An algorithm to generate random dense arrangements discs based on the triangulation J-F. Jerier, F.V. Donze, D.Imbault Laboratoire Sols, Solides, Structures et Risques, Grenoble, France
[email protected] [email protected] [email protected]
ABSTRACT Discrete element methods are emerging as useful numerical analysis tools for engineers concerned with granular materials such as soil, metallurgy powders, food grains, or pharmaceutical powders, etc... Initially, the first step in a discrete element simulation is the generation of the discs (2D) or spheres (3D) in the boundaries. Today, the complex shapes of study domain coupled to simulations more and more accurate induce a very time-consuming for DEM simulations. In spite of a variety of specimen generation methods for particulate materials, the difficulty is to obtain a dense, stable and homogeneous granular packing. We propose a geometric algorithm (in two dimensions) based on the triangulation of domain to fill up all type of geometries with a variety of assemblages. The concept is based on the triangulation of the domain and the filling of these triangles with discs at a local level. The main advantages of this approach are the small computational cost in the generation of granular packing and the different possible arrangements.
INTRODUCTION The first algorithms create dense random initial arrangement with elements of various sizes have been the Random Loose Packing algorithm (George et al. 1990) optimised by compression and vibration algorithm (Torquato et al. 2000). Now, the challenge is to create algorithms efficient that can produce quickly a random arrangement of millions of densely packed particles. The interest of this task is an application field well-known as soil and rock mechanics and foundation engineering and in other areas. These last years, the researches have developed dynamic or quasi-static or geometric approaches to simulate packing should be at least similar to those of the real structures investigated. This paper presents an algorithm geometric in 2D which can generate a variety arrangement that the engineers use in the DEM simulations by different optimisation level.
D1
With the current computing speed processing, it’s very fast to divide a region by different triangular mesh, then our idea has controlled the replenishment level to propose at users granular arrangements with various density differently oriented. In this review, firstly we present and discuss different methods which can recently be found in the DEM software and the literature. Then, we detail the main steps of the proposed twodimensional algorithm. In the last part, we talk about axes of development envisaged like the passage in three dimensions.
PRESENTATION AVAILABLE APPROACHES Dynamic techniques Most engineers use “the mechanic” to achieve the densification of their initial arrangements. This approach has many advantages to permit the filling up of all geometries with elements size predefine by the users and with a density controlled. One of these algorithms (hopper to mix) consists to generate a regular arrangement and to create a disorder or a random mixture with discs fall under the gravity force (Feng et al 2003) (cf Figure 1).
a)
b)
c)
Figure 1. Dynamic algorithm (a: Regular arrangement b: DEM simulations c: Dense random discs packing)
Others algorithms are using the DEM simulation instituted by CUNDALL (Cundall 1971, Cundall & Starck 1979) to realise the initial arrangement with any variants as the walls translation (P.Stroeven & M. Stroeven 2003) or the gradually increase of discs (LS-algorithm) (Lubachevsky & Stillinger 1990, Torquato et al. 2002) or the inevitable dropping method or ballistic method (RBD) (Radjai et al 2002, Jullien & Meakin 2000). In the aim to be very exhaustive on the dynamics techniques, we discuss on the large diversity of packing obtained, with the possibility to create an arrangement completely isotropic with a size distribution respected. To realize it, the method consists to start a loose packing, then to increase the size discs and to reduce parallel the coefficient friction (µ) with no gravity for at final that the user has a dense packing isotropic (Chareyre 2003). In spite of all packings which are able to be realized with this technique, the limitations of these techniques include a computational cost expensive for the time associated with the DEM simulations required to increase the density of the system.
D2
Quasi-Static techniques The quasi-static approach is composed algorithms and methods which haven’t used DEM simulations to generate assemblies but rather Monte-Carlo simulation. To support this hypothesis, we take like example the Random Loose Packing (RLP) (George et al. 1990) which is proposed by Itasca in his software (PFC2D). This approach consists of a random number generator to define a random size and location for each particle within problem domain and if the particle is not overlapping with an existing particle occurs; the radius and the location are retained. This last algorithm is simple and very fast, it’s often used to generate element within domain before the densification because it present a density very low (ρ=0.5 in 2D ρ=0.3 in 3D). Then, the researchers decided to exploit the RLP’s advantages by the development of an algorithm which generates discs with a random distribution of centers and which eliminates overlaps (Jodrey & Tory 1985, Bagiel & Moscinski 1991). The packing generated is homogenous and dense but in reason of the elimination of overlaps it requires a very long simulation times.
Geometric techniques Another type of approaches, called constructive algorithms, may provide advantageous alternatives to the dynamical methods or to the quasi-static methods. The basic feature of these algorithms is that the assemblies are directly created with the help of purely geometrical calculations, without simulating the dynamics of particle motions. The granular literature contains several methods to produce random sets of equal or non-equal discs or spheres; we can mention for the 2 dimensions the excellent algorithm based on the advancing front approach developed by FENG & al (Feng et al. 2003). This algorithm consists to start generating the disks from the (bottom) boundary in starting from the left corner. Every disc placed constitutes the front and all the others layers will be built upon the first layer following a similar procedure (cf Figure 2).
Figure 2. Advancing front (a: first layer discs, b: all discs generated, c: the disks inside the domain)
It’s necessary to know that an ameliorated version of the advancing front approach has been developed by K.BAGI (Bagi 2005), this amelioration permits to have an optimal filling of all close form. This method presents a various positives points to the dynamics techniques like the time of simulation, the size distribution controlled by users, an isotropic orientation and a density which sometime is the same. This method developed by K.BAGI is very efficient and can fill up all polyhedral shape but its main disadvantage is the passage in 3 dimensions with spheres which stays a real challenge. The only ones in 3 dimensions which concurrence the dynamic
D3
algorithms are theses developed by S. R Williams & A. P Philipse (Williams & Philipse 2003) and R.M Kadushnikov & E.Yu (Kadushnikov & Yu 2001) which is the algorithm of S3D SpheroPack software. But the hope developing other algorithms is feasible from the algorithm of L.Cui & C.O’Sullivan (Cui & O’Sullivan 2003) for 2D and 3D assemblies discs or spheres (cf Figure 3). This method is based on filling up by indiscs or inspheres of triangular or tetrahedral meshes in the domain of interest. The resulting systems for the density and the average coordination number are may be low but the advantages of this algorithm are, its simplicity, its easy generalization for 3D, and the very small computational cost associated with the preparation of an arrangement.
a)
b)
Figure 3. Algorithm L.Cui & C.O’Sullivan (a :indiscs insertion b : insphere insertion)
In more the interest to persevere in the development and in the optimisation of this method is the possibility to evolve at a local scale and on a simple geometry which is the elementary mesh (triangle or tetrahedron). In this paper is proposed any ways optimisation for the density and the average coordination number in the case in 2D while keeping a short simulation times.
OPTIMISATION OF THE TRIANGULATION BASED APPROACH During years the discs packing have used by the mesh algorithm to subdivide boundaries and to create and to ameliorate the quality of mesh (Bern & Eppstein 1997). Since, the researchers in granular materials have noted the progress mesh algorithm and they exploited the VornoiDelauney cells to analyse and characterise the granular assemblies (Alinchenko et al 2004). Today, the triangulation is the support constructive methods (Cui & O’Sullivan 2003) and its permits to traverse the study domain in scanning every mesh triangles. Then, the interest of the algorithm proposed is to give any steps a solution to optimise the filling up of domains The process consists of the following main steps: Level 1: 1) Mesh the domain with the GMSH software (cf Figure 4) 2) The users choose the diameter discs 3) List every triangle1, every segments of triangle1, as well as the points associated (number, coordinates) with file text of GMSH. 4) Place a first serie of discs tangent to the boundaries nodes and on the domain nodes (cf Figure 5) 4) List the discs placed at nodes corresponding 5) Place a second serie of discs at the middle of the segment in the domain (cf Figure 6) 6) Verification 1 7) Density 1 (no yet integrated in the code) 8) List the discs placed associated at segments correspondent 9) Selection every triangle one by one 10) Set up one disc on the center of incircle at the triangle1 (cf Figure 7) 11) Verification 2 D4
12) Density 2 (no yet integrated in the code) Level 2: 1) Divide the triangle in six little triangles (triangle2) composed vertex, center of incircle and the segment middle 2) Set up one disc on the center of incircle at the triangle2 3) Verification 3 4) Density 3 (no yet integrated in the code)
Figure 4. Domain meshing
Figure 5. Setting up discs at nodes
Figure 6. Setting up one disc at segment middle
Figure 7. Setting up one disc on the incircle
The first step in our algorithm is to put discs on the domain nodes and tangent at boundaries nodes, then at middle domain segments. Every discs placed will be associated at one node or at one segment in a matrix, so when the triangles are scanning for the filling up, the program know the discs neighbours. Every disc placed later like these on the center of incircle will be notified. Once that the disc is putting on the middle segment or in the triangle, there is a step of verification which consists to eliminate the overlaps with the discs very near. For it, there is an identification of maximal overlapping then the reduction of diameters two spheres concerned. The next step after Verification may be Density which permits to finalise the arrangement in increasing the discs diameter until contact other disc or boundaries. Due to at this step in algorithm, the code can propose at user different arrangements (porosity, coordination number, distribution size). Knowledge that more the program is running, more D5
the arrangement will be dense with a distribution size near diameters selected at start. For the moment, the step Density and the level 2 isn’t again implemented in the code. But we can suppose that the results will be convincing.
CONCLUSION This paper proposes an algorithm in the specimen generation method which consists to fill up a study domain in 2D with discs in various levels due to the triangular mesh. The triangular mesh of domain is a tool very efficient to develop a constructive algorithm because the method which manages to fill up triangles with elements can fill up any domain shapes with the support of this tool. Then, the idea kept by us is to put discs in the domain with easy geometrics features and to implement this feature at different algorithm levels. This protocol coupled at the different mesh orientation gives the advantage to generate a multitude of arrangements for one domain. Actually, the aim to reach is to finish the algorithm and to exploit the results (orientation tensor, coordination number, density and distribution size) obtained to realize the comparison with other methods. In the future, our method has for final objective to pass in 3D. The performance realized in 2D can be reproducible with one dimension in more due to the same tool which will be a tetrahedral mesh of domain. In spite of fact that for the moment the tetrahedral approach generates a density as low as Random Loose Packing in 3D [18], we think with the same approach that in 2D, we arrive to fill up the voids.
REFERENCES Torquato, Truskett & P. G. Debenedetti “Is Random Close Packing of Spheres Well Defined?” Phys. Rev. Lett. 84, number 10 (2000) Feng, Han & Owen, “Filling domains with disks: An advancing front approach” Methods in Engineering 56(5) (2003), p. 699–731 Cundall, Proc. Symp. ISRM, Vol. 1, II-8, Nancy, France, 1971. Cundall & Strack, Geotechnique 29 (1979) 47–65. P.Stroeven & M.Stroeven “Dynamic computer simulation of concrete on different levels of the microstructure-Part1” Computer simulation of concrete 2003 Lubachevsky and F.H. Stillinger, J. Stat. Phys. 60, 561 (1990). Torquato, Kansal & Frank H. Stillinger “Diversity of order and densities in jammed hard-particle packings” Physical Rev E 66, 041109 (2002) Radjai, Bratberg & Hansen “Dynamic Rearrangements and Packing Regimes in Randomly Deposited Two-Dimensional Granular Beds“ Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Sep Jullien & Meakin “Computer simulations of steepest descent ballistic deposition” Colloids and Surfaces A: Physicochemical and Engineering Aspects Volume 165, (2000), p 405-422
D6
Chareyre « Thesis : Modélisation du comportement d’ouvrages composites sol-geosynthetique par éléments discrets- application aux ancrages en tranchées en tête de talus » (2003) p50-51 George, Onoda & Liniger “Random loose packings of uniform spheres and the dilatancy onset” Phys. Rev. Lett. 64, 2727 - 2730 (1990) Jodrey & Tory “Computer simulation of close random packing of equal spheres” Physical Review A, Volume 32 number 4 (1985) Bagiel & J.Moscinski “C-language program for the irregular close packing of hard spheres” Computer Physics Communications 64 (1991) 183-192 Feng, Han & Owen “Filling domains with disks: an advancing front approach” Int. J. Numer. Meth. Engng 2003; 56:699–713 Bagi “An algorithm to generate random dense arrangements for discrete element simulations of granular assemblies” Granular Matter Volume 7, Number 1 p 31-43 (2005) Williams & Philipse “Random packings of spheres and spherocylinders simulated by mechanical contraction” Phys Rev E 67, 051301 (2003) Kadushnikov & Yu “Investigation of the Density Characteristics of Three-Dimensional Stochastic Packs of Spherical Particles Using a Computer Model” Powder Metallurgy and Metal Ceramics Volume 40, Numbers 5-6 (2001) Cui & O’Sullivan :”Analysis of a triangulation based approach for specimen generation for discrete element simulations”. Granular Matter 5(3), 135–145 (2003) Bern & Eppstein, “Quadrilateral meshing by circle packing” Proc. 6th Internat. Meshing Roundtable, Park City, UT (1997), pp. 7–19. Alinchenko, Anikeenko, Medvedev, Voloshin, Mezei & Jedlovszky “Morphology of voids in molecular systems. A Voronoi-Delaunay analysis of a simulated DMPC membrane” J. Phys. Chem. B 2004 108:19056–19067 (2004)
D7
D8
