An iterative Bayesian filtering framework for fast and automated calibration of DEM models. (English) Zbl 1441.74233
Summary: The nonlinear, history-dependent macroscopic behavior of a granular material is rooted in the micromechanics between constituent particles and irreversible, plastic deformations reflected by changes in the microstructure. The discrete element method (DEM) can predict the evolution of the microstructure resulting from interparticle interactions. However, micromechanical parameters at contact and particle levels are generally unknown because of the diversity of granular materials with respect to their surfaces, shapes, disorder and anisotropy.
The proposed iterative Bayesian filter consists in recursively updating the posterior distribution of model parameters and iterating the process with new samples drawn from a proposal density in highly probable parameter spaces. Over iterations the proposal density is progressively localized near the posterior modes, which allows automated zooming towards optimal solutions. The Dirichlet process Gaussian mixture is trained with sparse and high dimensional data from the previous iteration to update the proposal density.
As an example, the probability distribution of the micromechanical parameters is estimated, conditioning on the experimentally measured stress-strain behavior of a granular assembly. Four micromechanical parameters, i.e., contact-level Young’s modulus, interparticle friction, rolling stiffness and rolling friction, are chosen as strongly relevant for the macroscopic behavior. The a priori particle configuration is obtained from 3D X-ray computed tomography images. The a posteriori expectation of each micromechanical parameter converges within four iterations, leading to an excellent agreement between the experimental data and the numerical predictions. As new result, the proposed framework provides a deeper understanding of the correlations among micromechanical parameters and between the micro- and macro-parameters/quantities of interest, including their uncertainties. Therefore, the iterative Bayesian filtering framework has a great potential for quantifying parameter uncertainties and their propagation across various scales in granular materials.
The proposed iterative Bayesian filter consists in recursively updating the posterior distribution of model parameters and iterating the process with new samples drawn from a proposal density in highly probable parameter spaces. Over iterations the proposal density is progressively localized near the posterior modes, which allows automated zooming towards optimal solutions. The Dirichlet process Gaussian mixture is trained with sparse and high dimensional data from the previous iteration to update the proposal density.
As an example, the probability distribution of the micromechanical parameters is estimated, conditioning on the experimentally measured stress-strain behavior of a granular assembly. Four micromechanical parameters, i.e., contact-level Young’s modulus, interparticle friction, rolling stiffness and rolling friction, are chosen as strongly relevant for the macroscopic behavior. The a priori particle configuration is obtained from 3D X-ray computed tomography images. The a posteriori expectation of each micromechanical parameter converges within four iterations, leading to an excellent agreement between the experimental data and the numerical predictions. As new result, the proposed framework provides a deeper understanding of the correlations among micromechanical parameters and between the micro- and macro-parameters/quantities of interest, including their uncertainties. Therefore, the iterative Bayesian filtering framework has a great potential for quantifying parameter uncertainties and their propagation across various scales in granular materials.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 62F15 | Bayesian inference |
| 74E20 | Granularity |
Keywords:
iterative parameter estimation; sequential Monte Carlo; Dirichlet process mixture model; discrete element method; X-ray tomography; cyclic oedometric compressionReferences:
| [1] | Cundall, P. A.; Strack, O. D.L., A discrete numerical model for granular assemblies, Géotechnique, 29, 1, 47-65 (1979) |
| [2] | Li, X.; Wan, K., A bridging scale method for granular materials with discrete particle assembly - Cosserat continuum modeling, Comput. Geotech., 38, 8, 1052-1068 (2011) |
| [3] | Wellmann, C.; Wriggers, P., A two-scale model of granular materials, Comput. Methods Appl. Mech. Engrg., 205-208, 46-58 (2012) · Zbl 1239.74083 |
| [4] | Guo, N.; Zhao, J., 3D multiscale modeling of strain localization in granular media, Comput. Geotech., 80, 360-372 (2016) |
| [5] | Effeindzourou, A.; Thoeni, K.; Giacomini, A.; Wendeler, C., Efficient discrete modelling of composite structures for rockfall protection, Comput. Geotech., 87, 99-114 (2017) |
| [6] | Yuan, C.; Chareyre, B., A pore-scale method for hydromechanical coupling in deformable granular media, Comput. Methods Appl. Mech. Engrg., 318, 1066-1079 (2017) · Zbl 1439.74090 |
| [7] | Cheng, H.; Yamamoto, H.; Guo, N.; Huang, H., A simple multiscale model for granular soils with geosynthetic inclusion, (Proc. 7th Int. Conf. Discret. Elem. Methods (2017), Springer Singapore: Springer Singapore Singapore), 445-453 · Zbl 1381.74047 |
| [8] | Fuchs, R.; Weinhart, T.; Meyer, J.; Zhuang, H.; Staedler, T.; Jiang, X.; Luding, S., Rolling, sliding and torsion of micron-sized silica particles: Experimental, numerical and theoretical analysis, Granul. Matter, 16, 3, 281-297 (2014) |
| [9] | Gilson, L.; Kozhar, S.; Antonyuk, S.; Bröckel, U.; Heinrich, S., Contact models based on experimental characterization of irregular shaped, micrometer-sized particles, Granul. Matter, 16, 3, 313-326 (2014) |
| [10] | Iwashita, K.; Oda, M., Rolling resistance at contacts in simulation of shear band development by DEM, J. Eng. Mech., 124, 3, 285-292 (1998) |
| [11] | Jiang, M.; Yu, H.-S.; Harris, D., A novel discrete model for granular material incorporating rolling resistance, Comput. Geotech., 32, 5, 340-357 (2005) |
| [12] | Wensrich, C.; Katterfeld, A., Rolling friction as a technique for modelling particle shape in DEM, Powder Technol., 217, 409-417 (2012) |
| [13] | Kulatilake, P.; Malama, B.; Wang, J., Physical and particle flow modeling of jointed rock block behavior under uniaxial loading, Int. J. Rock Mech. Min. Sci., 38, 5, 641-657 (2001) |
| [14] | Coetzee, C. J., Review: Calibration of the discrete element method, Powder Technol., 310, 104-142 (2017) |
| [15] | Benvenuti, L.; Kloss, C.; Pirker, S., Identification of DEM simulation parameters by artificial neural networks and bulk experiments, Powder Technol., 291, 456-465 (2016) |
| [16] | Yoon, J., Application of experimental design and optimization to PFC model calibration in uniaxial compression simulation, Int. J. Rock Mech. Min. Sci., 44, 6, 871-889 (2007) |
| [17] | Johnstone, M. W., Calibration of DEM Models for Granular Materials using Bulk Physical Tests (2010), The University of Edinburgh, (Ph.D. thesis) |
| [18] | Hanley, K. J.; O’Sullivan, C.; Oliveira, J. C.; Cronin, K.; Byrne, E. P., Application of Taguchi methods to DEM calibration of bonded agglomerates, Powder Technol., 210, 3, 230-240 (2011) |
| [19] | Rackl, M.; Hanley, K. J., A methodical calibration procedure for discrete element models, Powder Technol., 307, 73-83 (2017) |
| [20] | Wilkinson, S. K.; Turnbull, S. A.; Yan, Z.; Stitt, E. H.; Marigo, M., A parametric evaluation of powder flowability using a freeman rheometer through statistical and sensitivity analysis: A discrete element method (DEM) study, Comput. Chem. Eng., 97, 161-174 (2017) |
| [21] | Lewis, R.; Zheng, Y., Coarse optimization for complex systems: An application of orthogonal experiments, Comput. Methods Appl. Mech. Engrg., 94, 1, 63-92 (1992) |
| [22] | Beyer, H.-G.; Sendhoff, B., Robust optimization – a comprehensive survey, Comput. Methods Appl. Mech. Engrg., 196, 33, 3190-3218 (2007) · Zbl 1173.74376 |
| [23] | Oden, J. T.; Moser, R.; Ghattas, O., Computer predictions with quantified uncertainty, Part I, SIAM News, 43, 9 (2010) |
| [24] | Oden, J. T.; Moser, R.; Ghattas, O., Computer predictions with quantified uncertainty, Part II, SIAM News, 43, 10 (2010) |
| [25] | Hadjidoukas, P.; Angelikopoulos, P.; Rossinelli, D.; Alexeev, D.; Papadimitriou, C.; Koumoutsakos, P., Bayesian Uncertainty quantification and propagation for discrete element simulations of granular materials, Comput. Methods Appl. Mech. Engrg., 282, 218-238 (2014) · Zbl 1423.74956 |
| [26] | Hadjidoukas, P.; Angelikopoulos, P.; Papadimitriou, C.; Koumoutsakos, P., \( \Pi 4\) U: A high performance computing framework for Bayesian uncertainty quantification of complex models, J. Comput. Phys., 284, 1-21 (2015) · Zbl 1352.65009 |
| [27] | Angelikopoulos, P.; Papadimitriou, C.; Koumoutsakos, P., Bayesian Uncertainty quantification and propagation in molecular dynamics simulations: a high performance computing framework, J. Chem. Phys., 137, 14, Article 144103 pp. (2012) |
| [28] | Farrell, K.; Oden, J. T.; Faghihi, D., A Bayesian framework for adaptive selection, calibration, and validation of coarse-grained models of atomistic systems, J. Comput. Phys., 295, 189-208 (2015) · Zbl 1349.62078 |
| [29] | Farrell, K.; Oden, J. T., CaLibration and validation of coarse-grained models of atomic systems: Application to semiconductor manufacturing, Comput. Mech., 54, 1, 3-19 (2014) · Zbl 1398.74487 |
| [30] | Kulakova, L.; Arampatzis, G.; Angelikopoulos, P.; Chatzidoukas, P.; Papadimitriou, C.; Koumoutsakos, P., Data driven inference for the repulsive exponent of the Lennard-Jones potential in molecular dynamics simulations, Sci. Rep., 7, 1, 16576 (2017) |
| [31] | Evensen, G., Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res.: Oceans, 99, C5, 10143-10162 (1994) |
| [32] | Nakano, S.; Ueno, G.; Higuchi, T., Merging particle filter for sequential data assimilation, Nonlinear Process. Geophys., 14, 395-408 (2007) |
| [33] | Manoli, G.; Rossi, M.; Pasetto, D.; Deiana, R.; Ferraris, S.; Cassiani, G.; Putti, M., An iterative particle filter approach for coupled hydro-geophysical inversion of a controlled infiltration experiment, J. Comput. Phys., 283, 37-51 (2015) · Zbl 1351.76278 |
| [34] | Rossi, M.; Manoli, G.; Pasetto, D.; Deiana, R.; Ferraris, S.; Strobbia, C.; Putti, M.; Cassiani, G., Coupled inverse modeling of a controlled irrigation experiment using multiple hydro-geophysical data, Adv. Water Resour., 82, 150-165 (2015) |
| [35] | Cheng, H.; Shuku, T.; Thoeni, K.; Yamamoto, H., CaLibration of micromechanical parameters for DEM simulations by using the particle filter, EPJ Web Conf., 140, 12011 (2017) |
| [36] | Cheng, H.; Shuku, T.; Thoeni, K.; Yamamoto, H., Probabilistic calibration of discrete element simulations using the sequential quasi-Monte Carlo filter, Granul. Matter, 20, 1, 11 (2018) |
| [37] | Ruiz, H.; Kappen, H. J., Particle smoothing for hidden diffusion processes: adaptive path integral smoother, IEEE Trans. Signal Process., 65, 12, 3191-3203 (2017) · Zbl 1414.94525 |
| [38] | Kappen, H. J.; Ruiz, H. C., Adaptive importance sampling for control and inference, J. Stat. Phys., 162, 5, 1244-1266 (2016) · Zbl 1338.93166 |
| [39] | Chavali, P.; Nehorai, A., Hierarchical particle filtering for multi-modal data fusion with application to multiple-target tracking, Signal Process., 97, 207-220 (2014) |
| [40] | Fan, Z.; Ji, H.; Zhang, Y., Iterative particle filter for visual tracking, Signal Process. Image Commun., 36, 140-153 (2015) |
| [41] | Yin, S.; Zhu, X., Intelligent particle filter and its application to fault detection of nonlinear system, IEEE Trans. Ind. Electron., 62, 6, 3852-3861 (2015) |
| [42] | I. Yoshida, T. Shuku, Particle filter with Gaussian mixture model for inverse problem, in: Proc. 6th Asian-Pacific Symp. Struct. Reliab. its Appl., 2016, pp. 643-648.; I. Yoshida, T. Shuku, Particle filter with Gaussian mixture model for inverse problem, in: Proc. 6th Asian-Pacific Symp. Struct. Reliab. its Appl., 2016, pp. 643-648. |
| [43] | Z. Medina-Cedina, H.D.V. Khoa, Probabilistic calibration of discrete particle models for geomaterials, in: Proc. of the 17th Int. Conference on Soil Mechanics and Geotechnical Engineering, 2009, pp. 704-707.; Z. Medina-Cedina, H.D.V. Khoa, Probabilistic calibration of discrete particle models for geomaterials, in: Proc. of the 17th Int. Conference on Soil Mechanics and Geotechnical Engineering, 2009, pp. 704-707. |
| [44] | Oden, J. T.; Prudencio, E. E.; Bauman, P. T., Virtual model validation of complex multiscale systems: Applications to nonlinear elastostatics, Comput. Methods Appl. Mech. Engrg., 266, 162-184 (2013) · Zbl 1286.62016 |
| [45] | Farrell-Maupin, K.; Oden, J. T., Adaptive selection and validation of models of complex systems in the presence of uncertainty, Res. Math. Sci., 4, 1, 14 (2017) · Zbl 1377.92036 |
| [46] | Oden, J. T.; Babuška, I.; Faghihi, D., Predictive computational science: computer predictions in the presence of uncertainty, (Stein, E.; Borst, R.; Hughes, T. J., Encyclopedia of Computational Mechanics (2017)) |
| [47] | Šmilauer, V., Yade Documentation (2015) |
| [48] | Bagi, K., An algorithm to generate random dense arrangements for discrete element simulations of granular assemblies, Granul. Matter, 7, 31-43 (2005) · Zbl 1094.74058 |
| [49] | Magnanimo, V.; Ragione, L. L.; Jenkins, J. T.; Wang, P.; Makse, H. A., Characterizing the shear and bulk moduli of an idealized granular material, Europhys. Lett., 81, 3, 34006 (2008) |
| [50] | Vlahinić, I.; Andò, E.; Viggiani, G.; Andrade, J. E., Towards a more accurate characterization of granular media: extracting quantitative descriptors from tomographic images, Granul. Matter, 16, 1, 9-21 (2014) |
| [51] | Zhao, B.; Wang, J., 3D quantitative shape analysis on form, roundness, and compactness with uCT, Powder Technol., 291, January, 262-275 (2016) |
| [52] | Särkkä, S., Bayesian Filtering and Smoothing. Institute of Mathematical Statistics Textbooks (2013), Cambridge University Press · Zbl 1274.62021 |
| [53] | Halton, J. H., Sequential monte carlo techniques for the solution of linear systems, J. Sci. Comput., 9, 2, 213-257 (1994) · Zbl 0844.65023 |
| [54] | Gerber, M.; Chopin, N., Sequential quasi Monte Carlo, J. R. Stat. Soc. Ser. B Stat. Methodol., 77, 3, 509-579 (2015) · Zbl 1414.62109 |
| [55] | van Leeuwen, P. J., Nonlinear data assimilation in geosciences: an extremely efficient particle filter, Q. J. R. Meteorol. Soc., 136, 653, 1991-1999 (2010) |
| [56] | Kitagawa, G.; Sato, S., Monte Carlo Smoothing and Self-Organising State-Space Model, 177-195 (2001), Springer New York: Springer New York New York, NY · Zbl 1056.93581 |
| [57] | Blei, D. M.; Jordan, M. I., Variational inference for dirichlet process mixtures, Bayesian Anal., 1, 1, 121-143 (2006) · Zbl 1331.62259 |
| [58] | Rasmussen, C. E., The infinite gaussian mixture model, (Proceedings of the 12th International Conference on Neural Information Processing Systems. Proceedings of the 12th International Conference on Neural Information Processing Systems, NIPS’99 (1999), MIT Press: MIT Press Cambridge, MA, USA), 554-560 |
| [59] | Bishop, C. M., (Pattern Recognition and Machine Learning. Pattern Recognition and Machine Learning, Information Science and Statistics (2006), Springer-Verlag: Springer-Verlag Berlin, Heidelberg) · Zbl 1107.68072 |
| [60] | Caron, F.; Davy, M.; Doucet, A.; Duflos, E.; Vanheeghe, P., Bayesian Inference for linear dynamic models with dirichlet process mixtures, IEEE Trans. Signal Process., 56, 1, 71-84 (2008) · Zbl 1391.62144 |
| [61] | McAuliffe, J. D.; Blei, D. M.; Jordan, M. I., Nonparametric empirical bayes for the dirichlet process mixture model, Stat. Comput., 16, 1, 5-14 (2006) |
| [62] | Görür, D.; Edward Rasmussen, C., Dirichlet process gaussian mixture models: Choice of the base distribution, J. Comput. Sci. Tech., 25, 4, 653-664 (2010) |
| [63] | Ferguson, T. S., A bayesian analysis of some nonparametric problems, Ann. Statist., 1, 2, 209-230 (1973) · Zbl 0255.62037 |
| [64] | Sethuraman, J., A constructive definition of dirichlet priors, Statist. Sinica, 4, 2, 639-650 (1994) · Zbl 0823.62007 |
| [65] | Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; Vanderplas, J.; Passos, A.; Cournapeau, D.; Brucher, M.; Perrot, M.; Duchesnay, E., Scikit-learn: Machine learning in Python, J. Mach. Learn. Res., 12, 2825-2830 (2011) · Zbl 1280.68189 |
| [66] | Arganda-Carreras, I.; Kaynig, V.; Rueden, C.; Eliceiri, K. W.; Schindelin, J.; Cardona, A.; Sebastian Seung, H., Trainable weka segmentation: a machine learning tool for microscopy pixel classification, Bioinformatics, 9, 15, 676-682 (2017) |
| [67] | Legland, D.; Arganda-Carreras, I.; Andrey, P., Morpholibj: integrated library and plugins for mathematical morphology with ImageJ, Bioinformatics, 13, 22, btw413 (2016) |
| [68] | Tengattini, A.; Andò, E., Kalisphera: an analytical tool to reproduce the partial volume effect of spheres imaged in 3D, Meas. Sci. Technol., 26, 9, Article 095606 pp. (2015) |
| [69] | Cheng, H.; Yamamoto, H.; Thoeni, K., Numerical study on stress states and fabric anisotropies in soilbags using the DEM, Comput. Geotech., 76, 170-183 (2016) |
| [70] | Cheng, H.; Yamamoto, H.; Thoeni, K.; Wu, Y., An analytical solution for geotextile-wrapped soil based on insights from DEM analysis, Geotext. Geomembranes (2017) |
| [71] | De Rainville, F.-M.; Gagné, C.; Teytaud, O.; Laurendeau, D., Evolutionary optimization of low-discrepancy sequences, ACM Trans. Model. Comput. Simul., 22, 2, 9:1-9:25 (2012) · Zbl 1386.65162 |
| [72] | O’Sullivan, C., Particulate Discrete Element Modelling: A Geomechanics Perspective (2011), Taylor & Francis: Taylor & Francis Hoboken, NJ |
| [73] | Do, H. Q.; Aragón, A. M.; Schott, D. L., A calibration framework for discrete element model parameters using genetic algorithms, Adv. Powder Technol., 29, 6, 1393-1403 (2018) |
| [74] | Chernatynskiy, A.; Phillpot, S. R.; LeSar, R., Uncertainty quantification in multiscale simulation of materials: a prospective, Annu. Rev. Mater. Res., 43, 1, 157-182 (2013) |
| [75] | Zhao, C.; Yin, Z.; Misra, A.; Hicher, P., Thermomechanical formulation for micromechanical elasto-plasticity in granular materials, Int. J. Solids Struct., 138, 64-75 (2018) |
| [76] | Feng, Y.; Zhao, T.; Kato, J.; Zhou, W., Towards stochastic discrete element modelling of spherical particles with surface roughness: A normal interaction law, Comput. Methods Appl. Mech. Engrg., 315, 247-272 (2017) · Zbl 1439.74217 |
| [77] | Matouš, K.; Geers, M. G.; Kouznetsova, V. G.; Gillman, A., A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials, J. Comput. Phys., 330, 192-220 (2017) |
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