Vol. 21 No. 2/3/4, 2004 pp. 278-303 q Emerald Group Publishing Limited. 0264-4401 ... computational mechanics is given by Belytschko et al. (2000). Belytschko et al. ... Explicit methods are popular for a number of reasons: they allow ...... analysis”, J. Eng. Mechanics, ASCE, Vol. 121 No. 11, pp. 1234-43. EC. 21,2/3/4. 302 ...
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EC 21,2/3/4
278 Received February 2003 Revised June 2003 Accepted June 2003
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Selecting a suitable time step for discrete element simulations that use the central difference time integration scheme Catherine O’Sullivan Department of Civil Engineering, University College Dublin, Dublin, Ireland
Jonathan D. Bray Department of Civil and Environmental Engineering, University of California, Berkeley, California, USA Keywords Simulation, Time measurement Abstract The distinct element method as proposed by Cundall and Strack uses the computationally efficient, explicit, central difference time integration scheme. A limitation of this scheme is that it is only conditionally stable, so small time steps must be used. Some researchers have proposed using an implicit time integration scheme to avoid the stability issues arising from the explicit time integrator typically used in these simulations. However, these schemes are computationally expensive and can require a significant number of iterations to form the stiffness matrix that is compatible with the contact state at the end of each time step. In this paper, a new, simple approach for calculating the critical time increment in explicit discrete element simulations is proposed. Using this approach, it is shown that the critical time increment is a function of the current contact conditions. Considering both two- and three-dimensional scenarios, the proposed refined estimates of the critical time step indicate that the earlier recommendations contained in the literature can be unconservative, in that they often overestimate the actual critical time step. A three-dimensional simulation of a problem with a known analytical solution illustrates the potential for erroneous results to be obtained from discrete element simulations, if the time-increment exceeds the critical time step for stable analysis.
Introduction As outlined by Bardet (1998) and Kishino and Thornton (1999), a number of different algorithms have been developed to model the granular materials discretely. The most widely used method is the distinct element method as Engineering Computations Vol. 21 No. 2/3/4, 2004 pp. 278-303 q Emerald Group Publishing Limited 0264-4401 DOI 10.1108/02644400410519794
Funding for this work has been provided by The O’Reilly Foundation and David and Lucile Packard Foundation. Professor Ng of the University of New Mexico provided the source code for the program Ellipse3D. Two excellent reviews by anonymous reviewers significantly helped the authors to address a number of shortcomings in the first draft of this paper and the sharing of their insight and expertise is gratefully acknowledged.
proposed by Cundall and Strack (1979). Kishino and Thornton (1999) outline the assumptions that are common to particle-based discrete element method (DEM) codes developed to model the granular materials. In the absence of damping, all these DEM algorithms are concerned with solving the equilibrium equations at discrete time intervals for the system of particles: Ma þ KDx ¼ Df
ð1Þ
Selecting a suitable time step 279
where M is the mass matrix, a is the acceleration vector, K is the stiffness matrix, Df is the incremental force vector, and Dx is the incremental displacement vector. The elements of the stiffness matrix represent primarily the shear and normal springs that are present at the contact points. The stiffness matrix (K) changes during the analysis as contacts are formed and broken. Discrete element simulations can therefore be classified as non-linear, dynamic analyses. The principal difference between the available DEMs is the time integration algorithm used to solve equation (1) (O’Sullivan and Bray, 2001). Equation (1) is similar to the global equation considered in continuum finite element modeling. An analogy can therefore be drawn between a discrete element framework and a finite element framework; discrete element particles corresponding to the finite element nodes and inter-particle contacts corresponding to finite elements as shown in Figure 1. Recognition of this analogy is important as literature relating to finite element analysis is useful when discussing time integration issues in discrete element modeling. In this paper, the motivation for using a conditionally stable, explicit, time-integration algorithm is clarified first. Then, a new approach for calculating the critical time increment is proposed. Considering both two- and three-dimensional systems, it is shown that the critical time increment is
Figure 1. Schematic diagram illustrating DEM-FEM analogy. (a) Elements and nodes (FEM); and (b) particles and contacts (DEM)
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related to the current state of particle contact. The critical time steps for several particle arrangements are found to be significantly lower than those proposed by others (Itasca, 1998). The simulation of a problem with a known analytical solution illustrates the potential for erroneous results to be obtained from DEM simulations as a consequence of numerical instability due to the use of an insufficiently small time step. The importance of monitoring the energy balance of the system is highlighted. Time integration scheme A primary consideration in developing a discrete element procedure is the selection of the time integration scheme. Typically, in geotechnical continuum modeling for linear analyses, an implicit time integration scheme is employed. The choice between an implicit or explicit scheme for time integration is not clear for non-linear problems. A recent general discussion of the relative merits of implicit and explicit schemes in computational mechanics is given by Belytschko et al. (2000). Belytschko et al. advise that the selection of an integration method depends on the type of partial differential equation, the smoothness of data, and the response of interest. For hyperbolic systems, explicit schemes are generally preferable for wave propagation and inertial problems, where interest lies in the high frequency response. Belytschko et al. also argue that the explicit time integration is well suited to dynamic contact/impact problems as the time steps are small because of stability requirements, so the discontinuities due to contact-impact pose less challenges and neither linearization nor a non-linear solver is needed. It is interesting to recognize that the explicit time integration is frequently used in non-linear computational analysis and an explicit time integrator is typically used in the “mesh-free” finite element methods (Li and Liu, 2000). Explicit methods are popular for a number of reasons: they allow the evaluation of non-linear constitutive laws using historical information, they are easy to implement, and are robust. Furthermore, they allow for element-by-element evaluation (or in the case of DEM, particle-by-particle evaluation) of the internal force vector and hence do not require a global stiffness matrix. O’Sullivan and Bray (2001) discussed the relative merits of implicit and explicit schemes for discrete element modeling of particle systems. As a consequence of non-linearity of the problem, implicit discrete element formulations require a series of iterations to determine the stiffness matrix. One of the better-known implicit particle-based approaches is the discontinuous deformation analysis for disks (DDAD) algorithm. The DDAD algorithm was developed by Ke and Bray (1995) and is based on the discontinuous deformation analysis (DDA) algorithm proposed by Shi (1988). The approach in a DDA analysis is to construct the stiffness matrix based on the state of contact of the particles at the end of the previous time
step. Using this trial stiffness, the displacement increments are determined, and the stiffness matrix is modified, depending on whether contacts form or are broken over the course of the time increment. This can be thought of as being analogous to the return map algorithm in plasticity. The first iteration assumes linear or elastic behavior. If some plastic behavior is observed (in the form of contact bonds being broken or formed), the stiffness matrix is adjusted. This system of adjustment can only work if a relatively small number of contacts change status during the time increment. If convergence is not attained within a specified number of iterations (typically six), the analysis for that time step is repeated using a smaller time step. This iterative process increases the number of matrix iterations required per time step and varies the time increment used over the course of an analysis. The increase in computational time involved in the implicit method is not merely related to the cost of a single matrix inversion per time step. It is anticipated that multiple matrix inversions may be required during each time step, especially in systems where a large number of contacts are breaking and forming. It is also difficult to prove conclusively that the convergence is actually achieved in the last iteration of each time step. Moreover, changing the time step during the calculation can lead to changes in the algorithmic damping, which is dependent on the magnitude of the time step. Even in the absence of contact damping for the implicit analysis commonly used in DDA, a significant amount of damping occurs. The amount of this algorithmic damping that a given time integration method will produce can be calculated by evaluating the eigenvalues of the amplification matrix for that method (O’Sullivan and Bray, 2001). The amount of algorithmic damping is dependent on the time integration method employed as well as the time increment and natural period of the system. This is a significant limitation of using the implicit time integration scheme used in DDAD for particulate media simulations. In explicit algorithms, for each time step the equations of motion for each particle are first integrated completely independently, as if not in contact. The uncoupled update indicates, which parts of the body are in contact at the end of the time step and then the contact conditions are imposed. Within each time step, the iterations required in an implicit scheme are therefore avoided. Based on these considerations, and realizing that to be realistic simulations of granular materials will need to include millions of small discrete particles, there is a strong case to adopt an explicit time integration scheme in particle-based discrete element codes.
Stability of central difference time integration As discussed earlier, DEM uses the central difference time integration scheme. Using linear stability analysis, it can be shown that this scheme is
Selecting a suitable time step 281
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conditionally stable and the critical time-increment required for stable analyses can be calculated. If it is assumed that this linear stability analysis also holds for non-linear cases, then the maximum stable time increment (Dtcrit) is a function of the eigenvalues of the current stiffness matrix (Belytschko, 1983; Hughes, 1983). For a linear, undamped system the relationship is given by: Dtcrit ¼
2
vmax
ð2Þ
The maximum frequency, vmax, is related to the maximum eigenvalue (lmax) of the M 2 1K matrix as: pffiffiffiffiffiffiffiffiffi ð3Þ vmax ¼ lmax Calculating the eigenvalues of a large matrix is an expensive operation. In explicit finite element analysis, the following relationship is frequently used (this is an extension of Rayleigh’s theorem; a proof is given by Belytschko (1983)):
lmax # lemax
ð4Þ
where lemax is the maximum eigenvalue of the M e2 1K e matrix for element “e”, (M e¼element mass matrix, K e¼element stiffness matrix). An estimate for the critical time increment can then be made by applying equations (2) and (3), once lemax is known. For a single degree of freedom system, i.e. considering a single DEM particle, for the central difference time integration scheme, the critical time step (Dtcrit) is simply given by: Dtcrit
rffiffiffiffiffiffiffiffi T m ¼ ¼2 K eff p
ð5Þ
where T is the period of free vibration of the degree of freedom, m is the particle mass, and Keff is the effective stiffness governing the particle motion. Itasca (1998) describes a method for estimating the critical time step to avoid unstable behavior for an assembly of particles using an equivalent single degree of freedom system. In this approach, the critical time step is evaluated by considering the system to be an infinite series of point masses and springs as shown in Figure 2. The critical time increment (Dtcrit ) calculated by Itasca (1998) is then Dtitasca crit ¼
pffiffiffiffiffiffiffiffiffiffi m=K ;
Selecting a suitable time step 283
Figure 2. Multiple mass-spring system considered by Itasca (1998) in calculating the critical time step
where K is the contact spring stiffness and m is the particle mass. Itasca (1998) suggests that the critical time increment should be multiplied by a safety factor, with the default safety factor being 0.8. For the quasi-static type of analysis that is frequently considered in the geotechnical discrete element simulations, the time step constraint can be relaxed rather by using density or mass scaling. When mass scaling is used, it is assumed that the response of the system is not sensitive to inertia effects. The particle density can then be increased to increase the critical time step as calculated from equation (5). Mass scaling is frequently used in discrete element analyses (Thornton, 2000). If high frequency response is important, mass scaling is not recommended. Calculating the critical time step The approach for calculating the critical time step for DEM analyses proposed here considers the response of uniform disks and spheres with regular packing arrangements. The regular packing arrangement, especially at the densest possible state, leads to a more restrictive case for calculating the critical time step for this case, and hence can be used to estimate it for irregular packings of uniform disks and spheres. For cases, where the particles are not uniformly sized, or where the material fabric is irregular, a similar approach would need to be conducted to calculate the critical time step. However, the calculations would be complicated by the loss of symmetry for these non-uniform irregular cases. In this paper, the critical time step is calculated by considering a central disk or sphere in contact with a number of disks or spheres with equal dimensions. A compressive stress,
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acting in all directions, is applied to the system so that the overlap at each contact, Dtotal contact ; is non-zero and all the contact springs are engaged in compression. This stress condition would arise in the isotropic compression stage commonly employed initially in DEM simulations of laboratory tests (Lin and Ng (1997); Thornton (2000)). The current study differs from the earlier approach of Itasca (1998) for calculating the critical time step as a direct analogy is drawn between the nodes and elements of a finite element mesh and particles and contacts of a discrete element assembly. The approach adopted in this paper also differs from the approach proposed by O’Sullivan (2002), which was essentially an extension of that used by Itasca (1998). The new approach has the advantage of allowing the critical time step to be calculated directly by using Rayleigh’s theorem (equation (4)). It should be noted that the critical time steps calculated using the current approach are more restrictive than the results proposed by O’Sullivan (2002). Considering the finite element mesh shown in Figure 1(a), each element in the mesh is assigned a stiffness (using a constitutive model) and a mass (the product of the material density and element area or volume) so that the element mass matrices (M e) and element stiffness matrices (K e) can be formed. For the discrete element assembly, the stiffness of each contact is explicitly defined in the analysis, and the contact element stiffness matrix can be determined from the geometrical considerations. However, the contacts themselves do not have any mass; each contact links two particles with a finite mass. In this paper, some simple assumptions are adopted to distribute the mass of the particles of the contacts so that the contact element mass matrices can be constructed. Knowing the contact element stiffness and mass matrices, equations (2)-(4) can then be used to estimate the critical time increment for a discrete element assembly of particles. In the current study, the critical time steps are calculated for four cases: (1) two-dimensional disks with two translational degrees of freedom for each particle; (2) two-dimensional disks with two translational degrees of freedom and one rotational degree of freedom for each particle; (3) three-dimensional spheres with three translational degrees of freedom for each particle; and (4) three-dimensional spheres with three translational degrees and three rotational degrees of freedom for each particle. For each case, a number of regular symmetrical configurations of uniformly sized particles (disks and spheres) were considered. In each case, the eigenvalues l e for the elements were determined from the element stiffness and mass matrices and the critical time step was then calculated using equations (2)-(4).
Element stiffness matrices As shown in Figure 3 (in two dimensions), the contact elements in a discrete element model are very similar to strut elements used in structural analysis. Consequently, the element stiffness matrix for each contact element is very similar to the element stiffness matrix for a strut as detailed in numerous structural engineering texts (Sack, 1989). If a local coordinate system is assigned to each element and if each particle has two translational degrees of freedom (ui along the strut axis and v i orthogonal to the strut axis), then the element stiffness matrix for each contact (considering translational motion only) is given by: 2 K e;local
Kn
6 6 0 6 ¼6 6 2K n 6 4 0
0
2K n
Ks
0
0
Kn
2K s
0
0
Selecting a suitable time step 285
3
7 2K s 7 7 7 0 7 7 5 Ks
ð6Þ
where K e,local is the stiffness matrix for the local coordinate system, Kn the stiffness of the normal spring, and Ks the shear spring stiffness. This stiffness matrix can be transformed to the global coordinate system by: K e;global ¼ T T K e;local T
ð7Þ
where T is an orthogonal transformation matrix, given by:
Figure 3. (a) Rheological contact model for DEM analysis, and (b) strut element used to determine the contact element stiffness matrix
2
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cos u
6 6 2sin u 6 T¼6 6 0 4 0
286
sin u
0
0
cos u
0
0
cos u
0
2sin u
3
7 0 7 7 7 sin u 7 5 cos u
ð8Þ
and u is the inclination of the contact as shown in Figure 3. Most discrete element analysis programs allow both translational and rotational motions of the particles. In two dimensions, there is only one rotational degree of freedom. Assuming an assembly of uniform disks in an increment of rotation, Dfi, the moment induced on a disk with a radius r is given by r 2KsDfi. Then, when the rotational degrees of freedom are included in the analysis, the element stiffness matrix is:
K e;local
2 K n 6 0 6 6 6 0 6 ¼6 6 2K n 6 6 6 0 4 0
0
K sr
0
0
Kn
0
2K s
0
0
Ks
0
0
0
0
0 3 0 7 7 7 0 7 7 7 0 7 7 7 0 7 5 K sr 2
0
0
0
Ks
0
2K n
0
0
2K s
0
0
2
ð9Þ
and the transformation matrix is 2
cos u
6 6 2sin u 6 6 0 6 T¼6 6 0 6 6 6 0 4 0
sin u
0
0
0
cos u 0
0
0
0
1
0
0
0
0
cos u
sin u
0
0
2sin u
cos u
0
0
0
0
3
7 07 7 07 7 7 07 7 7 07 5
ð10Þ
1
Using the strut analogy in three dimensions, and again assigning a local coordinate system to each element, each particle has three translational degrees of freedom, u i ; along the strut axis (local x-axis), and v i ; and w i ; both orthogonal to the strut axis (local y- and z-axis). Considering translational motion only the local stiffness matrix is then:
2
K e;local
Kn
6 0 6 6 6 0 6 ¼6 6 2K n 6 6 0 4 0
0
0
2K n
0
Ks
0
0
2K s
0
Ks
0
0
0
0
Kn
0
2K s
0
0
Ks
0
2K s 0
0
0
0
3
0 7 7 7 2K s 7 7 0 7 7 7 0 7 5 Ks
Selecting a suitable time step ð11Þ
As earlier, the local stiffness matrix can be related to the global coordinate system by equation (7), T is an orthogonal transformation matrix, given by: 3 2 cos ax cos ay cos az 0 0 0 7 6 6 cos bx cos by cos bz 0 0 0 7 7 6 7 6 6 cos gx cos gy cos gz 0 0 0 7 7 6 ð12Þ T¼6 7 0 0 cos ax cos ay cos az 7 6 0 7 6 6 0 0 0 cos bx cos by cos bz 7 7 6 5 4 0 0 0 cos gx cos gy cos gz where ax is the angle between the local x-axis and global x-axis, ay the angle between the local y-axis and global x-axis, az is the angle between the local z-axis and global x-axis, bx is the angle between the local x-axis and global y-axis, etc. (Sack, 1989). There is no coupling between the translational and rotational kinematical degrees of freedom in two dimensions. The three-dimensional case is more complicated as the translational and rotational degrees of freedom are coupled for three-dimensional rigid body kinematics. However, if the reference axes for calculating angular momentum are coincident with the principal axes of inertia of the body, Euler’s equations can be used and the equilibrium equations can be decoupled. For the general case, where the moments of inertia around the three principal axes of inertia are not equal (i.e. I x – I y – I z ), a coupling remains between the three rotational degrees of freedom (about each of the principal axes of inertia). The central difference method cannot be used when considering the rotational kinematic equations for non-spherical three-dimensional particles, but suitable alternative time integration approaches have been proposed by Lin and Ng (1997) and Munjiza et al. (2003). In the case of perfect spheres (as considered here), I x ¼ I y ¼ I z ; this second level of coupling does not arise and the central difference approach is applicable. For these non-coupled cases, the element stiffness matrix is given by:
287
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2
3 Kn 0 0 0 0 0 2K n 0 0 0 0 0 6 7 6 0 7 Ks 0 0 0 0 0 2K s 0 0 0 0 6 7 6 7 6 0 7 0 Ks 0 0 0 0 0 2K s 0 0 0 6 7 6 7 0 0 K s r2eff;1 0 0 0 0 0 2K s r 2eff;1 0 0 6 0 7 6 7 6 7 2 2 6 0 7 0 0 0 0 0 2K s r eff;2 0 0 0 0 K s r eff;2 6 7 6 7 2 2 6 0 7 r 0 0 0 0 0 2K r 0 0 0 0 K s s eff;3 eff;3 7 6 7 K e;local ¼ 6 6 2K n 0 7 0 0 0 0 K 0 0 0 0 0 n 6 7 6 7 6 0 2K s 0 7 0 0 0 0 Ks 0 0 0 0 6 7 6 7 0 2K s 0 0 0 0 0 Ks 0 0 0 6 0 7 6 7 6 7 2 2 0 0 2K s r eff;1 0 0 0 0 0 K s r eff;1 0 0 6 0 7 6 7 6 7 6 0 7 0 0 0 0 0 K s r 2eff;2 0 0 0 0 2K s r 2eff;2 6 7 4 5 2 2 0 0 0 0 K s r eff;3 0 0 0 0 0 2K s r eff;3 0 ð13Þ
where reff,i is the orthogonal distance from the axis of rotation i to the contact point on the surface of the sphere. Note that in contrast to the two-dimensional case, in three dimensions r eff ;i – r in all cases, where r is the sphere radius. Consider, for example, the rhombic (or hexagonal close) packing configuration and rotation about a vertical axis through the central sphere as shown in Figure 4. For the contacts between the central sphere and spheres numbered 1-6, reff;i ¼ r: However, forpthe ffiffiffi contacts between the central sphere and spheres numbered 7-9, reff;i ¼ r= 3:
Figure 4. Rotation of uniform spheres with rhombic packing configuration
Mass submatrix As discussed by Zienkiewicz and Taylor (2000), in the finite element method either “consistent” or “lumped” mass matrices are typically used. A “lumped” mass matrix approach is popular as it yields a diagonal global mass matrix and is computationally efficient. In the lumped mass matrix approach, the nodal mass values are the sum of the mass contributions from the elements contacting at that node and the calculation details depend on the element shape. For example, in two dimensions for a simple three node triangular element with two degrees of freedom at each node, the element mass matrix is given by: 2
1
6 60 6 6 e60 rA 6 e M ¼ 3 6 60 6 60 4 0
0 0
0
0
1 0
0
0
0 1
0
0
0 0
1
0
0 0
0
1
0 0
0
0
0
3
7 07 7 07 7 7 07 7 7 07 5 1
ð14Þ
where r is the material density and A e the element area. Then, the nodal mass for each node in the system is given by: mi ¼
ne X rj Aej 3 j¼1
ð15Þ
where mi is the nodal mass for node i and ne the number of elements meeting at node i. In a discrete element assembly, the mass is explicitly “lumped” at the nodes (particles) as a mass value is assigned to each particle when the model is generated. In the current study, it is proposed to distribute the mass of each particle amongst the particle contacts so that a mass matrix can be developed for each contact element. When the particle assembly is made up of uniform disks or spheres with a regular, symmetrical packing configuration, the mass of the particles can be distributed equally amongst the contacts to develop the element mass matrices. If the particle mass is given by mi, the particle rotational inertia is given by Ii and the number of contacts for particle i is nci ; for the four scenarios considered in this study, the element mass matrices for the contact element linking particle i and j are then given by the following.
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Case I.
290
Two-dimensional system, no particle rotation allowed. 3 2 mi 0 0 0 nci 7 6 6 0 mi 0 0 7 c 7 6 ni 7 6 e 6 Mij ¼ 6 0 0 mj 0 7 7 ncj 7 6 7 6 4 mj 5 0 0 0 nc
ð16Þ
j
Case II. Two-dimensional system, particle 2 mi 0 0 0 nci 6 6 0 mci 0 0 ni 6 6 6 6 0 0 nI ic 0 6 i 6 Meij ¼ 6 0 0 0 mcj 6 nj 6 6 60 0 0 0 6 6 4 0 0 0 0
rotation allowed. 3 0 0 7 0 07 7 7 7 0 07 7 7 0 07 7 7 7 mj 07 ncj 7 7 Ij 5 0 nc
ð17Þ
j
Case III.
Three-dimensional system, no particle rotation allowed. 3 2 mi 0 0 0 0 0 nci 7 6 6 0 mi 0 0 0 0 7 c 7 6 ni 7 6 7 6 i 7 60 0 m 0 0 0 nci 7 6 7 6 Meij ¼ 6 0 0 0 mj 0 0 7 7 6 ncj 7 6 7 6 mj 6 0 0 0 0 nc 0 7 7 6 j 7 6 4 m 5 0 0 0 0 0 ncj j
ð18Þ
Three-dimensional system, particle rotation allowed. 3 2m i 0 0 0 0 0 0 0 0 0 0 0 nci 7 6 7 6 6 0 mnci 0 0 0 0 0 0 0 0 0 0 7 i 7 6 7 6 m i 60 0 0 0 0 0 0 0 0 0 07 nci 7 6 7 6 x 7 6 6 0 0 0 nI ic 0 0 0 0 0 0 0 0 7 7 6 i 7 6 7 6 I yi 6 0 0 0 0 nc 0 0 0 0 0 0 0 7 7 6 i 7 6 z 7 6 6 0 0 0 0 0 nI ic 0 0 0 0 0 0 7 7 6 i 7 6 e mj 6 Mij ¼ 6 0 0 0 0 0 0 nc 0 0 0 0 0 7 7 j 7 6 7 6 m j 60 0 0 0 0 0 0 0 0 0 07 ncj 7 6 7 6 7 6 m 6 0 0 0 0 0 0 0 0 ncj 0 0 0 7 7 6 j 7 6 7 6 I xj 6 0 0 0 0 0 0 0 0 0 nc 0 0 7 7 6 j 7 6 y 7 6 6 0 0 0 0 0 0 0 0 0 0 I jc 0 7 7 6 nj 7 6 6 z 7 I 4 0 0 0 0 0 0 0 0 0 0 0 jc 5 n
Selecting a suitable time step
Case IV.
291
ð19Þ
j
Realizing that the mass matrix is minimum for the most dense packing configurations (where nc is a maximum), it is important to ensure that the mass matrix is formulated correctly for these limiting cases. For uniform disks (two dimensions), hexagonal packing is the densest configuration that can be attained. As shown in Figure 5, this packing configuration is symmetrical so that the particle mass can be uniformly distributed amongst all six contacts. In three dimensions, two packing configurations can achieve the maximum packing density for uniform spheres; the rhombic (or hexagonal close) and face-centered-cubic (FCC) packing configurations (Figure 6). As noted by O’Sullivan (2002), while the void ratio and coordination number values are equivalent for both cases, there is a difference in the material fabric, resulting in differences in the mechanical response for spheres with these two packings.As shown in Figure 6, the rhombic and FCC packing configurations are not perfectly symmetrical. From the perspective of crystallography, these lattice packings can be described in terms of their symmetry. There are three types of symmetry: translational, rotational, and reflection (Kelly and Groves, 1980).
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292 Figure 5. Hexagonally packed uniform disks (Densest two-dimensional packing configuration)
Considering rotational symmetry, a lattice possesses an n-fold axis of rotational symmetry if it coincides with itself on rotation about the axis of 3608/n. Taking the axis of symmetry to pass vertically through the central sphere (Figure 6), the FCC packing has four-fold rotational symmetry, while the rhombic packing has three-fold rotational symmetry. Two approaches were then adopted to calculate the mass matrix for contact elements in the case of uniform spheres with rhombic and FCC packing configurations. In the first approach, the mass was simply distributed equally among all 12 contacts. In the second approach, the sphere was divided into three zones as shown in Figure 7. The mass of the upper zone was then distributed equally amongst the upper row of contacts (four for the FCC case and three for the rhombic case). The mass of the central zone was distributed equally amongst the upper row of contacts and central row of contacts (with a total of eight contacts for the FCC case and nine contacts for the rhombic case). Both mass matrices were used in calculating the critical time increments given below. Eigenvalue calculations The calculation of eigenvalues of the M e2 1K e,global can be simplified as follows. As M e is a diagonal matrix, then M e2 1 is also diagonal and M e21 T T K e;local T ¼ T T M e21 K e;local T
ð20Þ
T T ¼ T 21
ð21Þ
T is orthogonal so
and the matrices M e2 1K e,local and T TM e2 1K e,localT are similar and have the same eigenvalues (Golub and Van Loan, 1983). Then, for each contact element, the calculation of the eigenvalues of M e2 1K e,global is equivalent to calculating
Selecting a suitable time step 293
Figure 6. Orthogonal views of uniform spheres with FCC and rhombic packings
the eigenvalues of M e2 1K e,local. Recognizing this equivalence and assuming that the normal and shear spring stiffnesses are equal (i.e. K n ¼ K s ¼ KÞ; the eigenvalues were calculated for a number of symmetrical configurations of uniform disks and spheres. The critical time steps were then determined using equations (2)-(4). Figure 8 shows the critical time steps for a number of symmetrical arrangements of two-dimensional disks. These results indicate that the minimum critical timep increment in two dimensions for the case of translational ffiffiffiffiffiffiffiffiffiffi motion only is 0:577 m=K ; while the minimum critical time increment is pffiffiffiffiffiffiffiffiffiffi 0:408 m=K when particle rotation is also incorporated.
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Figure 7. Illustration of division of spheres for non-uniform distribution of inertia values to contact elements
Figure 9 shows the calculated critical time increments for a number of three-dimensional symmetrical arrangements of spheres. When compared with the two-dimensional case, the critical time increments for three-dimensional analysis are more restrictive. If the particle inertia is distributed pffiffiffiffiffiffiffiffiffiffi equally amongst all contacts, the minimum critical time step is 0:408 m=K for the case p of ffiffiffiffiffiffiffiffiffiffi translational motion only, and the minimum critical time step is 0:258 m=K when rotation is also allowed. If the particle inertia is distributed in a non-uniform manner according to Figure 7, then the minimum critical time pffiffiffiffiffiffiffiffiffiffi step isp0:348 m=K for translation only, and the minimum critical time step is ffiffiffiffiffiffiffiffiffiffi 0:221 m=K if rotation is also allowed. Hence, in general, for the three-dimensional case with uniform-sized spheres, the critical time step
Selecting a suitable time step 295
Figure 8. Critical time steps for various symmetrical configurations of uniform disks (two-dimensional)
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Figure 9. Critical time steps for various symmetrical configurations of uniform spheres (three-dimensional)
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi should be # 0:22 m=K if rotation is allowed and # 0:34 m=K if rotation is not allowed. Considering both two- and three-dimensional analyses, these calculated critical time steps are significantly more restrictive than the time step pffiffiffiffiffiffiffiffiffiffi recommended by Itasca (1998), which is m=K : Furthermore, the critical time increment for an assembly is a function of the packing density; as the coordination number (the number of contacts per particle) increases, the critical time step decreases. The material fabric is also important, reflected in the differences in the critical time steps calculated for the rhombic and FCC packings where the packing densities are equivalent.
Generalized equation for stability The particle assemblies typically analyses using DEM are statically indeterminate. It is therefore, difficult to establish a general expression for the mass matrix using to calculate the critical time increment. The results presented here clearly indicate that the recommendation by Itasca (1998) is not conservative. A p more ffiffiffiffiffiffiffiffiffiffiappropriate guideline would be to use a critical time increment # 0:3 m=K in two dimensions (includes a safety factor multiplier pffiffiffiffiffiffiffiffiffiffi of 0.75), while in three dimensions, the critical time increment # 0:17 m=K ; where m is the minimum particle mass and K is the maximum contact stiffness (includes a safety factor multiplier of about 0.75). Energy balance As discussed by Belytschko et al. (2000), numerical instabilities in explicit simulations can be detected by an energy balance check, as an instability results in the spurious generation of energy, which leads to a violation of the conservation of energy. Using the approach proposed by Belytschko et al. (2000), the energy balance requirement is given by: ðjW kin þ W int 2 W ext jÞ # 1 maxðW ext; W int ; W kin Þ
ð22Þ
where Wkin is the kinetic energy, Wint is the internal energy, Wext is the external energy, and 1 is a constant on the order of 102 2. Belytschko et al. (2000) recommend calculating the energy terms using an incremental approach. Guidance on the calculation of energy terms in DEM simulations is given by Itasca (1998). The kinetic energy term, Wkin, is given by: N
W kin ¼
p 1X ðv i ÞT m i v i 2 i¼1
ð23Þ
where Np is the number of particles in the domain, v i the velocity vector for particle i, and m i the mass of particle i. The internal energy term, Wint, at time t þ Dt; is given by: s 2 N c n 2 F X F 1 i i tþDt tþDt tþDt ¼ W 2 W W ¼ þ W tþDt int strain friction strain 2 i¼1 K n Ks ð24Þ N c X
tþDt t s s W friction ¼ W friction þ F i Dui i¼1
where Wstrain represents the strain energy, Wfriction is the energy dissipated by frictional sliding, Nc is the number of contacts in the domain, F ni is the normal force at contact i, F si is the shear force at contact i, vector usi is the incremental tangential displacement at the contact, and as earlier, Kn and Ks are the normal and tangential spring stiffnesses respectively. The external energy term is given by:
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tþDt tþDt tþDt W tþDt ext ¼ W body force þ W external applied forces þ W rigid wall forces Np X t W tþDt ¼ W þ mi b i D i body force body force i¼1
298
W tþDt external applied forces
¼
W texternal applied forces
þ
Np X
F applied Di i
ð25Þ
i¼1 t W tþDt rigid wall forces ¼ W rigid wall forces þ
Nb X
F i Dui
i¼1
where Wbody forces is the energy associated with the body forces, such as gravity, acting on the particles, Wexternal applied forces is the energy associated with any eternally applied loads, Wrigid wall forces is the work done on the system by the rigid boundaries, bi is the vector of body forces acting on i, vector Di is the incremental displacement of particle i, vector F applied is the applied external i force acting on i, vector Fi is the contact force at contact i, and vector Dui is the incremental displacement at contact i. The incremental displacements are the displacements over the current time increment, t ! t þ Dt: Example numerical simulation The three-dimensional simulations described here were performed using a modified version of the code Ellipse3D, which was provided by Professor Ng of the University of New Mexico. O’Sullivan (2002) summarizes the modifications that were made to this code. A simple three-dimensional simulation, considering the plane strain compression of 150 spheres with a FCC packing, was chosen to examine the implications of the stability issues discussed. This problem is considered in detail by Thornton (1979) and is also discussed by Itasca (1998). The analysis parameters are given in Table I, and the initial particle arrangement of the specimen is shown in Figure 10. These parameters are similar to the parameters used in the simulation of Rowe’s triaxial test by Cundall and Strack (1979). The sensitivity of the results to the simulation time-increment are shown in Figures 11 and 12. The stress ratio szz =sxx is plotted against the axial strain for p various ffiffiffiffiffiffiffiffiffiffi a values in Figure 11. The parameter, a, is defined by Dtanalysis ¼ a m=K ¼ a Dtitasca crit : Initially, a small amount of damping was applied, the coefficient of friction was set to be zero, and the system was cycled until the kinetic energy term became very small ð, 1025 Þ: Then, the damping coefficient was set to zero, friction was set to 0.3, and top cap was moved downwards at a constant velocity rate. For the prescribed coefficient of surface friction, the theoretical peak ratio, szz =sxx ; is given by Thornton (1979) as 4.88. As shown in Figure 11, when a ¼ 0:75; the results are clearly unstable. However, for a ¼ 0:45; while the results are incorrect, it would be difficult to
determine that the results are erroneous in the absence of knowledge of the correct theoretical solution. Examining the normalized energy difference values in Figure 12 (i.e. maxðW ext ; W int ; W kin Þ=ðjW kin þ W int 2 W ext jÞ), it can be seen that for the analysis with an a value of 0.45, the steady normalized energy difference is 1, which is unreasonably excessive. Re-examining Figure 11 for the analysis with an a value of 0.4, the results appear to be initially close to the theoretical value. However in this case, the steady normalized energy difference is excessive (i.e. larger than 102 2) and the system becomes unstable at very large strain values. When a ¼ 0:05 and 0.35, the results shown in Figures 11 and 12 indicate that the values close to the theoretically correct value of peak strength are
Parameter Normal spring stiffness Shear spring stiffness Density Radius Coefficient of friction
Selecting a suitable time step 299
Value Table I. 1.0 £ 1011 M/T2 Analysis parameters for sensitivity analysis 1.0£1011 M/T2 (M ¼ unit of mass, 2,000 M/L3 L ¼ unit of length, and 20 L T ¼ unit of time) 0.3
Figure 10. Initial particle configuration for sample simulation
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Figure 11. Sensitivity of the results to a values – stress ratio (a11/a33) vs axial strain
obtained and that the normalized energy difference at steady state are below the proposed limit of 102 2. Note that an a value of 0.35 is larger than the minimum critical time step for the FCC packing case when both translation and rotation are allowed (Figure 9). However, the lattice configuration considered in this analysis does not allow significant particle pffiffiffiffiffiffiffiffiffiffirotation, so for this case, the transition-only critical time step of # 0:37 m=K is more appropriate. The results of this analysis clearly indicate that an a value with an acceptable margin of safety must be used in the DEM analyses. Furthermore, the energy balance of the simulation should be monitored to ensure that the erroneous results are not calculated due to numerical instabilities associated with too large a time step. A further observation can be made in relation to these findings. Where stiffness proportional damping is included, a further reduction is required, and the recommendations outlined in this paper will not be conservative for this case. Belytschko (1983) and Belytschko et al. (2000) considered the case where the equilibrium equation for the system is given by:
Selecting a suitable time step 301
Figure 12. Sensitivity of the results to a values – energy balance
Ma þ Cv þ KDx ¼ Df
ð26Þ
where v is the velocity vector and the damping matrix C is given by: C ¼ a1 M þ a2 K
ð27Þ
Here, a1 and a2 are arbitrary parameters (i.e. Rayleigh damping). For this case, the critical time increment is given by: qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Dt crit ¼ ð28Þ zI þ 11 2 zI 4I where vI is the maximum frequency of the system and the damping ratio, zI (#1), is given by: a1 a2 4 I zI ¼ þ : 24 I 2 Conclusion The explicit, central difference time integration scheme used in many DEMs is conditionally stable. A new, simple method for calculating the critical time increment for numerical stability was proposed in this paper. Using the
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proposed method, the stability of systems of uniform particles with regular packing configurations was examined. By examining the cases of uniform circular and spherical particles with regular packing configurations, earlier pffiffiffiffiffiffiffiffiffiffi estimates of the critical time step, i.e. Dt itasca ¼ m=K ; were shown to be crit unconservative for these cases. A smaller critical time step of Dt crit # pffiffiffiffiffiffiffiffiffiffi 0:17 m=K is more appropriate for three-dimensional DEM simulations pffiffiffiffiffiffiffiffiffiffi involving uniform-sized particles and Dtcrit # 0:3 m=K should be used for the two-dimensional DEM simulations involving uniform-sized disks. The most critical scenario considered herein was the rotational motion of uniform spheres in FCC and rhombic configurations in the three dimensions. The critical time step was shown to be a function of the packing configuration and number of contacts per particle. This finding is especially significant for systems of non-convex particles, where there can be potentially far more contacts for each particle than when using convex particles. DEM analysts using “clusters” of particles that are “glued together” to form non-spherical particles should be particularly careful, as there is a significant increase in coordination number (i.e. number of contacts per particle) when these non-spherical particles are used (O’Sullivan and Bray, 2002). Additionally, simulations involving particles of vastly different sizes can lead to significantly larger coordination numbers, which would in turn also require smaller time steps. Therefore, it is necessary to monitor the energy balance of the system to detect potential numerical instabilities. This procedure was illustrated by considering the simple, well-understood case of uniform spheres in a FCC packing configuration in plane strain compression. References Bardet, J.-P. (1998), “Introduction to computational granular mechanics”, in Cambou, B. (Ed.), Behaviour of Granular Materials, CISM Courses and Lectures, No. 385, Springer-Verlag, Wien, New York, NY. Belytschko, T. (1983), “An overview of semidiscretization and time integration procedures”, in Belytschko, T. and Hughes, T.J.R. (Eds), Computational Methods for Transient Analysis, Computational Methods in Mechanics Series, Vol. 1, North Holland, New York, NY. Belytschko, T., Liu, W.K. and Moran, B. (2000), Nonlinear Finite Elements for Continua and Structures, Wiley, New York, NY. Cundall, P.A. and Strack, O.D.L. (1979), “A distinct element model for granular assemblies”, Geotechnique, Vol. 29, pp. 47-65. Golub, G.H. and Van Loan, C.F. (1983), Matrix-Computations, North Oxford Academic, Oxford. Hughes, T.J.R. (1983), “Transient algorithms and stability”, in Belytschko, T. and Hughes, T.J.R. (Eds), Computational Methods for Transient Analysis, Computational Methods in Mechanics Series, Vol. 1., North Holland, New York, NY. Itasca Consulting Group (1998), PFC2D 2.00 Particle Flow Code in Two Dimensions, Itasca Consulting Group, Inc., Minneapolis, Minnesota, MN. Ke, T.-C. and Bray, J.D. (1995), “Modeling of particulate media using discontinuous deformation analysis”, J. Eng. Mechanics, ASCE, Vol. 121 No. 11, pp. 1234-43.
Kelly, A. and Groves, G.W. (1980), Crystallography and Crystal Defects, Longman, London. Kishino, Y. and Thornton, C. (1999), “Discrete element approaches”, in Oda, M. and Iwashita, K. (Eds), Introduction to Mechanics of Granular Materials, Balkema, Rotterdam, The Netherlands, pp. 147-221. Li, S. and Liu, W.K. (2000), “Numerical simulations of strain localization in inelastic solids using meshfree methods”, International Journal for Numerical Methods in Engineering, Vol. 48, pp. 1285-309. Lin, X. and Ng, T.-T. (1997), “A three-dimensional discrete element model using arrays of ellipsoids”, Geotechnique, Vol. 47 No. 2, pp. 319-29. Munjiza, A., Latham, J.P. and John, N.W.M. (2003), “3D dynamics of discrete element systems comprising irregular discrete elements – integration solution for finite rotations in 3D”, International Journal for Numerical Methods in Engineering, Vol. 56 No. 1, pp. 35-55. O’Sullivan, C. (2002), “The application of discrete element modeling to finite deformation problems in geomechanics”, PhD thesis, Department of Civ. Eng., University of California, Berkeley, CA. O’Sullivan, C. and Bray, J.D. (2001), “A comparative evaluation of two approaches to discrete element modeling of particulate media”, in Bicanic, N. (Ed.), Proceedings of the Fourth International Conference on Discontinuous Deformation, University of Glasgow, Scotland, UK, pp. 97-110. O’Sullivan, C. and Bray, J.D. (2002), “Relating the response of idealized analogue particles and real sands”, Proceedings of the Numerical Modeling in Micro-Mechanics via Particle Methods, First International PFC Symposium, November 2002. Sack, R.L. (1989), Matrix Structural Analysis, Waveland Press, Illinois, IL. Shi, G.-H. (1988), “Discontinuous deformation analysis, a new numerical model for the statics and dynamics of block systems”, PhD thesis, Department of Civ. Eng., University of California, Berkeley, CA. Thornton, C. (1979), “The conditions for failure of a face-centered cubic array of uniform rigid spheres”, Geotechnique, Vol. 29 No. 4, pp. 441-59. Thornton, C. (2000), “Numerical simulations of deviatoric shear deformation in granular media”, Geotechnique, Vol. 50 No. 4, pp. 43-53. Zienkiewicz, O. and Taylor, R. (2000), The Finite Element Method, Volume 1, The Basis, 5th ed, Butterworth Heinemann, Oxford. Further reading Rowe, P.W. (1962), “The stress-dilatancy relation for static equilibrium of an assembly of particles in contact”, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 269 No. 1339, pp. 500-27. Thomas, P.A. (1997), “Discontinuous deformation analysis of particulate media”, PhD thesis, Department of Civ. Eng., University of California, Berkeley, CA.
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