Estimating DEM microparameters for uniaxial compression simulation with genetic programming

https://doi.org/10.1016/j.ijrmms.2019.03.024Get rights and content

Abstract

Among the steps in modeling with the Discrete Element Method (DEM), one of the most important is parameter calibration. The commonly used trial-and-error approach brings drawbacks such as user dependence and high computational cost. As an alternative, artificial intelligence methods, such as neural networks and genetic algorithms, have been adopted. In this work, a new methodology based on Genetic Programming (GP) is presented as an alternative to calibrate DEM microparameters. From DEM models, GP provides functions relating microparameters and macro-properties. Given target macro-properties, the microparameters are obtained by an optimization procedure. The calibration procedure was evaluated for a uniaxial compression simulation and showed good accuracy for data sets with a reduced number of models. In addition, GP is less user dependent and less computationally intensive than other calibration methods. The methodology proved to be effective for DEM calibration and can be extended to other multiscale models.

Introduction

The Discrete Element Method (DEM), developed by Cundall & Strack1 is generally adopted when analyzing granular materials. In contrast to continuum approaches, such as the Finite Element Method (FEM), the DEM enables simulating large deformations along with being capable of naturally representing fracture initiation and propagation. The latter is achieved in association with bonded contact models.2 Bonded contact models are responsible for representing the cohesive behavior observed in real rocks. Nevertheless, each contact model has different microparameters, which must be carefully chosen in order to accurately simulate a specific rock behavior. The process of selecting the microparameters that relate to the expected macro-responses is called calibration.3 These input parameters were first considered as representative of the behavior at the microscale, i.e. an actual representation of the physical properties of the particles at the micro level. Therefore, the most straightforward approach is to directly obtain them from laboratory experiments. However, although some microparameters are easily obtained experimentally, others are not. Even though this approach has been adopted by some authors, it is mainly applied to particles at the millimeter scale.4

Alternatively, microparameters can be obtained by matching DEM numerical results to the bulk behavior obtained in laboratory experiments. This approach, referred to as bulk calibration by Coetzee,3 is the most commonly used, not without bringing some potential drawbacks. Coetzee3 points out that different combinations of microparameters may result in the same bulk behavior (e.g.5). In this case, a set of microparameters may be accurate for one application and not necessarily for another. Also, the task of matching numerical and experimental bulk behaviors may be time-consuming.

Amongst the bulk calibration approaches, trial-and-error is the most widely used2,6: several models with different input values are carried out in order to match laboratory and modeling results. This can be very laborious and computationally intensive because meeting satisfactory results may require running a great number of models. In addition, this calibration procedure strongly influences the comparison of different contact models.

In order to reduce the computational cost of performing numerous DEM simulations, authors have proposed the use of Design of Experiments (DOE). Yoon7 used two designs in order to calibrate seven microparameters for a bonded contact model of uniaxial compressive tests in rocks. Firstly, the author determined the upper and lower limits of the microparameters by conducting a sensitivity analysis using ranges observed for Young's Modulus (E), Poisson's ratio and Uniaxial Compressive Strength (UCS) for common rocks. The author then used a Plackett-Burman (PB) design as a screening procedure to select the most significant microparameters for each macro response. Then, the author conducted a response surface analysis using a Central Composite Design (CCD). From the PB and CCD analyses, linear and non-linear relations between microparameters and macro-properties were obtained. An optimization method using those relations provided optimum sets of microparameters for different rocks. A total of 51 simulations were necessary and the calibration procedure achieved good results when the microparameters were within the pre-established ranges.

A similar approach was employed by Deng et al.,8 who used DOE techniques to calibrate a heat transfer DEM model. The authors used a small CCD, considering three microparameters and three macro-responses. A small CCD is a variation of the CCD, in which fewer tests are required but the end result may be less accurate. In this case, 15 simulations were run in order to obtain the relations between microparameters and bulk behavior. A similar optimization procedure was applied and the calibration results were in good agreement with the laboratory results. Chehreghani et al.9 also used a variation of the CCD to calibrate a Bonded Particle Model (BPM) simulation of uniaxial compressive tests. They used a half CCD with 32 simulations for five microparameters and two macro-properties. Matching the calculated macro-responses with the values measured experimentally from a rock resulted in the calibrated microparameters.

Rackl & Hanley10 applied an automatic procedure based on Latin hypercube sampling and Kriging in order to calibrate a model of a system used to measure bulk density and angle of repose of loose glass beads. According to the authors, the automated calibration procedure provided satisfactory input sets, especially when considering the experimental scatter generally observed in those measurements. An average of 51 DEM simulations was necessary for this calibration procedure.

In the search for an automated calibration method, machine learning techniques, such as Artificial Neural Networks (ANN) and Genetic Algorithms (GA), have been used by some authors. Do et al.11 employed genetic algorithms and DIRECT optimization in order to calibrate DEM angle of repose tests of sand grains in different set-ups. DIRECT is an algorithm that finds global minimums of multivariable functions. Both methods were used to minimize the difference between experimental and numerical macroscopic responses. The obtained microproperties showed good agreement with the experimental references. In another study, Do et al.12 used a multi-objective genetic algorithm to calibrate sandglass, ledge and conical pile simulations. Two objective functions were defined for minimization: one to relate microparameter and macro-responses and the other to reduce simulation time. The proposed calibration procedure successfully estimated the microparameters for all tests in both studies. In order to save computational time, multi-dimensional grids of simulated macro-responses were created and used as a database for interpolating values of macro-outputs for arbitrary microproperties.

Tawadrous et al.13 used artificial neural networks to predict microproperties for uniaxial compression models for the DEM commercial software PFC3D. Multi-layer perceptron networks were trained with 3125 DEM models. Validation followed with 35 other cases. The estimation of sets of three microparameters led to an overall prediction error of 17% and of 8% when borderline cases were not considered. However, when one more microparameter was considered, the overall prediction error increased to 31.3%.

Also using ANN, Benvenuti et al.14 presented a study for DEM models simulating the angle of repose test (AoR) and the Schulze shear cell test (SSC). In both tests, non-cohesive materials were used, and a set of parameters from the DEM model was given to the ANN in order to train the network. This ANN had as outputs the material bulk behavior. First, the authors trained different ANN with the DEM parameter combinations and their corresponding bulk properties. By comparing the output values from the ANN and the DEM simulations, the authors picked the best network. Then, they used the selected ANN to identify the DEM input parameters. This procedure was carried out by comparing the bulk behavior from the selected ANN with the one from the macroscopic experiments. The authors found close agreement between the DEM and the ANN values. Regarding the calibration of the DEM model, the authors were able to find 3884 valid parameter combinations from the total of 6250000, confirming the viability of using the ANN for parameter identification.

Another technique available in the literature for DEM calibration is dimensional analysis, as presented by Fakhimi & Villegas.15 The authors first performed a sensitivity analysis, using dimensionless parameters. After finding the relation between the parameters and the bulk behavior, it was possible to calibrate a specific material by comparing the behavior from the rock with the synthetic material. In order to validate the calibration procedure, the authors carried out a uniaxial compressive test, a Brazilian splitting test, a triaxial test and two non-conventional triaxial tests on a Pennsylvania Blue Sandstone. The results of the numerical and experimental tests showed good agreement.

Behraftar et al.16 have also adopted the dimensional analysis for the calibration of DEM microparameters. The authors used the cracked chevron notched Brazilian discs (CCNBD) with different inclination angles for a sensitivity analysis. The peak force and the crack mouth opening displacement (CMOD) provided the macro-response used for calibration, after defining a specific load and inclination angle. The parameters found by the authors were then successfully validated using a different inclination angle.

The present work proposes a new method for the calibration of DEM parameters based on Genetic Programming (GP). GP has been applied in engineering for other purposes, such as estimating the uniaxial compressive strength of a rock formation based on its properties17 and predicting stress-dependent soil water retention curves for a given unsaturated soil.18 By adopting GP in this work, the aim is to reduce the number of models required for calibration and user dependency during the analysis. According to Koolivand-Salooki et al.,17 among the artificial intelligence algorithms, GP is the one that uses fewer models for training. In addition, to compare the accuracy and efficiency of the proposed method, a procedure using design of experiments was employed.

Section snippets

Calibration techniques

The discrete element method, developed by Cundall & Strack1 is based on Newton's second law and force-displacement law. The normal (Fn) and tangential (Fs) forces between two particles are evaluated according to equations (1), (2).Fn=knunFs=Fs+ΔFsΔFs=ksΔus

Here, kn is the normal stiffness, un is the relative displacement between particles, ks is the shear stiffness and Δus is the relative incremental tangential displacement. These equations are the force-displacement laws used by Cundall &

Discussions

As previously mentioned, the calibration of DEM models may follow different approaches. When the DEM discretization corresponds to the real material pore-grain structure, the DEM material model parameters can correspond to microproperties at grain contact. However, in many cases, the DEM discretization does not reflect grain structure and model calibration takes place by the comparison of the experimental and numerical bulk behaviors. In such cases, the DEM microparameters do not represent

Conclusions

This work proposes a new calibration procedure for discrete element models aiming at computational efficiency and reducing user interference. It combines genetic programming and an optimization method. The calibration was applied to a uniaxial compression test, aiming at obtaining microparameters, which would match global UCS and Young's Modulus laboratory results. With the proposed GP procedure, we obtain functions relating microparameters and macro-properties for a DEM discretization.

The

Acknowledgements

The authors gratefully acknowledge support from Shell Brazil through the “Coupled Geomechanics” project at Tecgraf Institute (PUC-Rio) and the strategic importance of the support given by ANP "Compromisso com Investimentos em Pesquisa e Desenvolvimento" through the R&D levy regulation.

References (32)

  • M. Koolivand-Salooki et al.

    Application of genetic programing technique for predicting uniaxial compressive strength using reservoir formation properties

    J Pet Sci Eng

    (2017)
  • K.J. Berger et al.

    Challenges of DEM : II . Wide particle size distributions

    Powder Technol

    (2014)
  • D.O. Potyondy et al.

    A bonded-particle model for rock

    Int J Rock Mech Min Sci

    (2004)
  • C.D. Martin et al.

    The progressive fracture of Lac du Bonnet granite

    Int J Rock Mech Min Sci

    (1994)
  • P.A. Cundall et al.

    A discrete numerical model for granular assemblies

    Geotechnique

    (1979)
  • M. Marigo et al.

    Discrete element method (DEM) for industrial applications: comments on calibration and validation for the modelling of cylindrical pellets

    KONA Powder Part J

    (2015)
  • Cited by (31)

    • Exploring hydraulic fracture behavior in glutenite formation with strong heterogeneity and variable lithology based on DEM simulation

      2023, Engineering Fracture Mechanics
      Citation Excerpt :

      However, these parameters cannot be directly obtained from physical experiments. In this paper, the trial-and-error method is used to determine the micro-mechanical parameters through adjusting the micro-mechanical parameters to ensure that the numerical simulation results are consistent with the experimental results [45]. The triaxial compression experiment, Brazil disc test and straight-notched Brazilian disk test are conducted to calibrate and obtain the most matching micro-mechanical parameters.

    • A novel genetic expression programming assisted calibration strategy for discrete element models of composite joints with ductile adhesives

      2022, Thin-Walled Structures
      Citation Excerpt :

      Genetic programming (GP) algorithm is considered as an originally evolutionary based predictive machine learning method. It has been widely used in finding the optimized regression model in a wide range of topics, e.g., geotechnics related problems [40]. Reports concerning its use in predicting the failure strength [52] and mixed mode behaviours of adhesive joints [53] can also be found.

    • A strain energy-based elastic parameter calibration method for lattice/bonded particle modelling of solid materials

      2022, Powder Technology
      Citation Excerpt :

      Instead of developing a comprehensive surrogate mapping relation, optimisation based methods are goal-oriented with the single aim to discover the optimal microscopic parameters for macroscopic targets. Prevalent optimisation methods that have been employed in parameter calibration include: (1) statistic-based optimisation, such as simulated annealing [35] and Gaussian process regression [36]; (2) population-based optimisation, e.g. Ant-colony optimisation [37], particle swarm optimisation [38,39], genetic algorithms [40,41]; (3) surrogate based optimisation [42,43] and (4) gradient-based optimisation [44]. Recently, Qu et al. [45] developed a physics-informed gradient-based optimisation framework for the calibration of elastic parameters and cohesive bond strength parameters.

    • Response surface methodology calibration for DEM study of the impact of a spherical bit on a rock

      2022, Simulation Modelling Practice and Theory
      Citation Excerpt :

      The second approach finds a mathematical model before optimization, DEM simulations to be executed are defined at the beginning, which limits the computational cost and simplifies the optimization of the mathematical model. Among the studies that use this calibration strategy are [14–16,34–36]. The response surface methodology (RSM) has been used to calibrate the microparameters of the discrete element method using the quasi-static uniaxial compression test as a calibration experiment [35–37].

    View all citing articles on Scopus
    View full text