Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 341893, 15 pages
http://dx.doi.org/10.1155/2015/341893
Research Article

Discontinuous Galerkin Method for Material Flow Problems

Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany

Received 11 August 2015; Accepted 30 September 2015

Academic Editor: Yuming Qin

Copyright © 2015 Simone Göttlich and Patrick Schindler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Hoher, P. Schindler, S. Göttlich, V. Schleper, and S. Röck, “System dynamic models and real-time simulation of complex material flow systems,” in Enabling Manufacturing Competitiveness and Economic Sustainability, H. A. ElMaraghy, Ed., part 3, pp. 316–321, Springer, Berlin, Germany, 2012. View at Publisher · View at Google Scholar
  2. G. Reinhart and F.-F. Lacour, “Physically based virtual commissioning of material flow intensive manufacturing plants,” in Procedings of the 3rd International Conference on Changeable, Agile, Reconfigurable and Virtual Production (CARV ’09), H. A. Zaeh and M. F. ElMaraghy, Eds., pp. 377–387, Munich, Germany, 2009.
  3. S. Röck, “Hardware in the loop simulation of production systems dynamics,” Production Engineering, vol. 5, no. 3, pp. 329–337, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Gaugele, F. Fleissner, and P. Eberhard, “Simulation of material tests using meshfree Lagrangian particle methods,” Proceedings of the Institution of Mechanical Engineers Part K: Journal of Multi-Body Dynamics, vol. 222, no. 4, pp. 327–338, 2008. View at Publisher · View at Google Scholar · View at Scopus
  5. S. Göttlich, S. Hoher, P. Schindler, V. Schleper, and A. Verl, “Modeling, simulation and validation of material flow on conveyor belts,” Applied Mathematical Modelling, vol. 38, no. 13, pp. 3295–3313, 2014. View at Publisher · View at Google Scholar · View at Scopus
  6. P. A. Langston, U. Tüzün, and D. M. Heyes, “Discrete element simulation of granular flow in 2D and 3D hoppers: dependence of discharge rate and wall stress on particle interactions,” Chemical Engineering Science, vol. 50, no. 6, pp. 967–987, 1995. View at Publisher · View at Google Scholar · View at Scopus
  7. V. L. Popov, Contact Mechanics and Friction: Physical Principles and Applications, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar
  8. H. P. Zhu and A. B. Yu, “Averaging method of granular materials,” Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, vol. 66, no. 2, Article ID 021302, 2002. View at Publisher · View at Google Scholar · View at Scopus
  9. H. P. Zhu and A. B. Yu, “Micromechanic modeling and analysis of unsteady-state granular flow in a cylindrical hopper,” Journal of Engineering Mathematics, vol. 52, no. 1–3, pp. 307–320, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. M. Garavello and B. Piccoli, “Conservation laws models,” in Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, Ill, USA, 2006. View at Google Scholar
  11. C. D’Apice, S. Göttlich, M. Herty, and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  12. R. M. Colombo, M. Garavello, and M. Lécureux-Mercier, “A class of nonlocal models for pedestrian traffic,” Mathematical Models and Methods in Applied Sciences, vol. 22, no. 4, Article ID 1150023, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. R. M. Colombo and M. Lécureux-Mercier, “Nonlocal crowd dynamics models for several populations,” Acta Mathematica Scientia—Series B: English Edition, vol. 32, no. 1, pp. 177–196, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. J. H. Evers, S. C. Hille, and A. Muntean, “Mild solutions to a measure-valued mass evolution problem with flux boundary conditions,” Journal of Differential Equations, vol. 259, no. 3, pp. 1068–1097, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  15. A. Aggarwal, R. M. Colombo, and P. Goatin, “Nonlocal systems of conservation laws in several space dimensions,” SIAM Journal on Numerical Analysis, vol. 53, no. 2, pp. 963–983, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Göttlich, A. Klar, and S. Tiwari, “Complex material flow problems: a multi-scale model hierarchy and particle methods,” Journal of Engineering Mathematics, vol. 92, pp. 15–29, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. S. Canic, B. Piccoli, J.-M. Qiu, and T. Ren, “Runge-Kutta discontinuous Galerkin method for traffic flow model on networks,” Journal of Scientific Computing, vol. 63, no. 1, pp. 233–255, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. B. Cockburn and C. W. Shu, “The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems,” Journal of Computational Physics, vol. 141, no. 2, pp. 199–224, 1998. View at Publisher · View at Google Scholar
  19. V. Gowda and J. Jaffré, “A discontinuous finite element method for scalar nonlinear conservation laws,” Rapport de Recherche INRIA, Institut National de Recherche en Informatique et en Automatique, Rocquencourt, France, 1993. View at Google Scholar
  20. J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods-Algorithms, Analysis, and Applications, Springer, Berlin, Germany, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  21. H. Hoteit, P. Ackerer, R. Mosé, J. Erhel, and B. Philippe, “New two-dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes,” International Journal for Numerical Methods in Engineering, vol. 61, no. 14, pp. 2566–2593, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. S. Sun and M. Dong, “Continuum modeling of supply chain networks using discontinuous Galerkin methods,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 13-16, pp. 1204–1218, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. S. Zhang, S. Sun, and H. Yang, “Optimal convergence of discontinuous Galerkin methods for continuum modeling of supply chain networks,” Computers & Mathematics with Applications, vol. 68, no. 6, pp. 681–691, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  25. F. Bassi and S. Rebay, “High-order accurate discontinuous finite element solution of the 2D Euler equations,” Journal of Computational Physics, vol. 138, no. 2, pp. 251–285, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. B. J. Block, M. Lukáčová-Medvid’ová, P. Virnau, and L. Yelash, “Accelerated GPU simulation of compressible flow by the discontinuous evolution Galerkin method,” European Physical Journal: Special Topics, vol. 210, no. 1, pp. 119–132, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. B. Cockburn, G. Karniadakis, and C. Shu, Discontinuous Galerkin Methods. Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Springer, 2000.
  28. D. Gottlieb and C.-W. Shu, “On the Gibbs phenomenon and its resolution,” SIAM Review, vol. 39, no. 4, pp. 644–668, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  29. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer, Berlin, Germany, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  30. J. S. Hesthaven and R. M. Kirby, “Filtering in legendre spectral methods,” Mathematics of Computation, vol. 77, no. 263, pp. 1425–1452, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus