Modeling Imperfect Interfaces in the Material Point Method ... · [2007]) or specialized cohesive...

Copyright c© 2013 Tech Science Press CMES, vol.1, no.1, pp.1-15, 2013

Modeling Imperfect Interfaces in the Material Point Method using MultimaterialMethods

J. A. NAIRNWood Science & Engineering, Oregon State University, Corvallis, OR, 97331, USA

Abstract: The “multimaterial” version of the materialpoint method (MPM) extrapolates each material to itsown velocity field on a background grid. By reconcilingmomenta on nodes interacting with two or more materi-als, MPM is able to automatically handle contact withoutany need for special contact elements. This paper extendsmultimaterial MPM to automatically handle imperfectinterfaces between materials as well. The approach isto evaluate displacement discontinuity on multimaterialnodes and then add internal forces and interfacial energydetermined by an imperfect interface traction law. Theconcept is simple, but implementation required numer-ous corrections to make the analysis mesh independent,to work for any stiffness interfaces, and to find the in-terfacial normal vector. Several examples illustrated theneed and demonstrated the validity of the various correc-tions. A composite mechanics problem found the bulkmodulus of a particulate filled composite as a function in-terface quality. This calculation revealed a scaling effect— interfaces in nanoparticle composites must be betterthan interfaces in the corresponding microparticle com-posites for the nanocomposite mechanical properties tobe as good as the conventional composite properties.

keyword: Material point method, MPM, Imperfect in-terfaces, multimaterial, contact, nanocomposites

1 Introduction

Modeling interfaces, which are often finite-thickness in-terphases, in composite materials is difficult. In numeri-cal modeling, it would seemingly be straight-forward todiscretize interphase zones and thereby explicitly modelall effects. This approach has two problems. First, inter-phase zones may be much smaller than the bulk materi-als. Resolving both bulk materials and a thin interphasewould require a highly refined model, which may ex-ceed computational capacity. Second, interphase proper-ties may be unknown and/or may vary within a transition

zone from one material to another. Furthermore, findinginterphase properties is not just a matter of extra experi-ments. More commonly, an interphase is not accessibleas a bulk material, which means its properties would beunmeasurable. New methods for interphase modeling areneeded to overcome these challenges. This need is espe-cially important in nanocomposites because the amountof interphase per unit volume of reinforcement greatlyexceeds the amount of interphase in composites with mi-cron or larger reinforcement phases. As a consequence,interphases are expected to play a larger role, good orbad, in nanocomposite properties.

One way to model interphases is to abandon attemptsfor explicit modeling and instead replace 3D interphaseswith 2D interfaces (Hashin [1990]). The interphase ef-fects are reduced to modeling the response of 2D inter-faces due to tractions normal and tangential to the interfa-cial surface, which can be modeled by interface tractionlaws. Elimination of 3D interphases removes the resolu-tion problem. The use of interface traction laws replacesnumerous unknown and potentially unmeasurable inter-phase properties with a much smaller number of interfaceparameters. If interface traction laws can be determined,one can potentially model interphase effects well. Thisapproach to interphase modeling was developed for an-alytical modeling of interface effects in composite ma-terials (Hashin [1991a,b], Nairn and Liu [1997], Nairn[2004]) and for wave transmission at damage planes (An-gel and Achenbach [1985]).

Numerical modeling of interfaces requires methods forimplementing interface traction laws. In finite elementanalysis (FEA), one can use interface elements (Nairn[2007]) or specialized cohesive elements (Needleman[1987]). In the material point method (MPM) with ex-plicit cracks (Nairn [2003]), it is similarly possible to im-plement interfaces (Nairn [2007]) or cohesive law trac-tions on crack surfaces (Nairn [2009]). In other words,the crack is modeling an interface instead of a crack.

2 Copyright c© 2013 Tech Science Press CMES, vol.1, no.1, pp.1-15, 2013

Both FEA and crack-based MPM methods require ex-plicit introduction of interface paths prior to the start ofthe analysis, which complicates discretization and maylimit the range of possible simulations. Another ap-proach available in MPM is the so-called “multimaterial”approach where each material extrapolates to its ownfield of variables on a common grid and multimaterialalgorithms adjust the response on every node containingparticle information from two or more materials. Previ-ously, this method was used to model contact physics,which, to date, has only considered Coulomb frictionor stick conditions (Bardenhagen and Brackbill [1998],Bardenhagen, Guilkey, Roessig, Brackbill, Witzel, andFoster [2001]). This paper extends multimaterial MPMto modeling interfaces with traction laws. A crucial stepin the algorithm is finding the normal to the interface.This paper examines the influence of normal vector cal-culations and offers several strategies for best accuracy.These new normal-vector methods are recommended forboth contact and interface calculations.

2 Multimaterial Contact and Interfaces

2.1 Imperfect Interface Theory

Replacing a 3D interphase with a 2D interface in com-posite mechanics is known as “imperfect interface the-ory” (Hashin [1990]). To model 3D interphase effects,the 2D interface is allowed to develop displacement dis-continuities. For an isotropic interphase, it suffices to re-solve interfacial traction into normal and tangential trac-tions (Tn and Tt) and assume they are functions of normaland tangential displacement discontinuities ([un] and [ut ])at the interface:

Tn = fn([un]) and Tt = ft([ut ]) (1)

Interfaces in composite materials may develop poten-tial energy that is needed for effective property analysis(Hashin [1992]). For an elastic interface, interfacial po-tential energy is

φi =∫

Si

(∫ uuu

0TTT · [uuu]

)dS (2)

where Si is the interfacial area.

Most analytical modeling and all examples below makethe simplest assumption that traction laws are linear andelastic. The tractions become:

Tn = Dn[un] and Tt = Dt [ut ] (3)

and interfacial energy becomes

φi =12

∫Si

(Dn[un]

2 +Dt [ut ]2)dS (4)

Notice that all 3D interphase properties have been re-duced to just two interface parameters, Dn and Dt . Theseinterface stiffnesses range from zero, for a debonded in-terface with no tractions, to infinity, for a perfect interfacewith no displacement discontinuity.

2.2 Detecting Multimaterial Interfaces and Contact

In MPM, bodies are discretized into particles on a back-ground grid (Sulsky, Chen, and Schreyer [1994]). Ineach time step, particle mass and momenta along withother quantities needed for algorithms, are extrapolatedto nodes on the grid. In multimateral MPM, each mate-rial extrapolates to its own field. For example, material-specific mass and momenta would:

mi, j = ∑p∈ j

Sipmp (5)

pppi, j = ∑p∈ j

Sippppp (6)

where i is a node, j is a material type, and summationsare are over all particles of each type. Sip are generalizedshape functions for particle p at node i (Bardenhagen andKober [2004]) and mp and pppp are the mass and momen-tum of particle p. After extrapolation, nodes containingonly a single material type proceed by standard MPMmethods, but nodes with two or more interacting materi-als have to detect if the materials are in contact, imple-ment contact laws (if needed), and, in this new algorithm,implement imperfect interfaces.

Figure 1 shows an interface between two materials, a andb, with a normal vector (that varies along the interface),n, directed from a to b. Nodes indicated with a dot, suchas node i, are nodes involving both materials. Two keytasks at such nodes are to determine if the two materialsare in contact and to determine their separation. Previ-ous multimaterial MPM assumed materials are in con-tact when they are approaching each other, i.e., when∆vvvi · n < 0, where ∆vvvi = vvvi,b−vvvi,a. This condition is nec-essary, but not sufficient for accurate interface modeling;additional calculations are needed.

First define, ∆pppi,a as the momentum change applied tomaterial a on node i for its velocity to change to the cen-

Multimaterial Imperfect Interfaces 3

Material a

Material b

n

i

Figure 1 : Material points in a 2D MPM model havingtwo different material types on either side of a contactor interfacial surface. The normal vector n is defined asdirected from material a to material b. The black dotsindicate nodes (such as node i) that have multimaterialfields.

ter of mass velocity:

∆pppi,a = mi,avvv(c)i − pppi,a (7)

where vvv(c)i is the center of mass velocity:

vvv(c)i =∑ j pppi, j

m(c)i

=∑ j mi, jvvvi, j

m(c)i

(8)

and m(c)i = ∑ j mi, j is the total nodal mass. To anticipate

handling nodes with more than two materials, we replacematerial b by a virtual material b that lumps all materialsexcept a (i.e., mi,b = m(c)

i −mi,a and pppi,b = ppp(c)i − pppi,a).Using Eq. (7), the velocity difference is:

∆vi = vvvi,b−vvvi,a =ppp(c)i − pppi,a

m(c)i −mi,a

− pppi,a

mi,a

=m(c)

i

mi,a(m(c)i −mi,a)

∆pppi,a (9)

The first calculation is to determine if ∆vvvi · n < 0, whichis equivalent to determining if ∆pppi,a · n < 0. If this check

fails, the two materials are assumed to be separated (i.e.,not in contact). When a multimaterial MPM simulationis only modeling contact, the velocity field for material ais left unchanged. When implementing imperfect inter-faces, however, tractions may exist for both contact andseparated conditions, so this algorithm continues.

The next task is to calculate the separation between thetwo materials, which is done by extrapolating particle po-sition to the grid using standard MPM methods:

mi, jxxxi, j = ∑p∈ j

Sipmpxxxp (10)

Again lumping all materials except a into virtual b, theseparation vector is:

δδδi = xxxi,b−xxxi,a =m(c)

i

m(c)i −mi,a

(xxx(c)i −xxxi,a

)(11)

where xxx(c)i = ∑ j mi, jxxxi, j/m(c)i is the center of mass posi-

tion. Contact is now determined by the component of theseparation vector normal to the surface or

δn = δδδi · n (12)

If δn < 0, the materials are in contact, otherwise they areseparated. But, inherent imprecision of surfaces in MPM(or any particle method) causes a problem. Consider thetwo surfaces in Fig. 1. When calculations start, theirnodal positions extrapolated to common nodes will showa positive separation. Calculations with MPM shapefunctions show that the calculated separation based onpositions of two materials precisely in contact is δn ≈0.8∆x rather than zero (Lemiale, Hurmane, and Nairn[2010]). Two approaches can resolve this issue:

1. Extrapolate displacements instead of position using

mi, jxxxi, j = ∑p∈ j

Ni(xxxp)mp

(xxxp−xxx(0)p

)(13)

where xxx(0)p is the initial position for particle p. Now theseparation for two materials in contact when calculationsstart is zero and subsequent δn < 0 will indicate contact.

2. Define a contact-position offset, such as δcon = 0.8∆x,and correct the normal separation using:

δ′n = δδδi · n−δcon (14)

(see below for revised correct in irregular meshes).


Both options should be available in multimaterial cal-culations. The first (or displacements) method requiresmaterial contact situations to exist in the initial state andrequires relative displacements between materials to bemodest. This situation is common in interface modeling.The second (or offset) method must be used when theserequirements are not met, which is common in many con-tact problems. All calculations in this paper involved in-terfaces and small displacements and thus used the dis-placements method.

If both ∆pppi,a · n < 0 and δn < 0, the two materials are de-termined to be in contact; otherwise they are separated.Some codes implement an additional screening basedon total nodal volume at a multimaterial node (Guilkey[2010]). If that volume is less than a chosen cutoff frac-tion of the element volume, then both contact and inter-face calculations are skipped. These nodes appear at in-terfaces near free edges. Although they might actually byin contact or have an interface, in can be better to ignorethem (see results and discussion section for examples).

2.3 Imperfect Interface and/or Contact Forces

The next task is to implement imperfect interface or con-tact laws. In multimaterial simulations, each pair of ma-terials can be described by contact (e.g., friction) or as animperfect interface, but not both. If a pair is interactingonly by contact and is separated, no changes are made atthat node; otherwise calculations proceed as follows:

1. Find the tangent vector in the direction of the sur-face opening displacement. Given the normal sepa-ration, δn, and normal vector, n, the tangent vectoris

δt t = δδδi−δnn (15)

where δt = δδδi · t is displacement difference in thetangential direction.

2. If two materials are interacting by contact, changethe momentum for material a. If they interact bystick, change material a’s momentum by ∆pppi,a. Ifthey interact by frictional contact and ∆pppi,a · t >−µδn, where µ is the coefficient of friction, thematerial pair is sliding and the momentum changeshould be

∆ppp′i,a = δn(n−µt) (16)

otherwise the material pair is stuck and material a’smomentum changes by ∆pppi,a. This step is iden-tical to previously-derived MPM contact (Barden-hagen, Guilkey, Roessig, Brackbill, Witzel, andFoster [2001])

3. If the two materials are connected by an interface,this algorithm applies an internal force to the noderather then changing the nodal momentum. Thelogic is that an interface is a material propertyleading to interfacial tractions and should behavesimilarly to material constitutive laws that lead tostresses and internal forces. The internal force formaterial j on node i, which is added to other MPMinternal forces, is:

f (int)i, j = (Dnδnn+Dtδt t)Ai (17)

and the interfacial energy at node i is

U (int)i = (Dnδ

2n +Dtδ

2t )Ai (18)

where Ai is interfacial contact area. These termsassume a linear interface law; any other interfacialtraction law could be substituted at this step.

Calculation of Ai needs to account for the number of in-teracting nodes along the interface, which is influencedby location and orientation of the interface as illustratedin Fig. 2 for 2D calculations. The needed correction isfound by extrapolating particle volume (or particle areain 2D):

Ωi, j =

∑p∈ j SipVp 3D∑p∈ j SipAp 2D

(19)

where Vp or Ap are volume or area of the deformed par-ticle domain. Consider a 2D interface close to a grid linewhen particles are close to their starting positions (1/4points of cells with Sip as indicated; see Fig. 2A). Thenodal domains at nodes i and j would be Ωi,a = Ωi,b =Acell/2, Ω j,a = 0, and Ω j,b = Acell . Here Acell = ∆x∆y,where ∆x and ∆y are grid spacings in the x and y direc-tions. The total nodal domains are Ωi = Ωi,a +Ωi,b =Ω j = Acell . The contacting area is obviously Ai = t∆y,where t is thickness, or Ai = tΩi/∆x. Furthermore, nodei is a multimaterial node, but node j is not. In con-trast, when the interface is at element midplanes, bothnodes i and j are multimaterial nodes with nodal do-mains of Ωi,a = Ω j,b = 7Acell/8, Ωi,b = Ω j,a = Acell/8,


and Ωi = Ω j = Acell (see Fig. 2B). If contacting areas areunchanged, i.e., Ai = tΩi/∆x and A j = tΩ j/∆x, the to-tal interfacial force would be double the force in Fig. 2Abecause it involves twice as many nodes. For these twocases, however, total interfacial force should be the same.One solution is to scale the contact area by the distancefrom the interface to the node as estimated by the mis-match in nodal domains for the two materials. A reason-able function that gets the correct scaling for both Fig. 2Aand B, and correctly drops to zero as one material van-ishes is:

Ai =t√

2Ωi min(Ωi,a,Ωi,b)

∆x(20)

This scaling results in Ai ≈ tAcell/∆x and A j ≈ 0 forFig. 2A and Ai ≈ A j ≈ 2Acell/(2∆x) for Fig. 2B; the totalinterfacial force becomes independent of interface loca-tion. In 3D problems, Ωi is volume and therefore t isdropped in Eq. (20). In axisymmetric MPM (Nairn andGuilkey [2013]), the t is replaced by ri, which is the ra-dial position of the node, to provide interface forces perradian.

Figure 2C shows a tilted interface, which sometimesneeds an additional correction. Imagine an ellipsoid cen-tered on a node and consider the plane through that nodewith normal vector n. Assume that the volume associatedwith the interfacial contact matches the cell volume suchthat

Vcell = ∆x∆y∆z = h⊥Ac (21)

where Ac is the contacting area and h⊥ is an effectivethickness of the contacting volume perpendicular to thecontacting surface. Once h⊥ is known, Ai can be adjustedby replacing ∆x with h⊥. Three h⊥ methods were tried:

1. Let h1 and h2 be distances from the origin to theellipsoid along the tangential vector, t, and alonga second tangent defined by n × t, then h⊥ =∆x∆y∆z/(h1h2).

2. Let h⊥ be the distance along the normal vector tothe ellipsoid.

3. Let h⊥ = ∆x∆y∆z/Ac where Ac is the cross sectionalarea that the plane with normal n makes with theentire element.

i

ij

j

h 1

∆x

∆y

n = (nx, ny)

1/16

3/16

3/16

9/16

A.

B.

C.

Figure 2 : Various orientations of an interface includingat a grid line (A), an elements mid plane (B), and titledwith respect to global x-y-z axes. The numbers on parti-cles in A are values of shape functions when particles arenear their initial positions. C shows an ellipse around thecenter node circumscribed by neighboring cells.


Calculations (see results and discussion) showed thatmethod 1 best accounts for tilted planes. That methodevaluates to

h⊥ =

√

n2x∆x2 +n2

y∆y2 2D

∆x∆y∆z√(

a21 +a2

2 +a23

)(b2

1 +b22 +b2

3

)3D

(22)

where nx and ny are normal vector components, ai =t2i /∆x2

i , bi =(n× t)2i /∆x2

i , and i = x, y, and z. For a regularmesh with ∆x = ∆y = ∆z, h⊥ is always equal to ∆x; i.e.,this extra correction is only needed for meshes with un-equal cell lengths. The 2D result applies to axisymmetricMPM as well. Note that h⊥ is also useful for correctingextrapolated positions by changing to δcon = 0.8h⊥.

Finally, as Dn → ∞ and Dt → ∞, the interface becomesperfect with zero displacement discontinuity, but in ex-plicit integrations, large Dn or Dt will eventually be toostiff to converge. A practical solution is to monitor in-ternal forces. If they exceed the force required to movethe particle to the center of mass, that node is processeddifferently. Integrating nodal velocity over time step k,nodal position for the center of mass updates by midpointrule to:

uuu(k+1)i,c = uuu(k)i,c +

∆t

m(k)i,c

(ppp(k)i,a + ppp(k)i,b +

∆t2( fff (k)i,a + fff (k)i,b )

)(23)

where fff (k)i, j are MPM forces for each material besides in-terface forces. The center of mass evolves independentlyof interface tractions. Materials a and b at the interface,however, evolve separately by:

uuu(k+1)i,a = uuu(k)i,a +

∆t

m(k)i,a

(ppp(k)i,a +

∆t2( fff (k)i,a + fff (int,k)

i,a )

)(24)

uuu(k+1)i,b = uuu(k)i,b +

∆t

m(k)i,b

(ppp(k)i,b +

∆t2( fff (k)i,b − fff (int,k)

i,a )

)(25)

where force conservation was used to find interface forceon material b as opposite of the force on material a.The interface force that would cause materials a and bto evolve to the center of mass position (i.e., uuu(k+1)

a =

uuu(k+1)b = uuu(k+1)

c ) is:

fff (int,k)i,a =

2mred

(∆t)2

(uuu(k)b −uuu(k)a

)+

2∆pppi,a

∆t+∆ fff i. (26)

where

mred =m(k)

i,a m(k)i,b

m(k)i,a +m(k)

i,b

and ∆ fff i =m(k)

i,a fff (k)i,b −m(k)i,b fff (k)i,a

m(k)i,a +m(k)

i,b

(27)

If calculated interfacial force in the normal or tangentialdirection exceeds the normal or tangential components ofthe force in Eq. (26), that node direction should be con-verted to a node with stick contact and having zero inter-facial energy (i.e., a perfect interface); otherwise the in-terfacial force is added to internal forces. It is convenientin MPM codes to implement interface calculations af-ter extrapolating material velocity fields, but before otherforces are known (i.e., along with contact calculations).In this location, ∆ fff i will be unknown, but because it issecond order (when ∆t is small), it can be ignored.

A code refinement for efficiency is to “flag” either Dn orDt as being a infinite or perfect. When these flags areset, forces are always limited and therefore that direc-tion reverts to stick contact with zero interfacial energy.Because the forces are always limited, the above calcula-tions for maximum force can be skipped.

Prior work on imperfect interfaces in MPM implementedthem on explicit cracks (Nairn [2007]). That paper didnot address the need to correct contact area by relativevolumes, to account for tilt in irregular meshes, or to limitforces for stiff interfaces. Those corrections, which arederived above, should be used for interfaces on cracksas well. Because the examples in Nairn [2007] alwaysused cracks along grid lines in regular meshes with eithersufficiently low stiffness or “flagged” perfect interfaces,those examples did not need these new corrections.

2.4 Rigid Materials

A useful feature in MPM is to allow rigid material par-ticles that move at a prescribed velocity. Because theyare rigid, their velocity cannot be changed by forces, butthey can apply forces to nonrigid particles through con-tact or interfacial interactions. For interactions betweennon-rigid material a and rigid material b, the above equa-tions change to:

vvv(c)i = vvvi,b (28)

∆pppi,a = −mi,a(vvvi,a−vvvi,b

)(29)

∆v =∆pppi,a

mi,a(30)

δδδi = xxxi,b−xxxi,a (31)


The remaining equations are the same except momentumchanges and internal forces are applied only to materiala.

2.5 Finding the Normal Vector

Experience in contact and interface simulations supportsthis assertion — MPM contact and interface calculationsare very accurate provided the normal vector is accu-rate (Lemiale, Hurmane, and Nairn [2010]). OriginalMPM contact found the normal from the mass gradi-ent for each material (Bardenhagen, Guilkey, Roessig,Brackbill, Witzel, and Foster [2001]). Because methodsdescribed below compare gradients between materials,which might have different densities, it is preferable touse a domain gradient:

‖nnni,a‖nnni,a =−∇Ωi,a (32)

where nnni,a is an unnormalized, outward directed normalfor material a at node i. In MPM, a good way to finddomain gradients is to extrapolate using shape functiongradients:

gggi, j =

∑p∈ j GGGipVp 3D∑p∈ j GGGipAp 2D

(33)

where GGGip are generalized shape function gradients (Bar-denhagen and Kober [2004]). In axisymmetric MPM,gggi, j would treat nodes at r = 0 as being on the edgewith a non-zero r component. But in realty, such nodesare at the core of the object and therefore by symme-try, the r component must be set to zero (Nairn andGuilkey [2013]). Curiously, gggi, j = −Ωa(xxxi) leading to||ni,a||ni,a = gggi, j (Bardenhagen [2012]).

Unfortunately, MPM calculations for material a basedsolely on gggi,a frequently give poor results. The follow-ing replacement options are proposed:

Maximum Volume Gradient (MVG): find the normal frommaterial a or b, whichever has the domain gradient withthe largest magnitude. The logic is that higher gradientsare likely closer to an interface and likely more accuratethan when the magnitude is smaller.

Average Volume Gradient (AVG): find the normal from adomain-weighted average of gradients from the two ma-terials — (Ωi,agggi,a−Ωi,bgggi,b)/(2Ωi)

Specific Normal (SN): if the normal can be predeter-mined, used that specified result rather than domain gra-dients.

Rigid Material Volume Gradient (RMVG): find the nor-mal from the rigid material’s domain gradient. In prob-lems with rigid materials, especially fixed rigid particles,the gradient from that material is often more reliable thanthe gradient from non-rigid particles.

Results below compare calculations using methodsMVG, AVG, and SN. Unfortunately, the best method de-pends on the problem. MPM code therefore needs toallow all these options, and potentially may need moreoptions for new problems.

2.6 Two or More Materials

Contact and interfaces in MPM work best when only twomaterials interact at multimaterial nodes. This situationis most likely to get accurate normals. For efficiency,contact or interface calculations only need to be doneonce. Once completed for material a, the negative mo-mentum and force changes can be applied to material b’sfield variables without any additional calculations. Thisapproach conserves momentum, balances forces, and ismore efficient. When calculating interfacial energy (e.g.,Eq. (4)), that energy should be added only once for eachpair.

When more than two materials are present, the followingalgorithm allows calculations to proceed:

1. Iterate through each non-rigid material and find mo-mentum changes and internal forces for each one us-ing the method above that lumps all remaining ma-terials as a virtual material b. Apply changes to eachmaterial separately, but increment interfacial energyonly once for each multimaterial node.

2. If a simulation has different properties between dif-ferent material pairs (e.g., different coefficients offriction or interfacial properties), use the propertiesfor interaction between material a and the remainingmaterial with the most volume.

3. If one of the materials is a rigid material, implementrigid contact between each non-rigid material andthe rigid material. In this case, include interfacialenergy for each interaction.

Handling nodes with more than two materials will alwaysbe approximate. In some situations, the approximationsare due to resolution, i.e., the model is not resolving thematerials. But no resolution increase can avoid multiple


b

c

a i

Figure 3 : Three different materials, a, b, and c, interact-ing at a single multimaterial node i.

material nodes such as node i in Fig. 3. At such nodes,the above algorithm adjusts momenta and applies internalforces, but it does not conserve momentum, apply bal-anced forces, or find accurate interfacial energy. The sep-arate material fields simply have insufficient informationfor highest accuracy calculations. In general increasedresolution should help. As resolution increases, the num-ber of nodes along interfaces increases, but the number ofnodes similar to Fig. 3 remains constant. The goal shouldbe to have the number of multimaterial nodes with morethan two materials be much less than the number withtwo materials.

3 Results and Discussion

Figure 4A shows two identical layers connected by animperfect interface. The isotropic elastic layers hadE = 2400 MPa, ν = 0.33, and were 4×4×20 mm. Onelayer was loaded to constant 0.5 MPa stress on top whilethe entire bottom was fixed. The interface was assumedto be perfect in the normal direction (Dn = ∞) while Dt

was varied. The normal vector used the SN method withn = (1,0) (normal vector effects are discussed later). Inthese dynamic MPM calculations, the load was rampedup linearly over 0.02 ms and then held constant; griddamping was used to converge to static results after about0.4 ms.

The results for total interfacial energy per unit volume(by Eq. (4)) are given in Fig. 5. The FEA curve is a fi-nite element analysis result using imperfect interface ele-ments (Nairn [2007]) with a regular grid of 0.25 mm ele-ments. The open symbols are 2D MPM results for back-

A B C

Figure 4 : Three sample specimens. A. Two layer spec-imens loaded on top of one layer. B. A double lap shear(DLS) specimen with loading on two ends. C. 1/8 of a3D model for a spherical particle embedded in a matrixin a cubic array. The dotted circles in A and B show edgeregions that cause problems when calculating interfacenormals.

ground cells of 0.5 mm (squares), 0.25 mm (triangles),and 0.1 mm (circles). The MPM results changed muchless between 0.25 mm and 0.1 mm than between 0.5 mmand 0.25 mm suggesting they are closed to converged for0.25 mm. The converged results agree reasonably wellwith FEA calculations. The differences were probablycaused by differences in the way FEA and MPM applyboundary conditions on the top of the specimen. Thesolid symbols are 3D calculations to verify 3D interfacemethods. In 3D, a single tangential force is applied in thedirection of tangential motion. The dotted curve is a priorMPM method for MPM imperfect interfaces by usingtraction laws on an explicit crack surface along the inter-face (Nairn [2007]). The crack calculation used 0.25 mmgrid and was essentially identical to the 0.25 mm gridresults here based on multimaterial MPM methods.

Total interfacial energy density in two layer specimenswas a small fraction of the total energy — < 2% at most— but the interface had a large affect on axial displace-ments. Figure 6 shows that the y direction displacementsfrom FEA and MPM for Dt = 5 MPa/mm agree well. Theinterface is sharper by FEA due to nature of element-based solution compared to a particle-based solution in


1/Dt (mm/MPa)

Inte

rface

Ene

rgy

Den

sity

(N/m

2 )

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

FEA

MPM Crack

3D Model

2D Model

Figure 5 : Interfacial energy density in the two layer ex-ample as a function of Dt with Dn = ∞. The open sym-bols are MPM results for three different resolutions (0.5,0.25, and 0.1 mm cells). The solid symbols are 3D MPMresults. The dashed curve is MPM with an imperfect in-terface on an explicit crack. The solid line is finite ele-ment analysis.

MPM.

A second example modeled a double lap shear (DLS)specimen (see Fig. 4B). The interface plays a larger rolein this specimen, which prompted recent work to suggestits use for measuring Dt (Le and Nairn [2012]). By nu-merical modeling, total energy U can be evaluated as asum of the strain energy and the interfacial energy. Thisenergy is related to specimen stiffness by k = P2/(2U),where P is total applied load. By measuring k and com-paring to modeling results for U where Dt is the onlyunknown, one can, in principle, measure Dt . The keyspecimen requirement is that k changes by measurableamounts in the range of actual interfacial properties. Thisprocess worked reasonably well for wood strands with aninterface of wood glue (Le and Nairn [2012]).

The calculations here modeled a DLS specimen withwood layers, which were transversely isotropic with theaxial wood grain in the y direction. The properties wereEA = 10000 MPa, ET = 1000 MPa, GA = 500 MPa,νA = 0.2, and νT = 0.3. Each layer was 1 mm thick, thebond line was 25 mm long, and free ends on top and bot-tom were 5 mm long. The two tops were loaded to 1 MPawhile the bottom was loaded to 2 MPa (to have balancedtotal force). The interface was assumed to be perfect inthe normal direction (Dn = ∞) while Dt was varied. Thenormal vector used the SN method with n = (1,0). Theload was ramped up linearly over 0.04 ms and then held

Figure 6 : Axial displacements in two layer specimensby MPM or FEA when Dt = 5 MPa/mm. The shadingfrom red to purple shows 11 contours from -0.0005 mmto 0.005 mm.

constant; grid damping was used to converge to static re-sults after about 0.22 ms.

Figure 7 plots DLS stiffness for wood specimens. Theopen circles are MPM results using the SN normalmethod. The filled squares are FEA results. MPM andFEA results are nearly identical. The solid line is ananalytical solution based on shear-lag analysis (Nairn[2007]). The theory agrees well with numerical re-sults, which both validates the theory and further vali-dates the numerical implementation of imperfect inter-faces. The open diamonds are MPM results using ex-plicit cracks rather than multimaterial interfaces. Theseresults show again that multimaterial mode MPM can ac-curately model imperfect interfaces. Notice also that overthe range analyzed, DLS stiffness changes by over a fac-tor of five. In other words, the interface plays a majorrole in DLS specimen stiffness. The total interfacial en-ergy increased to 91% of total energy as Dt decreased to1 MPa/mm. If DLS stiffness can be measured accurately,inversion of the curve in Fig. 7 provides a measure of Dt

(Le and Nairn [2012]).

Figure 8 show results for shifted interfaces in two-layerspecimens to verify the correction term in Eq. (20). Thesquare symbols show results for the interface along the


1/Dt (mm/MPa)

Stiff

ness

(N/m

m)

0.0 0.2 0.4 0.6 0.8 1.0 0

100

200

300

400

500

FEA

CracksMVG

AVG

SN

Shear lag Theory

Figure 7 : Calculated stiffness for the DLS specimen.The symbols are numerical results using three differentnormal methods (SN, MVG, and AVG), MPM with ex-plicit cracks (Cracks), and finite element analysis (FEA).The solid line is an analytical shear lag solution.

grid line (see Fig. 2A); the open squares are correctedusing Eq. (20) while solid squares use the uncorrectedcontact area of tΩi/∆x. Although corrected and uncor-rected Ai are the same for nodes close to the interface,as the solution evolves a few particles migrate resultingin small, non-zero shape functions on nodes one elementaway from the interface (e.g., node j in ig. 2A). The cor-rected area appropriately results in very small force onsuch nodes because the minimum volume is low. In con-trast, the uncorrected area gives force equal to nodes onthe grid line causing interfacial energy to be too high.

The circle symbols in Fig. 8 show results for an inter-face along cell midplanes (see Fig. 2B); the open circlesare corrected using Eq. (20) while solid circles use theuncorrected contact area of tΩi/∆x. When Eq. (20) isnot used the contact area is twice as large resulting ininterface results that effectively have Dt twice as large.This doubling of effective Dt causes the x axis in a 1/Dt

plot to shift right by a factor of 2. Clearly a contact areacorrection is needed regardless of the location of the in-terface; furthermore the correction in Eq. (20) is accuratebecause the results are independent of interface locationand always close to FEA results.

Imperfect interface modeling needs to work for any inter-facial orientation, and not just for the shifting discussedabove. To confirm this ability, the DLS example was re-run by rotating the specimen with respect to the x andy axes. The solid line in Fig. 9 is for a regular mesh(∆x = ∆y) and the results are independent of interface

1/Dt (mm/MPa)

Inte

rface

Ene

rgy

Den

sity

(N/m

2 )

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

FEA

grid line (corrected)

midplane (uncorrected)

midplane (corrected)

grid line (uncorrected)

Figure 8 : Interfacial energy density in two layer speci-mens with the interface located either at a grid line or atelement midplanes. The open symbols correct the con-tact area using Eq. (20) while the solid symbols do not.The FEA curve is corresponding result by finite elementsanalysis for reference.

angle. For an irregular mesh, however, the results needan additional correction to replace ∆x in Eq. (20) with h⊥in Eq. (22). The dotted lines in Fig. 9 show DLS resultsas a function of angle for a mesh with ∆y = 2∆x using“No Correction” (i.e., h⊥ = ∆x) and the three h⊥ meth-ods described above. The “No Correction” result is poor.Method 1 (see Eq. (22)) worked best and is therefore therecommended method. Even the best method, however,is not as good as the regular mesh result, which may bebecause the irregular mesh was at a lower resolution inthe y direction for this example.

Figure 10 shows the importance of the method usedto find the contact normal for two-layer specimens(Fig. 4A) using 0.25× 0.25 mm grids. First, look atthe “grid” curves, which are when the interface is neara grid line. For this geometry, a specified normal (SNusing n = (1,0,0)) is the control and is the same resultas in Fig. 5. If the normals are found by the MVG (grid)method instead, the results are poor. The inaccuracies arecaused by the single node on the top of the specimen (seecircled region in Fig. 4A). At this position the normals forthe two materials are at 45 to the interface rather than at(1,0,0). This error can be fixed by averaging the nor-mals because the two 45 normals average to the correct(1,0,0) normal. This fix is confirmed by the AVG (grid)results, which agree with the control.

Next, look at the “mid” results, which are for the inter-face at element midplanes. Again the SN (mid) results


Angle (degrees)

Stiff

ness

(N/m

m)

0 10 20 30 40 50 60 70 80 90 140

160

180

200

220

240

260

280 No Correction

Method #1

Method #2

Regular Mesh

Method #3

Figure 9 : Calculated stiffness for the DLS specimenfor tilted specimen as a function of rotation angle. Thesolid line is for a regular grid. The dotted lines are foran irregular grid with ∆y = 2∆x and various methods forcorrecting for interface tilt.

1/Dt (mm/MPa)

Inte

rface

Ene

rgy

Den

sity

(N/m

2 )

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

MVG, AVG (mid, Ωmin=0.75Acell)

SN (mid)

MVG, AVG (mid)

MVG (grid)

SN, AVG (grid)MVG, AVG (mid, Dn=Dt, Ωmin=0)

Figure 10 : Interfacial normal energy density in two-layer specimens with the interface at a “grid” line or at el-ement “mid” planes. The discrete numerical results weresmoothed by a spline interpolant to make it easier to vi-sualize differences between similar results.

gave the correct result (as already discussed for Fig. 8).But now, neither MVG (mid) nor AVG (mid) gave ac-ceptable results. The problem was again the contactingnodes on the top edge, but now the errors were not fixedby averaging. This example illustrates a potential prob-lem in all MPM contact or interface methods — the nor-mals are often wrong when the contact surface is at afree edge. In many calculations, errors caused by suchnodes might be small, but in this two layer specimenthey happen to be large. A solution for this specimen(and maybe many MPM calculations) is to ignore thoseproblem nodes. In other words, it can be better to ig-nore edge nodes then to include them with potentiallyerroneous normals (Guilkey [2010]). A method to ignoreedge nodes is to screen them out based on total nodaldomain, Ωi. For the edge nodes in two layer specimensΩi ∼ Acell/2, while for all other nodes Ωi ∼ Acell . Edgenodes can therefore be screened out by ignoring interfa-cial forces and energy for multimaterial nodes with Ωi <0.75Acell . The MVG and AVG (mid, Ωmin = 0.75Acell)results in Fig. 10 confirms this approach agrees well withthe control.

This two-layer specimen may be a pathological specimenthat exaggerates the importance of normals (an hence agood example for developing interface methods). The re-sults are very sensitive to the normal because total inter-facial energy is very small. Any inaccuracies can lead tolarge changes in interfacial energy. The two-layer prob-lem was further exacerbated by its perfect interface inthe normal direction. Because a perfect interface corre-sponds to a large Dn, any errors in n get amplified by thehigh interfacial stiffness. Figure 10 shows calculationswith Dn = Dt (i.e., when both directions are imperfect)by MVG and AVG (mid) and without volume screening.These results agree with controls, despite inclusion of in-accurate edge nodes. Because Dn is smaller, those inac-curacies have much less affect on the results. The rea-son they are close to controls that had Dn = ∞ are thatthe normal direction plays little role in this shear-loadedspecimen. The slightly higher peak energy, however, islikely the net effect of an imperfect, rather then perfect,normal interface.

Figure 7 shows calculation of DLS stiffness for SN (thecontrol with n = (1,0)), MVG, and AVG. Now SN andMVG agree well, while AVG has small errors. Theproblems for this specimen are nodes where one layerstops while the other one continues (see circled regions


in Fig. 4B). At these locations, the continuous layer hasthe correct normal and is the material with the maximumvolume gradient. Hence, the MVG method worked well.In contrast, the AVG method averages a correct normalwith an inaccurate normal at 45 and get less accurate re-sults The effect is much smaller in this specimen becausethe interfacial energy for other nodes is large. Never-theless, for maximum accuracy, it is important to haveaccurate normals.

The results in Fig. 7 are for the interface at grid lines.The normals effect is larger when the interface is atcell midplanes, as shown in Fig. 11. Now both MVGand AVG gave poor results. These results can be fixedby the same two methods used for the two layer spec-imen — screen out edge nodes by ignoring nodes withΩi < 0.85Acell (because those nodes are expected to haveΩi ∼ 0.75Acell) or set Dn = Dt (i.e., allow the interface tobe imperfect in the normal direction rather than perfect).Using either method along with any normal method (SN,MVG, or AVG) all gave identical results, which agreedwith shear lag theory (see “All Corrected Methods (mid)”symbols).

In summary, the most accurate results are always by SN.When SN is not possible, which happens whenever in-terfaces might reorient during a calculation, both MVGand AVG can work, but require care. If neither Dn norDt are too stiff, both MVG and AVG usually work. AsDn and Dt get higher, however, problems frequently ariseat edge nodes. Therefore, in problems with one or morestiff direction and with edge nodes, it is probably essen-tial to ignore edge nodes by ignoring interfacial force andenergy at nodes with Ωi less then a carefully selectedcritical value. These recommendations apply to MPMcontact calculations as well as interface calculations. Infact, MPM contact can be characterizes as Dn = 0 in ten-sion but Dn = ∞ during contact. In other words, contactcalculations always have one stiff direction. Those cal-culations should use MVG or AVG along with volumescreening to ignore edge nodes.

A final example looked at a 3D composites mechanicsproblem and evaluated the algorithm to limit forces whenDn or Dt get high. The problem, shown in Fig. 4B, is 1/8of a spherical particle embedded in a cubic array. Theparticle had radius of 700 µm in a cube of side 1000 µmto model a particulate filled composite with particle vol-ume fraction Vp = 17.96%. The particles were glass(E = 70,000 MPa, ν = 0.25) embedded in a polymer

1/Dt (mm/MPa)

Stiff

ness

(N/m

m)

0.0 0.2 0.4 0.6 0.8 1.0 0

100

200

300

400

500

600

Shear lag Theory

AVG (mid)MVG (mid)

All “Corrected” Methods (mid)

Figure 11 : The stiffness of a DLS specimen with the in-terface at element “mid” planes. The symbols are MPMresults. The solid line is an analytical shear lag solution.

matrix (E = 2400 MPa, ν = 0.33). The interface wasimperfect in both directions with Dn = Dt . The normalwas found by the SN method; if the origin is at the centerof the particle, the normal at node i is in the direction (xi,yi, zi). The cube had zero displacement on the symme-try planes and was loaded to applied strain of ε0 = 0.5%strain in all three directions on the cube’s outer surfaces.The surfaces were loaded at a rate of about 0.1% of thepolymer wave speed and then held fixed until grid damp-ing settled on a static result.

By composite variational mechanics, the bulk modulus isrelated to total energy, U , by

U =V2

K(

∆VV0

)2

(34)

With an imperfect interface, U is a sum of strain energyand interfacial energy (kinetic energy is zero in this staticlimit). For this problem, ∆V/V0 = 3ε0 and V = l3 (wherel is the side of the cube). The bulk modulus can thereforebe found from:

K =2(Ustrain +Uinter f ace)

9l3ε20

(35)

Figure 12 shows K found by energy analysis using vari-ous element sizes and with or without the recommendedmethod to limit forces when Dn or Dt are high. Thehorizontal line labeled “CSA” is the predicted modulusfor a particulate composite with a perfect interface us-ing the “composites spheres assemblage” model (Hashin[1962]). When forces are not limited, K rises above thetheoretical limit and keeps rising until the calculations


1/Dt = 1/Dn (mm/MPa)

Bulk

Mod

ulus

(MPa

)

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 1000

2000

3000

4000

5000

6000

7000

8000

100 µm

50 µm

25 µm

100 µm (shorter Δt)50 µm (no force limits)

CSA

Vf = 17.96%rp = 700 µm

Figure 12 : Bulk modulus in a cubic array of sphericalparticles (i.e., a particulate composite) of microparticles(rp = 700 µm) and nanoparticles (rp = 700 nm) deter-mined by energy methods and using various cell sizes,time steps, and with or without force limitation (as indi-cated). The dashed horizontal line is the analytical com-posite spheres assemblage (CSA) result for a perfect in-terface.

become unstable. All other results limited forces and re-mained stable for any values of Dn and Dt , but the re-sults for a crude mesh (100 µm cells) had an erroneouspeak in the transition region around 1/Dt = 1/Dn =10−6 mm/MPa. The reason for the peak is that 100 µmcells do not resolve a spherical surface of radius 700 µmwell enough for accurate energy results. The calculationswere improved by either a reduction the time step (whichshifts the peak to higher Dt) or by smaller cells. Smallercells worked much better because the peak is nearly gonefor 50 µm cells and is entirely gone for 25 µm cells.These converged results are composite mechanics analy-sis for bulk modulus of a particulate composite as a func-tion of interface properties. The results are close to thetheoretical limit for high Dt = Dn, decrease as Dt and Dn

decreases, and reach a low plateau value for a very poorinterface (i.e., debonded interface for Dt = Dn→ 0)

The particulate composite results have uniform averagestress, 〈σxx〉 = 〈σyy〉 = 〈σzz〉 = P, where P is pressure;thus K can also be found from

K =P

3ε0(36)

Figure 13 shows K found from pressure for various ele-ment sizes and two different particle sizes. For micropar-ticles (700 µm) all element sizes worked well, although100 µm was rather crude and had minor errors in the

1/Dt = 1/Dn (mm/MPa)

Bulk

Mod

ulus

(MPa

)

10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 1000

1500

2000

2500

3000

3500

4000

Nanoparticle

Microparticle

100 µm50 µm

25 µm

25 nm

Vf = 17.96%

CSA

Figure 13 : Bulk modulus in a cubic array of spheri-cal particles (i.e., a particulate composite) of microparti-cles (rp = 700 µm) and nanoparticles (rp = 700 nm) de-termined by average stress over applied volumetric stainand using various cell sizes (as indicated). The solid linesare analytical results using a composite spheres assem-blage (CSA).

transition region around 1/Dt = 1/Dn = 10−6 mm/MPa.The results for 50 µm or 25 µm were nearly identical andgave smooth curves. The solid lines are analytical model-ing for a composites spheres assemblage with an imper-fect interface (Hashin [1991b]). The numerical modelingagrees with the analytical modeling.

The “Nanoparticle” curve shows the modulus for a par-ticulate composite of nanospheres (with radius 700 nm)as a function of interface properties using 25 nm cells.Hashin [1991b] suggests that Dn =(Ki+4Gi/3)/ti whereKi and Gi are bulk and shear modulus of the interphaseand ti is its thickness. Physically, Ki, Gi, and t1 are deter-mined by interactions between the phases (e.g., betweenglass and polymer), but they should not be affected by ra-dius of curvature of the surface. In other words, Dn andDt should be independent of particle size. When this rela-tion holds, Fig. 13 shows that bulk modulus of a nanopar-ticle composite is always lower than the correspondingmicroparticle composite. Nanocomposites research fre-quently notes a scaling effect that nanocomposites havemuch more interface per unit volume reinforcement thanconventional composites and nanocomposites hype as-sumes this extra interface must be beneficial. Contraryto that hype, the straightforward composite mechanicsin Fig. 13 shows that extra interface is always detrimen-tal for mechanical properties. A better picture emergesby realizing the interface in composites is burdened withtransferring stress between phases. The extra interfacial


area in nanocomposites means their interfacial propertiescarry an extra burden. In other words, when making newnanocomposites, the interface better be excellent, oth-erwise the mechanical properties are likely to be poor.Furthermore, interface-quality needs are much higherfor nanocomposites then when making a correspondingcomposite with larger phases. For example, consider aglass-polymer system with Dt = Dn = 106 MPa/mm inFig. 13. For a nanocomposite, this material would havevery poor interface with K near the plateau for a com-pletely debonded interface. In contrast, a composite withmicroparticles would have K near the theoretical limit fora perfect interface.

All examples in this paper used a linear elastic inter-face (e.g., Tn = Dn[un]) and calculated elastic interfacialenergy (see Eq. (4)). Obviously it is trivial to replacethese assumptions with any non-linear interfacial tractionlaws and energy integral. For example, the normal di-rection might use a bilinear law where Dn is very largeor perfect in compression (to avoid interpenetration) buthas a lower stiffness in tension. But, interpenetrationdoes not need to be prohibited. A 2D imperfect inter-face represents an actual 3D interphase and it is possi-ble to compress such an interphase, which would corre-spond to a small negative [un] when loaded in compres-sion. An interfacial traction law that is history depen-dent, such as a plasticity based load with hardening oran interface that fails at some critical discontinuity, re-quires new methods. In multimaterial MPM used here,interacting nodes are re-calculated on each time step andcarry no history. Such methods do not allow history de-pendent traction laws. One solutions is to use explicitcracks in MPM with tractions laws. By this approach,traction laws are carried on crack particles and thereforecan track their history. Those traction laws can imple-ment history-dependent plasticity and failure at criticalcrack opening displacements. In other words, they canbe used as a particle-based approach to cohesive zone el-ements in finite element analysis.

4 Conclusions

Extension of multimaterial MPM methods for imperfectinterfaces is simple in concept — calculate displacementdiscontinuities at interfaces and add internal forces deter-mined by interfacial tractions laws. But, implementationdetails are subtle and important. If these details are ig-nored, the results can be poor. When they are included,

however, imperfect interface modeling is a robust exten-sion of MPM. An advantage over other numerical meth-ods (e.g., FEA) is that imperfect interfaces are dynami-cally evaluated on each time step, thereby avoiding theneed to use contact or interface elements in the mesh.The key details needed for accurate MPM interface mod-eling are:

Grid Independence: For results to be independent of in-terface location within the MPM grid, interfacial contactarea has to be corrected by comparing nodal domains forthe two material using Eq. (20). For an irregular mesh(i.e., ∆x 6= ∆y or 6= ∆z), an additional correction is neededto account for angled interfaces, which is done by replac-ing ∆x in Eq. (20) with h⊥ in Eq. (22).

Stiff Interface Issues: As Dn or Dt approach ∞, their highstiffness would make it difficult to get converged results.A good solution is to limit interfacial forces to the maxi-mum allowed forces defined by Eq. (26).

Interface Normals: Calculation of interfacial normal iscrucial to accurate results and the optimum method isproblem specific. The SN method is always best androbust, but only works for specialized problems. Forother problems, MVG and AVG provide two alterna-tives. If neither Dn nor Dt are too high, both workwell. If either Dn or Dt become stiff, however, prob-lems can arise. These problems are usually confined toedge nodes, which can be screened out by ignoring in-terface calculations at nodes with Ωi below some cutoffvalue. These conclusions apply to MPM contact methodsas well, which can be regarded as always having Dn = ∞

during contact.

Interface Energy: Interfacial energy has no role in evolu-tion of MPM results, but it is useful to track for compos-ite mechanics calculations. The examples in this paperassumed a linear elastic imperfect interface with energygiven by Eq. (4). If different traction laws are used, theinterfacial energy calculation has to be changed, other-wise the a sum of strain energy, kinetic energy, and in-terfacial energy would not correspond to the actual totalenergy in the object.

Acknowledgement: This work was supported in partby the National Research Initiative of the United StatesDepartment of Agriculture (USDA) Cooperative Re-search, Education and Extension Service, Grant 2006-35504-17444 and by a grant from the National ScienceFoundation (award 1161305). The author also thanks


Scott Bardenhagen and Jim Guilkey for many helpfuldiscussions.

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