Granular materials are heterogenous grains in contact, which are ubiquitous in many scientific and engineering applications such as chemical engineering, fluid mechanics, geomechanics, pharmaceutics, and so on. Granular materials pose a great challenge to predictability, due to the presence of critical phenomena and large fluctuation of local forces. In this paper, we consider the quasi-static simulation of the dense granular media, and investigate the performances of typical minimization algorithms such as conjugate gradient methods and quasi-Newton methods. Furthermore, we develop preconditioning techniques to enhance the performance. Those methods are validated with numerical experiments for typical physically interested scenarios such as the jamming transition, the scaling law behavior close to the jamming state, and shear deformation of over jammed states.
Citation: Haolei Wang, Lei Zhang. Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions[J]. Electronic Research Archive, 2020, 28(1): 405-421. doi: 10.3934/era.2020023
Granular materials are heterogenous grains in contact, which are ubiquitous in many scientific and engineering applications such as chemical engineering, fluid mechanics, geomechanics, pharmaceutics, and so on. Granular materials pose a great challenge to predictability, due to the presence of critical phenomena and large fluctuation of local forces. In this paper, we consider the quasi-static simulation of the dense granular media, and investigate the performances of typical minimization algorithms such as conjugate gradient methods and quasi-Newton methods. Furthermore, we develop preconditioning techniques to enhance the performance. Those methods are validated with numerical experiments for typical physically interested scenarios such as the jamming transition, the scaling law behavior close to the jamming state, and shear deformation of over jammed states.
[1] | Some remarks on preconditioning molecular dynamics. SMAI J. Comput. Math. (2018) 4: 57-80. |
[2] | K. Bagi, Stress and strain in granular assemblies, Mechanics of Materials, 22 (1996), no. 3,165–178. doi: 10.1016/0167-6636(95)00044-5 |
[3] | A. Brandt, S. McCoruick and J. Ruge, Algebraic multigrid (AMG) for sparse matrix equations, in Sparsity and its Applications, Cambridge Univ. Press, Cambridge, 1985, 257–284. |
[4] | E. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices, in Proceedings of the 1969 24th National Conference, Association for Computing Machinery, New York, NY, 1969,157–172. doi: 10.1145/800195.805928 |
[5] | J. E. Jones, On the determination of molecular fields II. From the equation of state of a gas, Proc. R. Soc. Lond. A, 106 (1924), no. 738,463–477. doi: 10.1098/rspa.1924.0082 |
[6] | F. Göncü, O. Durán and S. Luding, Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres, Comptes Rendus Mécanique, 338 (2010), no. 10-11,570–586. doi: https://doi.org/10.1016/j.crme.2010.10.004 |
[7] | Function minimization by conjugate gradients. Comput. J. (1964) 7: 149-154. |
[8] | M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), no. 6,409–436. doi: 10.6028/jres.049.044 |
[9] | (1987) Contact Mechanics.Cambridge University Press. |
[10] | Granular materials flow like complex fluids. Nature (2017) 551: 360-363. |
[11] | Simulating granular materials by energy minimization. Comp. Part. Mech. (2016) 3: 463-475. |
[12] | S. Luding, Introduction to discrete element methods: Basic of contact force models and how to perform the micro-macro transition to continuum theory, European Journal of Environmental and Civil Engineering, 12 (2008), no. 7-8,785–826. doi: 10.1080/19648189.2008.9693050 |
[13] | Contact force measurements and stress-induced anisotropy in granular materials. Nature (2005) 435: 1079-1082. |
[14] | L. Mones, C. Ortner and G. Csnyi, Preconditioners for the geometry optimisation and saddle point search of molecular systems, Scientific Reports, 8 (2018), Art. 13991. doi: 10.1038/s41598-018-32105-x |
[15] | J. Nocedal, Updating quasi-Newton matrices with limited storage, Math. Comp., 35 (1980), no. 151,773–782 doi: 10.1090/S0025-5718-1980-0572855-7 |
[16] | J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer, New York, 2006. doi: 10.1007/978-0-387-40065-5 |
[17] | R. W. Ogden, Non-linear Elastic Deformations, Dover Publications, Mineola, NY, 1997. |
[18] | C. S. O'Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Jamming at zero temperature and zero applied stress: The epitome of disorder, Phys. Rev. E, 68 (2003), 011306. doi: 10.1103/PhysRevE.68.011306 |
[19] | D. Packwood, J. Kermode, L. Mones, N. Bernstein, J. Woolley, N. Gould, C. Ortner and G. Csányi, A universal preconditioner for simulating condensed phase materials, AIP J. Chem. Phys., 144 (2016), no. 16, 164109. doi: 10.1063/1.4947024 |
[20] | R. Pytlak, Conjugate Gradient Algorithms in Nonconvex Optimization. Nonconvex Optimization and its Applications, Vol. 89, Springer-Verlag, Berlin, 2009. |
[21] | R. J. Speedy, Glass transition in hard disc mixtures, AIP J. Chem. Phys., 110 (1999), no. 9, 4559–4565. doi: 10.1063/1.478337 |
[22] | A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO – the Open Visualization Tool, Modelling and Simulation in Materials Science and Engineering, 18 (2009), no. 1, 015012. doi: 10.1088/0965-0393/18/1/015012 |
[23] | Preconditioning. Acta Numer. (2015) 24: 329-376. |
[24] | H. Zhang and H. A. Makse, Jamming transition in emulsions and granular materials, Phys. Rev. E, 72 (2005), 011301. doi: 10.1103/PhysRevE.72.011301 |
[25] | L. Zhang, Y. Wang and J. Zhang., Force-chain distributions in granular systems, Phys. Rev. E, 89 (2014), 012203. doi: 10.1103/PhysRevE.89.012203 |