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Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation
Towards an optimization theory for deforming dense granular materials: Minimum cost maximum flow solutions
1. | Department of Mathematics and Statistics, Curtin University, GPO Box U1987 Perth, Western Australia 6845 |
2. | Department of Mathematics and Statistics, University of Melbourne, Melbourne, Australia 3010, Australia |
References:
[1] |
R. Arévalo, I. Zuriguel and D. Maza, Topology of the force network in the jamming transition of an isotropically compressed granular packing,, Physical Review E, 81 (2010). Google Scholar |
[2] |
D. P. Bertsekas, "Network Optimization: Continuous and Discrete Models (Optimization, Computation, and Control),", Athena Scientific, (1998). Google Scholar |
[3] |
J. A. Bondy and U. S. R. Murty, "Graph Theory,", Graduate Texts in Mathematics, (2008).
doi: 10.1007/978-1-84628-970-5. |
[4] |
I. Cavarretta and C. O'Sullivan, The mechanics of rigid irregular particles subject to uniaxial compression,, Géotechnique, 62 (2012), 681. Google Scholar |
[5] |
J. Duran, "Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials,", Springer-Verlag, (2000). Google Scholar |
[6] |
J. Edmonds and R. M. Karp, Theoretical improvements in algorithmic efficiency for network flow problem,, Journal of the Association for Computing Machinery, 19 (1972), 248. Google Scholar |
[7] |
A. Garg and R. Tamassia, A new minimum cost flow algorithm with applications to graph drawing,, Graph Drawing, 1190 (1997), 201. Google Scholar |
[8] |
M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints,, Journal of Industrial and Management Optimization, 4 (2008), 247.
doi: 10.3934/jimo.2008.4.247. |
[9] |
F. S. Hillier and G. J. Lieberman, "Introduction to Operations Research,", McGraw-Hill, (2005). Google Scholar |
[10] |
D. Jungnickel, "Graphs, Networks and Algorithms,", Third edition. Algorithms and Computation in Mathematics, (2008).
doi: 10.1007/978-3-540-72780-4. |
[11] |
Q. Lin and A. Tordesillas, Granular rheology: Fine tuned for optimal efficiency?, Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics, (2012). Google Scholar |
[12] |
R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica J. IFAC, 45 (2009), 2250.
doi: 10.1016/j.automatica.2009.05.029. |
[13] |
H. B. Mühlhaus and I. Vardoulakis, The thickness of shear bands in granular materials,, Géotechnique, 37 (1987), 271. Google Scholar |
[14] |
M. Oda and H. Kazama, Microstructure of shear bands and its relation to the mechanisms of dilatancy and failure of dense granular soils,, Géotechnique, 48 (1998), 465. Google Scholar |
[15] |
M. Oda, J. Konishi and S. Nemat-Nasser, Experimental micromechanical evaluation of strength of granular materials: Effects of particle rolling,, Mechanics of Materials, 1 (1982), 269. Google Scholar |
[16] |
A. Ord and B. E. Hobbs, Fracture pattern formation in frictional, cohesive, granular material,, Philosophical Transactions of the Royal Society A, 368 (2010), 95. Google Scholar |
[17] |
J. Paavilainen and J. Tuhkuri, Pressure distributions and force chains during simulated ice rubbling against sloped structures,, Cold Regions Science and Technology, 85 (2013), 157. Google Scholar |
[18] |
J. M. Padbidri, C. M. Hansen, S. D. Mesarovic and B. Muhunthan, Length scale for transmission of rotations in dense granular materials,, Journal of Applied Mechanics, 79 (2012). Google Scholar |
[19] |
F. Radjai, D. E. Wolf, M. Jean and J. J. Moreau, Bimodal character of stress transmission in granular packings,, Physical Review Letters, 80 (1998), 61. Google Scholar |
[20] |
A. L. Rechenmacher, S. Abedi, O. Chupin and A. D. Orlando, Characterization of mesoscale instabilities in localized granular shear using digital image correlation,, Acta Geotechnica, 6 (2011), 205. Google Scholar |
[21] |
A. Tordesillas, Force chain buckling, unjamming transitions and shear banding in dense granular assemblies,, Philosophical Magazine, 87 (2007), 4987. Google Scholar |
[22] |
A. Tordesillas, A. Cramer and D. M. Walker, Minimum cut and shear bands,, Powders & Grains AIP Conference Proceedings 1542 (2013), 1542 (2013), 507. Google Scholar |
[23] |
A. Tordesillas, Q. Lin, J. Zhang, R. P. Behringer and J. Shi, Structural stability and jamming of self-organized cluster conformations in dense granular materials,, Journal of the Mechanics and Physics of Solids, 59 (2011), 265. Google Scholar |
[24] |
A. Tordesillas, D. M. Walker, E. Andò and G. Viggiani, Revisiting localised deformation in sand with complex systems,, Proceedings of the Royal Society of London Series A, (2013). Google Scholar |
[25] |
A. Tordesillas, D. M. Walker, G. Froyland, J. Zhang and R. P. Behringer, Transition dynamics and magic-number-like behavior of frictional granular clusters,, Physical Review E, 86 (2012). Google Scholar |
[26] |
A. Tordesillas, D. M. Walker and Q. Lin, Force cycles and force chains,, Physical Review E, 81 (2010). Google Scholar |
[27] |
D. M. Walker, A. Tordesillas, S. Pucilowski, Q. Lin, A. L. Rechenmacher and S. Abedi, Analysis of grain-scale measurements of sand using kinematical complex networks,, International Journal of Bifurcation and Chaos, 22 (2012).
doi: 10.1142/S021812741230042X. |
[28] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications,, Journal of Industrial and Management Optimization, 5 (2009), 705.
doi: 10.3934/jimo.2009.5.705. |
[29] |
Y. Zhao and M. A. Stadtherr, Rigorous global optimization for dynamic systems subject to inequality path constraints,, Industrial and Engineering Chemistry Research, 50 (2011), 12678. Google Scholar |
show all references
References:
[1] |
R. Arévalo, I. Zuriguel and D. Maza, Topology of the force network in the jamming transition of an isotropically compressed granular packing,, Physical Review E, 81 (2010). Google Scholar |
[2] |
D. P. Bertsekas, "Network Optimization: Continuous and Discrete Models (Optimization, Computation, and Control),", Athena Scientific, (1998). Google Scholar |
[3] |
J. A. Bondy and U. S. R. Murty, "Graph Theory,", Graduate Texts in Mathematics, (2008).
doi: 10.1007/978-1-84628-970-5. |
[4] |
I. Cavarretta and C. O'Sullivan, The mechanics of rigid irregular particles subject to uniaxial compression,, Géotechnique, 62 (2012), 681. Google Scholar |
[5] |
J. Duran, "Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials,", Springer-Verlag, (2000). Google Scholar |
[6] |
J. Edmonds and R. M. Karp, Theoretical improvements in algorithmic efficiency for network flow problem,, Journal of the Association for Computing Machinery, 19 (1972), 248. Google Scholar |
[7] |
A. Garg and R. Tamassia, A new minimum cost flow algorithm with applications to graph drawing,, Graph Drawing, 1190 (1997), 201. Google Scholar |
[8] |
M. Gerdts and M. Kunkel, A nonsmooth Newton's method for discretized optimal control problems with state and control constraints,, Journal of Industrial and Management Optimization, 4 (2008), 247.
doi: 10.3934/jimo.2008.4.247. |
[9] |
F. S. Hillier and G. J. Lieberman, "Introduction to Operations Research,", McGraw-Hill, (2005). Google Scholar |
[10] |
D. Jungnickel, "Graphs, Networks and Algorithms,", Third edition. Algorithms and Computation in Mathematics, (2008).
doi: 10.1007/978-3-540-72780-4. |
[11] |
Q. Lin and A. Tordesillas, Granular rheology: Fine tuned for optimal efficiency?, Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics, (2012). Google Scholar |
[12] |
R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica J. IFAC, 45 (2009), 2250.
doi: 10.1016/j.automatica.2009.05.029. |
[13] |
H. B. Mühlhaus and I. Vardoulakis, The thickness of shear bands in granular materials,, Géotechnique, 37 (1987), 271. Google Scholar |
[14] |
M. Oda and H. Kazama, Microstructure of shear bands and its relation to the mechanisms of dilatancy and failure of dense granular soils,, Géotechnique, 48 (1998), 465. Google Scholar |
[15] |
M. Oda, J. Konishi and S. Nemat-Nasser, Experimental micromechanical evaluation of strength of granular materials: Effects of particle rolling,, Mechanics of Materials, 1 (1982), 269. Google Scholar |
[16] |
A. Ord and B. E. Hobbs, Fracture pattern formation in frictional, cohesive, granular material,, Philosophical Transactions of the Royal Society A, 368 (2010), 95. Google Scholar |
[17] |
J. Paavilainen and J. Tuhkuri, Pressure distributions and force chains during simulated ice rubbling against sloped structures,, Cold Regions Science and Technology, 85 (2013), 157. Google Scholar |
[18] |
J. M. Padbidri, C. M. Hansen, S. D. Mesarovic and B. Muhunthan, Length scale for transmission of rotations in dense granular materials,, Journal of Applied Mechanics, 79 (2012). Google Scholar |
[19] |
F. Radjai, D. E. Wolf, M. Jean and J. J. Moreau, Bimodal character of stress transmission in granular packings,, Physical Review Letters, 80 (1998), 61. Google Scholar |
[20] |
A. L. Rechenmacher, S. Abedi, O. Chupin and A. D. Orlando, Characterization of mesoscale instabilities in localized granular shear using digital image correlation,, Acta Geotechnica, 6 (2011), 205. Google Scholar |
[21] |
A. Tordesillas, Force chain buckling, unjamming transitions and shear banding in dense granular assemblies,, Philosophical Magazine, 87 (2007), 4987. Google Scholar |
[22] |
A. Tordesillas, A. Cramer and D. M. Walker, Minimum cut and shear bands,, Powders & Grains AIP Conference Proceedings 1542 (2013), 1542 (2013), 507. Google Scholar |
[23] |
A. Tordesillas, Q. Lin, J. Zhang, R. P. Behringer and J. Shi, Structural stability and jamming of self-organized cluster conformations in dense granular materials,, Journal of the Mechanics and Physics of Solids, 59 (2011), 265. Google Scholar |
[24] |
A. Tordesillas, D. M. Walker, E. Andò and G. Viggiani, Revisiting localised deformation in sand with complex systems,, Proceedings of the Royal Society of London Series A, (2013). Google Scholar |
[25] |
A. Tordesillas, D. M. Walker, G. Froyland, J. Zhang and R. P. Behringer, Transition dynamics and magic-number-like behavior of frictional granular clusters,, Physical Review E, 86 (2012). Google Scholar |
[26] |
A. Tordesillas, D. M. Walker and Q. Lin, Force cycles and force chains,, Physical Review E, 81 (2010). Google Scholar |
[27] |
D. M. Walker, A. Tordesillas, S. Pucilowski, Q. Lin, A. L. Rechenmacher and S. Abedi, Analysis of grain-scale measurements of sand using kinematical complex networks,, International Journal of Bifurcation and Chaos, 22 (2012).
doi: 10.1142/S021812741230042X. |
[28] |
L. Y. Wang, W. H. Gui, K. L. Teo, R. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications,, Journal of Industrial and Management Optimization, 5 (2009), 705.
doi: 10.3934/jimo.2009.5.705. |
[29] |
Y. Zhao and M. A. Stadtherr, Rigorous global optimization for dynamic systems subject to inequality path constraints,, Industrial and Engineering Chemistry Research, 50 (2011), 12678. Google Scholar |
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