# Probabilistic calibration of discrete element simulations using the sequential quasi-Monte Carlo filter

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## Abstract

The calibration of discrete element method (DEM) simulations is typically accomplished in a trial-and-error manner. It generally lacks objectivity and is filled with uncertainties. To deal with these issues, the sequential quasi-Monte Carlo (SQMC) filter is employed as a novel approach to calibrating the DEM models of granular materials. Within the sequential Bayesian framework, the posterior probability density functions (PDFs) of micromechanical parameters, conditioned to the experimentally obtained stress–strain behavior of granular soils, are approximated by independent model trajectories. In this work, two different contact laws are employed in DEM simulations and a granular soil specimen is modeled as polydisperse packing using various numbers of spherical grains. Knowing the evolution of physical states of the material, the proposed probabilistic calibration method can recursively update the posterior PDFs in a five-dimensional parameter space based on the Bayes’ rule. Both the identified parameters and posterior PDFs are analyzed to understand the effect of grain configuration and loading conditions. Numerical predictions using parameter sets with the highest posterior probabilities agree well with the experimental results. The advantage of the SQMC filter lies in the estimation of posterior PDFs, from which the robustness of the selected contact laws, the uncertainties of the micromechanical parameters and their interactions are all analyzed. The micro–macro correlations, which are byproducts of the probabilistic calibration, are extracted to provide insights into the multiscale mechanics of dense granular materials.

## Keywords

Discrete element method Calibration Data assimilation Sequential Monte Carlo Triaxial compression## 1 Introduction

The discrete element method (DEM) captures the collective behavior of a granular material by tracking the kinematics of the constituent grains [9]. From just a few micromechanical parameters, DEM can provide comprehensive cross-scale insights [2, 6, 15, 42] that are difficult to obtain in either state-of-the-art experiments or sophisticated continuum models. However, automated and systematic calibration of these parameters against macroscopic experimental measurements is still lacking, and often takes significant effort from DEM analysts.

Assuming homogeneous macroscopic deformation, the effective elastic properties of an ideal granular packing can be derived analytically from contact mechanics theories, micromechanical parameters and the microstructure of the packing [26, 31, 35, 46], which to some extent facilitates the calibration of DEM models with analytical *macro–micro* relations. Experimental studies were also conducted to improve the theories based on measured micromechanical [13, 37] and microstructural behaviors [19, 36] of grains and granular assemblies. Nevertheless, calibrating contact laws using microscopic measurements to reproduce the micromechanics at contacts does not necessarily recover the mechanical behavior at the macro-scale. Furthermore, in industrial applications, physically based contact laws (e.g., the classical linear (CL) [9] and Hertz–Mindlin (HM) no-slip contact laws [23]) are often coupled with fictitious non-physical parameters, e.g., rolling and twisting stiffnesses [21, 22, 52], which makes the calibration a daunting task. To tackle the issues mentioned above, an “inverse-modeling calibration” approach [1, 27] is needed to infer the micromechanical parameters from macroscopic experimental measurements.

The conventional calibration procedure for DEM simulations of granular materials employs a “one at a time” analysis of the micromechanical parameters. Many parametric studies were conducted to derive micro–macro interpolation charts for various contact laws and materials using the “one at a time” approach. For DEM simulations of rocks [7, 27, 50] in which intergranular forces depend linearly on relative displacements between adjoining grains, a linear correlation between the bulk Young’s modulus and the normal contact stiffness was identified, whereas a nonlinear correlation was found for sandy soils [38]. Given a constant value for the normal contact stiffness, both Young’s modulus and Poisson’s ratio were found to be linearly related to the tangential contact stiffness [4, 38], despite the fact that normal and tangential stiffnesses can jointly affect the deformability of a granular material. The micromechanical parameters that characterize deformability (e.g., contact stiffnesses) and yield (e.g., intergranular friction angle), however, are generally believed to be uncoupled, and thus can be calibrated separately. This has led to many parametric studies into the macroscopic internal friction angle which was proved to have a nonlinear dependency on the intergranular friction angle [7, 45, 49], irrespective of the contact stiffnesses.

Because several micromechanical parameters can collectively determine the bulk behavior of a granular material, a parameter set identified for a DEM model with a known grain configuration can be regarded as one of the numerous solutions to the parameter identification problem. Among a handful of systematic approaches to calibrating DEM models, the design of experiments (DOE) methods are efficient in searching for possible solutions in the multi-dimensional parameter space, with a manageable number of DEM simulations [18, 24, 39, 54]. Hanley et al. [18] applied DOE to calibrate DEM models of crushable agglomerates. The interaction between the key parameters was considered by the orthogonal arrays designed using the Taguchi methods. Yoon [54] developed a two-step optimization process in which a DOE method (Plackett–Burman design) was first applied to select the parameters that have the largest impacts on macroscopic material properties. The statistical micro–macro correlations were then determined by running additional DEM simulations. Despite a good agreement on the instantaneous macroscopic material properties (e.g., compressive strength), the transient response in the simulation (e.g., stress–strain curves) did not agree well with the experimental data. The efficiency of DOE methods largely depends on the prior knowledge of the interaction between parameters, which is needed to prepare DOE samples, but is also generally limited due to the diversity of granular materials.

The aforementioned calibration approaches aim to calibrate micromechanical parameters against the macroscopic material properties (e.g., Young’s modulus, peak- and critical-state friction angles), rather than the transient behavior of the bulk material. This is very likely to hinder the predictive capacity of DEM models. Local rather than global solutions to the parameter identification problem might be discovered and adopted in the models. This leads to deviations from the transient experimental response in the simulations. In addition, the dependency of granular materials on stress and fabric history cannot be accounted for in these approaches. To capture the elastoplasticity of granular materials, the experimental data measured throughout the entire loading history needs to be fully considered within the parameter identification procedure [41].

The sequential data assimilation techniques [12, 34] can be applied to solve the “inverse-modeling calibration” problems and to overcome the above-mentioned difficulties. For the current inverse problem, the sequential quasi-Monte Carlo (SQMC) filter^{1} and sequential importance sampling (SIS) are implemented, which can jointly account for the effects of loading history on the elastoplastic behavior of granular materials [43]. The SQMC filter applies the recursive formula of sequential Bayesian estimation. Experimental data obtained step by step during a loading process are assimilated into DEM models to approximate the evolution of posterior PDFs in the corresponding parameter space. This “inverse-modeling calibration” approach is expedient, because the synergy of the SQMC and SIS algorithms is well-justified for nonlinear and non-Gaussian geomechanical problems, as demonstrated in the applications of these methods to inverse finite element analyses [33, 43]. A few innovative calibration methods for DEM or molecular dynamics simulations were recently proposed using Bayesian approaches [3, 40, 53]. Hadjidoukas et al. [16, 17] employed the Transitional Markov Chain Monte Carlo algorithm to find optimal parameter values in DEM simulations of silo discharge. By assuming uniform prior distributions for all micromechanical parameters, the parameter uncertainties were quantified and a Bayesian model selection framework was provided for two-dimensional (2D) monodisperse granular systems. The SQMC filter implemented in this study requires no assumptions about the prior distributions of the model states. The Bayesian calibration is conducted for three-dimensional (3D) DEM simulations of polydisperse granular materials under triaxial compression, which in turn reveals the posterior PDFs that help in assessing the robustness of the contact laws. To the authors’ knowledge, this work is the first attempt to develop a sequential data assimilation procedure for calibrating DEM models over the transient behavior of bulk granular materials. For the current implementation, both SQMC and SIS algorithms are implemented in the open-source DEM package YADE [44].

The remainder of this paper is organized as follows. Section 2 outlines the contact laws and grain configurations which are the key ingredients of the DEM models. The fundamentals of SQMC and SIS, and their implementation, are detailed in Sect. 3, followed by the descriptions of the posterior PDFs resulting from the transient experimental data in Sects. 4 and 5. Section 6 discusses the robustness of the proposed method. Conclusions are drawn in Sect. 7.

## 2 DEM simulations of granular materials

The open-source DEM package YADE [44] is applied to perform 3D DEM simulations of dense granular materials under drained triaxial compression. The stochastic responses obtained from the simulations using quasi-randomly sampled parameter values, i.e., *samples*, are processed through the SQMC and SIS algorithms to approximate the posterior PDFs, conditioned to the experimental data. YADE models granular materials as packings of solid grains with simplified geometries and vanishingly small intergranular overlaps. It tracks the trajectory of each grain within the explicit time integration scheme, based upon the net force resulting from the intergranular forces. In the present work, the robustness of the SQMC filter for calibrating DEM models is examined by modeling a representative volume of cohesionless granular soil using dense packings of polydisperse spherical grains. The simple HM and CL contact laws are selected to describe the contact force–displacement relationships in the normal and tangential directions. To account for the effects of grain shape and roughness, rolling and twisting moment transfer is allowed at the contact points of adjoining spheres. Both intergranular tangential forces and rolling/twisting moments are assumed to be bounded by Coulomb type yield criteria. Each DEM simulation of the representative volume of the granular soil under drained triaxial compression is performed in a strain-controlled quasistatic manner. For each calculation cycle of an incremental loading, a global damping ratio of 0.2 is adopted to mimic the presence of a background fluid and ensure numerical stability. In the subsequent cycles, which are run to dissipate the total kinetic energy before the next incremental loading, the global damping ratio is raised to 0.9 until the quasistatic macroscopic variables (see Sect. 3.1) can be extracted.

### 2.1 Micromechanical contact laws and parameters

Expressions for intergranular normal, tangential and rolling stiffnesses and corresponding plastic limit conditions

Contact law | Contact stiffnesses | Plastic limit condition |
---|---|---|

CL | \( k_n = 2E_c r^* \) | |

\( k_s = 2E_c \nu _c r^* \) | \( \Vert \varvec{\mathrm {F}}_s\Vert =\tan {\phi _{\mu }} \Vert \varvec{\mathrm {F}}_n\Vert \) | |

\( k_m=\beta _m r_1 r_2 k_s\) | \( \Vert \varvec{\mathrm {M}}\Vert =\eta _m \Vert \varvec{\mathrm {F}}_n\Vert \min (r_1, r_2) \) | |

HM | \( k_n = \frac{2E_c}{3(1-\nu _c^2)}\sqrt{r^*\Vert \varvec{\mathrm {u}}_n\Vert } \) | |

\( k_s = \frac{2E_c}{(1+\nu _c)(2-\nu _c)}\sqrt{r^*\Vert \varvec{\mathrm {u}}_n\Vert } \) | \( \Vert \varvec{\mathrm {F}}_s\Vert =\tan {\phi _{\mu }} \Vert \varvec{\mathrm {F}}_n\Vert \) | |

\( k_m=\beta _m \) | \( \Vert \varvec{\mathrm {M}}\Vert =\eta _m \Vert \varvec{\mathrm {F}}_n\Vert (r_1+r_2)/2\) |

Because the grain shapes are simplified as rigid spheres, rolling resistance between adjoining grains is needed to mimic the effects of grain shape and surface roughness. One additional phenomenological parameter, namely the rolling stiffness \( k_m \), is thus introduced, which linearly relates the rolling/twisting moment in the contact area with the relative rotation/torsion between two contacting grains. Note that rolling and twisting are unified in one moment–rotation equation for simplicity. In YADE [32, 44], the rolling stiffness defined in the implementations of the HM and CL force–displacement models varies slightly (see Table 1). A constant rolling stiffness \( k_m = \beta _m \) is implemented in the former, whereas the latter uses \( \beta _m \) as a dimensionless coefficient to calculate \( k_m \) from the radii and contact tangential stiffness of the two contacting grains.

Friction criteria are imposed on intergranular tangential forces and rolling/twisting moments to account for sliding between contacting grains and, on the macro-sale, plastic deformation in the granular soil. Both failure criteria adopt the Mohr–Coulomb type formulation. The maximum tangential force \( \Vert \varvec{\mathrm {F}}_s\Vert \) is calculated as the intergranular friction coefficient \( \tan (\phi _{\mu }) \) multiplied by the magnitude of the normal force \( \Vert \varvec{\mathrm {F}}_n\Vert \), whereas the rolling/twisting moment is constrained by the rolling friction coefficient \( \eta _m , \Vert \varvec{\mathrm {F}}_n\Vert \) and the characteristic length related to the radii (see Table 1). Despite being phenomenological, rolling stiffness and friction are generally needed to obtain internal friction angles close to experimentally measured values for dense granular soils, without making an extra computational effort to capture the kinematics related to irregular grain shapes and surface roughness. Note that although the contact laws employed in the current work are implemented differently, the micromechanical parameters to be identified in the Bayesian calibration are the same, i.e., the Young’s modulus \( E_c \) and Poisson’s ratio \( \nu _c \) of contacting grains, the rolling stiffness and friction coefficient \(\beta _m \) and \( \eta _m \), and the intergranular friction angle \( \phi _{\mu } \).

### 2.2 Scale and resolution of DEM granular packings

In the light of the above considerations, grain configurations consisting of various numbers of spherical grains are created and investigated. For each grain configuration, a cloud of spherical grains with the same density \( \rho _g = 2650\,\mathrm {kg/m}^3 \) is first generated in a cuboidal periodic cell. The diameters of the spherical grains are sampled from a scaled grain size distribution of Toyoura sand. Initial isotropic compression is then applied to consolidate each cloud into a dense cuboidal packing (50 mm \(\times \) 50 mm \(\times \) 100 mm) with an initial void ratio of \(e_0=0.68\), the same as those in the experiments [47]. During the consolidation stage, a very high Young’s modulus is adopted to create grain configurations with negligible overlaps between grains. While maintaining a low confining pressure (\( p=100 \) Pa) by using a servo-controlled periodic boundary condition, the *initial* intergranular friction is tuned down slightly whenever quasistatic states are reached. After a couple of cycles of reducing the friction and minimizing unbalanced forces, various “stress-free” dense packings, which are “identical” in the sense of the initial void ratio, are generated. Among all the randomly generated dense packings three (\( N_g=1000 \), 2000 and 5000) are selected as illustrated in Fig. 1, because of their relatively smooth and spherical contact orientation diagrams after the initial isotropic compaction. Note that \( N_g=1000 \) is the minimum number of spherical grains needed to create “stress-free” dense packings, whose contact orientation distributions are uniform and statistically stable.

## 3 Data assimilation for granular materials using the SQMC filter

### 3.1 State space model and state estimation

*t*, whereas the observation vector \(\varvec{\mathrm {y_t}}\) is directly measured in the drained triaxial compression experiments [47]; Open image in new window and Open image in new window are the system error and the observation error respectively, whose PDFs follow normal distributions with zero means. \(\mathbb {F}\) denotes the operator that represents the nonlinear dynamic model (i.e., the DEM model). The current state of \(\mathbb {F}\) depends on all preceding states of the dynamic model, which evolves with physical constraints like externally applied loads. The nonlinear observation operator \(\mathbb {H}\) is reduced to an identity matrix of size three in the current problem.

### 3.2 SQMC filter

*i*) indicates the

*i*th sample \( \varvec{\mathrm {x}}^{(i)}_{\varvec{\mathrm {t-1|t-1}}} \) and its associated weight \( w_{t-1}^{(i)} \). It should be noted that all the weights are no less than zero and the summation of them must be one, i.e., \(0\le w_{t-1}^{(i)}\le 1\) and \(\sum _{i=1}^{N_p} w_{t-1}^{(i)} = 1\). With the help of the ensemble sampled using the quasi-Monte Carlo method, the recursive formulas for approximating the one-step-ahead forecast distribution \(p(\varvec{\mathrm {x_t}}|\varvec{\mathrm {y_{1:t-1}}})\) and the filtered distribution \(p(\varvec{\mathrm {x_t}}|\varvec{\mathrm {y_{1:t}}})\) are simplified as follows:

*i*th sample reads:

*m*is the dimension of the state vector, \(\varvec{\mathrm {M_t}}\) is a predetermined covariance matrix of the observation error, and \(\varvec{\mathrm {H_t}}\) is the observation operator \(\mathbb {H}\) in matrix form. From the previous weight \( w_{t-1}^{(i)} \) and the current regulated weight \( \tilde{w}_t^{(i)} \) on each sample (see Eq. 12), the new weight \( {w_t}^{(i)} \) at the data assimilation step

*t*can be updated as:

### 3.3 Sampling method and SQMC filtering procedure

- 1.
Initialization:

Generate an ensemble of realizations \(\{\varvec{\mathrm {x_0}}^{(1)}, \varvec{\mathrm {x_0}}^{(2)}, \ldots , \varvec{\mathrm {x_0}}^{(N_p)}\}\) by initializing dynamic models (DEM simulations) using parameter sets sampled from a low-discrepancy sequence in \( \varvec{\mathrm {R}}^n \).

*n*is the number of the parameters to be identified and is equal to five in the present work. - 2.
Prediction:

Update the state of each realization \( \varvec{\mathrm {x_t}}^{(i)} \) from the corresponding DEM simulation run at the data assimilation step

*t*. - 3.
Filtering:

Given the experimental data \(\varvec{\mathrm {y_t}}\), calculate the weights on the ensemble \(\{w_t^{(1)}, w_t^{(2)},\ldots , w_t^{(N_p)}\}\) that represents the “fitness” of the dynamic models to the physical system using Eqs. (12) and (13).

- 4.
Weight updating:

Approximate filtered distribution \( p(\varvec{\mathrm {x_t}}|\varvec{\mathrm {\varvec{\mathrm {y_{1:t}}}}}) \) with the updated ensemble and the associated weights. Return to Step 2 with \( p(\varvec{\mathrm {x_t}}|\varvec{\mathrm {\varvec{\mathrm {y_{1:t}}}}}) , \{\varvec{\mathrm {x_t}}^{(1)}, \varvec{\mathrm {x_t}}^{(2)},\ldots , \varvec{\mathrm {x_t}}^{(N_p)}\}\) and \(\{w_t^{(1)}, w_t^{(2)},\ldots , w_t^{(N_p)}\}\) for prediction and filtering at the next data assimilation step.

## 4 Numerical and experimental triaxial compression tests: the dynamic model and the physical system

Means and standard deviations assumed for the micromechanical parameters

\(\log E_c \) (Pa) | \( \nu _c \) | \( \beta _m \) (Nm/rad) | \( \eta _m \) | \( \phi _{\mu } \) (\(^\circ \)) | |
---|---|---|---|---|---|

Mean | 9 | 0.25 | 0.5 | 0.5 | 30 |

SD | 2 | 0.25 | 0.5 | 0.5 | 30 |

Number of grains and confining pressures applied in the numerical simulations

\( N_g \) | \( \sigma _c \) (MPa) | 0.2 | 0.5, 1.0, and 2.0 | ||
---|---|---|---|---|---|

CL | HM | CL | HM | ||

1000 | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) | |

2000 and 5000 | \(\checkmark \) | \(\checkmark \) | – | – |

## 5 Posterior probabilities of parameters in two DEM models

### 5.1 Effect of grain configurations on the posterior probability distributions

#### 5.1.1 Identified micromechanical parameters

To keep track on how the expectations of the state variables converge to their true values, the quasi-randomly generated parameters are weighted by the updated posterior probabilities at each data assimilation step (axial strain increment \( \, d \varepsilon _a = 0.1\% \)). The evolutions of the weighted averages, referred as *identified parameters*, are shown in Fig. 2. The posterior probabilities obtained from the three DEM packings (\( N_g =\) 1000, 2000 and 5000) and the two contact laws are compared in Figs. 2 and 4, to investigate the effect of the DEM models on the calibration process. The DEM packings undergo the exact triaxial loading history as the experimental specimen at a confining pressure of \( \sigma _c = 0.2\) MPa.

#### 5.1.2 Posterior PDFs of micromechanical parameters

### 5.2 Effect of loading conditions on the posterior probability distributions

#### 5.2.1 Identified micromechanical parameters

It is known that the elastoplastic behavior of granular soils strongly depends on the loading history experienced. With DEM, the plastic deformation of granular soils can be easily captured via irreversible change of the microstructure subjected to external loads. The questions are how the loading history can be taken into account during the calibration process such that the correct elastoplastic behavior is reproduced at the macro-scale, and how the identified micromechanical parameters of both contact laws would be affected by a range of loading conditions. Figure 5 shows the evolution of the micromechanical parameters identified for the DEM simulations of Toyoura sand (\( N_g \) = 1000), under triaxial compression at various confining pressures \( \sigma _c = 0.2, 0.5, 1.0\) and 2.0 MPa. From both Fig. 5 CL(a) and HM(a), it can be observed that the identified values of \( E_c \) stabilize after approximately 30 data assimilation steps, which is in line with the identified parameters in Fig. 2 CL(a) and HM(a). At different \( \sigma _c \), the true values identified for the DEM simulations using the HM contact law stay relatively closer than those using the CL contact law, as can be seen in Fig. 6. Analogous to Fig. 2 CL(b)–(d) and HM(b)–(d), the evolutions of the identified values for \( \nu _c , \beta _m \) and \( \eta _m \) appear to fluctuate between two or multiple plateaus during the calibration. The fluctuations indicate frequent exchange of weights between certain samples that possess comparable posterior probabilities at various data assimilation steps. The evolutions of the identified \( \phi _\mu \) in Fig. 5 CL(e) and HM(e) show a clear dependency on the applied confining pressures, i.e., the curves that represent the evolutions of identified \( \phi _\mu \) obtained with a smaller \( \sigma _c \) lying above those obtained with a larger \( \sigma _c \). This dependency can be attributed to the crushability of the granular soil that develops with increasing confining pressure. Despite the fluctuations and the dependency on the loading conditions, the evolutions of all identified parameters becomes stationary after sufficient data assimilation steps.

#### 5.2.2 Posterior PDFs of micromechanical parameters

## 6 Discussion

### 6.1 Numerical predictions versus experimental data

The DEM simulation results reproduced with the most probable parameters identified by the SQMC and SIS algorithms are plotted with the experimental data in Figs. 7 and 8. Good agreement between the numerical predictions and experimental data is achieved as shown in Figs. 7 and 8, regardless of the contact laws and the numbers of spherical grains in the DEM simulations. The combinations of parameter values identified to be the most probable in reproducing the experimental data at their respective confining pressures are listed in Tables 4 and 5. As can be seen in the tables, the intergranular friction angle which gives the best agreement in Fig. 8 decreases with an increase of the confining pressure for both contact laws. This dependency can be attributed to the crushing behavior of Toyoura sand in the experiments, which neither of the two contact laws can mimic in DEM simulations using smooth spherical grains. It should be noted that although the SQMC filter can give the most optimal combinations of parameters as listed in Tables 4 and 5, it is fundamentally different from any optimization techniques, because the probabilistic calibration is conducted by evaluating the parameter and model uncertainties through the posterior PDFs within the Bayesian framework. Each optimal solution presented here is simply one of many possible solutions to the parameter identification problem. The following section aims to interpret the interactions between micromechanical parameters from the posterior PDFs in the five-dimensional parameter space. The statistical micro–macro correlations are also established following the same concept mentioned above.

### 6.2 Correlations between micro- and macro-mechanical parameters

Most probable sets of micromechanical parameters identified for the CL contact law

\( \sigma _c\) (MPa) | \( E_c \) (GPa) | \( \nu _c \) | \( \beta _m \) (Nm/rad) | \( \eta _m \) | \( \phi _{\mu } \) (\(^\circ \)) |
---|---|---|---|---|---|

0.2 | 0.545 | 0.080 | 0.564 | 0.194 | 30.203 |

0.5 | 0.545 | 0.080 | 0.564 | 0.194 | 30.203 |

1.0 | 1.252 | 0.439 | 0.076 | 0.726 | 22.043 |

2.0 | 1.046 | 0.160 | 0.589 | 0.331 | 18.076 |

Most probable sets of micromechanical parameters identified for the HM contact law

\( \sigma _c \) (MPa) | \( E_c \) (GPa) | \( \nu _c \) | \( \beta _m \) (Nm/rad) | \( \eta _m \) | \( \phi _{\mu } \) (\(^\circ \)) |
---|---|---|---|---|---|

0.2 | 4.123 | 0.064 | 0.148 | 0.202 | 30.387 |

0.5 | 5.882 | 0.304 | 0.344 | 0.567 | 24.027 |

1.0 | 8.324 | 0.049 | 0.363 | 0.480 | 24.951 |

2.0 | 6.793 | 0.482 | 0.998 | 0.171 | 20.06 |

Figure 9a shows that at a very early stage the posterior distributions and the peaks associated with \( E_c \) and \( \phi _\mu \) of both DEM models are already identifiable. While these two parameters seem to be correlated, the other parameters \( \nu _c , \beta _m \) and \( \eta _m \) show less pronounced or insignificant correlations with \( E_c \) and \( \phi _\mu \), irrespective of the contact laws. The 2D projections of the posterior PDFs associated with \(E_c\) are almost aligned into straight bands perpendicular to the \( E_c\) axis. The physical system, which evolves from the initial elastic to maximum volumetric-contraction states, causes the peaks over \( \phi _\mu \) to shift towards larger values for both contact laws, as shown in Fig. 9b. However, the *true* values of \( E_c \) are almost the same as those in the initial calibration stage, as shown in Fig. 9a. In Fig. 9a, the distributions in both 2D \( E_c \)–\( \phi _\mu \) projections become increasingly parallel to the \( E_c \) axis, forming into long tails as \( E_c \) increases. These long tails gradually disappear, as the physical system evolves from the purely elastic to maximum volumetric-contraction states, as can be seen for both contact laws in Fig. 9b. This indicates that if, by trial and error, the identified \( E_c \) happens to be located at these local optima, which could happen because \( E_c \) is commonly calibrated against the bulk Young’s modulus, the resulting numerical simulations will not yield good agreement with the experimental results, even though \( \phi _\mu \) could be well determined by the bulk shear strength. While the physical system is approaching the peak stress ratio state, the distributions keep shrinking towards the *true* values, with no evident shift in the parameter spaces. It should be noted that the calibrations at least for \( E_c \) and \( \phi _\mu \) of both contact laws have already finished at the maximum volumetric-contraction state. The exchange of importance weights between certain samples can be observed by comparing the posterior PDFs associated with \( \beta _m \) and \( \eta _m \) in the marginals and 2D projections of Fig. 9b, c.

*true*values for the rolling parameters \( \beta _m \) and \( \eta _m \) can be identified by tuning the evolution of post-peak stress ratio over axial or deviatoric strain. It is worth noting that this technique is often adopted in the literature, because rolling stiffness and rolling friction can significantly affect the bulk shear strength, but have a negligible influence on the initial elastic behavior and dilatancy of a dense granular material [38].

A very important byproduct of the proposed calibration procedure is the large number of simulations that are needed to capture the complete picture of the posterior PDFs over the explored parameter space. From the huge amount of simulation data, although many of them largely deviate from the experimental results, it is possible to extract meaningful universal trends between the micro- and macro-mechanical parameters. From the numerically predicted stress–strain curves, the bulk Young’s modulus and Poisson’s ratio, \( E_{bulk} \) and \( \nu _{bulk} \) are determined as the secant values at the deviatoric strain \( \varepsilon _d = 1\)%. The internal friction angle \( \varphi _{bulk} \) is simply calculated from the peak stress ratio. Figure 10a, b respectively show the samples in the parameter subspaces constructed with the micromechanical parameters of the CL and HM contact laws and the corresponding bulk properties calculated directly from the numerical stress–strain curves. The samples in Fig. 10 are colored by the probability density functions estimated with Gaussian kernels.

From the statistics in Fig. 10a, b clear micro–macro correlations between the micromechanical parameters and the bulk properties of granular materials can be identified. The correlations between \( E_{bulk} \) and \( E_c \) are almost linear and piecewise linear for the CL and HM contact laws respectively in log–log scales. Although the correlation between \( E_{bulk} \) and \( \phi _\mu \) is not as significant as that between \( E_{bulk} \) and \( E_c \), a clear dependency of the bulk Young’s modulus on the intergranular friction can be seen from the samples with high probability densities. As shown in Fig. 10a, the bulk Poisson’s ratio \( \nu _{bulk} \) predicted by the CL contact law might have unrealistic values above the physical limit. The statistics show that \( \nu _{bulk} \) is more likely to be unrealistic with the increase of \( E_c \) and \( \phi _\mu \). All the bulk Poisson’s ratios predicted by the HM contact law, on the other hand, fall within the physical range between 0 and 0.5. From the subplots that characterize the dependency of \( \nu _{bulk} \) on the micromechanical parameters in Fig. 10a, b, it can be observed that \( \nu _{bulk} \) is nonlinearly correlated to both \( E_c \) and \( \phi _\mu \) in the case of the HM contact law, whereas this trend is barely noticeable for the CL contact law. The \( \varphi _{bulk}\)–\(\phi _\mu \) correlations appear to be nonlinear and are very similar for both the CL and HM contact laws, except that the samples are more scattered in the former than in the latter. In both \( \varphi _{bulk} \)–\( \phi _\mu \) subplots, there exist upper bounds on the possible values of \(\varphi _{bulk} \) depending on the selection of \( \phi _\mu \). In the case of the HM contact law, a lower bound is also present which further narrows down the choices for the initial guess of \( \phi _\mu \). Nevertheless, the samples become more scattered and the two bounds diverge increasingly as \( \phi _\mu \) increases, meaning that the calibration against large \( \varphi _{bulk} \) is generally more difficult than that against small \( \varphi _{bulk} \). Because of the non-physical nature of the rolling parameters, no significant correlations can be found from the evenly distributed samples in the relevant 2D micro–macro parameter spaces. This means that it is impractical to establish meta-models relevant to the rolling parameters, which may undermine the efficiency of an optimization process. The SQMC filter can resolve this difficulty without initially knowing the interactions between the parameters being identified, which usually comes at the price of increased computational cost. Note that the SQMC filter or other sequential Monte Carlo methods can be applied iteratively using the knowledge of the interactions obtained in the filtering step. In such a way, the computational cost could be significantly reduced. It can also be observed in Fig. 10a, b that the data gathered from the DEM simulations using the HM contact law are less scattered that those using the CL contact law, which suggests that the HM contact law is more robust than the CL contact law within the parameter space currently explored.

## 7 Conclusions

A novel probabilistic calibration approach is proposed for the DEM simulations of granular soils. The SQMC and SIS algorithms are implemented within the open-source DEM package YADE. The micromechanical parameters of the contact laws are successfully calibrated against the stress–strain behavior of Toyoura sand in drained triaxial compression conditions at various confining pressures. Compared with general optimization methods, the synergy of the SQMC and SIS algorithms can estimate the evolution of posterior probability distribution over a loading history in a high dimensional parameter space. From the distribution of the estimated probability density functions, one can easily evaluate the robustness and uncertainties of DEM models and interactions between the micromechanical parameters.

The probabilistic calibrations of six DEM models (combinations of two contact laws and three grain configurations) are conducted considering confining pressures ranging from 0.2 to 2.0 MPa. Despite some fluctuations in the evolution of the identified parameters during the calibration, stabilized posterior probability distributions are obtained for both contact laws. The distributions of \( E_c \) and \( \phi _\mu \) evolve from being uniform to Gaussian-like, whereas the distributions of the other parameters can hardly be approximated by mixtures of Gaussians. The effect of grain configuration on the posterior PDFs obtained using either the CL or HM contact law under the same loading condition appears to be negligible. It is proved that the macroscopic behavior of a granular material can be recovered from a “prototype” DEM packing with a minimal number of grains, as long as the same bulk properties, such as initial void ratios and stress states, are ensured for the DEM packing. For a predetermined grain configuration (e.g., \( N_g \) = 1000), the posterior PDFs obtained using the HM contact law under various loading conditions generally collapse near the consistent peaks in the parameter space. These findings indicate that although the scaling rule holds for both contact laws (perfectly for the CL contact law), the HM contact law is more robust in predicting the triaxial behavior of a dense granular material at a wide range of confining pressures. The scaling rule is of great importance, because in most laboratory- and industrial-scale applications the number of physical grains can easily exceed several millions. One reasonable approach to DEM modeling at these scales is to apply the scaling rule to reduce the computational effort. However, the up-scaling has to be carried out in such a way that the model resolution is still sufficient with negligible effects on the predicted bulk behaviors.

The SQMC and SIS algorithms are implemented as a separate calibration toolbox independent of the DEM package. It is straightforward to apply the proposed calibration approach to other DEM codes. Current work involves probabilistic calibration of dense granular materials under quasistatic loading conditions. In future research, it will be worth investigating the applicability of the SQMC and SIS to DEM calibrations against more complex behavior of granular materials, such as rockfalls [11], avalanches [51] and silo discharges [8]. Ongoing work on probabilistic calibration of DEM simulations is directed towards the development of an iterative SQMC filter, which is capable of resampling micromechanical parameters from prior probability distributions updated by the previously conducted probabilistic calibrations. The iterative version of the SQMC filter would allow the probabilistic calibration to learn from its preceding experience and explore a parameter space with different resolutions. By putting more samples in highly probable parameter subspaces, the computational cost can be greatly reduced.

## Footnotes

- 1.
The sequential Monte Carlo approximates posterior probability distribution functions (PDFs) by discretizing the state space with independent samples, and thus is also called “particle” filter. To avoid confusion between

*particles*referred to as Monte Carlo samples, physical grains or discrete elements in simulations, the term “particle” filter is not adopted in the context of Bayesian parameter estimation for DEM models.

## Notes

### Acknowledgements

The authors are grateful to Dr. Vanessa Magnanimo and Professor Stefan Luding for valuable discussions and suggestions.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

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