Complex material flow problems: a multi-scale model hierarchy and particle methods


Material flow simulation is in increasing need of multi-scale models. On the one hand, macroscopic flow models are used for large-scale simulations with a large number of parts. On the other hand, microscopic models are needed to describe the details of the production process. In this paper, we present a hierarchy of models for material flow problems ranging from detailed microscopic, discrete element method type, models to macroscopic models using scalar conservation laws with non-local interaction terms. Numerical simulations are presented at all levels of the hierarchy, and the results are compared to each other for several test cases.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. 1.

    Hoher S, Schindler P, Göttlich S, Schleper V, Röck S (2011) System dynamic models and real-time simulation of complex material flow systems. In: ElMaraghy HA (ed) Enabling manufacturing competitiveness and economic sustainability 2012, Part 3, pp 316–321

  2. 2.

    Jahangirian M, Eldabi T, Naseer A, Stergioulas LK, Young T (2012) Simulation in manufacturing and business: a review. Eur J Oper Res 203:1–13

    Article  Google Scholar 

  3. 3.

    Reinhart G, Lacour F (2009) Physically based virtual commissioning of material flow intensive manufacturing plants. In: Zaeh MF, ElMaraghy HA (eds) 3rd international conference on Changeable, Agile, Reconfigurable and Virtual production (CARV 2009), pp 377–387

  4. 4.

    Cleary PW, Sawley ML (2002) DEM modeling of industrial granular flows: 3D case studies and the effect of particle shape on hopper discharge. Appl Math Model 26:89–111

    Article  MATH  Google Scholar 

  5. 5.

    Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Geotechnique 29:47–65

    Article  Google Scholar 

  6. 6.

    Lätzel M, Luding S, Hermann HJ (2000) Macroscopic material properties from quasi-static, microscopic simulations of a two-dimensional shear-cell. Granular Matter 2:123–135

    Article  Google Scholar 

  7. 7.

    Campbell CS (1990) Rapid granular flows. Annu Rev Fluid Mech 22:57–92

    Article  ADS  Google Scholar 

  8. 8.

    Göttlich S, Hoher S, Schindler P, Schleper V, Verl A (2014) Modeling, simulation and validation of material flow on conveyor belt. Appl Math Model 38:3295–3313

    Article  Google Scholar 

  9. 9.

    Lun CKK, Savage SB, Jeffrey DJ, Chepurniy N (1984) Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield. J Fluid Mech 140:223–256

    Article  ADS  MATH  Google Scholar 

  10. 10.

    Carrillo JA, D’Orsogna MR, Panferov V (2009) Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models 2:363–378

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Etikyala R, Göttlich S, Klar A, Tiwari S (2014) Particle methods for pedestrian flow models: from microscopic to non-local continuum models. Math Models Methods Appl Sci 24:2503–2523

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Colombo R, Garavello M, Lecureux-Mercier M (2012) A class of non-local models for pedestrian traffic. Math Models Methods Appl Sci 22:1150023

    Article  MathSciNet  Google Scholar 

  13. 13.

    Babic M (1997) Average balance equations for granular materials. Int J Eng Sci 35:523–548

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Zhu HP, Yu AB (2002) Averaging method of granular materials. Phys Rev E 66:021302

    Article  ADS  Google Scholar 

  15. 15.

    Zhu HP, Yu AB (2005) Micromechanic modeling and analysis of unsteady-state granular flow in a cylindrical hopper. J Eng Math 52:307–320

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Braun W, Hepp K (1977) The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun Math Phys 56:101–113

  17. 17.

    Neunzert H (1977) The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles. Trans Fluid Dyn 18:663–678

    Google Scholar 

  18. 18.

    Cañizo JA, Carrillo JA, Rosado J (2011) A well-posedness theory in measures for some kinetic models of collective motion. Math Models Methods Appl Sci 21:515–539

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Chuang YL, D’Orsogna MR, Marthaler D, Bertozzi AL, Chayes L (2007) State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Phys D 232:33–47

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Topaz CM, Bertozzi AL (2004) Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J Appl Math 65:152–174

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Huang F, Marcati P, Pan R (2005) Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch Ration Mech Anal 176:1–24

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Tiwari S, Kuhnert J (2007) Modelling of two-phase flow with surface tension by finite point-set method (FPM). J Comp Appl Math 203:376–386

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. 23.

    Klar A, Tiwari S (2014) A multi-scale meshfree particle method for macroscopic mean field interacting particle models. Multiscale Model Simul 12:11671192

    Article  MathSciNet  Google Scholar 

  24. 24.

    Degond P, Mustieles FJ (1990) Approximation of diffusion equations by deterministic convections of particles. SIAM J Sci Stat Comput 11(2):293–310

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Aronson DG (1969) Regularity properties of flows through porous media. SIAM J Appl Math 17:461–467

    Article  MATH  MathSciNet  Google Scholar 

Download references


This work was supported by the German research foundation, DFG grants KL 1105/20-1, GO 1920/3-1 and the BMBF Project KinOpt.

Author information



Corresponding author

Correspondence to S. Göttlich.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Göttlich, S., Klar, A. & Tiwari, S. Complex material flow problems: a multi-scale model hierarchy and particle methods. J Eng Math 92, 15–29 (2015).

Download citation


  • DEM material flow simulation
  • Diffusive limits
  • Hydrodynamic limits
  • Interacting particles
  • Mean field equations