The Material Point Modeling of Granular Flow in a Silo
Zdzislaw A. Wieckowski, Chair of Mechanics of Materials, Technical University of Lodz, al. Politechniki 6, Lodz, Poland

The problem of flow of a granular material during the processes of silo discharge and silo filling is considered in the paper. The problem is interesting not only from the view-point of a designer that should calculate interactions between the flowing material and silo walls but also from the view-point of a user that should predict the flow pattern and the flow rate.

The analysis of the considered problems is of high complexity--the problem is dynamic, geometrically and materially non-linear. This complexity is a reason that, in the past but also nowadays, various kinds of simplifications are used in the analysis of the problem. These simplifications have geometrical or physical nature and, usually, lead to violation of some relations of continuum mechanics. As examples of such simplified approaches, the papers by Janssen, Jenike and Walters can be mentioned. In eighties of the previous century, some authors, namely: Eibl, Häussler and Rombach, solved the problem of silo discharge, stated correctly from the view-point of continuum mechanics, using the finite element method. They used Eulerian description of motion in their analysis and treated the flowing granular material as a fluid. Such an approach is very useful when the region occupied by the flowing material is constant in time which means that the silo is refilled continuously with the material. This approach was used also by other researchers, e.g. Runesson and Nilsson. On the other hand, the use of the finite element method formulated in the Lagrangian description of motion is useful when the entire process of silo discharge is analyzed [7]. In this approach, however, mesh re-zoning must be applied in order to restore proper shapes of elements which are subjected to large distortions. Remeshing has to be accompanied by mapping of the state variables from a deformed mesh to a newly-generated one. This mapping is not an easy task and introduces additional computational errors which deteriorate the robustness of the method. The entire process of silo discharge as well as the filling process can be solved by the discrete element method, in which the granular material is treated as a set of balls or cylinders of equal diameter. It seems, however, that this method is limited to the case when the ratio between the diameter of a silo outlet and the diameter of a material grain is not very large. In order to eliminate disadvantages of the Lagrangian approach to the finite element method related to large elements distortions, the material point method has been utilized to the silo discharge problem by the author of the present paper and his co-workers [8,9], and recently by Oger et al. [5] and Mühlhaus et al. [4]. The material point method, the technique well-known in fluid mechanics as the particle-in-cell method, belongs to the group of computational approaches known as point-based methods.

The point-based methods, also called particle methods or meshless methods, overcome the main drawback of the finite element method related to the mesh distortions. Their common feature is that the history of the state variables is traced at the points (particles) which are not connected with any element mesh the distortions of which is a source of numerical difficulties. With respect to an approximate technique used, the point-based methods can be classified in several basic groups: methods based on the moving weighted least square (MWLS) approximation, kernel methods, partition of unity methods, moving kriging interpolation method and the material point (particle-in-cell) method. An overview of the point-based methods can be found, e.g., in the article by Belytschko et al. [1].

The material point (MPM) method, introduced originally in fluid dynamics by Harlow and co-workers (see [3] and references therein) and known as the particle-in-cell (PIC) method, has been applied successfully to the problems of solid mechanics by Burgess, Sulsky, Brackbill, Chen and Schreyer [2,6]. The material point method can be regarded as the finite element method formulated in an arbitrary Lagrangian-Eulerian description of motion. State variables for the analyzed body are traced at the set of points (material points) defined independently of an Eulerian mesh (computational mesh) on which the equations of motion are formulated and solved. As the computational mesh can be defined in an arbitrary way, the problem of mesh distortion, which leads to difficulties in an Lagrangian formulation, is avoided.

In the present paper, the considered problem of granular flow in a silo is solved by the use of the material point method. Various constitutive models have been used in the analysis of the considered problem by the author of the present paper in order to describe the mechanical behavior of the granular material, namely: the non-associative elastic-plastic and elastic-viscoplastic models with the Drucker-Prager yield condition and several hypoplastic models. In the case of elastic-viscoplastic model, the pressure-volume change relation is non-linear elastic with accompanying viscoelastic term. The interactions between the flowing material and silo walls are described as the unilateral frictional contact problem; the Coulomb model of friction is implemented.

Several numerical examples of plane and axisymmetric silo discharge and filling problems have been analyzed. The method allows to analyze the problems with complex geometry; the examples of flows around silo inserts (see Figures) and flow in the silo of the ``silo-in-silo'' type have been also investigated. In the case of silo discharge problem, the flow rate has been calculated and compared with the value obtained from an empirical formula; good agreement of both the results has been obtained.

Figure 1: Plane flow in a silo with inserts
Image flow_profile.jpg
Figure 2: Wall tractions, normal tractions: red lines, tangential tractions: blue lines
Image tractions.jpg

References

[1] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments, Comp. Meth. Appl. Mech. Eng., 139, 3-47, (1996).

[2] D. Burgess, D. Sulsky and J.U. Brackbill, Mass matrix formulation of the FLIP particle-in-cell method, Journal of Computational Physics, 103, 1-15, (1992).

[3] F.H. Harlow, The particle-in-cell computing method for fluid dynamics, in B. Adler, S. Fernbach, M. Rotenberg (eds.), Methods for Computational Physics, Vol. 3, Academic Press, New York, 319-343, (1964).

[4] H.-B. Mühlhaus, H. Sakaguchi, L. Moresi and M. Fahey, Discrete and continuum modelling of granular materials, in P.A. Vermeer, S. Diebels, W. Wehlers, H.J. Herrmann, S. Luding and E. Ramm (eds.), Continous ans Discontinuous Modelling of Cohesive-Frictional Materials, Springer, 185-204, (2001).

[5] L. Oger, S.B. Savage and M. Sayed, Granular flow using particle-in-cell approach, Proc. 4th Euromech Conf., Metz, France, June 26-30, 2000.

[6] D. Sulsky, Z. Chen and H.L. Schreyer, A particle method for history-dependent materials, Comp. Meth. Appl. Mech. Eng., 118, 179-196, (1994).

[7] Z. Wieckowski, M. Klisinski, Finite deformation analysis of motion of granular materials in a silo, Arch. Mech., 47, 617-633, (1995).

[8] Z. Wieckowski, S.K. Youn and J.H. Yeon, A particle-in-cell solution to the silo discharging problem, Int. J. Numer. Meth. Eng., 45, 1203-1225, (1999).

[9] Z. Wieckowski, The material point method in large strain engineering problems, Computer Methods in Applied Mechanics and Engineering, 193, 4417-4438, (2004).

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Process Modeling of Bulk Solid Systems in Industrial Applications

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