Optimal parameters for a hierarchical grid data structure for contact detection in arbitrarily polydisperse particle systems

Transcription

1 Comp. Part. Mec. 04) : DOI 0.007/s Optimal parameters for a ierarcical grid data structure for contact detection in arbitrarily polydisperse particle systems Dinant Krijgsman Vitaliy Ogarko Stefan Luding Received: 0 January 04 / Revised: 0 April 04 / Accepted: 4 April 04 / Publised online: May 04 Te Autors) 04. Tis article is publised wit open access at Springerlink.com Abstract Te objective of tis paper is to find te optimum number of ierarcy levels and teir cell sizes for contact detection algoritms based on a versatile ierarcical grid data structure, for polydisperse particle systems wit arbitrary distribution of particle radii. Tese algoritms perform as fast as ON) for N particles, but te prefactor can be as large as N for a given system, depending on te algoritm parameters cosen, making a recipe for coosing tese parameters necessary. We estimate teoretically te calculation time of two distinct algoritms for particle systems wit various packing fractions, were te sizes of te particles are modelled by an arbitrary probability density function. We suggest several metods for coosing te number of ierarcy levels and te respective cell sizes, based on truncated power-law radii distributions wit different exponents and widts. Te teoretical estimations are ten compared wit simulation results for particle systems wit up to one million particles. Te proposed recipe for selecting te optimal ierarcical grid parameters allows to find contacts in arbitrarily polydisperse particle systems as fast as te commonly-used linked-cell metod in purely monodisperse particle systems, i.e., extra work is avoided in presence of polydispersity. Furtermore, te contact detection time per particle even decreases sligtly wit increasing polydispersity or decreasing particle packing fraction. Dinant Krijgsman and Vitaliy Ogarko ave contributed equally to tis study. D. Krijgsman B) Univeristy of Twente, PO Box 7, 7500 AE Enscede, Te Neterlands d.krijgsman@utwente.nl V. Ogarko vogarko@gmail.com S. Luding s.luding@utwente.nl Keywords Contact detection Collision detection Hieracical grid Polydisperse Computational cost Discrete element Introduction Collision detection is a basic computational problem arising in systems tat involve spatial interactions among many objects suc as particles, granules or atoms in many diverse fields suc as robotics, computer grapics, pysical simulations, clot modelling, computational surgery, crowd simulations, etc. All tese systems ave rater sort-ranged interaction in common. Particle based modelling tecniques like te discrete element metod, event-driven) molecular dynamics, Monte-Carlo simulations and smooted particle ydrodynamics, to name a few, play an important role for pysically based simulations of powders, granular materials, fluids, colloids, polymers, liquid crystals, proteins and oter materials. Te performance of te computation relies on several factors, wic include te pysical model, on te one and, and te contact detection algoritm used, on te oter. Te contact detection of pairwise interactions between particles can be one of te most time-consuming tasks in calculations wen no suitable contact detection algoritm is used. Because te number of objects treated in simulations is often large, contact detection can become a computational bottleneck. For tis reason, te development of efficient contact detection algoritms is crucial to te overall performance of simulations. Wit te straigtforward all-to-all approac eac pair of particles is cecked for collision. Tis requires ON ) collision cecks for N particles, wic is computationally proibitively expensive. More efficient contact detection metods use a two-pase approac to reduce te computational 3

2 358 Comp. Part. Mec. 04) : costs: a broad pase and a narrow pase [6]. Te broad pase determines pairs of objects tat migt possibly collide. It is frequently done by dividing space into regions and testing if objects are close to eac oter in space. Because objects can only intersect or interact if tey occupy te same region of space, te number of pairwise tests can be reduced to ON). Te pairs tat survive te broad pase test are passed to te narrow pase, wic uses specialised tecniques to test eac candidate pair for a real contact [6,7,34]. Te latter is trivial for sperical particles, were one as to compare te distance between particle centres wit te sum of particle radii, but can be very costly for particles of arbitrary sape. For example, if tere are S surface points per particle, a naive sceme may take order OS ) operations. More sopisticated scemes, suc as te discrete function representation sceme, require on average OS / ) operations [34]. Since te broad pase basically acts as a filter for te narrow pase, coices for te two algoritms can usually be made independently. We distinguis tree types of broad pase contact detection metods/data structures: i) based on coordinate sorting or spatial sorting), e.g., sweep and prune, ii) based on Delaunay triangulation, and iii) based on spatial subdivision, e.g., ierarcical) grids or cell-based metods) and trees e.g., Octrees in 3D and Quadtrees in D). Below we briefly describe te above metods and teir advantages and weaknesses, wile for te detailed analysis see Refs. [4,5,6,8 0,7] and references terein. Contact detection algoritms based on coordinate sorting imply maintaining particles in a sorted structure along eac axis [3,8]. Tese metods are not sensitive to te particle sizes i.e., radii for sperical particles) and consume ON) memory; but tey require sorting, wic can range in effort from ON) to ON ), depending on te sorting metod used, volatility of te sorting lists over time and spatial distribution of objects. Te Delauney triangulation data structure consumes ON) memory and it is not sensitive to te particle sizes wen weigted triangulation is used [5]. However, it as te disadvantage tat building or re-building) te structure as a ig computational cost, especially for moving particles. Te use of flipping algoritms for maintaining and only incrementally updating te triangulation allows decreasing te overead of re-building te triangulation [0], but unfortunately in tree-dimensional system flipping can get stuck [7]. Furtermore, its parallelisation and maintaining of periodic boundary conditions wic are frequently used in particle simulations) is complicated. Te tree data structure for contact detection does not allow to coose cell sizes at every level of ierarcy independently, terefore, leaving no room for optimisation for various distribution of particle sizes [,3,3]. Moreover, accessing neigbour sub-cubes in te tree is not straigtforward since tey can be nodes of different tree brances; no more details are given ere since tis metod is not used any furter. Te single-level grid-based contact-detection metods, like for example te linked-cell metod [,9,6], are straigtforward, widely used and perform well for similarly sized objects. Te problem of suc metods is teir inability to efficiently deal wit particles of greatly varying sizes [0]. If te particles witin te system are polydisperse, te cell size of a grid would ave to conform to te largest particle size. Ten many small particles may occupy te same cell, wic increases te number of pairwise cecks, and terefore affects te computational performance a lot. Tis cell size problem can effectively be addressed by te use of multi-level ierarcical grids [3,4,8,0,5,6,,4]. Particles are positioned at different levels according to teir size) and collision cecks are performed in two steps: i) witin te level of insertion wic is usually performed in te same way as in te linked-cell metod), and ii) cross-level cecks. Te cell size at eac ierarcy level can be selected independently, terefore one can adapt grid cells according to a given particle size distribution. Several algoritms based on ierarcical grid data structures were employed, wic differ in te way in wic te above two steps are implemented. Te ierarcical grid data structure performs ON) for arbitrary polydisperse systems and uses ON) memory. Tis data structure is robust, can be easily parallelised, allows straigtforward andling of periodic boundary conditions and can easily deal wit unbounded systems. Moreover, it provides O) access to te particle data and to all particle nearest neigbours, and, more importantly, allows for O) particle insertion and removal from te system, wic is often needed in modelling of dynamical systems, like for example opper or granular flows. Finally, it provides a natural multiscale framework as particles from different ierarcy levels usually ave different pysical properties besides teir size. For example, small particles are often fast i.e., ave iger velocity/energy) and big ones are slow, so tey can ave different time scales at different ierarcy levels. Tese are te reasons wy we cose te ierarcical grid as our primary data structure for contact detection and analyse ow to optimise it for fastest contact detection in widely polydisperse particle mixtures. Te ierarcical grid data structure as many parameters to configure, i.e., an arbitrary coice of te number of ierarcy levels plus an arbitrary coice of te cell size at every level of ierarcy te cell sizes are, as convention, increasing wit increasing level of ierarcy). Te coice of parameters affects te number of contact cecks and te overead of te algoritm, i.e., te number of times te cells are accessed. Due to te many parameters involved, finding te optimal ones, i.e., tose wic minimize an average number of calculations, T, is a non-trivial problem. Tis involves multi- 3

3 Comp. Part. Mec. 04) : dimensional optimisation were te optimum dimension is unknown. We are not aware of any study were tis question was fully addressed, except for te study by Ogarko et al. [] in wic te autors tried to address tis problem by providing a ypotesis on te optimal coice of te ierarcical grid parameters, and ten comparing teir teoretical predictions wit te simulation results. In tis study we teoretically analyse te performance complexity of te ierarcical grid data structure for contact detection in polydisperse particulate systems. We provide detailed analysis on te average number of calculations, T, for two distinct algoritms based on te ierarcical grid data structure as applied to polydisperse systems of sperical particles wit a power-law distribution of radii, for various power-law exponents, and for various particle volume fractions. We compare several ways metods) of coosing te ierarcical grid parameters i.e., te number of levels and te cell sizes at eac level) and present te optimal parameter coice. We provide instructions on wic ierarcical grid contact detection algoritm sould be used and ow to coose te optimal parameters for a given arbitrary distribution of particle radii. Finally, we compare our teoretical predictions wit simulation results of realistic particle systems. In te next section we outline te two different algoritms based on te ierarcical grid data structure tat are used in tis study. We ten analyse te performance of te described algoritms and derive general estimates for te number of contact cecks per particle in Sect. 3. Section 4 presents te types of particle size distributions considered in tis study. In Sect. 5 we introduce several ways of coosing te ierarcical grid parameters and compare teir expected performance. For selected parameters tese expected performances are also compared wit real discrete particle simulations, using te MercuryDPM code mercurydpm.org) [,9,30,33]. Finally, te results are summarised and discussed, wit some conclusions in Sect. 6. Algoritm Te ierarcical grid HGrid) algoritm is designed to determine all pairs of particles, in a set of N particles in a d- dimensional Euclidean space, tat overlap or interact. Te split between local particle geometry and global neigbour searcing is acieved troug te use of a bounding volume. Tis way, te contact detection algoritm is able to treat all particle sapes in te same, simplified way. Wile any bounding volume can be used, te spere is cosen for tis implementation since it is represented simply by a position of its centre x p and its radius r p, and is rotationally invariant. For differently-sized speres, r min and r max denote te minimum and te maximum particle radius, respectively, and = r max /r min is te extreme size ratio. Te algoritm consists of two pases. In te first mapping pase all te particles are mapped into a ierarcical gridspace Sect..). In te second contact detection pase Sect..) te potential contact partners are determined for every particle in te system. Tis list of potentially contacting particle pairs is te output of te algoritm. Wit tis list one can perform geometrical intersection tests to ceck if particles are really in contact, i.e., if tey overlap. For sperical particles tis can be acieved easily by comparing te distance between two particles wit te sum of te radii. For non-sperical particles tese tests become more difficult and computational expensive, owever, tis is beyond te scope of tis paper. Requirements for te algoritm are: All pairs of particles tat are in contact must be in te list of potential contacts, i.e., te algoritm is not allowed to miss any pair. Te list of pairs of particles must be unique, i.e., no pair of contacts may appear twice in te list. Te list of pairs of particles sould be as small as possible. Te computational time of te algoritm sould be as small as possible, and for large N, tus must scale linearly wit te number of particles, i.e., ON). Te memory consumption of te algoritm must be proportional to te number of particles, i.e., ON).. Mapping pase Te d-dimensional HGrid is a set of L regular grids wit different cell sizes. Every regular grid is associated wit a ierarcy level [, L], were L is te integer number of ierarcy levels. Eac level as a different cell size s R, were te cells are d-dimensional cubes. Grids are ordered wit increasing cell size suc tat = corresponds to te grid wit smallest cell size, i.e., s < s +. For a given number of levels and corresponding cell sizes, te ierarcical grid-cells are defined by te spatial mapping, M, of points x R d to a cell at specified level : M : x, ) c = x /s,..., x d /s, ), ) were x denotes te floor function. Te first d components of a d + )-dimensional vector c represent cell indices integers), and te last one is te associated ierarcy level. It must be noted tat te cell size of eac level can be set independently, in contrast to contact detection metods wic use a tree structure for partitioning te domain [4,3,3], were te cell sizes are taken as double or triple) te size of te previous lower level of ierarcy, ence Te largest integer not greater tan x. 3

4 360 Comp. Part. Mec. 04) : s + = s or 3s ). Te flexibility of independent s allows one to select te optimal cell sizes, according to te particle size distribution, to improve te performance of te contact detection algoritm. Using te mapping M, every particle p can be mapped to its cell: c p = Mx p, p)), ) 4 6 y a.u.) B were p) is te level of insertion to wic particle p is mapped to. Te level of insertion p) is te lowest level, were te cells are big enoug to contain te particle p: { p) = min L : r p s }. 3) 8 3 In tis way te diameter of particle p is smaller or equal to te cell size at te level of insertion and terefore te classical linked-cell metod [] can be used to detect contacts among particles witin te same level of ierarcy. Figure illustrates a -dimensional two-level grid for te special case of a bi-disperse system wit r min = 3/, size ratio = 8/3, and cell sizes s = 3, and s = 8. Since te system contains particles of only two different sizes, two ierarcy levels are sufficient ere y a.u.) 6 4 x a.u.). Contact detection pase C After all particles are mapped to teir cells, te contact detection pase is able to calculate all potential contacts. Te contact detection is performed by looping over all particles p and searcing for possible contacts wit particles at te same ierarcy level and for possible contacts at different ierarcy levels. Searcing for contacts at te same ierarcy level is performed using te classical linked-cell metod []. Te searc is done in te cell were p is mapped to, i.e. c p, and in its neigbouring surrounding) cells. Only alf of te surrounding cells are searced, to avoid testing te same particle pair twice. Searcing for contacts at oter ierarcy levels can be performed in two ways. Te first one is te Top-Down metod, illustrated in Fig. top). In tis metod one searces for potential contacts only at levels j lower tan te level of insertion: j <. Tis implies tat te particle p will be cecked only against smaller particles, tus avoiding double cecks for te same pair of particles. Te second metod, te Bottom-Up metod, sketced in Fig. bottom), does exactly te opposite. Here potential contacts are only searced for at ierarcy levels j iger tan te level of insertion: < j L. Tis implies tat te particle p will be cecked only against larger particles, tus avoiding double cecks for te same pair of particles x a.u.) Fig. A -dimensional two-level grid for te special case of a bidisperse system wit cell sizes s = r min = 3 a.u.), and s = r max = 8 a.u.). Tefirst level grid isplotted witdased lines wile te second level is plotted wit solid lines. Top) Te Top-Down case: Te radius of particle B is r B = 4 a.u.) and its position is x B = 0.3, 4.4). Terefore, according to Eqs. ) and3), particle B is mapped to te second level to te cell c B =,, ). Te cross-level cells tat ave to be cecked for possible contacts wit particle B range from,3,) to 5,6,), and are marked in grey. Bottom) Te Bottom-Up case: A particle C is mapped to te cell c C = 4, 4, ). Te cross-level cells tat ave to be cecked for possible contacts wit particle C range from,, ) to,, ), and are marked in grey. Te particles located in te marked grey) cells are coloured dark green). Color figure online) Te details for bot metods are actually quite similar. Te algoritm to find potential contacts for particle p at ierarcy level wit oter particles at ierarcy level j is as follows: 3

5 Comp. Part. Mec. 04) : ) Define te cells c start and c end at level j as c start := Mxc, j), and c end := Mx c +, j), 4) were a searc box cube in 3D) is defined by x ± c = x p ±β d i= e i, wit β = r p +s j / and e i is te standard basis for R d. Any particle q from level j, wit centre x q outside tis box can not be in contact wit particle p, since te diameter of te largest particle at tis level can not exceed s j. ) Te searc for potential contacts is performed in every cell c = c,...,c d, j) for wic c start i c i c end i for all i [, d], 5) were c i denotes te i-t component of vector c. In oter words, eac particle wic was mapped to one of tese cells is tested for contact wit particle p. In te Top-Down metod, te small cells, defined in Eq. 5), wic are almost fully covered by big particles i.e., no small particles can reside in tose cells) can be excluded from te contact searc, like for example te cells 3, 4, ) and 3, 5, ) in Fig. top). However, we do not know ow to identify suc cells efficiently, and terefore, ave not implemented tis optimisation. 3 Performance analysis Te algoritm is applicable to arbitrary systems, owever, to estimate te performance of te algoritm, we restrict ourself to systems tat are omogeneous in time and in space. In suc a system accurate estimates can be obtained and optimal HGrid parameters can be found teoretically. To analyse te algoritm two time consuming effects are considered: ) T cd collision detection effort) Te number of possible contacts tat ave to be examined more closely. Te output of te HGrid-algoritm is a number of possible contacts. Optimum HGrid parameters lead to a low number of possible contacts, because for all tese possible contacts a computationally expensive exact geometrical intersection test as to be performed to ceck if te particles really are in contact. ) T ca cells access effort) Te number of times information is retrieved from a cell. Wile te goal of te HGrid is to obtain a list of all possible contacts, it comes at a computational) cost. Tis cost is estimated by te number of times information is obtained from a single cell. To calculate estimates for T cd and T ca, consider a system of N polydisperse particles wit: Random positions x p witin a d-dimensional box at packing fraction ν witout excluded volume effects). Random radii between r min and r max = r min, according to a normalised probability density function f r) for more details see Sect. 4). Wit tese properties te expected mean volume per particle V p can be calculated using: V p = V d r max r min r d f r) dr, 6) were V d is te volume of a d-spere of unit radius, i.e., V = π and V 3 = 4/3) π. So te total volume V of all particles becomes: V = NV p. 7) Given tis volume and te packing fraction ν, te size A of a d-dimensional box can be calculated as: A = ) V d. 8) ν Now define N as te expected number of particles at level and N c as te number of cells at tis level: N = N s s N c = A s ) d = N V p ν f r) dr, 9) ) d, 0) s were s 0 = r min and s L = r max. So te expected average number of particles per cell at level, m, becomes: m = N N c = νsd V p s s f r) dr. ) It must be noted tat te number of particles per cell m is independent of te total number of particles N. As described in Sect.., te algoritm cecks for possible contacts at te level of insertion and at te oter levels. For bot types of contacts estimates of te number of possible contacts and te number of cells tat ave to be accessed are made in te following two subsections. 3

6 36 Comp. Part. Mec. 04) : Level-of-insertion searc At te level of insertion, N particles are randomly distributed over N c cells. Terefore, te number of particles in a specific cell at tis level, X, is binomially distributed wit N te number of trials and /N c te probability of success. Wit tis assumption te expected number of potential contacts witin a single cell is obtained see Estimated number of contacts witin a cell in Appendix). Te number of cells tat ave to be processed is just equal to te number of cells at tis level. Terefore, we obtain: T cd = N N N c = m N ) m N, ) T ca = N, 3) were in te last step of Eq. ) it is assumed tat te number of particles at level is muc greater tan unity. For possible contacts between neigbouring cells one just as to square te expected numbers of particles in a cell, m, and multiply it by te number of neigbouring cells tat ave to be cecked, n c, and te total number of cells at te current level, N c : T cd = n c N c m = n c N m, 4) T ca = n c N c m = n c N, 5) wit n c = 3 d ), i.e., n c = 4 in D and n c = 3 in 3D. We obtain tat T cd, T ca, T cd and T ca are all linearly dependent on te number of particles N at level. 3. Cross-level searc To estimate te number of potential contacts for te crosslevel searc, first an estimate of te number of cross-cell cecks between particles at ierarcy level j = wit particles at level as to be made. In Number of cells for cross-level searc in Appendix, te number of cells at level j tat ave to be scanned for potential contacts wit particles at level is found to be: b j, ) = s d r s s j + ) f r) dr s s f r) dr wit expected) lower and upper limits: b lower j, ) = b upper j, ) =, 6) + s ) d, 7) s j + s s j ) d. 8) Te expected number of cross-level cecks and te number of cells tat ave to be accessed witin a cross-level ceck can easily be calculated. For te Top-Down algoritm: T cd3 T ca3 = N m j b j, ), 9) j= = N b j, ), 0) j= and for te Bottom-Up algoritm: T cd3 T ca3 = N = N L j=+ L j=+ m j b j, ), ) b j, ). ) Just as for te level-of-insertion searc, we obtain tat T cd3 and T ca3 are linearly dependent on te number of particles N at level, for bot algoritms. 3.3 Total computational work Te total computational work per level can now be calculated by just summing of its components. For te Top-Down algoritm: T cd T ca ) = N + n c m + m j b j, ), j= 3) = N + n c + b j, ), 4) j= and for te Bottom-Up algoritm: T cd T ca = N + n c = N + n c + ) m + L j=+ L j=+ m j b j, ), 5) b j, ). 6) Note tat bot T cd and T ca are linear in te expected number of particles at level, N, for bot metods, because it was sown in Eq. ) tat m is independent of te number of particles. Tis means tat te complexity of te total algoritm is linearly dependent on te total number of particles N, for any number of levels L used. However, depending on 3

7 Comp. Part. Mec. 04) : te packing fraction and te particle radii distribution function a uge pre-factor in front of N, even larger tan N, can appear wen coosing inappropriate HGrid parameters i.e., cell sizes and number of ierarcy levels). To find te optimal number of ierarcy levels and teir cell sizes, an estimate of te required computational time tat is associated wit bot types of effects, i.e., collision detection work and cell access work, is required. Terefore, te ratio K of time required for a single geometric contact detection over te time required to retrieve information from a cell is introduced. From simulations it is found to be close to K = 0. for sperical particles and not dependent on te particle volume fraction []. Terefore, an estimate of te total time required for a contact detection step is found to be: T = L = T cd ) + KT ca. 7) Tis result is general in te sense tat it describes every possible particle system. However, to get a feel for te optimal HGrid parameters, we limit ourself to a single type of particle size probability distribution function. 4 Particle size probability distribution functions In order to estimate te performance of te HGrid-algoritm, te distribution of particle radii as to be known. To account for all possible radii distributions te previous section used a normalised probability distribution function f r). Te probability to find a particle wit radius between r and r + dr is equal to f r) dr. Tis requires tat: 0 f r) dr =. 8) No general strategy as been found to determine te optimal HGrid-parameters for a general particle radii probability distribution function. Terefore, trougout te remainder of tis paper a truncated) power law size distribution wit a constant exponent α is used: f r) = Cr α for r min r r max, 9) wit normalisation factor C = r max r min r α dr. 30) Different values of α ave different pysical significance, for tree-dimensional systems tey represent see also Fig. ): fr) α= 0 α= α= α= r/r min Fig. Probability density function, Eq. 9), for different values of α wit = r max /r min = 3 α = 0: Uniform size rectangular) distribution, i.e., same number of bigger as smaller particles in intervals dr. α = : Uniform area distribution, i.e., te total surface area of particles wit radii between r and r +dr is equal to te total surface area of particles wit radii between r and r + dr,etc. α = 3: Uniform volume distribution, i.e., te total volume occupied by particles wit radii between r and r + dr is equal to te total volume occupied by particles wit radii between r and r + dr,etc. In general, α>0 not used furter) implies tat tere are more bigger particles, wereas α<0 implies tat tere are more smaller particles, wile α = 0 corresponds to a similar number of small and big particles. 5 Cell sizes distribution Having defined te HGrid algoritm, te last ting to do is to decide about te number of ierarcy levels L and te sizes associated wit tese levels s. In tis section four different cell size distributions are introduced, discussed and compared in order to find optimal HGrid parameters. In te end, te predicted performance of te algoritm is compared against real discrete particle metod DPM) simulations, using MercuryDPM mercurydpm.org) [9,33]. 5. Single-level grid As a simple reference, we consider te case of a single ierarcy level L = ), i.e., te linked-cell metod, and compute te total work T as a function of volume fraction ν,sizeratio, exponent of te size distribution α and dimension d.due to L =, we ave N = N and s = s L = r max = r min. Using te definition of te number of particles per cell m from Eq. ) and inserting te average particle volume of 3

8 364 Comp. Part. Mec. 04) : α= 0 α= α= α= 3 5. Multi-level cell size distribution 5.. Linear cell size distribution Fig. 3 Computational effort of te HGrid algoritm using a singlelevel grid i.e., te linked-cell metod) as a function of te widt of te particle size distribution,, for various exponents α using d = 3, ν = 0.7 andk = 0. Eq. 6) we obtain: m = r min ) d ν rmin r min f r) dr V rmin d r min r d f r) dr. 3) Now substituting te particle radii probability function of Eq. 9) and evaluating te integrals yields for =, α = and α = d): m = ) d ν V d + d + α + α +α +d+α. 3) We are interested in wat appens wen te polydispersity increases, and tus take te limit of going to infinity: lim m = d ν V d +d+α +α d α< d d ν V d +d+α +α α d <α< d ν V d +d+α +α α>. 33) Equation 33) sows tat for α< te number of particles per cell m increases wit increasing. Tis means tat te efficiency of te linked-cell algoritm is eavily dependent on. Tis result is also sown in Fig. 3, were te required computational effort per particle is plotted as a function of te widt of te distribution function, for different exponents α. All curves, except te one for α = 0 diverge. In general tis is true for α>, meaning tat te single-level approac is only appropriate for tese values of α.in te following subsections different distributions of te HGrid cell sizes using multiple ierarcy levels are tested to find parameters tat lead to minimal computation effort. Te easiest metod to define te HGrid cell sizes is to use a linear distribution: s = r min + ). 34) L Using tis cell size distribution te total work T as a function of te number of HGrid levels L can be calculated. Te number of levels were te required computational effort is minimal is cosen as te optimal level and is denoted by L. Te minimal work T and te optimal number of levels L are sown in Fig. 4 for uniform size α = 0) and uniform volume α = 3) particle size distributions. Comparing te work in Fig. 4a wit te work for te linked-cell metod Fig. 3), it becomes immediately clear tat te HGrid algoritm reduces te work significantly. For α = 0 te improvement is less significant, but still te computational effort is reduced by approximately 60 %, wile for α = 3 a speed-up of several orders of magnitude is acieved. Furtermore, wile for α>0 te Bottom-Up algoritm works sligtly better, for α<0 te Top-Down algoritm is preferred data for oter values of α not sown). However, in Fig. 4b te disadvantage of using a linear cell size distribution becomes clear. For te α = 3 case, te optimal number of levels increases significantly wit increasing. Terefore, a different cell size distribution migt give better results, as we sow in te following subsections. 5.. Exponential cell size distribution To reduce te optimal number of required levels for te HGrid algoritm an exponential cell size distribution is tested. Tis distribution stems from te ierarcical tree data structure, were te cell sizes are usually taken as double te size of te previous lower level of ierarcy. Tis can be generalised by taking cell sizes wic are defined as: s + = qs, 35) wit q >, to make sure tat iger level cell sizes are larger. If one substitutes te boundary conditions s 0 = r min and s L = r min ), te system of equations can be solved analytically: s = r min L. 36) As for te linear cell size distribution, te total work is calculated as a function of te number of HGrid levels and te optimum values are selected and plotted in Fig. 5 for differ- 3

9 Comp. Part. Mec. 04) : Top Down, α= 0 Top Down, α= 3 Bottom Up, α= 0 Bottom Up, α= 3 L * Top Down, α= 0 Top Down, α= 3 Bottom Up, α= 0 Bottom Up, α= a) b) Fig. 4 Computational effort and optimal number of levels for te HGrid algoritm wit a linear cell size distribution as a function of te widt of te particle size distribution,, for various exponents α, for bot te Top-Down and te Bottom-Up algoritms using d = 3, ν = 0.7andK = Top Down, α= 0 Top Down, α= 3 Bottom Up, α= 0 Bottom Up, α= Top Down, α= 0 Top Down, α= 3 Bottom Up, α= 0 Bottom Up, α= a) L * b) Fig. 5 Computational effort and optimal number of levels for te HGrid algoritm wit an exponential cell size distribution as a function of te widt of te particle size distribution,, for various exponents α, for bot te Top-Down and te Bottom-Up algoritms using d = 3, ν = 0.7and K = 0. ent values of α and. Forα = 0 te Bottom-Up algoritm works better, wile for α = 3 te Top-Down algoritm is preferred. Data for oter values of α not sown ere), indicate tat in general for α te Top-Down algoritm is preferred. Te optimal number of levels is not tat strongly dependent on as te linear cell size distribution, wen te Top-Down approac is used. However, tere is still a trend: te optimal number of levels increases wit increasing for te Bottom-Up algoritm Constant ratio of te number of particles per cell Anoter approac originates from te idea tat it may be beneficial to keep te number of particles per cell fixed at every ierarcy level []. Tis simple idea can easily be extended to a rule were te ratio of particles per cell over two adjacent levels is fixed: m + = qm. 37) Tis implies tat for q < te number of particles per cell is decreasing, for q > increasing and for q = constant, wit increasing ierarcy level. Using te definition of m from Eq. ) we can rewrite Eq. 37): s+ d Fr) s + / = qs d Fr) s /, 38) s / s / were dfr) = f r), 39) dr s 0 = r min and s L = r min. Tis system of equations can be at least numerically) solved in terms of s, once a size distribution function f r) is specified. Using tis cell size distribution we reduce te problem of selecting te number of levels and teir sizes to just coosing te number of levels L and te ratio of particles per cell for different ierarcy levels q. InRef.[] te ypotesis was tat it is optimal to keep te number of particles per cell at eac level constant, or equivalently using q =. To ceck 3

10 366 Comp. Part. Mec. 04) : q=0.5 q= q= q= q L Fig. 6 Computational effort of te HGrid algoritm wit a cell size distribution were te ratio of te numbers of particles per cell, q, is constant as a function of te widt of te particle size distribution,, for different numbers of levels L for te Top-Down metod using = 00, α = 3, d = 3, ν = 0.7, K = 0.. In te inset te minimum computational effort is sown for different values of q tis ypotesis te computational effort for different values of q as a function of te number of levels L is sown in Fig. 6 for a system wit uniform volume radii distribution α = 3) using = 00, ν = 0.7, d = 3 and K = 0.. For values of q =, and 5 te minima in te required computation effort are rougly equal. Tis is more clearly visible in te inset, were te minimum computational effort is plotted against q. Te optimal value is somewere between q = and q =. Te range and number of different levels, for wic te computational effort is acceptable, are muc bigger for q = tan for q = and tus it is advised to use q =, for te sake of simplicity. Tis is also confirmed for different system parameters and te Bottom-Up algoritm data not sown). Using q =, te optimal work and optimal number of levels L is sown in Fig. 7. For all values of α te Top- Down algoritm is preferred data for oter values of α not sown) Optimal cell size distribution In order to ceck if te constant number of particles per cell metod indeed gives a close-to-optimal result, a numerical optimisation metod is used to minimize Eq. 7) in terms of L and s under te conditions tat s + s, s 0 = r min and s L = r min. Tis is performed using te MATLAB [4] iterative optimisation function fmincon. In tis function, a quadratic programming subproblem is solved at eac iteration, were te Hessian of te Langrangian at eac iteration is calculated using te BFGS algoritm. Te minimal required computational effort T and te optimal number of levels L for tis metod are sown in Fig. 8. Note tat te results are quite comparable to tat of te constant number of particles per cell metod in Fig Comparison of te cell size distribution functions In tis subsection te four different cell size distributions are compared and best practices are given. From Fig. 3 and te analysis of te single-level reference case it becomes clear tat te HGrid algoritm is essential for α, owever, te required optimal parameters are yet to be determined. Te required computational effort for different particle distributions using te four cell size distribution functions is sown in Figs. 4, 5, 6, 7, 8. All of te used algoritms sow a significant decrease in computational effort over te singlelevel reference case for all parameters of te particle size distribution function. Even more important, all but one te Bottom-Up algoritm using a linear cell size distribution for α = 0) of te test cases sow tat for large polydispersities i.e., ig values of ) te optimal efficiency of te algoritm is independent on. Also te coice of te cell size distribution functions is not too important as long as te oter HGrid parameters are cosen optimally. However, in practice it is often difficult or impossible to calculate optimal parameters in advance, due to canging particles, density or geometries. Terefore, te sensitivity of te algoritm to different parameters becomes important. Te required work for all previously discussed cell size distributions is sown in Fig. 9 for te case = 00 and α = 3using te Top-Down algoritm. Again we see clearly tat te minima of te four curves are rougly equal.40,.57,.60 and, 58 respectively), owever, te location of te minimum L and te sensitivity of te work to using suboptimal parameters differ quite a lot. For te linear cell size distribution te location of te minimum is at L = 43 outside te domain of te figure), wic is significantly iger tan for te oter distributions. Suc a ig number of levels results in additional overead, especially for particles wit complex geometries, terefore it is not advised to use te linear cell size distribution. For te exponential cell size distribution te minimum is located at L = 4, owever coosing L = 3orL = 6 already decreases te performance by 43 and 4 % respectively. For te constant number of particles per cell distribution te optimum is located at L = and coosing L = 8orL = 9 reduces te performance just by 0 %. So, it is advised to eiter use te constant number of particles per cell or te truly optimal cell size distribution, wic is even less sensitive to L = L. Te values of te cell sizes s and te numbers of particles per cell m for te different cell size distributions are sown in Fig. 0. We observe tat in most cases m < is a good coice. 5.4 Comparison wit simulations Te estimated computational efficiency of te HGrid algoritm is compared against DPM simulations to ceck te 3

11 Comp. Part. Mec. 04) : Top Down, α= 0 Top Down, α= 3 5 Bottom Up, α= 0 Bottom Up, α= a) L * Top Down, α= 0 Top Down, α= 3 Bottom Up, α= 0 Bottom Up, α= b) Fig. 7 Computational effort and optimal number of levels for te HGrid algoritm wit a cell size distribution, were te number of particles per cell is te same at eac level as a function of te widt of te particle size distribution,, for various exponents α, for bot te Top-Down and te Bottom-Up algoritms using d = 3, ν = 0.7 and K = Top Down, α= 0 Top Down, α= 3 5 Bottom Up, α= 0 Bottom Up, α= a) L * Top Down, α= 0 Top Down, α= 3 Bottom Up, α= 0 Bottom Up, α= b) Fig. 8 Computational effort and optimal number of levels for te HGrid algoritm wit an optimal cell size distribution as a function of te widt of te particle size distribution,, for various exponents α, for bot te Top-Down and te Bottom-Up algoritms, using d = 3, ν = 0.7 and K = Linear Exponential Constant particles/cell Optimal wile finally a full DPM simulation as been performed wit optimal parameters and compared against a simulation using te linked-cell parameters Contact detection test Fig. 9 Computational effort of te HGrid algoritm as a function of te number of levels, L, for different cell size distributions, using α = 3, = 00, d = 3, ν = 0.7 andk = 0. assumptions used in te derivation. Tis is done in two steps, first only a single contact detection step as been performed on particle positions obtained from real DPM simulations, L For te single contact detection step, different packings of particles are generated, using a combination of event-driven and soft particle metods, for different packing fractions, particle size distributions and numbers of particles. More specific, we use omogeneous, isotropic, disordered systems of colliding elastic sperical particles in a cubical box wit ard walls or periodic boundary conditions for more details see Ref. []). In MercuryDPM mercurydpm.org) [9,33] a contact detection step as been run using te optimal cell size distribution, were bot te number of times a cell is accessed and te number of narrow pase contact detection steps ave been counted to compare against te teoretical predictions. 3

12 368 Comp. Part. Mec. 04) : Linear Exponential Constant particles/cell Optimal Linear Exponential Constant particles/cell Optimal s /r min 00 m /L * a) /L * b) Fig. 0 Cell sizes and number of particles per cell for te four different cell size distribution metods wit optimal HGrid parameters, using α = 3, = 00, d = 3, ν = 0.7andK = =0 =50 = ν 5 0 α= 0 α= N Fig. Comparison of te estimated HGrid computational effort lines) versus tat for a real DPM system symbols) for different packing fractions and different polydispersities, using α = 3, N =,000,00, d = 3, K = 0. and Optimal cell size distribution In Fig. te results are sown for one million particles using different packing fractions ν and different widts of te particle size distribution function. For lower packing fractions te results are extremely accurate, but for iger volume fractions te required work in real simulations becomes sligtly iger tan expected, owever, te overall trend is captured nicely. Te main reasons for tis deviation we attribute to te excluded volume and finite size effects. In te model derivation te particle centres are assumed to be randomly distributed trougout te domain, wereas in real DPM simulations particles are not allowed to ave large overlaps. Tis is already seen in Fig. top), were large particle B is so big tat it completely covers te small grid cells 3, 4, ) and 3, 5, ). So te number of cells were te small particles can be distributed is significantly reduced by te presence of large particles. In te test case, for α = 3, = 00 and ν = 0.6 te percentage of cells on te lowest ierarcy level were two or more particles reside is.6 % wit excluded volume and.6 % for fully random positions. Te excluded volume effect leads to a significant increase in te number of Fig. Comparison of te estimated HGrid computational effort lines) versus tat for a DPM system symbols) for different packing fractions and different exponents of te particle size distribution function, using = 0, ν = 0.6, d = 3, K = 0. and optimal cell size distribution. Open symbols correspond to simulations using solid walls, filled symbols represent systems wit periodic boundary conditions cells wit two or more particles, and tus also in te number of possible collisions tat ave to be furter examined. Tis effect increases wit increasing volume fraction. Te same conclusion olds for different numbers of particles and different sapes of te particle size distribution function, as sown in Fig.. Te required computational effort is estimated quite nicely, especially for large numbers of particles. For a small number of particles te computational work is sligtly less tan expected from te model, because a system of infinite size is assumed. Wen only a finite number of particles is used tere will be particles at te boundary of te domain, wic will ave less neigbouring particles tan particles in te middle of te domain. Wen using more particles, te ratio of particles at te boundary compared to particles in te central part will become lower and tus increasing te computational effort. Tis dependence as been tested and confirmed by creating and testing 3

13 Comp. Part. Mec. 04) : systems wit periodic boundary conditions solid circles in Fig. ) Full DPM test Trougout tis paper te performance of te HGrid algoritm is estimated and measured from te number of times a cell is accessed and te number of narrow pase contact detection steps tat are performed. In real DPM simulations, owever, additional computational work is required, for example, during te integration routines, for andling periodic boundaries or walls and for writing data. To sow te real improvement of using optimal HGrid parameters, a simple free cooling simulation wit moderate polydispersity as been run using different HGrid parameters [3]. More specifically, we performed D simulations using solid walls wit 0 4 particles over time steps. Te particle sizes are distributed according to te truncated) power law size distribution wit parameters = 0 and α = 3, wit a packing fraction of 0.4. According to our analysis te optimal parameters for tis system are: te Top-Down algoritm wit 5 levels wit sizes 4.0, 7.9, 5., 7. and 40 times r min, respectively. Tese settings sould give a speedup of te contact detection part of te simulation by a factor of 35 wen comparing against a linked-cell reference case i.e., one level wit size 40). Te full DPM simulation wit optimal parameters took about 7 min, wereas te reference case required 66 minutes. Tis speed-up by a factor of 9.9 is naturally lower tan te predicted speedup of te contact detection part due to force calculations, integration routines and wall interactions, but is still quite significant. 6 Summary and conclusion Contact detection is a fundamental problem tat occurs in many different kinds of simulation metods. Tis process is often computationally expensive, usually taking up a considerable proportion of CPU time, especially for systems wit non-uniform density or polydisperse particle sizes. In tis paper, we studied analytically te computational effort of two algoritms for contact detection i.e., Bottom- Up and Top-Down), based on te multi-level ierarcical grid data structure. Te basic idea of tese algoritms is te fact, tat usually tere are lots of particles in te system, wic cannot be in contact, as tey are too distant. Te presented metods save a lot of time by excluding suc particles from a detailed and time consuming contact examination and evaluation. Te performance of te neigbour searcing algoritm based on bot te number of particles and te widt of te particle size distribution, is of great importance. As an input for te algoritm, te number of ierarcy levels and teir cell sizes are required. Terefore, we tested four metods for coosing te ierarcical cell size distribution i.e., linear, exponential, constant number of particles per cell and optimal) and compare teir teoretical performance for a power law particle size distribution function wit exponent α. For almost all metods te performance of te algoritm becomes independent of te widt of te particle size distribution, in contrast to te linked-cell metod. Even better, te computational effort per particle, using te algoritm decreases wit increasing, or wit decreasing α, at constant system packing fraction. In general, wit optimal parameters, te algoritm is able to find contacts in arbitrarily polydisperse particle systems as fast as te linked-cell metod finds contacts in purely monodisperse particle systems, i.e., no extra work is required due to polydispersity. For te linear cell size distribution te optimal number of ierarcy levels is uge for systems wit large polydispersity and α<0 i.e., te systems dominated by small particles). Terefore, for tese kinds of systems, te linear cell size distribution as ig computational overead and in general does not perform well, especially for particles wit complex geometries. Te exponential cell size distribution performs better, owever, it is very sensitive to te number of ierarcy levels used. So it is not appropriate to use tis metod in dynamical systems, were te particle size distribution, density or system geometry is canging over time. Bot te constant number of particles per cell and te optimal cell size distribution metods perform well, are not too sensitive to te number of levels, and ave low overead. For α te use of a multilevel grid becomes extremely efficient i.e., orders of magnitude faster) as compared to te single level linked-cell metod, if optimal parameters are used. On te oter side, for α>, te use of a multilevel grid does not present a major advantage but can improve performance sligtly). In all our test cases wit optimum HGrid parameters), te contact detection time is estimated to be T 30N for tree-dimensional systems wit sperical particles were a unit of time is defined as te time required for a two-spere overlap test). For future researc, our analysis tecnique allows to investigate ow te algoritm performs for oter more realistic size distributions, e.g., log-normal. 7 Recommendations In tis section recommendations are given for setting te HGrid parameters. Use te Top-Down algoritm. If possible, perform your own minimisation using your exact system and te overead factor K applicable to 3

14 370 Comp. Part. Mec. 04) : your solver, to obtain te optimal number of levels and teir cell size distribution. Oterwise, use a cell size distribution were te number of particles per cell is approximately te same at eac level. Since te optimum number of levels L depends on te particle size distribution function and te packing fraction no general recommendations can be given. For distributions similar to te truncated) power law size distribution used trougout tis paper we refer to Fig. 7. Acknowledgments We would like to tank A. R. Tornton and S. González for elpful discussions. Tis researc is supported by te Dutc Tecnology Foundation STW, wic is te applied science division of NWO, and te Tecnology Programme of te Ministry of Economic Affairs, project number STW-MUST 00, STW-HYDRO 7, and STW-VICI grant 088. Open Access Tis article is distributed under te terms of te Creative Commons Attribution License wic permits any use, distribution, and reproduction in any medium, provided te original autors) and te source are credited. Appendix Estimated number of contacts witin a cell To calculate te estimated number of potential contacts witin a cell, we assume tat at level tere are N particles randomly divided over N c cells, suc tat te probability tat a certain particle goes to a certain cell is /N c. Te number of particles in cell i, X i, follows te binomial distribution wit parameters N and /N c suc tat te probability of finding n particles in cell i is equal to: P X i = n) = N n ) ) n N c ) N n N c, 40) were N n ) = N! n! N n)! 4) is te binomial coefficient. If n particles reside in a certain cell, we ave to ceck for n n ) / potential contacts in tat cell and tus we can estimate te number of potential contacts witin a single cell by calculating its weigted average. However, we are interested in te potential contacts witin cells for te wole ierarcy level and tus ave to multiply tis by te number of cells at tis level N c to obtain T cd. T cd = N c N n n ) P X i = n). 4) n=0 First, note tat for n = 0 and n = te rigt-and side equals zero, and tus we can cange te summation domain: T cd = N c N n= n n ) N n ) N c ) n ) N n N c. 43) Furtermore, by using te definition of te binomial coefficient we obtain: N n ) = ) N N n n. 44) And we can rewrite Eq. 4) as: T cd = N c N n= N N c N N c N n ) N c ) n ) N n N c. 45) Substituting n = a + and N = b + gives T cd = N c = N c N N c N N c N N c b a=0 ) ) b a a N c ) b a N c N N c, 46) were in te second step te definition of a probability density function is used. Number of cells for cross-level searc To estimate te number of potential contacts for te crosslevel searc, first an estimate of te number of cells tat ave to be cecked for possible cross-level contacts as to be made. Terefore, consider a particle p at position x p wit radius r p suc tat it resides at ierarcy level and we want to calculate te number of cells at ierarcy level j tat ave to be cecked for possible contacts. Tis can be calculated from Eq. 4), wic reads for te x-direction note tat te directions are statistically independent): c start = c end = x p r p s j / x p r p = s j s j x p + r p + s j / x p + r p = + s j s j, 47). 48) 3