# American Institute of Mathematical Sciences

March  2020, 28(1): 405-421. doi: 10.3934/era.2020023

## 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

* Corresponding author: Lei Zhang

Received  November 2019 Revised  January 2020 Published  March 2020

Fund Project: HW and LZ were partially supported by National Natural Science Foundations of China (NSFC 11871339, 11861131004, 11571314)

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. Electronic Research Archive, 2020, 28 (1) : 405-421. doi: 10.3934/era.2020023
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171731.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer, New York, 2006. doi: 10.1007/978-0-387-40065-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] R. Pytlak, Conjugate Gradient Algorithms in Nonconvex Optimization. Nonconvex Optimization and its Applications, Vol. 89, Springer-Verlag, Berlin, 2009.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] A. J. Wathen, Preconditioning, Acta Numer., 24 (2015), 329-376.  doi: 10.1017/S0962492915000021.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171731.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] J. Nocedal and S. J. Wright, Numerical Optimization, 2$^nd$ edition, Springer, New York, 2006. doi: 10.1007/978-0-387-40065-5.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] R. Pytlak, Conjugate Gradient Algorithms in Nonconvex Optimization. Nonconvex Optimization and its Applications, Vol. 89, Springer-Verlag, Berlin, 2009.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] A. J. Wathen, Preconditioning, Acta Numer., 24 (2015), 329-376.  doi: 10.1017/S0962492915000021.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Two particles in contact
$\ell^\infty$ norm of gradient vs iteration number for one energy minimization step of a granular system with particle number 4096, $\phi = 0.8389$, the radii of particular grow by $r \rightarrow (1+\delta)r, \delta = 5\times 10^{-5}$
Schematic of the evolution from non-jamming to jamming state of bi-disperse granular system using Algorithm 1, a Initial non-jamming configuration. b Intermediate configuration. c Jamming configuration, Visualization tool: OVITO [22]
pressure $p$ vs $\phi-{\phi_{\mathrm{cr}}}$ for a 2D bi-disperse granular system with particle number 4096
Shear modulus $G$ vs $\phi-{\phi_{\mathrm{cr}}}$ for a 2D bi-disperse granular system with particle number 4096
Average contact number $z$ vs $\phi$ for a 2D bi-disperse granular system with particle number 4096. $z_0$ is the average contact number of the granular system with volume fraction ${\phi_{\mathrm{cr}}}$. In our 2D simulation, particles are frictionless, $z_0 = 4.0$
Iteration numbers (average and variance) with respect to $\phi-{\phi_{\mathrm{cr}}}$. Red circles and blue stars are average iteration numbers of 10 quasi-static simulations using preconditioned L-BFGS method and un-preconditioned L-BFGS method, respectively. In addition, we plot the error bars for those simulations
Stress ($\sigma$) vs. strain and fabric ($F$) vs. strain curve for pure shear, $\gamma$ is the shear strain. The subscripts iso and dev, represent the isotropic and deviatoric parts of these tensors (stress, fabric), respectively. Let $\lambda_1$ and $\lambda_2$ be the eigenvalues of those tensors, the isotropic part is defined as $(\lambda_1+\lambda_2)/2$ and the deviatoric part is $|\lambda_1 - \lambda_2|/2$
 Algorithm 1 Generation of the Jamming Configuration. Input: particle number $N$; initial packing fraction ${\phi_{\mathrm{init}}}$; increment ${\delta \phi}$; prescribed accuracy $\varepsilon$. Output: critical volume fraction ${\phi_{\mathrm{cr}}}$; jamming configuration ${{\mathcal{C}}_{\mathrm{jam}}}$ with ${\phi_{\mathrm{cr}}}$. 1: Generate an initial configuration ${{\mathcal{C}}_{\mathrm{init}}}$ with $N$ particles and volume fraction ${\phi_{\mathrm{init}}}$; 2: let ${{\mathcal{C}}_{\mathrm{ref}}} = {\rm arg\;min} ({\phi_{\mathrm{init}}}, {{\mathcal{C}}_{\mathrm{init}}})$, with total energy ${E_{\mathrm{ref}}}=0$ and pressure ${p_{\mathrm{ref}}}=0$. 3: // Increment Step 4: while ${E_{\mathrm{ref}}}=0$ and ${p_{\mathrm{ref}}}=0$ do 5: Let $\phi = {\phi_{\mathrm{ref}}} + {\delta \phi}$; 6: Fix the position of the particles, and enlarge their radii uniformly to generate a new configuration ${{\mathcal{C}}_{\mathrm{ref}}}$ with volume fraction $\phi$; 7: Let ${{\mathcal{C}}_{\mathrm{eq}}}(\phi) = {\rm arg\;min} E(\phi, {{\mathcal{C}}_{\mathrm{ref}}})$, compute $E({{\mathcal{C}}_{\mathrm{eq}}})$ and $p({{\mathcal{C}}_{\mathrm{eq}}})$. 8: if $E({{\mathcal{C}}_{\mathrm{eq}}})>0$ & $p({{\mathcal{C}}_{\mathrm{eq}}})>0$ then 9: ${\phi_{\mathrm{cr}}}^{\mathrm{app}} = \phi$, ${{\mathcal{C}}_{\mathrm{jam}}} = {{\mathcal{C}}_{\mathrm{eq}}}$, break;10: else 11: ${\phi_{\mathrm{ref}}} = \phi$, ${{\mathcal{C}}_{\mathrm{ref}}} = {{\mathcal{C}}_{\mathrm{eq}}}$; 12: end if 13: end while 14: // Bisection Step 15:while ${\phi_{\mathrm{cr}}}^{\mathrm{app}} - {\phi_{\mathrm{ref}}} > \epsilon$ do 16: ${\phi_{\mathrm{m}}}= ({\phi_{\mathrm{cr}}} + {\phi_{\mathrm{ref}}})/2$; 17: Generate an intermediate configuration ${{\mathcal{C}}_m}$ with volume fraction ${\phi_{\mathrm{m}}}$, starting from ${{\mathcal{C}}_{\mathrm{ref}}}$. 18: Let ${{\mathcal{C}}_{\mathrm{eq}}} = {\rm arg\;min} ({\phi_{\mathrm{m}}}, {{\mathcal{C}}_m})$, and calculate $E({{\mathcal{C}}_{\mathrm{eq}}})$ and $p({{\mathcal{C}}_{\mathrm{eq}}})$. 19: if $E({{\mathcal{C}}_{\mathrm{eq}}})>0$ & $p({{\mathcal{C}}_{\mathrm{eq}}})>0$ then 20: ${\phi_{\mathrm{cr}}}^{\mathrm{app}} = {\phi_{\mathrm{m}}}$; 21: else 22: ${\phi_{\mathrm{ref}}} = {\phi_{\mathrm{m}}}$, ${{\mathcal{C}}_{\mathrm{ref}}} = {{\mathcal{C}}_{\mathrm{eq}}}$; 23: end if 24: end while 25: Denote ${\phi_{\mathrm{cr}}} = {\phi_{\mathrm{cr}}}^{\mathrm{app}}$, ${{\mathcal{C}}_{\mathrm{jam}}} = {{\mathcal{C}}_{\mathrm{eq}}}$. 26: return ${\phi_{\mathrm{cr}}}$, ${{\mathcal{C}}_{\mathrm{jam}}}$;
 Algorithm 1 Generation of the Jamming Configuration. Input: particle number $N$; initial packing fraction ${\phi_{\mathrm{init}}}$; increment ${\delta \phi}$; prescribed accuracy $\varepsilon$. Output: critical volume fraction ${\phi_{\mathrm{cr}}}$; jamming configuration ${{\mathcal{C}}_{\mathrm{jam}}}$ with ${\phi_{\mathrm{cr}}}$. 1: Generate an initial configuration ${{\mathcal{C}}_{\mathrm{init}}}$ with $N$ particles and volume fraction ${\phi_{\mathrm{init}}}$; 2: let ${{\mathcal{C}}_{\mathrm{ref}}} = {\rm arg\;min} ({\phi_{\mathrm{init}}}, {{\mathcal{C}}_{\mathrm{init}}})$, with total energy ${E_{\mathrm{ref}}}=0$ and pressure ${p_{\mathrm{ref}}}=0$. 3: // Increment Step 4: while ${E_{\mathrm{ref}}}=0$ and ${p_{\mathrm{ref}}}=0$ do 5: Let $\phi = {\phi_{\mathrm{ref}}} + {\delta \phi}$; 6: Fix the position of the particles, and enlarge their radii uniformly to generate a new configuration ${{\mathcal{C}}_{\mathrm{ref}}}$ with volume fraction $\phi$; 7: Let ${{\mathcal{C}}_{\mathrm{eq}}}(\phi) = {\rm arg\;min} E(\phi, {{\mathcal{C}}_{\mathrm{ref}}})$, compute $E({{\mathcal{C}}_{\mathrm{eq}}})$ and $p({{\mathcal{C}}_{\mathrm{eq}}})$. 8: if $E({{\mathcal{C}}_{\mathrm{eq}}})>0$ & $p({{\mathcal{C}}_{\mathrm{eq}}})>0$ then 9: ${\phi_{\mathrm{cr}}}^{\mathrm{app}} = \phi$, ${{\mathcal{C}}_{\mathrm{jam}}} = {{\mathcal{C}}_{\mathrm{eq}}}$, break;10: else 11: ${\phi_{\mathrm{ref}}} = \phi$, ${{\mathcal{C}}_{\mathrm{ref}}} = {{\mathcal{C}}_{\mathrm{eq}}}$; 12: end if 13: end while 14: // Bisection Step 15:while ${\phi_{\mathrm{cr}}}^{\mathrm{app}} - {\phi_{\mathrm{ref}}} > \epsilon$ do 16: ${\phi_{\mathrm{m}}}= ({\phi_{\mathrm{cr}}} + {\phi_{\mathrm{ref}}})/2$; 17: Generate an intermediate configuration ${{\mathcal{C}}_m}$ with volume fraction ${\phi_{\mathrm{m}}}$, starting from ${{\mathcal{C}}_{\mathrm{ref}}}$. 18: Let ${{\mathcal{C}}_{\mathrm{eq}}} = {\rm arg\;min} ({\phi_{\mathrm{m}}}, {{\mathcal{C}}_m})$, and calculate $E({{\mathcal{C}}_{\mathrm{eq}}})$ and $p({{\mathcal{C}}_{\mathrm{eq}}})$. 19: if $E({{\mathcal{C}}_{\mathrm{eq}}})>0$ & $p({{\mathcal{C}}_{\mathrm{eq}}})>0$ then 20: ${\phi_{\mathrm{cr}}}^{\mathrm{app}} = {\phi_{\mathrm{m}}}$; 21: else 22: ${\phi_{\mathrm{ref}}} = {\phi_{\mathrm{m}}}$, ${{\mathcal{C}}_{\mathrm{ref}}} = {{\mathcal{C}}_{\mathrm{eq}}}$; 23: end if 24: end while 25: Denote ${\phi_{\mathrm{cr}}} = {\phi_{\mathrm{cr}}}^{\mathrm{app}}$, ${{\mathcal{C}}_{\mathrm{jam}}} = {{\mathcal{C}}_{\mathrm{eq}}}$. 26: return ${\phi_{\mathrm{cr}}}$, ${{\mathcal{C}}_{\mathrm{jam}}}$;
Iteration number and computational time for L-BFGS, P-L-BFGS, FR-CG, and P-FR-CG method for three different cases ($A$ is the parameter in (19)). The granular system has 4096 particles and volume fraction $\phi = 0.8438$
 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
 [1] Roman VodiČka, Vladislav MantiČ. An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1539-1561. doi: 10.3934/dcdss.2017079 [2] Przemysław Górka. Quasi-static evolution of polyhedral crystals. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 309-320. doi: 10.3934/dcdsb.2008.9.309 [3] Christopher J. Larsen. Local minimality and crack prediction in quasi-static Griffith fracture evolution. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 121-129. doi: 10.3934/dcdss.2013.6.121 [4] Alice Fiaschi. Young-measure quasi-static damage evolution: The nonconvex and the brittle cases. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 17-42. doi: 10.3934/dcdss.2013.6.17 [5] Dorothee Knees, Andreas Schröder. Computational aspects of quasi-static crack propagation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 63-99. doi: 10.3934/dcdss.2013.6.63 [6] Irina F. Sivergina, Michael P. Polis. About global null controllability of a quasi-static thermoelastic contact system. Conference Publications, 2005, 2005 (Special) : 816-823. doi: 10.3934/proc.2005.2005.816 [7] M. Montaz Ali. A recursive topographical differential evolution algorithm for potential energy minimization. Journal of Industrial & Management Optimization, 2010, 6 (1) : 29-46. doi: 10.3934/jimo.2010.6.29 [8] K. A. Ariyawansa, Leonid Berlyand, Alexander Panchenko. A network model of geometrically constrained deformations of granular materials. Networks & Heterogeneous Media, 2008, 3 (1) : 125-148. doi: 10.3934/nhm.2008.3.125 [9] Xiangdong Du, Martin Ostoja-Starzewski. On the scaling from statistical to representative volume element in thermoelasticity of random materials. Networks & Heterogeneous Media, 2006, 1 (2) : 259-274. doi: 10.3934/nhm.2006.1.259 [10] Qun Lin, Antoinette Tordesillas. Towards an optimization theory for deforming dense granular materials: Minimum cost maximum flow solutions. Journal of Industrial & Management Optimization, 2014, 10 (1) : 337-362. doi: 10.3934/jimo.2014.10.337 [11] Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099 [12] Xavier Dubois de La Sablonière, Benjamin Mauroy, Yannick Privat. Shape minimization of the dissipated energy in dyadic trees. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 767-799. doi: 10.3934/dcdsb.2011.16.767 [13] María Teresa González Montesinos, Francisco Ortegón Gallego. The evolution thermistor problem under the Wiedemann-Franz law with metallic conduction. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 901-923. doi: 10.3934/dcdsb.2007.8.901 [14] Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447 [15] Anass Belcaid, Mohammed Douimi, Abdelkader Fassi Fihri. Recursive reconstruction of piecewise constant signals by minimization of an energy function. Inverse Problems & Imaging, 2018, 12 (4) : 903-920. doi: 10.3934/ipi.2018038 [16] Riccardo Adami, Diego Noja, Nicola Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1155-1188. doi: 10.3934/dcdsb.2013.18.1155 [17] Shummin Nakayama, Yasushi Narushima, Hiroshi Yabe. Memoryless quasi-Newton methods based on spectral-scaling Broyden family for unconstrained optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1773-1793. doi: 10.3934/jimo.2018122 [18] Sourabh Bhattacharya, Abhishek Gupta, Tamer Başar. Jamming in mobile networks: A game-theoretic approach. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 1-30. doi: 10.3934/naco.2013.3.1 [19] Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229 [20] Yifu Feng, Min Zhang. A $p$-spherical section property for matrix Schatten-$p$ quasi-norm minimization. Journal of Industrial & Management Optimization, 2020, 16 (1) : 397-407. doi: 10.3934/jimo.2018159

2018 Impact Factor: 0.263

## Tools

Article outline

Figures and Tables