Mathematical Problems in Engineering

Volume 2017, Article ID 6401835, 11 pages

https://doi.org/10.1155/2017/6401835

## Calibrating the Micromechanical Parameters of the PFC2D(3D) Models Using the Improved Simulated Annealing Algorithm

^{1}School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410083, China^{2}Hunan Provincial Key Laboratory of Shale Gas Resource Utilization, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China

Correspondence should be addressed to Min Wang; moc.kooltuo@703gnowleahcim

Received 15 January 2017; Revised 17 March 2017; Accepted 10 April 2017; Published 26 April 2017

Academic Editor: Yakov Strelniker

Copyright © 2017 Min Wang and Ping Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

PFC2D(3D) is commercial software, which is commonly used to model the crack initiation of rock and rock-like 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 contact-bond model and parallel-bond 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.

#### 1. Introduction

The discrete element method (DEM) was firstly proposed by Cundall in 1971 [1]. The DEM was investigated further over the following years [2–7]. It has been widely employed in modeling the damage and nonlinear behaviors of materials. The particle flow code in 2 or 3 dimensions (PFC2D(3D)) models the movement and interaction of circular (2D) or spherical (3D) particles using the DEM. It has many advantages [8–10]: It is efficient, as contact detection between circular objects is much simpler than contact detection between angular objects, it is possible for the blocks to break (because they are composed of bonded particles), among others. Thanks to these advantages of the PFC2D(3D), PFC2D(3D) is extensively utilized to solve rock mechanics and rock engineering problems [11–23]. However, the software also has its disadvantages: it requires calibration. In other words, some microparameters must be specified to result in a material with desired macroparameters such as the uniaxial compressive strength (UCS), Young’s modulus, Poisson’s ratio, and tensile strength.

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 [24]. While the drawbacks of the “trial and error” method are obvious, on the one hand, the calibration procedure is subjective, as it depends on the experiments and the experimenter; if the experimenter is not experienced at calibrating the microparameters, then the calibration process will take a very long time. On the other hand, calibrating the microparameters means modifying the microparameters in the command flow .txt file of the numerical simulations for each step of calibration, which would be hard work. In summary, calibrating the microparameters via the “trial and error” method does not illustrate how to calibrate the microparameters of PFC2D(3D) specifically, as it depends on the experiments or the experimenter. Despite these drawbacks of the “trial and error” approach, the method has been commonly adopted by many researchers [25–33] primarily because there is no other better way to determine the microparameters of PFC2D(3D) models.

To avoid the subjectivity in the process of calibrating the microparameters, Yoon [34] carried out a new approach for calibrating the microparameters of contact-bond models in PFC2D. The relationships between microparameters and UCS, Young’s modulus, and Poisson’s ratio were constructed. By combining the numerical simulation results, Plackett-Burman designed a central composite method. The optimum set of microparameters were determined, and the macroparameters obtained from the numerical simulation results were in good agreement with the laboratory results, whereas the interaction of different microparameters is not considered in the method presented by Yoon [34], and the approach can only be used in the contact-bond model of PFC2D. Additionally, the method can only determine the microparameters of rock materials with their physical properties falling within the following ranges: UCS (40–170 MPa), Young’s modulus (20–50 GPa), and Poisson’s ratio (0.19–0.25). In summary, the approach proposed by Yoon [34] has a limited application. Tawadrous et al. [35] conducted a large number of PFC3D numerical simulations. By combining the numerical simulation results and artificial neural networks, the microparameters of parallel-bond of PFC3D can be predicted. However, the numbers of determined microparameters are limited; namely, the parallel-bond and particle elastic modulus, normal-to-shear stiffness ratio, and parallel-bond strength can be determined by utilizing this method. Moreover, the key for the success of artificial neural networks is sufficient training data [36–38], which implies that a large number of numerical simulations should be conducted using this method. Its applicability to the other models of PFC2D(3D) also requires further investigation.

According to the references given above, the difficulty of screening out a proper set of microparameters can be classified into three categories: The subjectivity of calibrating the microparameters: the calibration process should be more objective and should not depend on the experiments or the experimenter; the hard work of calibrating the microparameters: during the calibration process, the microparameters must be changed by hand for each calibration step, which is time-consuming and tedious; the limited use of the microparameters calibration method: the calibration method should be applied to different kinds of numerical models in both PFC2D and PFC3D software.

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 contact-bond materials and parallel-bond 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.

#### 2. Calibration Process via the Improved Simulated Annealing Algorithm

##### 2.1. Introduction of the Simulated Annealing Algorithm

The simulated annealing algorithm was a stochastic search method that was first carried out by Metropolis et al. [46], and, then, the simulated annealing algorithm was successfully applied to solving the optimization problems by Kirkpatrick et al. [47]. The simulated annealing algorithm is analogous to the annealing process of materials. Boltzmann [48] reckoned that if a system was in thermal equilibrium at a temperature* T*, the probability of the system being in a given state could be expressed as follows:where is the energy of the state , is the Boltzmann constant (cooling coefficient), and is the set of all the possible states. However, (1) does not give any information on how the material reaches thermal equilibrium at a given temperature. Metropolis et al. [46] proposed an algorithm that simulated the process described by Boltzmann: when the system is in the original state with original energy , a random neighborhood state is selected, which leads to a new energy . Based on the Metropolis criterion, if , then the new state is accepted. If ), then the probability of accepting the new state can be written as follows:

To reach the thermal equilibrium completely, the process will be repeated* Markov* times at each temperature, and* Markov* is thus called the Markov chain [49]. For better understanding of the simulated annealing algorithm, its flowchart is illustrated in Figure 1.