diff options
Diffstat (limited to 'macros/latex/contrib/springer/svjour/granma/example/example.tex')
-rw-r--r-- | macros/latex/contrib/springer/svjour/granma/example/example.tex | 480 |
1 files changed, 480 insertions, 0 deletions
diff --git a/macros/latex/contrib/springer/svjour/granma/example/example.tex b/macros/latex/contrib/springer/svjour/granma/example/example.tex new file mode 100644 index 0000000000..39ec03ca46 --- /dev/null +++ b/macros/latex/contrib/springer/svjour/granma/example/example.tex @@ -0,0 +1,480 @@ +\documentclass[granma]{svjour} +\usepackage{graphicx} +%\input{totalin.grm} +%\idline{Granular Matter 1, 1-6}{1} +% +\begin{document} +\title{A Fast Model for the Simulation of Non-round Particles} +\author{Alexander V. Potapov and Charles S. Campbell} +\institute{Department of Mechanical Engineering \\ +University of Southern California \\ +Los Angeles, Ca. 90089-1453, USA} +\date{} +\maketitle + +\abstract{% +This paper describes a new, computationally efficient model for the discrete +element simulation of a certain class of non-round particles. The boundaries +of the particles in this model are constructed from the circular segments of +different radii in such a way that connections between these segments are +continuous. As such, the model does not permit the simulation of arbitrarily +shaped particles, but it does allow a wide enough variety of shapes to assess +the effects of non-round shapes (in particular, particle interlocking) in an +efficient maner. A direct test of the model's performance demonstrates that +the model is much more efficient than other models for non-round particles +currently available and is less than two times slower than models for the +same number of round particles.} + +\section{Introduction} + +In recent years, soft particle discrete element simulations have pro\-ven to be +a powerful tool for investigating the behavior of granular systems. All of +these simulations (see the reviews in \cite{camp86,camp97}) follow individual +particles as they move, rotate and interact with their neighbors. In a real +granular system the forces generated when particles are in contact result in +small changes in the particle shape, particularly around the points of +contact. However, in the simulation, it is generally assumed for simplicity's +sake, that individual particles do not change shape. Instead their surfaces +are allowed to overlap slightly and, in an approximation of the true elastic +response, generate a repulsive response which is a function of that overlap. +Virtually any imaginable dependence of the contact force on the overlap may +be incorporated into the model, e.g. simple linear elastic, Hertzian or some +other. + +At present time, the majority of the soft particle discrete element +simulations have been performed using round particles for both +two-dimensional and three-dimensional systems \cite{camp86,camp97}. The +round +shape is very easy to simulate in the sense that every point on its surface +can be determined simply by knowing the position of the center and its +diameter; in particular, there is no dependence on the orientation of the +particle. As a result, it is easy to determine the overlap between two +particles simply as the distance between their centers minus the sum of their +radii. Also, it reduces memory requirements as only the center position need +be stored. + +However the use of round particles has several limitations. Near\-ly all +natural particles are not round, a quality which is shared even by artificial +particles that are intended to be round (e.g. commercial glass beads). +Furthermore, non-round particle shape can create a vast difference in the +mechanical behavior because they do not roll easily. For example, bulk +materials composed of round particles have angles of repose that are much +smaller than natural materials (see, for example \cite{posch93}). One +solution to this problem is to not allow the particles to rotate at all, but +instead interact only frictionally. Realistic angles of repose may be +obtained using this technique, but it in no way clear that the resulting +material accurately simulates all the effects of non-round particles. In +particular, realistic particles may interlock with one another and produce +behavior that cannot be approximated by a simple friction at the surface of +round particles. Another technique is to introduce the special force term +\cite{cund79} to account for the \ Coulomb's law behavior commonly +associated with granular materials, but this moves the simulation even +farther from real situation. + +The most natural solution for this problem is to use particles that are not +round. Several such model have been proposed. In general they can be divided +into two classes. In the first, an analytic representation of the shape of a +particle is used for which it is possible to simplify the contact and overlap +determinations. For example, algorithms have been developed for particles +with elliptic (e.g. \cite{roth91,roth92,ting92,kohr93}) or superquadric (e.g. +\cite{will89,must93}) shapes (the superquadric particle shape is described by the +same equation as elliptic one with the only difference is that the +degree is above two). Such particle can interlock like real materials, +but they +do not resemble natural materials. Also, aspect ratios for elliptical +particles must be significantly different from unity in order to obtain +realistic angles of repose, which may present other problems as the majority +of natural granular materials have aspect ratios that are close to unity. \ \ +The second type of model assumes that the particles are in the shapes of +arbitrary convex polygons \cite{hogue93,hop92,kohr95,till95,posch95}. (Such a +model forms the basis +of the fracture simulations presented in \cite{pota95a,pota95b,pota96a,pota96b}.) More +realistic looking particles may be created in this manner, but at tremendous +computational costs. Polygonal simulations can take up to an order of +magnitude more computer time than their round counterparts. For analytically +represented particle shapes, (elliptic or superquadric cases) the majority +of the computer time is spent in iterative solutions of the non-linear +equations used to calculate the particle overlap. In the polygonal +simulations, the majority of the computer time is spent calculating +intersections of the sides of contacting polygons which is a necessary part +of the overlap determination. + +\begin{figure} +\includegraphics[width=\hsize]{p1.ps} +\caption{The picture of a typical oval. The +boundary of this oval consists +of four arches drawn around points $O_1$, $O_2$, $O_3$ and $O_4$, arches +drawn around points $O_1$ and $O_3$ have the same radius and angles +$\alpha_1$ and $\alpha_3$ are the same. The same is true about the arches +drawn around points $O_2$ and $O_4$ and angles $\alpha_2$ and $\alpha_4$. +Points $O_1$ and $O_3$ are situated in such a way that line connecting these +points pass through the center of an oval, the same is true about the line +connecting the points $O_2$ and $O_4$.}\label{fig1} +\end{figure} + + +\section{The Description of the Model} + + +Here we propose a soft particle discrete element model for the simulation of +non-round particles which exhibits a performance that is comparable to the +performance of round particle models and is able to accommodate a much wider +variety of shapes (and more realistic shapes) than the elliptic or +superquadric models. The idea for this model was born from the way the +ellipses are sometimes approximated in technical drawings; when appropriate +elliptical templates are not available, an oval with the same aspect ratio is +often used to approximate the ellipse. An oval is geometric figure whose +boundary is determined by four circular arches of two different radii which +are joined together in a continuous way (i.e. the first derivative is +continuous between the arches) and thus can be drawn with compasses and other +standard tools used to draw circles. However, it is possible to generate more +complex shapes, many nearly polygonal in shape, from circular arches. At the +same time, the determination of contact and overlap between two circular +arches segments is nearly identical to the same procedures for circles. Thus +a simulation of particles generated from circular arches will allow a variety +of shapes to be studied at only a small computational cost beyond that of +round particles. + + +To illustrate this technique, consider the typical oval presented in +Fig. \ref{fig1}. +The boundary consists of four arches drawn around points $O_1$, $O_2$, +$O_3$ and $O_4$. In a classical oval, the arches drawn around points $O_1$ +and $O_3$ have the same radius (we shall call this radius ``large'') and the +angles $\alpha_1$ and $\alpha_3$ are identical. The same is true about the +arches drawn around points $O_2$ and $O_4$ (we shall call this radius +``small'') and the angles $\alpha_2$ and $\alpha_4$. Points $O_1$ and $O_3$ +are situated in such a way that line connecting these points passes through +the center of an oval, as does the line connecting the points $O_2$ and $O_4$. + +\begin{figure} +\includegraphics[width=\hsize]{p2.ps} +\caption{This picture illustrates the procedure of the calculation of the +contact parameters between a pair of ovals. After the contacting arches are +determined (arches corresponding to angles $\alpha_1$ and $\alpha_2$), the +calculation of the contact parameters can be carried out in the same way as +it is done for the round particles.}\label{fig2} +\end{figure} + +Now suppose that we have a granular medium consisting of oval shaped +particles which will be simulated using an otherwise standard soft-particle +discrete element model. To do this, we must at every time step find the +particles which are overlapping, calculate the distance of this overlap and +determine the center of this overlap in order to know the point at which +contact forces are to be applied to the contacting particles. Let us suppose +that we have a contact between two ovals as is shown in Fig. \ref{fig2}. +Any possible +contact must occur between the arch of one oval and the arch of the other +oval. This type of contact can be treated in the same way as the contact of +the two circular particles. Thus the overlap distance is simply the sum of +the two radii of the contacting arches minus distance between the points +around which these arches are drawn (i.e. distance between the points $O_1$ +and $O_2$ in Figure \ref{fig2}). The center of the overlap lies in the +line connecting +$O_1$ and $O_2$ \ midway between the points of intersection of this line and +the two arches (point $C$ in the Figure \ref{fig2}). + +Compared to the case of round particles, the only additional operation which +is required here is to determine which of the arches may be involved in the +contact. This can be carried out in very simple and effective way. Note that +two ovals can have the contact along a pair of arches if and only if the line +connecting the points around which these arches are drawn lies within the +limits of the angle of both arches (in Figure \ref{fig2} this requires +determining +that the \ line $O_1O_2$ lies between angles $\alpha_1$ and $\alpha_2$). This +is easy to check by calculating the dot products of the unit vector +connecting the points around which the arches are drawn with unit vectors of +the bisectors of the angles of the arches. Plus, the dot product of the +vector connecting the centers of gravity of the ovals and the vector +connecting the points around which the arches are drawn should be calculated +to ensure, by its sign, that the potential contacting arches are oriented +towards rather than away from each other. The calculation of these three dot +products is an additional computational step with respect to the case of the +round particles. However, it is not necessary to perform all three +calculations at each time step. Since the particles do not drastically change +their orientation during a single time step, in the vast majority of cases, +it will only be necessary to check at each time step whether the same two +arches remain in contact. This means the calculation of only the first two +dot products, which is rather fast and straightforward procedure. Also, it is +quite likely that any two arches that are in contact, were in contact during +the previous time step. Consequently, much computer time can be saved by +simply verifying that the contact continues on subsequent time steps. The +calculation of the contacts of the ovals with the straight boundary walls is +performed in much the same way. \enlargethispage{3pt} + +\begin{figure} +\includegraphics[width=\hsize]{p3.ps} +\caption{Examples of the construction of the particles out of the +arches continuously joined to each other (the first derivative is the +same on the both sides of the arch to arch transition point): (a) +approximation of a triangle, (b) approximation of a square, (c) +approximation of a pentagon and (d) approximation of a heptagon.}\label{fig3} +\end{figure} + +The model described above is easily extended beyond the case of ovals. Ovals, +like ellipses, have the disadvantage of requiring relatively large aspect +ratios to obtain realistic values of the angles of repose. Much more +complex particles may be constructed in the same way the ovals are +constructed. If the boundary of a particle consists of any number of arches +which are continuously connected together, one can see that the algorithm for +determining contact as described above still holds true. Some examples of +this sort of particles are presented in Fig. \ref{fig3}. For these +particles, a large +radius is used to construct the ``sides'' of the particles and a smaller +radius is used to connected them smoothly together at the ``vertices.'' +Clearly, one can easily construct an approximation to a triangle, square, +pentagon and so on, and one can get very close to an actual polygonal shape +as one wishes by making the corresponding radius of curvature of the arches +that form the sides very large and those that form the corners, very small. +(This process is limited only by roundoff errors in the determination of the +overlap. I.e. as the overlap is computed by taking differences, high accuracy +requires that its magnitude be large enough relative to the radius to be +represented by several number significant digits in its floating point +representation within the computer.) If these particles are constructed of +the arches with only two values of radii as it was done for the oval and all +of the shapes in Fig. \ref{fig3}, no additional memory (beyond that +required for an +oval) need be used for the storage of the information about position and +orientation of the particles. + +One additional advantage and one additional disadvantage of the model +proposed here should be mentioned. The advantage is that this model has a +fixed value of the radius of curvature at every point of the surface of the +particles that is bounded by continuously joined arches. This makes it +possible to use the Hertzian contact model (which describes the contact +between particles with arbitrary, but finite and non-zero, radii of +curvature) without any additional assumptions - which would be a questionable +for the case of the angular particles. The disadvantage of the model is that +for the moment we can not see its simple three-dimensional implementation. + + +\section{Demonstration of the Performance of the Mo\-del} + +It is clear from the material presented above that the performance of the +proposed model should be only marginally worse than a round particle model as +only a few additional calculations are required. A direct test of this fact +is described in the following. + +For the test, we have chosen the problem of settling the particles into a +rectangular box under the influence of the gravity. In the first case, the +particles are round, in the second case, the particles are the same number of +quasi-triangles (Figure 3a), in the third case the particles are +quasi-squares (Figure 3b), in the forth case, particles are of superquadric +shape with superquadric degree 3.0 which are simulated by the technique +described in \cite{must93} and finally, in the fifth case +particles are +square and simulated by the algorithm described in \cite{hop92}. +Linear spring contact laws are assumed for all cases except for the case of +square particles for which the normal contact force was based on the area of +overlap (see \cite{hop92}). The geometry of the boundary walls, +the +particle masses, particle densities, the time step and the coefficient of the +restitution of the particles were the same in all cases. The models were run +on the same Sparcstation 20 computer (75 MHZ processor), for the same period +of simulation time (4.69$\cdot 10^4$ time steps), and the amount of CPU time +expended was determined and compared. The same contact search routine for +near neighbors \cite{hop92} was used in all simulations. + +\begin{figure} +\includegraphics[width=\hsize]{p4.ps} +\caption{Results of the simulation of gravity settling round particles. +This simulation has been performed in order to compare the efficiencies of +the models. (a) Initial positions of the particles; (b) positions of the +particles at the end of the simulation.}\label{fig4} +\end{figure} + +\begin{figure}[!t] +\includegraphics[width=\hsize]{p5.ps} +\caption{Results of the simulation of the gravity settling of the +quasi-triangular particles in a rectangular box. It was performed for the +same particle mass, same time stepand same time elapsed in the system (same +number of time steps) as the simulation for round particles presented on +Figure 4. (a) Initial positions of the particles; (b) positions of the +particles at the end of the simulation.}\label{fig5} +\end{figure} + +\begin{figure}[!t] +\includegraphics*[height=7.3cm,bb=118 219 494 572,angle=-90]{p6.ps} +\caption{Same as Figure 5, but for quasi-square particles}\label{fig6} +\end{figure} + +\begin{figure}[!t] +\includegraphics[width=\hsize]{p7.ps} +\caption{Results of the simulation of gravity settling of the same number of +superquadric particles \cite{will89} as on Figure 4. It was +performed +for the same particle mass, same time step and same time elapsed in the +system (same number of time steps) as the simulation for round particles +presented on Figure 4. \ (a) Initial positions of the particles; (b) +positions of the particles at the end of the simulation}\label{fig7} +\end{figure} + +\begin{figure}[!t] +\includegraphics*[height=7.1cm,bb=118 219 494 572,angle=-90]{p8.ps} +\caption{Results of the simulation of gravity settling of the same number of +square particles as on Figure 4. It was performed for the same particle mass, +same time step and same time elapsed in the system (same number of time +steps) as the simulation for non-round particles presented on Figure 4. (a) +Initial positions of the particles; (b) positions of the particles at the end +of the simulation. +}\label{fig8} +\end{figure} + +The initial positions of the particles are presented in Figure +\ref{fig4}a for the +case of round particles, in Figure \ref{fig5}a for the case of +quasi-triangular +particles, in Figure \ref{fig6}a for quasi-square particles, in Figure +\ref{fig7}a for the +superquadric particles and in Figure \ref{fig8}a for square particles. +Figures +\ref{fig4}b, +\ref{fig5}b and \ref{fig6}b, \ref{fig7}b and \ref{fig8}b depict the +final positions of the particles for these five cases. + +\begin{table} +\caption{CPU time spent on SPARC-20 for performimg of $4.69 +\cdot +10^4$ time steps}\label{tab1} +\begin{tabular}[t]{|l|l|} +\hline +Particle shape&CPU$\,$time (seconds)\\ +\hline +Round &29\\ +\hline +Quasi-triangle&48\\ +\hline +Quasi-square&51\\ +\hline +Superquadric&347\\ +\hline +Square&114\\ +\hline +\end{tabular}% +\end{table} + +The results of the simulation of these cases on a SPARC-20 workstation are +presented in Table \ref{tab1}. These results show the CPU difference of +about +66\% +between the case of round particles and case of the quasi-triangular +particles composed of arches. The case of quasi-square particles is just 4\% +slower than the case of quasi-triangular particles, this time difference is +due to slightly longer procedure of the determination of contacting arches +for quasi-square particles than for quasi-triangular particles due to larger +number of arches. The case of superquadric particles was more than one +order of magnitude slower than the case of round particles. (It is hard to do +exact performance comparisons for this case since the procedure is +iterational and ``exact'' solution is never reached. We used four iterations +per time step here, different number of iterations may produce different +results.) Finally, the model with square particles was almost four times +slower than the model for round particles and thus more than two times slower +that the model proposed in this paper. Thus, the model presented here easily +outperforms every other model for non-round particles available at the +present time. + + +\section{Conclusions} + +In this paper, a model has been proposed for a soft-particle discrete element +simulation of the motion of non-round particles. This model is based on the +construction of a particle boundary out of a set of arches which are +connected to each other so that the first derivative at the point of +transition from one arch to another is the same from both sides. The model +allows us to vary considerably the degree of roughness of a particle surface, +and it performs approximately 60\% slower than a model for the round +particles, thus easily outperforming any other model currently available for +non-round particles. + +The model is capable of efficiently simulating nearly polygonal particles. +The main use of this type of simulation is not necessarily to simulate the +actual shapes of the particles in naturally occurring materials, but to +provide a way of adding the effects of non-round shapes (in particular, +particle interlocking) into existing simulations. + + +\begin{thebibliography}{19} +\bibitem{camp86} Campbell, C.S., Computer simulation of rapid granular +flows, Proc. 10$^{th}$ US National Congress of Applied Mechanics, Austin +Texas, June 1986, ASME, New York, p.327-338. + +\bibitem{camp97} Campbell, C.S., Computer simulation of powder flows, to +appear in Powder Techology Handbook, Second Edition. (Gotoh et al. eds) +Dekker, New York, 1997, 777-793. + +\bibitem{posch93} Poschel, T. \& Buchholtz, V., Static +Friction Phenomena in +Granular Materials: Coulomb Law versus Particle Geometry, Physical Review +Letters 71 (1993), p. 3963-3966. + +\bibitem{cund79} Cundall, P.A. \& Strack, O.D.L., A discrete numerical model +for granular assemblies, Geotechnique 29 (1979), p.47-65. + +\bibitem{roth91} Rothenberg, L. \& Bathurst, R.J., Numerical Simulation of +Idealized Granular Assemblies with Plane Elliptical Particles, Computers and +Geotechnics 11 (1991), p.315-329. + +\bibitem{roth92} Rothenburg, L. \& Bathurst, R.J., Micromechanical features +of granular assemblies with plane elliptical particles, Geotechnique 1 +(1992), p.79-95. + +\bibitem{ting92} Ting, J.M., A robust algorithm for ellipse-based discrete +element modeling of granular materials, Computers and Geotechnics 13 (1992), +p.175-186. + +\bibitem{kohr93} Kohring, G.A., Computer simulations of +sintering via granular dynamics, Physica A 195 (1993), p.1-11. + +\bibitem{will89} Williams, J.R. \& Pentland, A.P., Superquadric and Modal +Dynamics for Discrete Elements in Concurrent Design, Proc. $1^{st}$ US +Conference on Discrete Element Methods, Golden Colorado, October 1989, (Eds. +Mustoe, G.G.W., Henriksen, M. \& Huttelmaier, H.P.) + +\bibitem{must93} Mustoe, G.G.W. \& DePooter, G., A numerical model for the +mechanical behavior of particulate media containing non-circular shaped +particles, Powders and Grains 93, 1993, (Ed. Thornton, C.) Balkena, +Rotterdam, p.421-427. + +\bibitem{hogue93} Hogue, C. \& Newland, D.E., Efficient +computer modeling of +the motion of arbitrary grains, Powders and Grains 93, 1993, (Ed. Thornton, +C.) Balkena, Rotterdam, p.413-419. + +\bibitem{hop92} Hopkins, M.A., The Numerical Simulation of Systems of +Multitudinous Polygonal Blocks, US Army Cold Regions Research and Engineering +Laboratory, USACRREL Report CR 99-22, 1992. + +\bibitem{kohr95} Kohring, G.A., Melin, S., Puhl, H., +Tillemans, H.J., +Vermohlen, W., Computer simulations of critical, non-stationary granular flow +through a hopper, Computer Methods in Applied Mechanics and Engineering 124 +(1995), p.273-281. + +\bibitem{till95} Tillemans, H.J. \& Herrmann, H.J., Simulating deformations +of granular solids under shear, Physica A 217 (1995), p.261-288. +\newpage + +\bibitem{posch95} Poschel, T. \& Buchholtz, V., Molecular Dynamics of +Arbitrarily Shaped Granular Particles, Journal de Physique I 5 (1995), +p.1431-1455. + +\bibitem{pota95a} Potapov, A.V., Hopkins, M.A. \& Campbell, C.S., A +Two-dimensional Dynamic Simulation of Solid Fracture. +Part I: Description of the Model, International Journal of Modern +Physics C 6 (1995), p.371-398. + +\bibitem{pota95b} Potapov, A.V., Campbell, C.S. \& Hopkins, M.A., A +Two-dimensional Dynamic Simulation of Solid Fracture. Part II: Examples, +International Journal of Modern Physics C 6 (1995), p.399-425. + +\bibitem{pota96a} Potapov, A.V. \& Campbell, C.S., A Hybrid +Finite-element +simulation of Solid Fracture, International Journal of Modern Phy\-sics C 7 +(1996), p.155-180. + +\bibitem{pota96b} Potapov, A.V. \& Campbell, C.S., A Three-dimensional +Dynamic Simulation of Solid Fracture, International Journal of Modern Physics +C 7 (1996), p.717-730. + +\end{thebibliography} +\end{document} |