Multi-sphere approximation of real particles for DEM

Powder Technology 286 (2015) 478–487

Contents lists available at ScienceDirect

Powder Technology journal homepage: www.elsevier.com/locate/powtec

Multi-sphere approximation of real particles for DEM simulation based on a modified greedy heuristic algorithm Cheng-Qing Li a, Wen-Jie Xu a,⁎, Qing-Shan Meng b a b

State Key Laboratory of Hydroscience and Hydraulic Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China

a r t i c l e

i n f o

Article history: Received 16 June 2015 Received in revised form 9 August 2015 Accepted 15 August 2015 Available online 20 August 2015 Keywords: Discrete element method (DEM) Particle shape Multiple overlapping spheres (MOS) Set-covering problem (SCP) Modified greedy heuristic (MGH)

a b s t r a c t In this paper, a new algorithm to approximate real particles using multiple overlapping spheres as numerical models for the discrete element method is introduced. First, we convert the issue of approximating particles with a cluster of multiple overlapping spheres to a set-covering problem. Then, we use an algorithm to solve the set-covering problem in detail. This manuscript presents three different solution schemes based on a modified greedy heuristic algorithm, namely, a body-covering scheme, a surface-covering scheme and a triangular surface-covering scheme. To evaluate the algorithm, we calculated the amount of multiple overlapping spheres, the intersection error of volume or area, the difference set error of volume or area between multiple overlapping spheres and the real particles, and the error of the moment of inertia of the multiple overlapping spheres and the real particles. The parameters used to evaluate the precision of the three different schemes indicated that all three schemes are excellent. It is understood that different schemes offer different levels of precision for different particles with different numbers of multiple spheres. Therefore, it is important to choose the scheme best suited to represent a particular objective. Besides, the computational time is considered as the efficiency of the algorithm. In general, the body-covering scheme approximates complicated particles with the fewer spheres and better accuracy, while the surface-covering scheme realizes the representation with less time for a few particles. However, if a particle is generated by fewer than several thousand triangles, the triangular surface-covering scheme may finish the approximation in shorter time and with fewer multiple spheres. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Cundall and Strack [1,2] introduced the discrete element method (DEM) to solve rock mechanics in 1971. The method further evolved to solve soil mechanics in 1979. The DEM has been widely used in many areas, especially in granular mechanics. The first DEM element type was proposed by Cundall and Strack [1,2]. Various kinds of DEM elements have been developed during the last three decades. Cundall and Strack [1,2] published a DEM program named BALL based on the disc element, and TRUBAL based on the spherical element, which evolved into PFC2D and PFC3D, a popular commercial DEM software. Algorithms incorporating ellipses in the DEM for solving 2-D problems were proposed by Rothenburg and Bathurst [3] and by Ting et al. [4,5], whereas Lin and Ng [6] and Vu-Quoc et al. [7] applied ellipsoidal particles to simulate particle mechanics within the DEM. Moreover, the use of algorithms of artificial particles as elements of the DEM for simulating complex problems was realized by Hogue [8]. The DEM element representation was implemented in the form of polygons representing mechanical properties by Mirghasemi et al. [9]. Höhner et al. [10,11] performed a comparison of ⁎ Corresponding author. E-mail address: [email protected] (W.-J. Xu).

http://dx.doi.org/10.1016/j.powtec.2015.08.026 0032-5910/© 2015 Elsevier B.V. All rights reserved.

the multi-sphere approach with the polyhedral approach to simulate non-spherical particles within the DEM. As contact detection algorithms for ellipses, ellipsoids, artificial particles or polygons are much more complex than the contact detection algorithms for discs in 2-D or spheres in 3-D, disc and sphere elements are the mainstream elements currently utilized in the DEM to analyze mechanical behavior of a single particle or particle assemblies because of their high computational efficiency. However, for 3-D DEM problems, it is not accurate to represent a particle with only one sphere, because one sphere on its own cannot cover all the characteristics of the complex geometrical shape of most granular particles. Favier et al. [12] introduced a multi-sphere approach in which a particle is represented by a cluster of connected spheres whose radiuses may vary or even overlap each other. Potyondy et al. [13] and Walton et al. [14] approximated complex real particles with non-overlapping sphere clusters. Hubbard [15] proposed an algorithm for approximating any strangely shaped objects. However, several thousand spheres are required for adequate approximation of one particle. And it is required millions of spheres to represent a numerical model with thousands of particles. Consequently, mechanical process simulation by the DEM program is time-consuming and the complicated engineering problem cannot be solved. Therefore, urgently solutions are required for simulating particles with fewer spheres and better accuracy,

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487

while computational time and precision taken into consideration. Wang et al. [16] presented an algorithm, named burning algorithm, for approximating particles with fewer non-uniform and nonoverlapping sphere clusters. This algorithm has already been employed widely by PFC3D [17]. As the author of this model noted, the number of spheres can be reduced by 90%, leading to a reduction in required memory and computation time with the burning algorithm. Nevertheless, a few thousand spheres will still be required if the investigators want to accurately represent a particle. Therefore, the burning algorithm is not acceptable because too many spheres will reduce the computational efficiency of DEM. Interestingly, Price et al. [18] realized an automatic method for approximating real particles with a given number of overlapping spheres. LIGGGHTS [19], an open-source discrete element software, implements the multi-sphere approach for approximating real particles. Amberger et al. [20] and Kloss et al. [21] illustrated this approach in detail and applied LIGGGHTS to simulate aggregate discharging and particles packing. Garcia et al. [22] proposed the cluster overlapping spheres (COS) algorithm for approximating particles with only 100 to 200 overlapping spheres. Kruggel-Emden et al. [23] has validated the effectiveness of the multiple overlapping spheres method by applying several DEM software packages to simulate the mechanics problem. Damping taken into consideration, the numerical error will be less if the particle is approximated by fewer spheres with the same approximation accuracy. Based on the idea of the cluster overlapping spheres algorithm, this manuscript presents a modified greedy heuristic (MGH) algorithm. First, the issue of approximating a particle with a cluster of multiple overlapping spheres (MOS) is converted to a set-covering problem (SCP), a classical mathematical problem. Next, the MGH algorithm is applied to represent a particle with 30 to 100 non-uniform MOS. Considering the specificity of the multi-sphere problem, we propose three different schemes in this manuscript. To evaluate the precision of the algorithm, several parameters were used to represent the approximation error between the real particle and the corresponding cluster of MOS. Notably, from the results observed, we found that the MOS approximation schemes provided in this paper can very closely represent real particles. 2. Converting multi-sphere approximation to a set-covering problem The point clouds, which generate the triangular network with topological structures that characterize the particle surface, can be obtained with an army of established tools. Lanaro et al. [24,25] used a 3-D laser to acquire the shape of a coarse aggregate, while Garboczi [26] and Taylor et al. [27] used X-ray computed tomography to scan the particle. Moreover, Rocchini et al. [28] and Hall-Holt and Rusinkiewicz [29] presented an approach for low cost real-time structured light scanning. The primary objective is to produce a cluster of MOS with spheres as few as possible to match the particle's surface as closely as possible. For multi-sphere approximation, the particle is discretized with numerous uniform small cells, which essentially implies that the particle will be filled with that many cells. The cells can be small spheres, cubes, or even just points, as long as the cells fill in the particle uniformly and completely. Therefore, a collection of cells can finely characterize the particle in terms of volume and surface. And in this paper the cells are spherical cells. Once the cell cluster is well represented by a cluster of MOS, the particle is essentially well approximated. Therefore, the target is to find a cluster of MOS that covers the entire cell collection and does not exceed the boundary of the cell collection. Specifically, we can generate a cluster of candidate spheres whose center lies on the center of the cell and whose radius is related to the minimum distance between the sphere's center and the cell's center. This means that the number of spheres will equal to the number of cells, and the sphere on the boundary cell's center is as large as the cell. One particle is filled with a pack of n cells, and n spheres are generated to contain all of the cells. The cell collection will be covered

479

perfectly by the sphere assembly made up with n spheres. Notably, this sphere assembly is not what we are interested in, because the number of spheres is quite large. The core aim of this exercise is to select a cluster of MOS that covers the cell collection completely with the least number of spheres from the sphere assembly. If each cell is regarded as an element, the cell collection will constitute a universal set U = {C1, C2, C3, ⋯, Cn} in which Ci, i = 1, 2, 3, …, n represents each cell. As mentioned above, each sphere contains several cells, such as C2, C35, C59, …, Ck, 1 ≤ k ≤ n. All cells contained in one sphere constitute a subset of U, denoted by Si ¼ fC i1 ; C i2 ; C i3 ; …; C i j g, in which 1 ≤ i1, i2, i3, …, ij ≤ n, Si ⊆ U, such as Si = {C2, C35, C59, …, Ck}, 1 ≤ k ≤ n. Moreover, the union of all Si is equal to U : S1 ∪ S2 ∪ S3 ∪ … ∪ Sn = U. Simply put, all Si constitute another universe set P = {S1, S2, S3, …, Sn}. Therefore, the problem that a cluster of MOS contain all of the cell collection is equal to the issue that a number of Si contain all the Ci, for both i = 1, 2, 3, …, n. That is, we need to find a subset P i ¼ fSi1 ; Si2 ; Si3 ; …; Si j g; 1≤ i1 ; i2 ; i3 ; …; i j ≤n of P to make Si1 ∪ Si2 ∪ Si3 ∪…∪ Si j ¼ U. Therefore, the optimal solution is to find the set Pi, whose number of elements is the least. This is a typical SCP [30,31]. SCP is a notoriously difficult optimization problem. It is known as one of Karp's NPcomplete problems [32], shown to be NP complete in 1972. For an NPcomplete problem, a polynomial-time algorithm with input size n, the worst-case running time O(nk) is observed for a constant k, and its exact solution has not been discovered thus far. With increasing size n, the complexity of the perfect solution may grow extremely quickly. One of the most popular solutions for a SCP is the greedy heuristic algorithm, which will return a set that is not too much larger than the optimal set. The greedy heuristic algorithm with logarithmic approximation error was introduced in detailedly by Chvatal [33] and Cormen et al. [34]. The greedy heuristic algorithm for the SCP can be denoted by the following pseudo-code: Greedy–Set–Cover (U, P) K=U L=∅ while K ≠ ∅ select an S ∈ P that maximizes |S ⋂ K| K=K−S L = L⋃{S} return L. The symbols U, P, S are the same with above meaning. K is an intermediate variable while finding the optimal result. K equals to the universal set U at first and will turn to smaller and smaller as the program runs. L is used to store all the S that are chosen. The program will stop and return L while K is empty. It has been proven that the greedy heuristic algorithm is a polynomial-time (ln|X| + 1)-approximation algorithm by Cormen et al. [34]. Importantly, the computational time consumption of the approximation solution by the greedy heuristic algorithm grows much more slowly with the size of instance, relative to the optimal solution. Based on the greedy heuristic algorithm, this manuscript provides the MGH algorithm for multi-sphere approximation while considering the particularity of the problem. 3. A modified greedy heuristic algorithm for multi-sphere approximation To explain the MGH algorithm for multi-sphere approximation clearly, the algorithm is segmented into the following four steps: 3.1. Discretizing the particle with a cell collection To discretize the particle with a cell collection, an oriented bounding box covering the particle just right is generated, with length (L) as X

480

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487

axis, width (W) as Y axis and height (H) as Z axis. The number of cells along X axis is assumed to be NX. This NX is directly correlated with the approximation accuracy. The larger NX is, the more accurate the approximation is. Then, the cell number along the Y axis and Z axis is NY ¼ WL NX , NZ ¼ HL N X , respectively. Hence, the radius (r) of the cell is r ¼ 2NL X . At the plane of XOY, first we generate a line segment along the X axis, and then the line segments are generated where Z = 0, Y = 2 × r × i, i = 0,1,2,3,…, NY. At other planes that are parallel to the XOY plane, Z = 2 × r × (0, 1, 2, ⋯, Nz), line segments are generated at each plane. All the line segments start from a point on the YOZ plane and end up at a point on the plane opposite of the YOZ plane of the oriented bounding box. Moreover, the distance along the Y or Z orientations between any two adjacent line segments equals the cell's diameter. As a result, the oriented bounding box is packed with a group of line segments parallel to each other with a separation distance of the cell's diameter. Fig. 1a shows how to generate the line segments group. We can place several points along each line segment that split the line segment with uniform length, each part equal to the cell's diameter. Then, a cell is placed at

Fig. 1. The generation of uniform cells: (a) group line segments; (b) uniform cells are generated to fill the particle.

each point of the line segment. Thus, line by line, a cell collection finally fills the box completely. The majority of the lines pass through the particle inside the oriented bounding box with two intersection points or more. Generally, there are two intersection points if the particle is simply connected space and four, six or even more intersection points, if the particle is complicatedly connected domain. The intersection points are in sequence named P1, P2, P3, P4 …. The cells between P1 and P2, P3 and P4 are all inside the particle. Consequently, these cells should be added to the cell collection. If the particle is split homogeneously by the line segments, then the particle will be completely packed with a cluster of uniform cells. Fig. 1b shows how the particle is packed with uniform cells. 3.2. Marking the boundary of the cell collection It is extremely necessary to mark the boundary of the cell collection, because it characterizes the particle surface, which is the most quintessential feature. The cell collection is generated one by one, just like the image formation procedure of an oscilloscope. The boundary can be sought out while generating the cell collection simultaneously. Fig. 2a shows a 2-D illustration, where a clump of circles approximate the complex polygon. The outer-most layer circles are the boundary of the

Fig. 2. Boundary of the cell collection: (a) 2-D; (b) 3-D.

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487

polygon. Each of the inner circles is surrounded by four circles: the upper one, the bottom one, the left one and the right one. Moreover, the boundary circle is surrounded by fewer than four circles. For the 3-D particle, the inner cells are surrounded by six cells: the upper one, the bottom one, the left one, the right one, the front one and the back one. Fig. 2b shows the boundary of the cell collection and the inner cells. In terms of the code, each cell has a tag with an initial value of zero. Once a cell is added into the cell collection, it is necessary to check if there is a cell on the left. If it is true, both the tags' values of this cell and the left cell will be added by one. As for the other directions, the tags' values will be added by one if there is a cell. Thus, all the cells with tag's value fewer than six are boundary cells and all others are inner cells. 3.3. Computing the radius of the spheres and the set-covering matrix For each cell, a sphere is generated as the candidate of the cluster of overlapping spheres. Once the boundary of the cell collection has been ascertained, the radius of each sphere candidate can be easily calculated. The distance between the center of the sphere and the center of each boundary cell can be computed easily as well. The smallest one will be the radius of the sphere, this is to make sure that the sphere does not exceed the boundary and covers the cells as many as possible. Fig. 3(a) and (b) show how the sphere radius is computed. For each inner cell, a sphere that is tangent to the boundary is placed there. The next step is to find all the cells that are inside their respective spheres by judging whether the distance between the cell's center and the sphere's center is less than the sphere's radius. For each sphere, a column vector with 1 if cell is inside the sphere and 0 if cell is outside

481

the sphere describes the inclusion relationship of all cells and one sphere. As a result, for all the large spheres and cells, a 0–1 matrix well represents the inclusion relationship. For example, C represents the cells and S represents the spheres in the matrix that can be called the SCP matrix.

C1 C2 C3 C4 C5 C6 C7 C8 C9 C 10 C 11 C 12 C 13 C 14

S1 1 0 1 1 0 0 0 0 1 1 0 1 1 0

S2 0 1 0 0 0 1 0 1 1 0 1 0 1 1

S3 1 1 1 0 0 0 1 0 1 0 1 1 0 0

S4 0 0 0 0 0 1 0 0 0 1 0 0 0 0

S5 0 0 1 0 1 1 0 0 0 0 1 0 0 1

S6 0 0 0 0 1 0 1 0 0 1 0 0 0 1

S7 0 0 0 1 0 0 0 0 0 1 1 1 1 0

S8 1 0 1 0 1 0 1 0 1 0 0 0 0 1

ð1Þ

Taking matrix (1) as example, the cross value of Ci and Sj is 1 if Ci is inside Sj, otherwise it is 0. As the matrix (1) shows, the cell C1 is inside sphere S1, S3, S8 and sphere S3 contains C1, C2, C3, C7, C9, C11, C12. As both S1 and S3 contain C1, C3, C9, C12, S1 and S3 are overlapping each other. The purpose is to find as few spheres as possible that can contain all of the cells. Therefore, now the problem is translated into identifying as few columns as possible to form a new matrix, which makes the sum of each row of the new matrix larger than 1. Once the sum of the row is more than one, the cell is contained by at least one sphere. The column No. (number) of the new matrix is the No. of spheres in the spheres cluster. 3.4. Realizing the modified greedy heuristic algorithm

Fig. 3. Calculation of the sphere radius: (a) 2-D; (b) 3-D.

Now the problem is to solve the matrix similarly to matrix (1). If there is only one cell inside one sphere, the sphere which contains that cell must be added to the MOS cluster. Otherwise, that cell will not be contained in any sphere. The MGH algorithm is described with a flow chart in Fig. 4. In the MGH algorithm, the selection criterion is that the newly added sphere will contain the maximum number of new contained cells. If the sum of one row is equal to one, that means the cell is just contained by only one sphere, which should be added to the MOS cluster. Once one column is chosen, all the rows whose value at that column is 1 should be deleted, because those cells (denoted by the rows) have been covered by spheres (denoted by the column). A new matrix with fewer rows is generated. The column of the new matrix with the largest sum is added to the MOS cluster until all the rows are deleted. Till now, all the cells are covered by spheres. The SCP matrix is solved and the MOS cluster is also determined. A multi-sphere approximation of a rounded particle is shown in Fig. 5. Furthermore, this algorithm can be used to approximate all kinds of particles. Fig. 6 shows another four quintessential particles, both simply connected particles and complicatedly connected particles. For all these five examples, NX is set as 50. If NX varies, the approximation result will also change. Additionally, Fig. 7 shows the approximation results of cubic particles with different NX and the result of cubic particles with NX = 50 can be found at Fig. 6(a). Approximation of particle surfaces is vital in many applications. Discretization of the surface is much more desirable than the discretization of the whole particle. Therefore, it is much more reasonable to ensure that all the boundary cells are covered by spheres. Additionally, it is not necessary to judge whether the inner cells are in the large spheres. As a result, the rows of the SCP matrix are the boundary cells, and the solution for the matrix is the same, with varying

482

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487

Fig. 4. Flow chart of the modified greedy heuristic algorithm.

results. In this manuscript we call this approach a surface-covering (SC) scheme and the last approach a body-covering (BC) scheme. For particle surface representation by the triangulation network, the surface can be discretized by many non-uniform cells, which will better characterize the particle surface than the two ways described above. A cell is placed at each triangle of the triangulation network. In this manuscript, the radius of the cell is defined as the mid-side of the triangle, and the center is defined as the inner normal of the triangle with distance of radius to the center of the triangle. Fig. 8 shows a particle represented by non-uniform cells. Each cell is given a weighting coefficient, which is equal to the area of the triangle. This weighting coefficient reasonably represents the surface area of the particle. Using this system we can obtain a matrix similar to that of the BC and SC schemes. The only difference is that a column vector constituting the weighting coefficient accompanies the matrix. This means for each row, there is a coefficient that is equal to the triangle area. This problem is called a weighted SCP. While solving the matrix, the sum of the product of element value for each column and weighting coefficient is the criterion rather than the sum of the column. The column with the maximum sum is chosen, and this approach is named the triangular surface-covering (TSC) scheme. Fig. 9 shows the cubic particle approximation result of different schemes and the result of BC scheme can be found at Fig. 6(a).

4. Analysis of results The aim of the multi-sphere approximation is to represent the real particle as accurately as possible, with as few MOS as possible. Hence,

Fig. 5. Multi-sphere approximation results for rounded particle (RP) with body-covering (BC) scheme and NX = 50 by 36 spheres: (a) the particle; (b) the multi-sphere; (c) the particle with multi-sphere. The blue is the particle and the gray is the multi-sphere.

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487

Fig. 6. Approximation for another four types of particles with body-covering (BC) scheme and NX = 50: (a) cubic particle (CP), 83 spheres; (b) slender particle (SP), 44 spheres; (c) multangular particle (MAP), 73 spheres; (d) complicatedly connected particle (CCP), 118 spheres.

483

Fig. 8. Surface representation of cells generated by surface triangles.

the first parameter to evaluate the approximation scheme is the number of MOS. The fewer the spheres are, the better the approximation scheme is. Accuracy implies that the cluster of MOS covers the particle as completely as possible and exceeds the particle as little as possible. Therefore, it's necessary to calculate the intersection set and difference set of the particle and the MOS. The intersection set describes how much the particle is covered by the MOS and the difference set describes

Fig. 7. Cubic particle (CP) approximation with body-covering scheme and different NX: (a) NX = 20, 18 spheres; (b) NX = 30, 38 spheres; (c) NX = 40, 63 spheres; (d) NX = 60, 109 spheres.

Fig. 9. Cubic particle (CP) approximation result with different schemes and NX = 50: (a) SC scheme, 90 spheres; (b) TSC scheme, 85 spheres.

484

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487

how much the MOS exceeds the particle. The intersection error of volume (IEV) is defined as one minus the volume of the intersection set divided by the volume of the particle and the difference error of volume (DEV) is defined as the volume of the difference set divided by the volume of the particle. Similarly, the intersection error of area (IEA) and difference error of area (DEA) are defined. Additionally, the average error of volume (AEV) and area (AEA) can overall characterize how well the particle is approximated by the MOS.   volume of intersection set 1−  100% volume of particle

IEV ¼



 area of intersection set  100% area of particle

ð2Þ

IEA ¼

1−

DEV ¼

volume of difference set  100% volume of particle

ð4Þ

DEA ¼

area of difference set  100% area of particle

ð5Þ

AEV ¼ ðIEV þ DEV Þ=2

ð3Þ

ð6Þ

Fig. 11. The error of moment of inertia of different particles with different schemes and NX = 50.

AEA ¼ ðIEA þ DEAÞ=2

ð7Þ

The principal moment of inertia is a big influence in the mechanical simulation within DEM. The principal moments of inertia of both the particle and MOS are calculated along X axis, Y axis, Z axis. We defined the error of moment of the inertia (IXE, IYE, IZE) as the moment of the union of the inertia of the MOS divided by the particle's moment of inertia minus one. The closer the IXE, IYE, IZE are to 0%, the more accurate the approximation is. The IXE, IYE, IZE are defined by Eqs. (8)–(10). It's necessary to calculate the union of the MOS.

IX E ¼

  moment of inertia of the union of MOS along X axis −1  100% moment of inertia of the particle along X axis

ð8Þ

IY E ¼

  moment of inertia of the union of MOS along Y axis −1  100% moment of inertia of the particle along Y axis

ð9Þ

Fig. 10. The error of volume and area of different particles with different schemes and NX = 50. (a) IEV, DEV, AEV; (b) IEA, DEA, AEA.

Fig. 12. The parameters that characterize how well the particle is approximated with different NX, cubic particle (CP) with body-covering (BC) scheme.

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487

IZ E ¼

  moment of inertia of the union of MOS along Z axis −1  100% moment of inertia of the particle along Z axis

ð10Þ

Consequently, the intersection error of volume, the difference error of volume, the intersection error of area, the difference error of area, the average error of volume, the average error of area and the error of moment of inertia of X axis, Y axis, Z axis are the nine other parameters for the evaluation of the approximation scheme. For the five particles, the rounded particle (RP), the cubic particle (CP), the slender particle (SP), the multangular particle (MAP), and the complicatedly connected particle (CCP), the triangular network of the particles constitutes 6000 triangles, except for the rounded sphere, which constitutes 1974 triangles.

485

Fig. 10 shows intersection error, difference error, average error of volume and area of different particles with different schemes and NX = 50. Most of the parameters are between 0% ~ 4%, which means that the particle is covered well by the MOS, and the MOS exceeds the particle very little. However, it does not indicate which scheme is the best, as all of the results are very favorable and the distinction is not obvious. For some particles, the BC scheme offers better results and the SC scheme approximates some other particles with better accuracy. For most kinds of particles, the results of the TSC scheme are not as good as that of the other two schemes. Moreover, the results of the complicatedly connected particles are poorer than those of other particles on account of the complex shape. It seems that the TSC scheme may not yield an improved result. However, if the cells are generated with an improved scheme in the surface triangular network, the TSC scheme

Fig. 13. The number of overlapping spheres and the computational time of different particles by different schemes with different NX. (a) Rounded particle (RP); (b) cubic particle (CP); (c) slender particle (SP); (d) multangular particle (MAP); (e) complicatedly connected particle (CCP).

486

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487

may provide a much better result. Furthermore, the TSC scheme gives a direction to improve the MGH algorithm to obtain a better result. Fig. 11 shows error of moment of inertia of different particles with different schemes and NX = 50. For the five particles, most of the IXE, IYE, IZE are between − 3.0%–3.5% that indicates the error of moment of inertia is very small with BC scheme. However, the error of moment of inertia of TSC scheme is a little larger than the other two schemes but acceptable. Furthermore, NX will influence the accuracy of the multisphere approximation. If NX is set smaller, the approximation will be less accurate. Taking the cubic particle within the BC scheme as an example, NX influences the approximation results. Fig. 12 shows that the average error of the volume and area and the error of the moment of inertia reduces as the NX grows. It can be seen that the accuracy will be enough when NX = 40 with respect of the volume, area and moment of inertia. The time of the representation and the number of the spheres of different particles with different schemes and NX is shown in Fig. 13. And the computer configuration is i7-2700 k CPU 3.50 GHz. From Fig. 13, it indicates that the time ~ NX curves are similarly linear. That means that the computational time is exponential relationship with NX. The time of TSC scheme increases slower as NX grows than the other two schemes. Except for the rounded particle (RP), the SC scheme finished the approximation with the shortest time which means that the SC scheme is much more efficient. However, the TSC scheme approximated the rounded particle (RP) with the shortest time because the triangle number of this particle is just 1974, rather than 6000. For the BC scheme and the SC scheme, the number of the spheres grows linear with NX growing. Nevertheless, the number of spheres with TSC scheme doesn't show obvious regulation. Generally, the BC scheme approximated the particle with the fewest number of spheres. However, for the rounded particle, the number of spheres of the TSC scheme is just 13 because the triangle number of this particle is just 1974. Therefore, for most particles, the approximation will result in few spheres and good accuracy of the BC and SC schemes for a short time if NX is set as 40. While NX is set as 40, less than 60 spheres are needed to approximate the simply connected particles and about 80 spheres are needed to approximate the complicatedly connected particles.

5. Conclusion It is very important to approximate the real particle with multiple overlapping spheres while simulating the mechanics problem with DEM. To realize the approximation, this paper transforms the multisphere approximation problem of real particles to a popular SCP and uses a 0–1 matrix to describe it. Based on the MGH algorithm, this manuscript proposes three different schemes to solve it with excellent accuracy and acceptable computational time. The computational time is from less than one second to several minutes. Importantly, the algorithm can solve not only simply connected particles, but also complicatedly connected particles. Five typical particles were approximated with the three schemes and quantitative analysis shows that the number of spheres is few and the average error of both volume and area are 0% ~ 4%. Besides, the error of moment of inertia is less than 3.5%. That's to say that the algorithm can solve the multi-sphere approximation problem almost perfectly. In addition, all the three schemes can represent the particles within a short time. The BC scheme can approximate most particles with fewer spheres and better accuracy and the SC scheme can approximate the particles with better time consumption. However, for some specific particles, the TSC scheme can approximate the particle with fewer spheres, albeit with less accuracy, if the triangles of the triangular network are fewer. Therefore, for most particles, the author advises the BC scheme to approximate particles. Moreover, as the problem has been converted to the SCP, the research advances of SCP by mathematicians and algorithm researchers can be applied to solve the multi-sphere approximation problem.

Acknowledgements The authors would like to acknowledge the project of “Natural Science Foundation of China (51479095)”, “Natural Science Foundation of China (51323014)”, “State Key Laboratory of Hydroscience and Engineering Project (2013-KY-4)”, and “Natural Science Foundation of China (41372316)”. References [1] P.A. Cundall, A computer model for simulating progressive, large-scale movements in blocky rock systems, Proc. Int. Symp. on Rock Fracture 1971, pp. 11–18. [2] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1979) 47–65. [3] L. Rothenburg, R.J. Bathurst, Numerical simulation of idealized granular assemblies with plane elliptical particles, Comput. Geotech. 11 (1991) 315–329. [4] J.M. Ting, M. Khwaja, L.R. Meachum, J.D. Rowell, An ellipse‐based discrete element model for granular materials, Int. J. Numer. Anal. Methods Geomech. 17 (1993) 603–623. [5] J.M. Ting, L. Meachum, J.D. Rowell, Effect of particle shape on the strength and deformation mechanisms of ellipse‐shaped granular assemblages, Eng. Comput. 12 (1995) 99–108. [6] X.S. Lin, T.T. Ng, A three-dimensional discrete element model using arrays of ellipsoids, Geotechnique 47 (1997) 319–329. [7] L. Vu-Quoc, X. Zhang, O. Walton, A 3-D discrete-element method for dry granular flows of ellipsoidal particles, Comput. Methods Appl. Mech. Eng. 187 (2000) 483–528. [8] C. Hogue, Shape representation and contact detection for discrete element simulations of arbitrary geometries, Eng. Comput. 15 (1998) 374–390. [9] A. Mirghasemi, L. Rothenburg, E. Matyas, Influence of particle shape on engineering properties of assemblies of two-dimensional polygon-shaped particles, Geotechnique 52 (2002) 209–217. [10] D. Höhner, S. Wirtz, H. Kruggel-Emden, V. Scherer, Comparison of the multi-sphere and polyhedral approach to simulate non-spherical particles within the discrete element method: Influence on temporal force evolution for multiple contacts, Powder Technol. 208 (2011) 643–656. [11] D. Höhner, S. Wirtz, V. Scherer, A numerical study on the influence of particle shape on hopper discharge within the polyhedral and multi-sphere discrete element method, Powder Technol. 226 (2012) 16–28. [12] J.F. Favier, M.H. Abbaspour-Fard, M. Kremmer, A.O. Raji, Shape representation of axi-symmetrical, non-spherical particles in discrete element simulation using multi-element model particles, Eng. Comput. 16 (1999) 467–480. [13] D. Potyondy, P.A. Cundall, C.A. Lee, Modelling rock using bonded assemblies of circular particles, Rock Mech. (1996) 1937–1944. [14] O. Walton, R. Braun, Simulation of rotary-drum and repose tests for frictional spheres and rigid sphere clusters, Joint DOE/NSF Workshop on Flow of Particulates and Fluids, Ithaca, NY, Sept 1993. [15] P.M. Hubbard, Approximating polyhedra with spheres for time-critical collision detection, ACM Trans. Graph. (TOG) 15 (1996) 179–210. [16] L.B. Wang, J.Y. Park, Y.R. Fu, Representation of real particles for DEM simulation using X-ray tomography, Constr. Build. Mater. 21 (2007) 338–346. [17] PFC3D User's Manual, Itasca Consulting Group, Inc., Minneapolis, Minnesota, USA, 1995. [18] M. Price, V. Murariu, G. Morrison, Sphere clump generation and trajectory comparison for real particles, Proc. Discret. Elem. Model. 2007 (2007). [19] C. Kloss, C. Goniva, LIGGGHTS: a new open source discrete element simulation software, Proceedings of The Fifth International Conference on Discrete Element Methods, London, UK 2010, pp. 25–26. [20] S. Amberger, M. Friedl, C. Goniva, S. Pirker, C. Kloss, Approximation of objects by spheres for multisphere simulations in DEM, ECCOMAS 20122012. [21] C. Kloss, P. SEIL, F. Hauzenberger, S. Amberger, C. Feimayr, S. Pirker, C. Goniva, Simulation of particle segregation in metallurgical furnaces for iron production, 2012. [22] X. Garcia, J. Xiang, J.P. Latham, J.P. Harrison, A clustered overlapping sphere algorithm to represent real particles in discrete element modelling, Geotechnique 59 (2009) 779–784. [23] H. Kruggel-Emden, S. Rickelt, S. Wirtz, V. Scherer, A study on the validity of the multi-sphere Discrete Element Method, Powder Technol. 188 (2008) 153–165. [24] F. Lanaro, L. Jing, O. Stephansson, 3-D-laser measurements and representation of roughness of rock fractures, Proc. Mechanics of Jointed and Faulted Rock, Vienna, AA Balkema, Rotterdam 1998, pp. 185–189. [25] F. Lanaro, P. Tolppanen, 3D characterization of coarse aggregates, Eng. Geol. 65 (2002) 17–30. [26] E.J. Garboczi, Three-dimensional mathematical analysis of particle shape using X-ray tomography and spherical harmonics: Application to aggregates used in concrete, Cem. Concr. Res. 32 (2002) 1621–1638. [27] M.A. Taylor, E.J. Garboczi, S.T. Erdogan, D.W. Fowler, Some properties of irregular 3-D particles, Powder Technol. 162 (2006) 1–15. [28] C. Rocchini, P. Cignoni, C. Montani, P. Pingi, R. Scopigno, A low cost 3D scanner based on structured light, Computer Graphics Forum. Wiley Online Library, 2001 299–308.

C.-Q. Li et al. / Powder Technology 286 (2015) 478–487 [29] O. Hall-Holt, S. Rusinkiewicz, Stripe boundary codes for real-time structured-light range scanning of moving objects, Computer Vision, 2001, ICCV 2001, Proceedings. Eighth IEEE International Conference on, IEEE 2001, pp. 359–366. [30] U. Feige, A threshold of ln n for approximating set cover, J. ACM 45 (1998) 634–652. [31] R.S. Garfinkel, G.L. Nemhauser, Integer programming, Wiley, New York, 1972. [32] R.M. Karp, Reducibility among combinatorial problems, Springer, 1972.

487

[33] V. Chvatal, A greedy heuristic for the set-covering probelm, Math. Oper. Res. 4 (1979) 233–235. [34] T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, MIT press, Cambridge, 2001.