol | Unit | Value |
---|---|---|---|

Median diameter of PSD | ${d}_{50}$ | mm | 11.16 |

Standard Deviation of PSD | ${\sigma}_{ln}$ | mm | 0.26 |

Coefficient of friction particle-wall | ${\mu}_{pw}$ | - | 0.363 |

Shear angle | $\phi $ | ° | 35.53 |

Bulk density | $\rho $ | kg/m³ | 1478.8 |

Constraint |
---|

${e}_{pp}=0.6$ |

${e}_{pw}=0.6$ |

${\mu}_{r,pp}=0.8$ |

${\mu}_{r,pw}=0.8$ |

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

${\mu}_{pp}$ | 0.1852 |

${\rho}_{p}$ | 2608.4 |

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**MDPI and ACS Style**

Richter, C.; Will, F.
Introducing Metamodel-Based Global Calibration of Material-Specific Simulation Parameters for Discrete Element Method. *Minerals* **2021**, *11*, 848.
https://doi.org/10.3390/min11080848

**AMA Style**

Richter C, Will F.
Introducing Metamodel-Based Global Calibration of Material-Specific Simulation Parameters for Discrete Element Method. *Minerals*. 2021; 11(8):848.
https://doi.org/10.3390/min11080848

**Chicago/Turabian Style**

Richter, Christian, and Frank Will.
2021. "Introducing Metamodel-Based Global Calibration of Material-Specific Simulation Parameters for Discrete Element Method" *Minerals* 11, no. 8: 848.
https://doi.org/10.3390/min11080848