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Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system
Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions
School of Mathematical Sciences, Institute of Natural Sciences, and MOE-LSC, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240, Shanghai, China |
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.
References:
[1] |
H. AlRachid, C. Ortner and L. Mones,
Some remarks on preconditioning molecular dynamics, SMAI J. Comput. Math., 4 (2018), 57-80.
doi: 10.5802/smai-jcm.29. |
[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] |
R. Fletcher and C. M. Reeves,
Function minimization by conjugate gradients, Comput. J., 7 (1964), 149-154.
doi: 10.1093/comjnl/7.2.149. |
[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] |
K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9781139171731.![]() |
[10] |
B. Kou, Y. Cao, J. Li, C. Xia, Z. Li, H. Dong, A. Zhang, J. Zhang, W. Kob and Y. Wang,
Granular materials flow like complex fluids, Nature, 551 (2017), 360-363.
doi: 10.1038/nature24062. |
[11] |
D. Krijgsman and S. Luding,
Simulating granular materials by energy minimization, Comp. Part. Mech., 3 (2016), 463-475.
doi: 10.1007/s40571-016-0105-8. |
[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] |
T. Majmudar and R. Behringer,
Contact force measurements and stress-induced anisotropy in granular materials, Nature, 435 (2005), 1079-1082.
doi: 10.1038/nature03805. |
[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. Google Scholar |
[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] |
A. J. Wathen,
Preconditioning, Acta Numer., 24 (2015), 329-376.
doi: 10.1017/S0962492915000021. |
[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. |
show all references
References:
[1] |
H. AlRachid, C. Ortner and L. Mones,
Some remarks on preconditioning molecular dynamics, SMAI J. Comput. Math., 4 (2018), 57-80.
doi: 10.5802/smai-jcm.29. |
[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] |
R. Fletcher and C. M. Reeves,
Function minimization by conjugate gradients, Comput. J., 7 (1964), 149-154.
doi: 10.1093/comjnl/7.2.149. |
[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] |
K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9781139171731.![]() |
[10] |
B. Kou, Y. Cao, J. Li, C. Xia, Z. Li, H. Dong, A. Zhang, J. Zhang, W. Kob and Y. Wang,
Granular materials flow like complex fluids, Nature, 551 (2017), 360-363.
doi: 10.1038/nature24062. |
[11] |
D. Krijgsman and S. Luding,
Simulating granular materials by energy minimization, Comp. Part. Mech., 3 (2016), 463-475.
doi: 10.1007/s40571-016-0105-8. |
[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] |
T. Majmudar and R. Behringer,
Contact force measurements and stress-induced anisotropy in granular materials, Nature, 435 (2005), 1079-1082.
doi: 10.1038/nature03805. |
[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. Google Scholar |
[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] |
A. J. Wathen,
Preconditioning, Acta Numer., 24 (2015), 329-376.
doi: 10.1017/S0962492915000021. |
[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. |







Algorithm 1 Generation of the Jamming Configuration. |
Input: |
particle number initial packing fraction increment prescribed accuracy |
Output: |
critical volume fraction jamming configuration 1: Generate an initial configuration 2: let 3: // Increment Step 4: while 5: Let 6: Fix the position of the particles, and enlarge their radii uniformly to generate a new configuration 7: Let 8: if 9: 10: else 11: 12: end if 13: end while 14: // Bisection Step 15:while 16: 17: Generate an intermediate configuration 18: Let 19: if 20: 21: else 22: 23: end if 24: end while 25: Denote 26: return |
Algorithm 1 Generation of the Jamming Configuration. |
Input: |
particle number initial packing fraction increment prescribed accuracy |
Output: |
critical volume fraction jamming configuration 1: Generate an initial configuration 2: let 3: // Increment Step 4: while 5: Let 6: Fix the position of the particles, and enlarge their radii uniformly to generate a new configuration 7: Let 8: if 9: 10: else 11: 12: end if 13: end while 14: // Bisection Step 15:while 16: 17: Generate an intermediate configuration 18: Let 19: if 20: 21: else 22: 23: end if 24: end while 25: Denote 26: return |
Method | L-BFGS | P-L-BFGS ($A=0$) |
P-L-BFGS ($A=3$) |
FR-CG | P-FR-CG | |
Case 1 | n_iter | 2003 | 705 | 813 | 4574 | 1101 |
time/s | 24.3 | 11.8 | 16.2 | 58.9 | 18.5 | |
Case 2 | n_iter | 2749 | 986 | 1351 | 7306 | 2013 |
time/s | 29.2 | 15.8 | 27.1 | 90.1 | 33.5 | |
Case 3 | n_iter | 602 | 380 | 432 | 940 | 528 |
time/s | 7.5 | 6.8 | 9.1 | 11.9 | 9.2 |
Method | L-BFGS | P-L-BFGS ($A=0$) |
P-L-BFGS ($A=3$) |
FR-CG | P-FR-CG | |
Case 1 | n_iter | 2003 | 705 | 813 | 4574 | 1101 |
time/s | 24.3 | 11.8 | 16.2 | 58.9 | 18.5 | |
Case 2 | n_iter | 2749 | 986 | 1351 | 7306 | 2013 |
time/s | 29.2 | 15.8 | 27.1 | 90.1 | 33.5 | |
Case 3 | n_iter | 602 | 380 | 432 | 940 | 528 |
time/s | 7.5 | 6.8 | 9.1 | 11.9 | 9.2 |
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