an improved contact algorithm for the material point...

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2001 Tech Science Press CMES, vol.2, no.4, pp.509-522, 2001

An Improved Contact Algorithm for the Material Point Method and Applicationto Stress Propagation in Granular Material

S.G. Bardenhagen1 and J.E. Guilkey2 and K.M. Roessig3 and J.U. Brackbill4 and W.M. Witzel5 and J.C.Foster6

Abstract: Contact between deformable bodies is a dif-ficult problem in the analysis of engineering systems.A new approach to contact has been implemented us-ing the Material Point Method for solid mechanics, Bar-denhagen, Brackbill, and Sulsky (2000a). Here two im-provements to the algorithm are described. The first isto include the normal traction in the contact logic tomore appropriately determine the free separation crite-rion. The second is to provide numerical stability byscaling the contact impulse when computational grid in-formation is suspect, a condition which can be expectedto occur occasionally as material bodies move throughthe computational grid. The modifications describedpreserve important properties of the original algorithm,namely conservation of momentum, and the use of globalquantities which obviate the need for neighbor searchesand result in the computational cost scaling linearlywith the number of contacting bodies. The algorithm isdemonstrated on several examples. Deformable body so-lutions compare favorably with several problems which,for rigid bodies, have analytical solutions. A much moredemanding simulation of stress propagation through ide-alized granular material, for which high fidelity data hasbeen obtained, is examined in detail. Excellent quali-tative agreement is found for a variety of contact con-ditions. Important material parameters needed for morequantitative comparisons are identified.

1 Dept. of Mechanical Engineering, University of Utah, Salt LakeCity, UT 84112, USA. Presently on leave from Group ESA–EA,MS P946, Los Alamos National Laboratory, Los Alamos, NM

2 Dept. of Mechanical Engineering, University of Utah, Salt LakeCity, UT 84112, USA.3 Air Force Research Laboratory, Munitions Directorate, EglinAFB, FL 32542, USA.4 Theoretical Division, Los Alamos National Laboratory, LosAlamos, NM 87545, USA

5 Dept. of Computer Science, University of Utah, Salt Lake City,UT 84112, USA.6 Air Force Research Laboratory, Munitions Directorate, EglinAFB, FL 32542, USA.

keyword: Contact Algorithm, Material Point Method,Wave Propagation, Granular Material.

1 Introduction

With continued growth of computational power, numer-ical analysis techniques are being extensively applied toengineering systems, rather than just individual compo-nents. The promise of reducing costs during design andtesting is enticing, but remains difficult to realize in prac-tice. Extensive effort has been expended to develop ac-curate material models and solution techniques for com-plex boundary value problems involving individual com-ponents and pinned or welded structures. In the moregeneral arena of engineering systems, however, compo-nents often contact and slide against one another. Contactmechanics is remarkably difficult, both due to potentiallyapplicable physics, and mathematical complexities asso-ciated with solution constraint inequalities, as discussedin the review article by Barber and Ciavarella (2000).

When engineering systems function within designed op-erating conditions severe contact is usually avoided, butit is of paramount interest in evaluating system responseduring severe loading and/or failure. Classic examplesinclude car crashes, aircraft engine fan blade contain-ment, and earth penetrators. Contact and impact have re-ceived substantial attention over the past several decades,as witnessed by a review of the subject by Zhong andMackerie (1994), which lists nearly 500 papers. Themajority of these papers describe numerical modelingapproaches and/or applications using the finite elementmethod. The problem is a very difficult one, as contactmust be sensed, surface normals constructed, and inter-action forces imposed to prevent interpenetration with-out making the system of equations to be solved ill–conditioned.

For large scale engineering simulations contact is fre-quently the aspect which must be tweaked by the ana-

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lyst, using various loosely physical parameters, just toget the simulation to run. In addition the algorithms aretraditionally expensive. Effective numerical simulationof engineering systems is in need of further develop-ment of accurate, efficient contact algorithms. Here wedescribe an alternate approach using a particle–in–cell(PIC) numerical technique for solid mechanics, the Ma-terial Point Method, Sulsky, Chen, and Schreyer (1994);Sulsky, Zhou, and Schreyer (1995); Sulsky and Schreyer(1996).

The Material Point Method (MPM) is one of the lat-est developments in several decades of particle–in–cellmethods, originally used to model highly distorted fluidflow, Harlow (1963). Subsequent developments ad-vanced the understanding of the algorithm and broughtmodifications to reduce numerical diffusion, Brackbill,Kothe, and Ruppel (1988); Burgess, Sulsky, and Brack-bill (1992). Fundamental aspects of PIC methods in-clude the interpolation of information between grid andparticles, and precisely which solution variables will beascribed to the grid, and which to the particles. Sev-eral variants have been tried, with a general trend towardkeeping more properties on particles. Most recently, themethod has been applied to solid mechanics, where theability of the particles, or “material points”, to advect nat-urally Lagrangian state variables, has been exploited inMPM.Recently a new approach to material contact has beendeveloped and used with MPM, Bardenhagen, Brack-bill, and Sulsky (2000a). This approach takes advan-tage of the Arbitrary Lagrangian/Eulerian (ALE) formu-lation of MPM which tracks Lagrangian particle motionthrough an Eulerian grid. Coulomb friction contact con-ditions are applied via the grid. The approach has beendemonstrated using two–dimensional calculations of col-lisions and the shearing of granular material, Barden-hagen, Brackbill, and Sulsky (2000a,b,c). Here an es-sential modification to the physics of the algorithm is de-scribed, as well as a modification for numerical stabil-ity. The algorithm is demonstrated on simple problemsin three–dimensions with analytical solutions. Finally itis applied to stress wave propagation in simple granularmaterials.

2 Approach

MPM is briefly reviewed here for completeness. La-grangian bodies are discretized into material points,

which carry all information required to specify the cur-rent state and advance the solution. This information in-cludes constitutive parameters (such as moduli and in-ternal variables), stress, strain, velocity and temperature.The MPM algorithm also uses a computational grid. Thegoverning equations are solved on the grid, providing acomputational savings and as well as a regular, structuredgrid on which to apply solution techniques. See Fig. 1for an example of the discrete representation of a disk,where material points and mesh are shown. The cou-pling between the material points and the mesh is a keyingredient in the solution algorithm. Quantities are inter-polated between the mesh and the material points suchthat total mass and momentum are conserved. Advan-tages of the MPM algorithm include the absence of meshtangling problems, error–free advection of material prop-erties (internal variables in particular) via the motion ofthe material points, and an efficient setting for the imple-mentation of material contact.

Figure 1 : Simple example of a material point discretiza-tion of a disk.

Bodies deform according to continuum mechanics con-stitutive models and conservation laws. If ρ � x � t � is themass density at point x at time t, and v � x � t � is the veloc-ity field, then conservation of mass is

dρdt �� ρ∇ � v � (1)in which the time derivative is the material derivative

ddt � ∂∂t v � ∇ (2)

An Improved Contact Algorithm for the Material Point Method... 511

Conservation of mass is satisfied implicitly in MPM. Ma-terial points are assigned fixed masses during the initialdiscretization, and grid masses are determined using aninterpolation scheme which conserves mass. Conserva-tion of momentum is

ρdvdt � ∇ � σσσ (3)

where σσσ is the Cauchy stress tensor. Different mate-rials are modeled via constitutive equations that gen-erate stress based on both the history and current me-chanical state. Versions of hyperelasticity, hypoelastic-ity, plasticity and viscoelasticity have been implemented,Sulsky, Chen, and Schreyer (1994); Sulsky, Zhou, andSchreyer (1995); Sulsky and Schreyer (1996); Barden-hagen, Brackbill, and Sulsky (2000b); Bardenhagen andBrackbill (1998); Bardenhagen, Harstad, Maudlin, Gray,and Foster (1998). Conservation of momentum is solvedon the grid and changes are interpolated to the materialpoints such that the change in momentum is the sameon the grid and on the material points. Interpolatingonly changes in momentum reduces numerical diffusion,Brackbill, Kothe, and Ruppel (1988). Energy conser-vation errors are proportional to the square of the timestep, Brackbill and Ruppel (1986); Bardenhagen (2001).Details of the explicit computational algorithm may befound in the references, Sulsky, Chen, and Schreyer(1994); Sulsky, Zhou, and Schreyer (1995); Sulsky andSchreyer (1996).

Interactions between bodies are modeled using a con-tact algorithm which forbids inter–penetration, but al-lows separation, sliding with friction, and rolling. A pre-vious version of the algorithm, in which the logic wasbased completely on kinematics, was detailed in, Barden-hagen, Brackbill, and Sulsky (2000a). Here an essentialmodification to the original formulation is described. Inaddition, a useful means of screening out the applicationof large contact forces, numerical errors due to unfor-tunate registration of material point information on thecomputational grid, is described. These changes providefor a more accurate and robust algorithm.

2.1 Contact Algorithm Logic Refinements

Recall that contact is modeled on the computational gridin MPM. The mass and momentum from material pointsare interpolated to the computational grid for each body.Individual body velocities are computed by taking the ra-

tio of momentum, pα, to mass, mα, at the grid nodes, i.e.,

vα � pαmα α � 1 2 �� N (4)where α indexes the bodies and N is the number of bodiesin a computation. The average velocity of all materialpoints in the vicinity of a grid node is termed the centerof mass velocity and denoted vcm

vcm � N∑α � 1 pαM (5)where

M � N∑α � 1mα (6)is the total mass contribution from all bodies in the vicin-ity of a grid node. The center of mass velocity and thetotal mass are the global quantities which enable the de-termination of contact grid nodes and precisely what thecontact constraints should be. Nodes in the vicinity ofmore than one body are detected by looking at the differ-ence between individual body and center of mass veloc-ities. Interface computational grid nodes are defined asthose for which

vcm � vα �� 0 � (7)Resolution of the interface is commensurate with thecomputational grid cell size.

Once a contact grid node has been identified, furtheranalysis is required to determine what contact conditionshould be applied (sticking or slipping contact, or freeseparation). Spatial differentiation of the individual bodymasses on the grid, mα, provides a computation of thebody surface normals nα. Using the surface normals, ap-proach and departure can be distinguished. A body isapproaching its neighbor(s) when�vα � vcm � � nα � 0 � (8)

For the algorithm described previously, Bardenhagen,Brackbill, and Sulsky (2000a), sticking or slipping con-tact constraints are applied if approach is detected, other-wise free separation is allowed.

It is the use of normal traction information which dis-tinguishes this algorithm from that described previously,Bardenhagen, Brackbill, and Sulsky (2000a). The nor-mal surface traction, tαn , is computed at a contact grid

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node by interpolating individual material point contribu-tions using the surface normals (and the relation tαn �nα � σσσα � nα). In the special case where contacting bodiesare stress free (e.g. when first coming into contact) it issufficient to determine which contact condition to applybased strictly on whether or not bodies are approaching.When the normal tractions are non–zero, it is importantto distinguish between compressive and tensile interfacetractions. For the sign convention used here compressivestress is negative and the normal traction is compressivewhen

tαn � 0 � (9)The essential modification to the contact algorithm logicis to apply frictional contact when the normal tractionis compressive, allowing free separation otherwise. Fric-tional contact is applied via constraints on the body ve-locities vα. Free separation requires that no constraintsbe prescribed. Note that the efficiency of the algorithmis not compromised by the additional logic involving thenormal traction. Calculation of contact conditions in thecenter of mass frame eliminates a separate contact de-tection step, achieves a solution with one sweep throughthe computational grid, and yields a linear scaling of thecomputational cost with the number of bodies, Barden-hagen, Brackbill, and Sulsky (2000a). It is also worthnoting that the complexity of the contact algorithm doesnot increase if the shape of the bodies are varied, so com-plex initial geometries and large deformations can easilybe modeled.

Using the simple example of a deformable body collidingwith a rigid boundary, it may be seen that simply mon-itoring approach and departure is insufficient to modelcontact correctly. As the body approaches the bound-ary the compressive traction builds, and by either kine-matic or normal traction criteria, frictional contact con-ditions would be applied. Application of the frictionalcontact constraints is equivalent to an instantaneous in-elastic collision, serving to exchange kinetic and strainenergies. The essential difference occurs during rebound.If free separation is prescribed as soon as departure be-gins, the equivalent scenario is instantaneous removal ofthe boundary. Rather, the normal traction must be mon-itored and frictional contact with the boundary appliedwhile it remains compressive. Free separation is allowedonce the normal traction is non–negative. This modifica-tion provides for the extraction of strain energy on depar-

ture, much like it provides for strain energy buildup dur-ing approach. Without this logic modification, friction-less collisions were found to slightly increase system en-ergy. Although the logic is not described in detail there,the modified algorithm was found to be slightly dissipa-tive Bardenhagen, Brackbill, and Sulsky (2000c).When applied to contact between deformable bodies thesituation is very similar to that described for a rigidboundary when both have the same material response.Enforcement of no interpenetration during contact is ap-plied by constraining the body velocity component nor-mal to the surface to be equal to the center of massvelocity in that direction. This is equivalent to a uni-form stretch assumption in the surface normal direction.Because different materials respond to the same stretchwith different stresses, the normal tractions which de-velop during contact will likely be different for differ-ent bodies. It is even possible that uniform un–strainingwill result in bodies of different materials experiencingfree separation at different times during departure, par-ticularly when material response allows for permanentset (e.g. plasticity). Although traction equilibrium is notmaintained at contact nodes, interfaces are unloaded totheir stress free state. This source of error could be elim-inated in an implicit formulation of MPM where the ap-propriate partitioning of strain increments between bod-ies for traction equilibrium would be determined.

2.2 Numerical Considerations

A practical consideration arises in the application of anMPM algorithm which uses grid node variables. Namely,interpolated values can be very small when a grid nodefirst begins to represent material point data. This can oc-cur when a body’s material points first cross into cellswhich previously contained no material points from thatbody. Because this happens at the edges of bodies, it isan especially important consideration in the applicationof contact conditions.

Frictional contact conditions are applied by assigningnew grid velocities, ṽα, after contact

ṽα � vα � ∆vαn � nα � µ � tα "! µ � � min # µ ! ∆vαt∆vαn $ (10)where µ is the interfacial coefficient of friction, tα is theunit tangent in the direction of sliding (t � ωωω % n in Bar-denhagen, Brackbill, and Sulsky (2000a)), and

∆vα � vα � vcm ! (11)


∆vαn & ∆vα ' nα ( (12)∆vαt & ∆vα ' tα ( (13)as described in detail in Bardenhagen, Brackbill, and Sul-sky (2000a).

Resolution of the topology of contacting surfaces on thecomputational grid requires no more than two bodies berepresented at a contact grid node. The remaining contactnode analyses are specialized for N & 2. It should benoted that not all of the properties derived below hold forN ) 2. The definition of the center of mass velocity, vcm,Eqn. 5, gives the useful identity

2

∑α * 1mα∆vα & 0 ( (14)from which it can be seen that ∆v1t + ∆v1n & ∆v2t + ∆v2n, i.e.either both bodies stick or both bodies slip.When the effective coefficient of friction, µ , , in Eqn. 10is not limited by the interfacial friction coefficient, µ, thesurfaces stick. It is easy to show that for this case thecontact algorithm conserves momentum exactly. The to-tal momentum change imposed by applying sticking con-tact is

2

∑α * 1mα - ṽα . vα / & . 2∑α * 1mα∆vα & 0 0 (15)where Eqn 14 has been used. For slipping contact, thesum of momenta on the grid gives

2

∑α * 1mα - ṽα . vα / & . 2∑α * 1 mα∆vα ' nαnα. µ 2∑

α * 1 mα∆vα ' nαtα 0(16)

While in general slipping contact does not conserve mo-mentum, the errors are associated with non–collinearityin the calculation of body normals and tangents, i.e. poorresolution of interface curvature. For n2 & . n1 and t2 &. t1, momentum is conserved exactly (from Eqn. 14).A common scenario as two bodies approach and departis for the grid mass of one body to approach zero whilethe other remains finite. If, through unfortunate registra-tion of material point mass on the computational grid, abody’s grid mass is very small, the resulting change invelocity prescribed by the contact algorithm may be very

large. This result can be seen from Eqn. 15, for exam-ple. Working with momenta, rather than velocities, isan elegant solution for updating material point positionsand velocities in this case, Sulsky, Zhou, and Schreyer(1995). However, calculations of material point incre-mental strains require velocities on the grid in order todifferentiate it there, and remain problematic. In prac-tice these unfortunate registration events are marked byabrupt changes in the kinematics of the contacting bod-ies, the occurance of which is time step size dependent.

An effective stability criterion has been developed whichis loosely analogous to the Courant explicit time steppingcriterion. The Courant condition demands that solutioninformation not be allowed to propagate across more thanone cell in a single time step. Similarly, for the contactalgorithm the instantaneous change in velocity must notbe large enough to collapse (or invert) a computationalcell. Define the instantaneous strain rate imposed by thecontact algorithm

ε̇̇ε̇εαj & ṽαj . vαj∆x j (17)where ∆x j is the grid spacing in each coordinate direc-tion. Then the condition that the contact strain incrementnot collapse a neighboring cell in one time step is

ε̇̇ε̇εαmax∆t 1 1 (18)where ε̇̇ε̇εαmax & max j 2 ε̇̇ε̇εαj 2 . Using Eqn. 14, the grid strainrates for each body may be related, e.g.

ε̇̇ε̇ε2j & . m1m2 ε̇̇ε̇ε1j & m1M . m1 ε̇̇ε̇ε1j 0 (19)These relations allow the most severe grid strain rate, foreither material, to be calculated independently

ε̇̇ε̇εmax & maxj 3 max 3 mαM . mα ( 1 4 2 ε̇̇ε̇εαj 2 450 (20)Only global grid quantities and quantities specific to thebody under consideration are needed in the calculation.

In practice, Eqn. 18 is modified to provide a safety factor,

ε̇̇ε̇εmax∆t 1 γ 0 1 γ 6 1 0 (21)Implementation of the collapsed cell stability conditionis achieved by scaling the velocity change imposed bythe contact algorithm when necessary

ṽαscaled & vα . max 3 γε̇̇ε̇εmax∆t ( 1 4 ∆vαn - nα 7 µ , tα / 0 (22)

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The most severe constraint, for either body, can be de-termined using only quantities specific to the body underconsideration, and global quantities, retaining the effi-ciency of the original formulation. In addition, the scal-ing preserves the momentum conservation properties ofthe algorithm, Eqn.s 15 and 16.Note that large grid strain increments occur at grid nodesfor which the value of the interpolating functions aresmall (resulting in a small grid mass there). Because thesame weighting is used to interpolate strain incrementsfrom the grid back to the material points, material pointstrain increments will be much smaller. If grid veloci-ties are unaltered by the contact algorithm, interpolationfrom material points to grid and back again scales out,and material point strains are well behaved regardless ofthe registration of material point information on the grid.It is only when the grid velocities are adjusted that parti-cle strain calculations can be problematic, Sulsky, Zhou,and Schreyer (1995).

The approach described meets the cell collapse con-straint, Eqn. 21, by systematically limiting the maximumgrid strain rate. Consideration was also given to the ob-vious alternative, simply reducing the current time stepsize, and restarting the time step. This alternative wasrejected because it is the initial registration of the parti-cle data on the grid which is ultimately responsible forlarge grid strain rates. These rates are computational ar-tifacts, traceable ultimately to the low order of the inter-polation scheme used to develop grid data. A remedy ofthis sort would more appropriately be applied to the pre-vious time step size. However, without a search to locatematerial points near grid boundaries, there is no way toguarantee avoiding the problem at all grid nodes simul-taneously. The above described algorithm provides anefficient means of improving accuracy by screening outcomputational artifacts.

3 Applications

To demonstrate the accuracy of the contact algorithm,numerical solutions are first compared to analytical onesfrom rigid body dynamics. A more complex example isprovided by simulating stress wave propagation in ide-alized granular material. For all cases, the contact algo-rithm scale factor γ was taken to be 0 9 5. Little sensitiv-ity to the scale factor was seen in the range 9 5 : γ : 1.However, for γ ; 0 9 5 bodies were observed to penetrateto varying degrees during contact. Based on this experi-

ence, a recommended range is 9 5 : γ : 1.3.1 Sphere Rolling on an Inclined Plane

For a simple test of the algorithm, a sphere rolling downa flat inclined plane is simulated. The simulation isperformed in 3–dimensions. A symmetry plane exists,and the simulation direction perpendicular to that planeserves only to distinguish the geometry (a sphere ratherthan a cylinder). See Fig. 2 for the problem setup. Thex– and z–directions lie in the plane of symmetry with xin the direction of rolling and z perpendicular. The radiusof the sphere, R, is 1.6 m, and the length of the planeon which it rolls is 20 m. The incline of the plane isdescribed by the angle between the z–direction and thatof gravity, θ. For all calculations the acceleration due togravity is taken to be 10 m/s2. The inclined plane prob-lem has exact solutions for a rigid sphere on a rigid plane.

g x

z

θ

Figure 2 : Schematic of the inclined plane problem andR < 4 cell size MPM discretization.In the simulations the sphere and plane are deformable.The grid is uniform with equal spacing in all directionsand eight material points per cell. The cell sides are oflength R < 4, providing 8 computational cells across thediameter of the sphere. This discretization is depicted inFig. 2. The material properties are chosen to allow forlarge time steps using an explicit code, and consequentlycorrespond to a rather soft material. Both sphere andplane are modeled as compressible Neo–Hookean hyper-elastic bodies, Simo and Hughes (1998). The sphere hasbulk modulus 6 MPa, shear modulus 3 MPa, and density1000 kg/m3, roughly approximating those of a naturalrubber. The plane has elastic constants and density anorder of magnitude larger, resulting in a much stiffer ma-terial with the same wave speeds. The longitudinal wavespeed is 100 m/s and a typical calculation for this resolu-tion requires 500 time steps.


Figure 3 : Sphere center of mass x–position as a functionof time for both the slip and no slip cases. The rigidsphere analytical solutions are shown for comparison.

Two cases are considered. For the first θ = π > 4 and thecoefficient of friction is .495. This case is referred toas the “no slip” case because the rigid sphere solutiondescribes rolling without slipping. For a rigid sphere thex–position of the center of mass is given by

x ? t @A= 25 B 214

t2 C (23)The center of mass position of the deformable sphere isdepicted in Fig. 3, along with the analytical solution forcomparison. For the second case θ = π > 3 and the co-efficient of friction is .286. In this case the analyticalsolution describes rolling while sliding and is referred toas the “slip” case. For a rigid sphere the x–position of thecenter of mass is given by

x ? t @A= 52? B 3 D 2

7@ t2 E (24)

The center of mass position for the deformable sphere,and the analytical solution for comparison, are also plot-ted in Fig. 3. For both cases the solutions are similar totheir rigid sphere counterparts, with the deformable disksrolling and sliding more slowly. The slower motion of thedeformable spheres is also seen in a plot of their centerof mass velocities in the direction of rolling, Fig. 4. The

abrupt changes in velocity, most evident in the no slipcase, are due to the application of the contact algorithm.

Figure 4 : Sphere center of mass x–velocity as a functionof time for both the slip and no slip cases. The rigidsphere analytical solutions are shown for comparison.

Attention is now focused on the no slip case. To investi-gate the effect of spatial resolution this case was run withcell sizes R > 2 and R > 8, corresponding to 4 and 16 cellsacross the sphere diameter. The effect of spatial resolu-tion on calculated position of the center of mass is de-picted in Fig 5, where the analytical solution for a rigidsphere appears on the graph for reference. Simulationsfor the deformable case are labeled by cell size. Thelowest resolution case, R > 2, is instructive. In this casethe resolution is too crude to resolve the geometry on arectangular grid and the sphere fails to roll because of aflat spot (the resolution is the three–dimensional equiva-lent of that depicted in Fig. 1). For the finer resolutions,R > 4 and R > 8, the geometry is sufficiently resolved to per-mit rolling and the solution quickly converges toward therigid sphere case. It appears that although the materialproperties correspond to a very soft material, it is resolu-tion of the geometry that is most important in this simu-lation.

The effect of spatial resolution on the sphere center ofmass velocity is shown in Fig. 6. For the lowest reso-lution there is an indication of vibration about the initial

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Figure 5 : No slip case sphere center of mass positionas a function of time for various spatial resolutions. Therigid sphere analytical solution is shown for comparison.

configuration, but no rolling, as expected. For finer res-olutions the sphere rolls, and the velocity converges to-ward the rigid sphere solution. The rigid sphere solutionappears to provide an upper bound on the velocity forthese simulations. This might be expected in part becausepotential energy is converted both to elastic (strain) en-ergy and kinetic energy for the deformable sphere, whilefor the rigid case all potential energy is converted to ki-netic energy. However, the sphere skips slightly, andwhile not in contact accelerates more quickly than whilerolling in contact. If the simulation was carried out longenough, the sphere would eventually travel more quicklyin the numerical simulation, due to skipping

3.2 Backspin Problem

A slightly more complex problem is the motion of an ini-tially stationary, spinning sphere on a flat plane (θ G 0)under gravity. Although spinning, the center of mass ve-locity is initially zero. This problem is termed the “back-spin” problem. The same material properties and acceler-ation of gravity are used for this problem, but the spherehas unit radius and only the higher resolutions consid-ered are used (cell sizes R H 4 and R H 8). The coefficient offriction is taken to be 0.5 and the initial angular velocity

Figure 6 : No slip case sphere center of mass velocityas a function of time for various spatial resolutions. Therigid sphere analytical solution is shown for comparison.

5 rad/s.

Once again, for a rigid sphere there is an analytical solu-tion. First the sphere slides and rolls while accelerating.When velocities at the contact point match it rolls with-out slipping at constant velocity. This solution for thecenter of mass position is given by

x I t JLK x0 GMI 5 H 2 J t2 t N 2 H 7 O (25)x I t JLK x0 GMI 5 H 2 JPI 2 H 7 J 2 Q I 10 H 7 JRI t K 2 H 7 J t S 2 H 7 O

(26)

where t G 2 H 7 is the time at which slipping stops. Numer-ical simulation results for deformable spheres, as well asthe rigid sphere solution, are shown in Figs. 7 and 8.

For both position and velocities, results converge towardthe rigid sphere results, indicating the importance of re-solving the geometry accurately. The deformability ofthe sphere plays a larger role at the higher angular ve-locities induced during this simulation. The combinationof deformability and skipping produces the strong oscil-lations in center of mass velocity seen in Fig. 8. Analo-gous to stick–slip like behavior, the sphere catches brieflycausing acceleration of the center of mass velocity, fol-


Figure 7 : Sphere center of mass position as a functionof time for R T 4 (labeled dx=.250) and R T 8 (dx=.125) cellsize MPM discretizations. The rigid sphere analytical so-lution is shown for comparison.

lowed by free spinning during which the velocity is es-sentially constant. The oscillations in velocity are due todeformability. When the sphere contacts with excessiveforward spin, the center of mass velocity is increased andthe sphere is compressed forward of the contact patch.As this strain energy is released during a skip (free flight),the sphere expands resulting in an increase in forwardvelocity at the following contact patch and an effectivebackspin, slowing the center of mass on contact.

3.3 Stress Waves In Granular Media

A significantly more demanding modeling problem ispresented by calculating stress waves in granular me-dia. This problem has been studied extensively by Shuklaand co–workers, Rossmanith and Shukla (1982); Shuklaand Damania (1987); Shukla (1991); Zhu, Shukla, andSadd (1996), who used detonators to load collections ofdisks. Using photoelastic disks and high–speed photog-raphy, they were able to temporally and spatially resolvethe stress state as the impulse traveled through variousassemblages. More recently similar experiments havebeen undertaken, but using a Hopkinson bar to dynam-ically load the disks, Roessig (2001). In this case the

Figure 8 : Sphere center of mass velocity as a functionof time for R T 4 (labeled dx=.250) and R T 8 (dx=.125) cellsize MPM discretizations. The rigid sphere analytical so-lution is shown for comparison.

loading results in a step change in velocity rather than anunsteady impulse of short duration. The latter is bettercharacterized, and a much easier boundary condition tosimulate.

Because frictional sliding is a defining characteristic ofgranular material, it is natural to test the contact algo-rithm by simulating granular material response. This wasdone previously for dynamic wave propagation, Barden-hagen and Brackbill (1998) and shearing deformations,Bardenhagen, Brackbill, and Sulsky (2000c,b) with verygood qualitative “macroscopic” agreement. However, adirect comparison of experimental and numerical resultsfor a specific microstructure was never made. Here re-sults for well characterized assemblies of disks and load-ing conditions are simulated and compared to experimen-tal measurements.

The experimental setup is facilitated by a mild steel load-ing frame which holds the disks. Dimensions adjust eas-ily, allowing various geometries to be assembled. One–quarter inch grooves in the sides of the frame hold thedisks in plane. Loading is applied via a split Hopkin-son bar. The input pulse is recorded and analyzed togive a striker velocity, which for the experiments reported

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here is 5.6 m/s. The disks are 2 inches in diameter, 1/4inch thick, and made of Plexiglas. Experimental mea-surements are made using a high speed camera and thetechnique of photoelasticity. The camera is triggered bywave propagation in the Hopkinson bar prior to reachingthe disks. The transfer of the stress wave to a loadingpin of equal thickness as the disks ensures plane stressloading conditions. Unfortunately, this results in someuncertainty ( V 10 µs) in the arrival time of the stress waveat the first disks (“impact”). However, interframe timesare much more precise, and provide the more exactingconstraint on the comparisons. Photoelasticity gener-ates dark fringes at contours of constant maximum shearstress. Because in the simulations the spatial variation ofthe stress state is known, it is straight–forward to do thecorresponding calculation. Specifically, the stress tensoris diagonalized. For isotropic response the the out–of–plane direction is a principal one, and decoupled fromthe in–plane response. The difference in in–plane prin-cipal stresses is then proportional to the maximum shearstress in–plane. Fringes are generated by taking the co-sine of the difference in in–plane principal stress dividedby a parameter to adjust fringe density.

Experiment

Simulation

Experiment

Simulation

Experiment

Simulation

Figure 9 : Stress wave propagation through a collectionof four disks with aligned centers. Matching frames arepresented in pairs with the experimental data on top andsimulation results below. Non–dimensional frame timesare 2.6, 6.6, and 9.2 for the experimental data and 2.2,5.5, and 7.7 for the simulation.

The numerical simulations use a linear hypoelastic con-stitutive model for the Plexiglas, Fung (1965). In partbecause the elastic constants are not known accuratelyfor the Plexiglas used (properties can vary with manu-facturer and to some degree even by material lot), andin part to facilitate a parameter study varying the elasticconstants, unit geometries and wave speeds were speci-fied in the simulations. The numerical simulations use 1cm diameter disks and material properties resulting in alongitudinal wave speed of 1cm/µs. Specifically the ma-terial properties are shear modulus 72 GPa, bulk modulus102 GPa and density 1900 kg/m3. If Plexiglas was ac-curately modeled using hypoelasticity with the selectedratio of shear to bulk moduli, then stress magnitudes andwave speeds could be scaled by the ratio of actual to sim-ulated moduli, and transit times could be scaled by theratio of disk diameters, to make precise comparisons be-tween simulations and experiments. However, the elasticconstants are not known very precisely yet and, in ad-dition, there is evidence for more complicated, rate de-pendent, material response, Roessig (2001). Still, purelyelastic response has been found to capture gross features,as measured photoelastically, remarkably well, as seen inFig. 9 for four collinear disks impacted on the right.

For this simple geometry, the computational resolutionserves primarily to resolve the geometry, as the contactconditions are generally no slip due to the compressiveloading and the geometric and material symmetry. Thesimulations use 80 cells across disk diameters and onematerial point per cell. Because of the material modelinguncertainties, the experimental results for this configura-tion of disks were used to determine the fringe densityparameter. Matching fringe patterns were chosen by eye,subject to the constraint of equal interframe times, as inthe experimental data. The same fringe density param-eter is then used in all subsequent photoelastic analysesfor the more complex geometries considered next.

To compare results more precisely, non–dimensionalframe times are reported. The non–dimensional frametime is defined as the time after impact divided by thelongitudinal wave transit time across one disk. For thesimulations the longitudinal wave transit time across onedisk is 1 µs. For the experiments, estimating the longi-tudinal wave speed as 0.27 cm/µs, Marsh (1980) resultsin a wave transit time of 19 µs. A selection of matchingframes is shown in Fig. 9. For the experiments, the non–dimensional frame times are 2.6, 6.6, and 9.2. For the


simulations, the non–dimensional frame times are 2.2,5.5, and 7.7. The computations accurately reproduce theexperimental fringe patterns. Note that a finite exposuretime results in some smoothing of the experimental im-ages not present in the simulation data (which is instan-taneous). The variation in fringe velocity through disksand across contacts is also accurately simulated. How-ever the non–dimensional frame times are different, thefringes propagate (relatively) more quickly in the simu-lation.

Experiment

Simulation

Experiment

Simulation

Experiment

Simulation

Figure 10 : Stress wave propagation through a collectionof five disks with zig–zagged centers. The simulationused a friction coefficient of 0.5. Matching frames arepresented in pairs with the experimental data on top andsimulation results below. Non–dimensional frame timesare 5.3, 10.5, and 15.8 for the experimental data and 3.0,6.0, and 9.0 for the simulation.

A geometry which promotes slipping contact was sim-ulated next. The disk centers were arranged such that

lines connecting the centers form a 90 degree zig–zagpattern. For the simulations a coefficient of friction of0.5 was used between the disks and contact with thewalls was frictionless. Experimental and numerical re-sults for this geometry are shown in Fig. 10 for match-ing fringe patterns (subject to the equal inter–frame timeconstraint). Matching fringe patterns were obtained atnon–dimensional times of 5.3, 10.5, and 15.8 in the ex-periments and 3.0, 6.0, and 9.0 in the simulation. Againthe fringe patterns and variations in fringe velocity areaccurately simulated. In addition, other gross features aresimilar. Because contacts can slide, load carrying pathsfirst develop between disk contacts and a wall. Only aftersufficient normal traction builds up and the next disk con-tact sticks do fringes propagate into the next disk in theseries. Note that the grooves in the experimental appara-tus, which serve to hold the disks in plane, obscure pho-toelastic measurements of disk contacts with the walls.The ability of contacts to slide hinders fringe propaga-tion across contacts and fringes propagate much moreslowly in this configuration, in both the experiment andthe simulation, than in the collinear disks configuration.However, the non–dimensional frame times are again inerror, with fringes propagating (relatively) faster in thesimulation than for the experiment, as for the collinearconfiguration.

To provide maximum contrast, the case was examinedwhere the disks were glued together in the experimentto eliminate slip all altogether. No slip contact condi-tions were implemented between disks in the simulationby taking the coefficient of friction to be infinite. Con-tact with the walls remained frictionless in the simula-tion. Results for this case are shown in Fig. 11. Match-ing fringe patterns (subject to the equal inter–frame timeconstraint) were obtained at non–dimensional times of1.6, 4.2, and 6.8 in the experiments and 1.4, 3.6, and 5.8in the simulation. Again the fringe patterns and vari-ations in fringe velocity are accurately simulated, andother gross features are similar. When contacts supportshear immediately, the primary load path across a diskfirst develops between disk contact points. Load pathsconnecting disk contact points to the boundary developlater. As a result, the fringes propagate through the as-sembly much faster than for the same configuration withslipping contact. However, the non–dimensional frametimes are again in error, with fringes propagating (rela-tively) faster in the simulation than in the experiment.

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Experiment

Simulation

Experiment

Simulation

Experiment

Simulation

Figure 11 : Stress wave propagation through a collec-tion of five disks with zig–zagged centers and stickingcontact. Matching frames are presented in pairs with theexperimental data on top and simulation results below.Non–dimensional frame times are 1.6, 4.2, and 6.8 forthe experimental data and 1.4, 3.6, and 5.8 for the simu-lation.

The above examples indicate that contact plays an im-portant role in stress wave propagation in collections ofdisks, and almost certainly plays in important role inmore general granular material as well. Fringe wavespeeds are strongly dependent on contact conditions,specifically the amount of slippage. The simulations arefound to be in excellent qualitative agreement with theexperiments. Some of the discrepancies in the detailsmay be attributable to the boundary conditions. The disksare initially in contact with each other and the walls in thesimulations, while for the experiments there are smalltolerances. The biggest differences however, the sys-tematic differences in non–dimensional frame times, are

most likely due to errors in modeling material response.

A parameter study was undertaken, using the collineardisks geometry, to determine the sensitivity of fringe ve-locity to the Poisson’s ratio, ν, keeping the longitudi-nal wave speed fixed. It was found that fringe velocityis strongly dependent on Poisson’s ratio, but fringe pat-terns much less so. For the elastic constants selected,ν X 0 Y 22. A reduction in fringe velocity by more than afactor of two was obtained by increasing the Poisson’s ra-tio from 0.22 to 0.48. This result is not unexpected, as thephotoelastic technique measures gradients in shear stress,and the shear modulus (and consequently the shear wavespeed) decreases with increasing Poisson’s ratio. Withboth contact conditions and elastic constants playing im-portant roles, a more quantitative assessment of the accu-racy of the calculations awaits a better characterizationof the Plexiglas’ material properties. At this point it issimply noted that for a given set of material constants,the same collective fringe velocity trends are exhibitedin the simulations as in the experimental data. In addi-tion, for ν Z 0 Y 22, an easily justifiable material modelingmodification for polymers, the decrease in fringe velocitywould tend to reduce the systematic differences in non–dimensional frame times. Better characterization of thePlexiglas used in these experiments is underway, and theresults will appear in conjunction with a study of morecomplicated disk assemblages, Roessig (2001).

4 Conclusions

The Material Point Method provides a convenient frame-work for the implementation of contact between de-formable bodies. The contact conditions are equivalentto a perfectly inelastic collision, providing for maximumexchange of kinetic and strain energy. For accurate appli-cation of contact under this scenario, the normal tractionsmust be monitored and included in the release logic. Thecontact calculations are computationally efficient, per-formed on the computational grid body by body, with-out requiring a search to identify neighbors. A potentialnumerical difficulty associated with unfortunate registra-tion of particle information on the computational grid hasbeen eliminated by a scaling which retains both the ef-ficient properties of the algorithm, and conservation ofmomentum during contact.

The algorithm is exercised on several simple problems,for which analogous rigid body problems have analyticalsolutions. The algorithm is found to compare well, even


for fairly coarse discretizations with resolution of geom-etry the most important factor in obtaining convergence.The algorithm is exercised on a more demanding sim-ulation involving stress propagation through collectionsof polymeric disks for which high fidelity experimen-tal data is available. Excellent qualitative agreement isfound for three examples with very different contact con-ditions. The importance of accurate simulation of ma-terial contact is demonstrated by the strong dependenceof load path development and collective wave propaga-tion speeds on the contact conditions. Other importantsimulation parameters were identified, including materialproperties and experimental tolerances. Future work willfocus on obtaining better measurements of these param-eters, comparing simulation to experiment quantitatively,and simulating more complex assemblages.

Acknowledgement: This work was supported by theU.S. Department of Energy through the Center for theSimulation of Accidental Fires and Explosions, undergrant W-7405-ENG-48.

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an improved contact algorithm for the material point...

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