PFC2D(3D) is commercial software, which is commonly used to model the crack initiation of rock and rocklike materials. For the PFC2D(3D) numerical simulation, a proper set of microparameters need to be determined before the numerical simulation. To obtain a proper set of microparameters for PFC2D(3D) model based on the macroparameters obtained from physical experiments, a novel technique has been carried out in this paper. The improved simulated annealing algorithm was employed to calibrate the microparameters of the numerical simulation model of PFC2D(3D). A Python script completely controls the calibration process, which can terminate automatically based on a termination criterion. The microparameter calibration process is not based on establishing the relationship between microparameters and macroparameters; instead, the microparameters are calibrated according to the improved simulated annealing algorithm. By using the proposed approach, the microparameters of both the contactbond model and parallelbond model in PFC2D(3D) can be determined. To verify the validity of calibrating the microparameters of PFC2D(3D) via the improved simulated annealing algorithm, some examples were selected from the literature. The corresponding numerical simulations were performed, and the numerical simulation results indicated that the proposed method is reliable for calibrating the microparameters of PFC2D(3D) model.
The discrete element method (DEM) was firstly proposed by Cundall in 1971 [
The relationship between the microparameters and the macroparameters is difficult to quantify and the microparameters cannot be directly determined according to the macroparameters obtained from the physical experiments. In practice, however, the microparameters of a numerical simulation model in PFC2D(3D) can be calibrated based on the macroparameters determined by the physical experiments, for example, UCS, Poisson’s ratio, Young’s modulus, and tensile strength. According to the difference between the macroparameters obtained from the physical experiments and the numerical simulation, the microparameters are calibrated until the macroparameters obtained from the numerical simulation are sufficiently closed to those from physical experiments. This calibration procedure is called the “trial and error” method [
To avoid the subjectivity in the process of calibrating the microparameters, Yoon [
According to the references given above, the difficulty of screening out a proper set of microparameters can be classified into three categories:
For convenience of singling out a proper set of microparameters of PFC2D(3D) based on some basic experimental macroparameters (UCS, Young’s modules, Poisson’s ratio, tensile strength, etc.), a new approach for calibrating microparameters is proposed in this paper. The method is based on the improved simulated annealing algorithm. In addition, Python scripts were developed to accomplish the calibration process automatically. The main merit of the proposed method is decreasing the difficulty of calibrating microparameters in calibration process. Additionally, it avoids the subjectivity in calibrating microparameters, and it can be applied to calibrate the microparameters of contactbond materials and parallelbond models in both PFC2D and PFC3D. Additionally, the numbers of microparameters and macroparameters can be increased or decreased according to the specific circumstances in the presented method, which is quite flexible in practical use.
The simulated annealing algorithm was a stochastic search method that was first carried out by Metropolis et al. [
To reach the thermal equilibrium completely, the process will be repeated
Flowchart of the simulated annealing algorithm.
In recent years, the simulated annealing algorithm has been widely applied in the field of optimization [
The simulated annealing algorithm has a drawback: the computation times are large [
In this paper, the simulated annealing algorithm would be improved from two aspects: the disturbance method and the cooling method.
For the original simulated annealing algorithm, the cooling process can be expressed as follows:
According to Ingber [
In practice use, (
In the traditional simulated annealing algorithm, the disturbance for the variable can be denoted as follows:
Based on the study by Ingber [
Before using the improved simulated annealing algorithm for calibrating the microparameters, some changes are required. After the PFC2D(3D) numerical simulations based on the set of microparameters, the corresponding macroparameters can be obtained. In the calibration process, the macroparameters determined by numerical simulations are defined as the state of the system. To implement the microparameters calibration process via the improved simulated annealing algorithm, the energy function (objective function)
To search for the microparameters completely within the determined range, the length of the Markov chain is utilized. The length of the Markov chain represents the iteration times at each temperature. With increasing length of the Markov chain, the corresponding iteration time will increase as well; in this paper, the length of the Markov chain is set to 100.
Another important factor is the bounding of the microparameters. The new microparameters should be generated within a certain range during the calibration, or the numerical simulations of PFC2D(3D) cannot proceed successfully. For example, all the microparameters in PFC2D(3D) should be larger than 0, or the numerical simulations cannot run successfully.
In this paper, the termination criterion of the calibration process is
By combining the improved simulated annealing algorithm and some basic parameters for the calibration process, the procedure of calibrating microparameters via the improved simulated annealing algorithm can be described stepbystep as follows.
The initial temperature
If
A new set of random neighborhood microparameters
If
If
End.
Because the microparameters need to be changed and the PFC2D(3D) software needs to be manipulated many times for each step of calibration, the process of calibrating microparameters is accomplished using Python script. Python (
The numerical simulation mentioned in this section includes four parts:
In practice, the command flow is frequently complied in a .txt file, which will be convenient for the PFC2D(3D) numerical simulations. Due to the change in the microparameters for each step of calibration, the command flow .txt file is written by the Python scripts, and the microparameters for each calibration are changed according to the improved simulated annealing algorithm.
Before the numerical simulation, a PFC2D(3D) assembly is generated, and the process involves particle generation, packing the particles, isotropic stress installation (stress initialization), floating particle elimination, and bond installation, which is described in detail by Itasca [
To obtain the macroparameters, the corresponding numerical simulations should be carried out. The UCS, Young’s modulus, and Poisson’s ratio can be obtained by the uniaxial compression test. Meanwhile, the tensile strength can be determined via the Brazilian disc test. Other numerical simulations can also be utilized to obtain the corresponding macroparameters. In practice, the numbers of macroparameters and microparameters can be increased or decreased according to the specific circumstances. The corresponding command flow .txt file should be changed accordingly.
The aim of the PFC2D(3D) numerical simulations is to obtain the numerical macroparameters. Thereafter, the macroparameters will be output in the log file when the numerical simulations terminate [
To illustrate the calibration process of microparameters more specifically, an example was selected. In [
In total, 10 microparameters determine the contactbonded model in PFC2D. The microparameters and their corresponding descriptions are listed in Table
Microparameters of a contactbonded material.
Parameter  Description 


Minimum ball radius [m] 

Ball size ratio, uniform distribution [ 

Ball density [kg/m^{3}] 

Ballball contact modulus [Pa] 

Ball stiffness ratio [ 

Ball friction coefficient 

Contactbond normal strength, mean [Pa] 

Contactbond normal strength, std. dev. [Pa] 

Contactbond shear strength, mean [Pa] 

Contactbond shear strength, std. dev. [Pa] 
As listed in Table
The range of the microparameters of the contactbonded material.
Parameter  Description  Range 


Minimum ball radius [m]  0.1 

Ball size ratio, uniform distribution [ 
1~20 

Ballball contact modulus [Pa]  0~100 

Ball stiffness ratio [ 
0~10 

Ball friction coefficient  0~1 

Contactbond normal strength, mean [Pa]  0~1000 

Contactbond normal strength, std. dev. [Pa]  0~100 

Contactbond shear strength, mean [Pa]  0~1000 

Contactbond shear strength, std. dev. [Pa]  0~100 
It should be noted that the minimum ball radius should be larger than
Some basic parameters of the improved simulated annealing algorithm need to be determined before the numerical simulations; these are listed in Table
Basic parameters of the improved simulated annealing algorithm.
Parameter  Value 

The length of the Markov chain  100 
Cooling efficient 
0.95 
The initial temperature 
100 
The number of microparameters 
9 
In the numerical simulation, the operation system is Linux Mint 18 Sarah (on a personal computer with an Intel Core i5, 3.30 GHz CPU, and 8 GB RAM); the operation system is open source. The Python version is 3.5. Additionally, to implement the improved simulated annealing algorithm and manipulate the PFC2D(3D) software, two extra Python packages were installed,
Based on the improved simulated annealing algorithm, after 2739 iterations, the termination criterion is satisfied and the numerical simulation is terminated. The UCS and Young’s modulus determined by the numerical simulation are 74.356 MPa and 14.824 GPa, respectively, which is quite close to the physical experiment results (UCS 82.1 MPa and Young’s modulus 14.3 GPa.). The corresponding microparameters are obtained and are listed in Table
Microparameters obtained by the improved simulated annealing algorithm.
Parameter  Value 


0.746 

1.567 

20.688 GPa 

1.833 

0.590 

88.916 MPa 

36.950 MPa 

83.768 MPa 

28.932 MPa 
The microparameters can successfully be determined by the improved simulated annealing algorithm. This saves a tremendous amount of hard work, as it is not necessary to change the microparameters and manipulate PFC2D by hand for each step of calibration, which is convenient for use.
From the analysis of the detailed example in Section
Macroparameters obtained by the improved simulated annealing algorithm.
Author  Material species  Numerical model  Macroparameters determined by the numerical simulations (error) 

Chang et al. 
Kimachi sandstone  PFC2D 





Chang 2002 
Hwangdeung granite  PFC2D 



Hong 2004 
Daejeon granite  PFC2D 



Wang et al. 
Rock cored from the rock mass in the Heishan open pit mine in Chengde city, China  PFC2D 



Yang et al. 
Red sandstone  PFC2D 



Zhou et al. 
Concrete  PFC2D 



Yang et al. 
Sandstone  PFC2D 



Fan et al. 
Rocklike materials  PFC3D 



Duan et al 
Beishan granite  PFC3D 

As listed in Table
PFC2D(3D) is a typical DEM, which is widely applied to model the damage and nonlinear behaviors of rock materials. However, the microparameters in PFC2D(3D) need to be determined before the numerical simulations. In theory, the microparameters can be calibrated on the basis of macroparameters determined by the physical experiments. In practice, determining the proper set of microparameters based on the macroparameters obtained from the physical experiments is very difficult but is critically important for the success of the PFC2D(3D) numerical simulations.
The microparameters are mainly calibrated based on the macroparameters determined by the physical experiments. In recent years, the most commonly used method for calibrating microparameters in PFC2D(3D) is called the “trial and error” method [
In this paper, the improved simulated annealing algorithm was applied to calibrate the microparameters based on the macroparameters determined from the physical experiments, and the calibration process was completely accomplished by Python scripts (the corresponding Python scripts in this paper will be published on GitHub (
However, the calibration process is realized in the Linux operation system in this paper; meanwhile, the PFC2D(3D) are mainly used in the Windows operation system. Thereafter, whether the complying Python scripts are suitable in the Windows operation system needs to be further investigated and will be our next work.
To determine a proper set of microparameters based on the macroparameters obtained by physical experiments, the improved simulated annealing algorithm was utilized to calibrate the microparameters of the PFC2D(3D) models. Moreover, the Python scripts were developed to accomplish the calibration process successfully. The main conclusions of this paper are as follows.
The authors declare that they have no conflicts of interest.
This study was funded by the National Natural Science Foundation of China (51174088, 51174228, 51274253, and 51474252); China Postdoctoral Science Foundation and the Postdoctoral Science Foundation of Central South University; the National Basic Research Program of China (013CB035401); the Fundamental Research Funds for the Central Universities of Central South University (2015zzts077, 2014zzts055); and the Open Research Fund Program of Hunan Provincial Key Laboratory of Shale Gas Resource Utilization, Hunan University of Science and Technology (E21527).