Examinationand Analysis of Implementation Choices within the Material Point Method...

Copyright c© 2008 Tech Science Press CMES, vol.31, no.2, pp.107-127, 2008

Examination and Analysis of Implementation Choices within the MaterialPoint Method (MPM)

M. Steffen1, P.C. Wallstedt2, J.E. Guilkey2,3, R.M. Kirby1 and M. Berzins1

Abstract: The Material Point Method (MPM)has shown itself to be a powerful tool in the sim-ulation of large deformation problems, especiallythose involving complex geometries and contactwhere typical finite element type methods fre-quently fail. While these large complex problemslead to some impressive simulations and solu-tions, there has been a lack of basic analysis char-acterizing the errors present in the method, evenon the simplest of problems. The large number ofchoices one has when implementing the method,such as the choice of basis functions and boundarytreatments, further complicates this error analysis.In this paper we explore some of the many choicesone can make when implementing an MPM al-gorithm and the numerical ramifications of thesechoices. Specifically, we analyze and demonstratehow the smoothing length within the General-ized Interpolation Material Point Method (GIMP)can affect the error and stability properties ofthe method. We also demonstrate how variouschoices of basis functions and boundary treat-ments affect the spatial convergence properties ofMPM.

Keyword: Material Point Method, GIMP,Meshfree Methods, Meshless Methods, ParticleMethods, Smoothed Particle Hydrodynamics,Quadrature

1 Introduction

The Material Point Method (MPM) [Sulsky,Chen, and Schreyer (1994); Sulsky, Zhou, and

1 School of Computing, University of Utah, Salt Lake City,UT, USA. {msteffen,kirby,berzins}@cs.utah.edu

2 Department of Mechanical Engineering, Uni-versity of Utah, Salt Lake City, UT, USA.{philip.wallstedt,james.guilkey}@utah.edu

3 Corresponding Author

Schreyer (1995)] is a mixed Lagrangian and Eu-lerian method utilizing Lagrangian particles tocarry history-dependent material properties andan Eulerian background mesh to calculate deriva-tives and solve the equations of motion.

MPM and its variants have been shown to beextremely successful and robust in simulating alarge number of complicated engineering prob-lems (see for example [Bardenhagen, Brydon, andGuilkey (2005); Nairn (2006); Sulsky, Schreyer,Peterson, Kwok, and Coon (2007)]). The mostwell known of these variants is the General-ized Interpolation Material Point (GIMP) Method[Bardenhagen and Kober (2004)], of which tradi-tional MPM is a special case. GIMP provides im-proved accuracy, stability and robustness to sim-ulations through the introduction of particle char-acteristic functions, which in most cases have theeffect of smoothing the grid basis functions. Theability to handle solid mechanics problems in-volving large deformations and/or fragmentationof structures, which are sometimes problematicfor finite element methods, has led, in part, to themethod’s success.

MPM, and later, GIMP, was chosen as the solidmechanics component for fluid-structure interac-tion simulations within the Center for the Sim-ulation of Accidental Fires and Explosions (C-SAFE). The goal of C-SAFE has been the devel-opment of a capability to simulate the responseof a metal container filled with explosives to alarge hydrocarbon pool fire, including heat up, ig-nition and rupture of the container. The pioneer-ing work of Kashiwa and co-workers [Kashiwa,Lewis, and Wilson (1996)] inspired this choiceas they had demonstrated many of the capabili-ties that would be required for such simulations,including material failure and solid-to-gas phase

108 Copyright c© 2008 Tech Science Press CMES, vol.31, no.2, pp.107-127, 2008

transition. To achieve the required level of par-allelization and to provide a platform for Adap-tive Mesh Refinement, C-SAFE investigators cre-ated the Uintah Computational Framework (UCF)[Parker, Guilkey, and Harman (2006)]. It is withinthis software environment that the implementa-tions of MPM and GIMP under consideration hereexist, along with components for fire simulation,compressible reacting flow, and fluid-structure in-teraction.

The main goal of this paper is to examine someof the implementation choices within GIMP in amulti-dimensional simulation setting and to un-derstand the algorithmic and numerical ramifica-tions of those choices. Specifically, we will focuson the smoothing length parameter (or the par-ticle characteristic function) and examine a fewchoices for evolving the smoothing length in timewhich have been implemented within the UCF.We will perform analysis and carry out simula-tions in both 1-D and 3-D in order to shed light onthe error and stability properties that result fromthe various choices.

This paper is organized as follows. Section 2provides background to give context concerningwhere and how MPM fits into the family of par-ticle and meshfree methods and introduces previ-ous analysis performed on MPM. Section 3 givesan algorithmic overview of MPM and GIMP, fo-cusing on some of the choices made when im-plementing MPM within the UCF. Section 4 pro-vides an analysis and interpretation of some of thespatial errors present in MPM and GIMP, build-ing on previous analysis by the authors. In theprocess, we investigate the relationship betweenGIMP as implemented in the UCF and MPM us-ing B-spline basis functions. Section 5 overviewsthe process for developing interesting problemswith analytical solutions which can be used to testour methods and measure errors in our solutions.In Section 6 we present numerical results and dis-cuss the differences that result from the aforemen-tioned choices. Lastly, Section 7 is a summary ofour findings and our conclusions.

2 Background

The Material Point Method was introduced bySulsky, Chen, and Schreyer (1994); Sulsky, Zhou,and Schreyer (1995) as a solid mechanics exten-sion to FLIP (Fluid-Implicit Particle) [Brackbilland Ruppel (1986); Brackbill, Kothe, and Ruppel(1988)], a “full-particle” Particle In Cell (PIC) flu-ids simulation method.

More recently, Bardenhagen and Kober (2004)generalized the development that gives rise toMPM and showed that MPM can be considereda subset of their “Generalized Interpolation Mate-rial Point” (GIMP) method.

Although not derived directly from what are clas-sically considered as meshfree or meshless meth-ods, MPM falls within a general class of mesh-free methods and is discussed within the mesh-free community since it has both many of thesame advantages and many of the same chal-lenges as other meshfree methods [Li and Liu(2004)]. Like many meshfree methods, the pri-mary partitioning of the material does not involvea polygonal tessellation (as in finite elements),but rather some alternative non-mesh-based un-structured representation. However, unlike fullymesh-free methods, such as the Meshless Lo-cal Petrov-Galerkin Method (MLPG) [Atluri andZhu (1998); Han, Rajendran, and Atluri (2005);Han, Liu, Rajendran, and Atluri (2006); Atluri(2006); Atluri, Liu, and Han (2006)], MPM uti-lizes a background mesh to perform differentia-tion, integration, and solve the equations of mo-tion. The use of a background mesh is still simi-lar to other meshfree methods such as the ElementFree Galerkin Method (EFGM) [Belytschko, Lu,and Gu (1994)]. While the background mesh isformally free to take any form, it is most oftenchosen for computational efficiency to be a Carte-sian lattice (i.e. segments, quadrilaterals and hex-ahedra in 1-D, 2-D and 3-D respectively). Thesefunctions are used, in essence, as a means of dis-cretizing the continuum equations, with the do-main of these functions being an alternative (inthe sense of versus particles) representation of thedeformed configuration of the material. Nodalintegration based upon particle positions as is

Implementation Choices within the Material Point Method 109

used in other particle methods such as PIC meth-ods[Grigoryev, Vshivkov, and Fedoruk (2002)] isemployed during the solution process.

Spatial integration errors were quickly deter-mined to be the limiting factor in the accuracyand stability of MPM. One proposed way to ame-liorate the convergence problems found in MPMwas to move away from the idea of nodal inte-gration and instead think of the particles as hav-ing extent within the quadrature scheme. Barden-hagen and Kober (2004) accomplished this withGIMP by adding particle characteristic functions.There were questions, however, on how to evolvethese functions in time within a multi-D simula-tion. Since the deformation gradient is only main-tained at one point within a particle’s voxel, it isunclear that the use of this information to deformparticles’ voxels is sufficient to maintain a par-tition of the deformed domain. And, if it weresufficient, it is even more unclear how to accom-plish the accurate spatial integration of these de-formed voxels. Ma, Lu, and Komanduri (2006)proposed another approach for evolving the par-ticle characteristic functions by adding masslesscorner particles to explicitly track the deforma-tion of a particle’s voxel, or integration domain.Steffen, Kirby, and Berzins (2008) analyzed thecase where nodal integration was still used, butlooked at how the use of smoother basis functionsdrastically reduced the nodal integration quadra-ture errors.

3 Overview of the Material Point Method

MPM is a mixed Lagrangian and Eulerianmethod with particles representing the discreteLagrangian state of a material. The history de-pendent properties of a material are carried andupdated on the particles. A background mesh isalso used, in part to solve the equations of motion.This background mesh can be non-uniform andbe comprised of elements of various shapes; how-ever for computational efficiency a uniform Carte-sian grid is almost always employed. Amongother benefits, a uniform Cartesian grid eliminatesthe need for computationally expensive neighbor-hood searches during particle-mesh interaction.Particle information is projected to this back-

ground mesh, from which gradients required forconstitutive model evaluation (at the particles) arecalculated and the equations of motion are solved.Using the solution to the equations of motion onthe grid, the material state, minimally velocitiesand positions, is then updated at the particles.

As the above procedure is similar for nearly allvariants of MPM, the main distinguishing fea-ture between the different MPM methods in thispaper is the choice of basis functions. We willstart by giving an overview of standard MPM, orMPM using standard piecewise-linear basis func-tions. Next, we will show how MPM can be eas-ily implemented using B-spline basis functionsand mention the benefits of these smoother basisfunctions. The Generalized Interpolation Mate-rial Point Method (GIMP) will then be reviewed.Lastly, various options for implementing kine-matic boundary conditions will be presented.

3.1 Standard Material Point Method

The MPM procedure begins by discretizing theproblem domain Ω with a set of material points,or particles. The particles are assigned initial val-ues of position, velocity, mass, volume, and de-formation gradient, denoted xp, vp, mp, Vp, andFp (subscript index p is used to distinguish parti-cle values versus an index of i for grid node val-ues). Alternatively, instead of velocity and mass,momentum and mass density may be prescribedat the particle location, from which mp and vp canbe calculated. Depending on the simulation, otherquantities may be required at the material pointsas well, such as temperature. The particles arethen considered to exist within a computationalgrid, which for ease of computation is usually aregular Cartesian lattice. Fig. 1 depicts a repre-sentation of a typical 2-D MPM problem.

At each time-step tk (all of the following quanti-ties will be assumed to be at time tk unless oth-erwise noted), the first step in the MPM compu-tational cycle involves projecting (or spreading)data from the material points to the grid. Specifi-cally, we are interested in projecting particle massand momentum to the grid to calculate mass and


Ω

xi

xp

Figure 1: Typical 2-D MPM problem setup.The dotted line represents the boundary of thesimulated object Ω and each closed point rep-resents a material point used to discretize Ω.The square mesh represents the background grid.Each square in the background grid is a grid cell,and grid nodes are located at the corners of gridcells.

velocity at the grid nodes in the following way:

mi = ∑p

φipmp (1)

vi = (∑p

φipmpvp)/mi, (2)

where φip = φi(xp) is the basis function centeredat grid node i evaluated at the position xp. Notethat Eq. 1 represents the mass-lumped version ofwhat Sulsky and Kaul (2004) describe as the con-sistent mass matrix Mi j = ∑p φipφ jpmp. Next, in-ternal force on the grid is found by taking the di-vergence of stress as a function of the constitutivemodel and the deformation gradient stored witheach particle.

σ p = σ(Fp) (3)

finti = −∑

p∇φ ip ·σ pVp, (4)

where ∇φ ip = ∇φ i(xp) and Vp = det(Fp)V 0p de-

notes the volume of the particle voxel (in its de-formed configuration). Combining the internalgrid force with any external forces fext

i , grid ac-celerations are then calculated as:

ai = (finti + fext

i )/mi. (5)

Next, grid velocities are updated with an appro-priate time stepping scheme. Implicit time step-ping schemes exist for MPM [Guilkey and Weiss(2003); Sulsky and Kaul (2004); Love and Sul-sky (2006b)], however we choose to use the ex-plicit Forward-Euler time discretization presentedwithin the original MPM algorithm:

vk+1i = vk

i +aiΔt. (6)

Velocity gradients are then calculated at the parti-cle positions using the updated grid velocities:

∇vk+1p = ∑

i

∇φ ipvk+1i . (7)

Lastly, the history-dependent particle quantitiesare time-advanced. Particle deformation gradi-ents, velocities, and positions are updated usingcalculated velocity gradients, grid accelerations,and grid velocities:

Fk+1p = (I+∇vk+1

p Δt)Fkp (8)

vk+1p = vk

p +∑i

φipaiΔt (9)

xk+1p = xk

p +∑i

φipvk+1i Δt. (10)

Eqs. 1-10 outline one time-step of MPM and as-sume initialization of particle values at time t0:x0

p, v0p, F0

p, and V 0p . Non-linear finite element

codes often use a staggered central differencemethod [Belytschko, Liu, and Moran (2000)]. Ifpossible, a simple change of initializing particlevelocities a half time step earlier, i.e. v−1/2

p , andusing the same MPM algorithmic procedure out-lined above leads to the following set of staggeredcentral-difference update equations:

vk+ 1

2i = v

k− 12

i +aiΔt (11)


∇vk+ 1

2p = ∑

i

∇φ ipvk+ 1

2i (12)

Fk+1p = (I+∇v

k+ 12

p Δt)Fkp (13)

vk+ 1

2p = v

k− 12

p +∑i

φipaiΔt (14)

xk+1p = xk

p +∑i

φipvk+ 1

2i Δt. (15)

A similar staggered central difference method isused for MPM by Sulsky, Schreyer, Peterson,Kwok, and Coon (2007), the benefits of whichare reviewed in detail by Wallstedt and Guilkey(2008).

The calculation of σ p involves a constitutivemodel evaluation and is specific for different ma-terial models. The neo-Hookean elastic consti-tutive model used in this paper is more fully de-scribed in Section 5.1.

Calculating fexti is another problem dependent pro-

cedure with several options. The first option isto calculate fext

i directly on the grid. This is eas-ily done with body forces such as gravity wherefexti = mig. Another option is to calculate fext

p onthe particles and project to the grid through thegrid shape functions:

fexti = ∑

pφipfext

p . (16)

Moving forces, such as surface tractions can beimplemented by associating the forces with a fi-nite set of “surface” particles in conjunction withEq. 16. Lastly, for performance reasons, MPMis typically implemented using a fixed, equallyspaced Cartesian lattice, however nothing in themethod requires the grid to be fixed. Movinggrid nodes can be implemented to track bound-ary forces, calculating fext

i directly on the bound-ary nodes, however implementing this for com-plex geometries in multiple dimensions is not triv-ial.

Most standard MPM implementations usepiecewise-linear basis functions for φi due totheir ease of implementation, small local support,and familiarity to those in the finite element

community. The 1-D form of the piecewise-linearbasis function is given by:

φ (x) =

{1−|x|/h : |x|< h

0 : otherwise,(17)

where h is the grid spacing. The basis functionassociated with grid node i at position xi is thenφi = φ (x−xi). The basis functions in multi-D areseparable functions, constructed using Eq. 17 ineach dimension. e.g., in 3-D,

φi(x) = φ xi (x)φ y

i (y)φ zi (z). (18)

3.2 B-Spline Material Point Method

As Bardenhagen and Kober (2004) described inthe development of GIMP, lack of regularity in∇φ is conjectured to be the root cause of whatis referred to as “grid cell crossing instabilities”.As can be seen in Fig. 2(a), piecewise-linear basisfunctions are only C0 continuous at cell bound-aries. Tran, Kim, and Berzins (2007) performeda detailed analysis concerning temporal errorswithin an MPM fluids framework in which thegrid crossing errors arising from use of piecewise-linear basis functions were precisely determined.An analysis by Steffen, Kirby, and Berzins (2008)shows how the lack of smoothness of the standardpiecewise-linear basis functions (Eq. 17) causessignificant spatial quadrature errors, and the useof smoother basis functions, such as B-splines,significantly reduces these errors.

A typical one-dimensional quadratic B-spline canbe constructed by convolving piecewise-constantbasis functions with themselves:

φ = χ ∗χ ∗χ/(|χ |)2 (19)

where χ(r) is the piecewise-constant basis func-tion:

χ(x) =

{1 : |x| < 1

2 l

0 : otherwise(20)

and l is width of χ . The B-spline basis functionassociated with node i, is then φi(x) = φ (x− xi).The multi-D B-spline basis functions in MPM arenot radial basis functions, as in other meshfree


methods, but rather are separable, constructed us-ing Eq. 19 in each dimension–the same way aswith piecewise-linear basis functions in Eq. 18.Evaluating Eq. 19 gives the 1-D B-spline basisfunction:

φ (x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

12h2 x2 + 3

2hx+ 98 : −3

2h ≤ x ≤−12 h

− 1h2 x2 + 3

4 : −12h ≤ x ≤ 1

2 h1

2h2 x2 − 32hx+ 9

8 : 12h ≤ x ≤ 3

2 h

0 : otherwise.

(21)

It is worth noting that a set of these quadraticB-spline basis functions maintain the partition ofunity property required by the mass-lumping im-plicit in the MPM projection functions such asEq. 1 and Eq. 2. An example set of these basisfunctions is shown in Fig. 2(b).

The above construction of B-splines basis func-tions by the convolution of piecewise-constantfunctions is helpful in showing the connection be-tween the various basis functions considered inthe paper. This convolution construction has anumber of consequences, including requiring theuse of extra, or ghost nodes, beyond the boundaryto maintain a partition of unity within the domain.This presents no problems when implementingperiodic boundary conditions, however Dirichletboundary conditions are non-trivial to implement.One such boundary treatment will be discussed inSection 3.4.

Another option, which requires a slight departurefrom the convolution construction, is to modifythe boundary basis functions to still maintain apartition of unity within the domain but enforcethat the boundary basis function evaluates exactlyto one on the boundary (which, by construction,would require all other basis functions to evaluateto zero at that boundary). One such set of basisfunctions was used by Steffen, Kirby, and Berzins(2008) in their simulation of a one-dimensionalbar with traction forces. The basis functions usedin that paper, however, can only represent func-tions which have zero slope on the grid bound-aries, which was the case in their simulation.

Another choice of B-spline basis constructionwhich allows for the same partition of unity prop-

(a) Piecewise-Linear

(b) Quadratic B-Spline

(c) GIMP

Figure 2: Example sets of 1-D basis functionsused in MPM. Each set of basis functions showsan accompanying set of particles (with height rep-resenting velocity) and the corresponding velocityfield on the grid after projecting particle values us-ing Eq. 2. Piecewise-linear basis functions resultin piecewise-linear velocity fields with a disconti-nuity of velocity gradients occurring at grid nodelocations. Both B-spline and GIMP basis func-tions result in smoother fields.

erty, the boundary basis functions to exactly eval-uate to one on the boundary, and which also al-lows representation of non-zero slope solutions isthe use of an open knot vector to describe the ba-sis functions. Again, for computational efficiency,


we choose the knot vector to be an open uniformknot vector with the end knots having a multi-plicity of k − 1 for a k-order B-spline. For ex-ample, if we discretize our 1-D domain of lengthl with N knots, the knot spacing would be h =l/(N −1), and the knot vector for a set of fourth-order B-splines (consisting of third-order polyno-mials) would look like

[x0,x0,x0,x1, . . . ,xi, . . .,xn−2,xn−1,xn−1,xn−1],

where xi = x0 + i · h. For a k-order B-spline, thisresults in N +k−2 basis functions which are cal-culated recursively as

φi,k = φi,k−1x−xi

xi+k−1 −xi+φi+1,k−1

xi+k −xxi+k −xi+1

(22)

φi,1 =

{1 xi ≤ x < xi+1

0 otherwise.(23)

If either denominator in Eq. 22 evaluates to zero(which only happens when knots are repeated inthe knot vector), the entire term is set equal tozero.

Note that this is a slight departure from the pre-vious MPM basis functions where there exists ex-actly one basis function for each grid node. Here,if grid nodes were used as knots, there are k− 2extra basis functions, or degrees of freedom in thesystem. However, the mechanics of the MPM al-gorithm remain exactly the same with the sub-script i representing values associated with de-grees of freedom, rather than values associatedwith grid points. Fig. 3 shows examples of thesemodified boundary B-spline basis functions.

3.3 Generalized Interpolation Material PointMethod

The Generalized Interpolation Material Point(GIMP) Method [Bardenhagen and Kober (2004)]is an extension to MPM which takes advantage ofthe fact that equations such as Eq. 1 take the form:

gi = ∑p

gpφip

= ∑p

gp

∫Ω φi(x)δ (x−xp)dΩ∫

Ω δ (x−xp)dΩ, (24)

1

0 0 1 2 3 4 5

(a) Quadratic B-Spline (k = 3)

1

0 0 1 2 3 4 5

(b) Cubic B-Spline (k = 4)

Figure 3: Example sets of modified boundary 1-DB-spline basis functions used in MPM.

with δ the Kronecker delta. GIMP then replacesδ with a general particle characteristic functionχp(x) centered at the particle position xp. Thisresults in new projection equations of the form:

gi = ∑p

gpφ ip, (25)

where φ ip is the weighting function given by

φ ip =1∫

Ω χp(x)dΩ

∫Ω

φi(x)χp(x)dΩ. (26)

Equations using ∇φ ip, such as Eq. 4 are similarlymodified to use a gradient weighting function:

∇φ ip =1∫

Ω χp(x)dΩ

∫Ω

∇φ i(x)χp(x)dΩ. (27)

GIMP is often implemented using standardpiecewise-linear grid basis functions (Eq: 17) andpiecewise-constant particle characteristic func-tions:

χp =

{1 : |x|< 1

2 lp

0 : otherwise,(28)

in which case the 1-D MPM and GIMP weightingfunctions can be grouped together in the follow-


ing general form:

φ =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1− (4x2 + l2p)/(4hlp) : |x|< lp

2

1−|x|/h : lp

2 ≤ |x| < h− lp

2(h+ lp

2 −|x|)2

/(2hlp) : h− lp

2 ≤|x|<h+ lp

2

0 : otherwise,

(29)

where lp is the width of the particle characteristicfunction χp. Again, the basis function associatedwith node i located at position xi is φi(x) = φ (x−xi) and the multi-D weighting function is con-structed as the tensor product of the 1-D weight-ing functions in each direction.

Since a particle moves and its voxel deforms intime, the question then becomes how to handlelp, the vector of widths of particle p’s voxel ina multi-D simulation. Ideally, we would like theparticles’ voxels to deform and tile space for alltime. In 1-D, this was accomplished by settinglp equal to the particle’s time-updated volumeVp. This scheme results in particle specific, time-dependent weighting functions φ ip and was re-ferred to as contiguous-particle GIMP (cpGIMP).For general multi-D simulations, however, the useof rectilinear χp will not allow a perfect tiling tooccur.

One choice for handling lp in a multi-D simula-tion, and what we will refer to as standard GIMP,is to leave the particle lengths unchanged for alltime, i.e. lp = l0p, where the superscript 0 indi-cates initial particle size. Standard GIMP weight-ing functions are then particle specific, but nottime-dependent. Another option is uniform GIMP(uGIMP), where a single smoothing length l isused for all particles, for all time. Note that inthe case where the initial discretization was per-formed using uniform particles, standard GIMPwould be the same as uGIMP. And lastly, whilespace cannot be tiled in a general multi-D simu-lation using rectilinear χp, updating lp in time togive a rough approximation to the particle’s de-formed voxel will still be referred to as cpGIMP.The cpGIMP approximation used in this paper islp = l0pdiag(F), such that the particle size varies

through time as dictated by the appropriate diago-nal term of the deformation gradient F. Note thatlp here refers to the full particle width, and not thehalf-width as used in the original GIMP formula-tion.

Fig. 4 shows an example 1-D GIMP weight-ing function φ ip and gradient weighting function

∇φ ip for a piecewise-constant χp with a character-istic length of l. Notice that that while φ ip lookssmooth in Fig. 4(a), the dashed lines show loca-tions of breaks in continuity which become ap-parent in ∇φ ip in Fig. 4(b). These breaks in con-tinuity will become important in later analysis.

−h − l/2 −h −h + l/2 −l/2 0 l/2 h − l/2 h h + l/20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xp

φ i (x p)

(a) GIMP Weighting Function

−h − l/2 −h −h + l/2 −l/2 0 l/2 h − l/2 h h + l/2

−1/h

0

1/h

xp

∇ φ

i(xp)

(b) GIMP Gradient Weighting Function

Figure 4: Example GIMP weighting function φ ip,

and gradient weighting function ∇φ ip centered at0 using piecewise linear grid basis functions andpiecewise constant particle characteristic func-tions χp. Dotted lines denote breaks in the con-tinuity of the functions.

3.4 Kinematic Boundary Conditions

One of the conveniences afforded by the use of aCartesian background grid is the ease of applica-


tion of kinematic boundary conditions. That is,Dirichlet or Neumann conditions, or a combina-tion, on the velocity field. Note that if no treat-ment is given to the boundary nodes, then parti-cles are able to freely advect from the computa-tional domain in what could be considered a zerogradient Neumann condition. This important partof the algorithm has received scant treatment inthe literature (although it is very relevant whenone is actually implementing MPM), so we turnattention to it here.

3.4.1 Traditional MPM

In traditional MPM, boundary conditions onlyneed be applied on those nodes which coincidewith the extents of the computational domain. Asillustrated in Fig. 2(a) nodes beyond those bound-aries are not influenced by particles within the do-main. This can be considered a result of the zerowidth of the Dirac delta characteristic functions.Boundary conditions must be applied to the ve-locity that has been projected to the nodes (Eq. 2),the time advanced velocity (Eq. 6), and the ac-celeration (Eq. 5). For Dirichlet conditions, thissimply means overwriting the calculated valuesfor the velocities with the prescribed values. Forthe acceleration, some debate exists regarding theproper means of treatment. The usual approachhas been to assume that a Dirichlet condition forvelocity implies that the acceleration should bezero on those boundary nodes. However, it is alsopossible that if Eq. 6 were solved for acceleration:

ai = (vk+1i −vk

i )/Δt, (30)

the proper value for the acceleration at the bound-ary nodes would be computed based on the dif-ference between the time advanced velocity (afterapplication of boundary conditions) and the pro-jected velocity (without the application of bound-ary conditions). Put another way, acceleration onthe boundary nodes can be considered to reflectthe force required to bring the velocity at thosenodes from the projected value to the prescribedvalue.

While these two approaches to the accelerationseem substantiallydifferent, the difference in sim-ulation results is very subtle. Indeed, when both

approaches are tested with the manufactured solu-tion described in Sec. 5.1, the superiority of eitheris not apparent. Currently, the UCF implementa-tion uses the boundary treatment given in Eq. 30.

In addition to prescribed velocity boundary con-ditions, “symmetry” boundary conditions are alsofrequently useful. Symmetry BCs are used to rep-resent a plane of symmetry, which allows the useof a reduced computational domain, or a friction-less surface. They are achieved by simply apply-ing a zero velocity Dirichlet condition on the com-ponent of velocity normal to a boundary, whileallowing the other components to remain at theircomputed values. Acceleration is handled in thesame manner, with the normal component eitherzeroed out, or computed as in Eq. 30.

3.4.2 GIMP and B-Spline MPM

When using GIMP or B-Spline MPM, there areadditional considerations in the applications ofthe boundary conditions. Namely, because oftheir increased extents, it is possible for particlesto influence, and be influenced by, nodes that lieoutside of the simulation domain, (see Figures2(b) and 2(c)). In the UCF, these are referred toas “extra” nodes, but may also be called “ghost”nodes by other investigators. Boundary condi-tion treatment of these nodes for Dirichlet con-ditions is the same as for the regular boundarynodes, namely, their computed values are replacedby prescribed values as described above.

In treating symmetry boundaries, the extra nodesrequire special care. In particular, the normalcomponent of velocity for these nodes is no longerset to zero, but rather should be set to the negativeof the value of the node opposite the boundary.The need to do so is apparent if one considers twoobjects approaching a collision plane symmetri-cally. The normal component of velocity on theopposite sides of that plane will have opposingsigns.

4 Analysis and Interpretation

MPM is a fairly new method and thus there hasbeen a recent push by the MPM community toprovide a more formal analysis of errors in the


method. Bardenhagen (2002) looked at energyconservation errors in MPM, focusing on the ef-fects of the choice of two time-stepping algo-rithms. Recently, Wallstedt and Guilkey (2008)expanded on the analysis of those time-steppingalgorithms. Love and Sulsky (2006a) and Loveand Sulsky (2006b) analyze an energy consis-tent implementation of MPM, the second of theseshowing an implicit implementation to be un-conditionally stable and energy-momentum con-sistent. Wallstedt and Guilkey (2007) focus onvelocity projection errors and present a schemewhich helps ameliorate these errors. Steffen,Kirby, and Berzins (2008) perform an analysison some of the spatial integration errors presentwithin MPM.

In this section, we continue adding a few morepieces to the error analysis of MPM. Specificallywe will look at integration errors which are af-fected by the smoothing of the piecewise-linearbasis functions.

4.1 The Relationship between GIMP and B-Splines

Taking a closer look at the weighting function(Eq. 26), we see that the construction is essen-tially a convolution of the grid basis functions φi

and the particle basis function χp. Since a stan-dard piecewise-linear φi can also be represented asthe convolution of piecewise-constant basis func-tions, we can rewrite the GIMP weighting func-tion as:

φ = χg ∗χg ∗χp/(|χg||χp|) (31)

where the width of χg is h (the grid spacing), andthe width of χp is lp, as described in the GIMPmethods. The equivalent GIMP basis functionwould then come from evaluating Eq. 31:

φ i(x) = φ (x−xi) (32)

with the GIMP weighting function equivalent toevaluating Eq. 32 at the particle position, xp. Thereason for rewriting the GIMP basis functions inthis way is to demonstrate the similarities be-tween the construction of GIMP basis functionsand the construction of B-spline basis functions

as in Eq. 19. Both basis functions are constructedby convolving piecewise-constant basis functionswith themselves; however all of the χ in the B-spline basis are of width h while one of the χfunctions used in the GIMP method has width lp.

In cpGIMP, the particle characteristic length lp

(of which there may be different lengths for dif-ferent directions) is updated in time, meaning thecpGIMP weighting function (Eq. 29) is time de-pendent, and is different for each particle p. Theideal case would be that the updating of lp in timewill cause the set of particle characteristic func-tions χp to perfectly tile space, but due to the rec-tilinear constraints of χp, this is not possible ingeneral multi-D simulations. Because of this in-ability to tile space, and the recognition that themajor benefit of GIMP is the smoother equiva-lent basis functions, a simplified standard GIMPis used in which lp = l0

p for all time. Further-more, if lp = l is constant for all particles p ina simulation (uGIMP), the effect is truly equiva-lent to using standard MPM with a smoother setof basis functions. In fact, if one were to disasso-ciate the smoothing length, l, from the particlesin a uGIMP formulation and instead leave l asa free parameter, the effect is to create quadraticB-spline-like basis functions, with l determiningthe maximum extent of the functions. Choos-ing l = l0

p would give standard GIMP. Choosingl = h would give quadratic B-spline basis func-tions. Choosing l = 0 would lead to the degen-erate case of χp = δ (x− xp), leaving us with thestandard piecewise-linear basis functions.

It has been our decision to leave the smoothinglength l as a free parameter the UCF, allowingfor various options when running simulations. Wewill explore various choices of l in the sections tofollow.

4.2 Smoothing Length Dependent IntegrationErrors

Spatial integrals within MPM are performed usingnodal integration – an approximation which takesthe form:

∫Ω

f (x)dΩ ≈ ∑p

f (xp)Vp. (33)


An analysis of errors in the above approximationwithin the MPM framework was performed bySteffen, Kirby, and Berzins (2008). There, thenodal integration approximations in MPM wereequated to non-uniformly-spaced midpoint inte-gration of functions with discontinuities in vari-ous derivatives. In particular, the analysis focusedon the errors when calculating the internal force(Eq. 4), which involves the following approxima-tion:

f inti = −

∫Ω

σ(x) ·∇φi(x)dΩ ≈ −∑p

σp ·∇φipVp.

(34)

The main result from that analysis showed that ifthe particle arrangements did not respect the dis-continuities which arise from the basis functions(i.e. a particle’s voxel overhangs node bound-aries), an extra integration, or “jump” error canarise in the above approximation which is of theorder C[[ f (p+1)]]Δxp+2, where the function, f , be-ing integrated is Cp continuous (with p = −1 fordiscontinuous functions). Here, [[·]] represents thejump in the p + 1 derivative of f at the discon-tinuity and Δx is the particle spacing. Note thatthe function ∇φi in Eq. 34 is discontinuous, thus ajump error of O(Δx) can arise in MPM when us-ing standard piecewise-linear basis functions, de-pending on particle spacing. Numerical examplesof this error were shown in Steffen, Kirby, andBerzins (2008).

The jump error for a single particle is calcu-lated as E jump =

∫Ωp

f (x)dx− f (xp)Δx, where Ωp

spans a discontinuity, or jump, and Δx is the widthof the particle. This consists of measuring themidpoint integration error for the single intervalspanning the jumps. Integration approximations,such as in Eq. 33, involve integration over thewhole domain, using multiple intervals, leading toa composite midpoint rule integration error whichis O(Δx2). These two errors are additive, giving atotal error of the form

Etotal =∫

Ωf (x)dΩ−∑

pf (xp)Vp = EMP +E jump,

(35)

where EMP is the composite midpoint error and

E jump is any errors arising from integrating acrossjumps. Note that if we assume particles are non-overlapping and fill space, this equation can alsobe written as

Etotal = ∑p[∫

Ωp

f (x)dΩ− f (xp)Vp]. (36)

Since the errors are additive and since EMP isalways O(Δx2), C0 and higher continuous func-tions exhibit an overall integration error whichis O(Δx2), while discontinuous functions havean error which is O(Δx). Again, this is impor-tant because the nodal integration for the internalforce calculation in Eq. 4 involves the gradientsof the basis functions, which are discontinuous atgrid cell boundaries when the standard piecewise-linear basis functions are used.

In uGIMP, we have the choice of a smoothing pa-rameter l (the width of our general particle charac-teristic function χ), independent of the individualparticle sizes, which ensures us C1 continuous ba-sis functions but leads to a situation which was notanalyzed in Steffen, Kirby, and Berzins (2008) –the case where the width of the particle is greaterthan the smoothing length. In such cases (as illus-trated in Fig. 5), a particle can span two jumps inthe continuity of the basis functions.

Consider a general case from Eq. 36 where a sin-gle particle, or Ωp, spans three regions of a piece-wise linear function. The first region (R1) is de-fined by the equation y1 = a1x + b1, the second(R2) will be y2 = a2x + b2, and the third (R3) isy3 = a3x + b3 with the particle located a distanceδ inside the second region (see Fig. 5). For this


(a) General Two-Jump Function

(b) GIMP Specific Two-Jump Situation

Figure 5: Example cases of a particle’s volume(the area between the square brackets) spanningtwo jumps in a piecewise-linear function: (a)shows a particle spanning two jumps in a gen-eral piecewise-linear function with ai and bi, i =1,2,3 the parameters describing the linear seg-ments, while (b) shows the specific piecewise-linear uGIMP gradient function ∇φ and a situa-tion where the particle size Δx is greater than thesmoothing length l.

case, the integration error is given by:

E jump =∫

Ωp

f (x)dx− f (xp)Vp (37)

=∫

Ωp∩R1

y1 dx+∫

Ωp∩R2

y2 dx+∫

Ωp∩R3

y3 dx−y2(xp)Δx (38)

=12(a3−a2)l2 +

12(a2−a3)lΔx+

(a2−a3)lδ +12(a3−a1)δ 2+

12(a1−2a2 +a3)δΔx+

18(a3−a1)Δx2

(39)

where l is the width of the center region and δ isthe particle offset into the center region. Since land δ are both less than Δx, this whole expres-sion appears to be O(Δx2). However, when weconsider the specific case of measuring the inte-gration error in internal force (Eq. 34 with σ = 1)when a particle spans the center region of ∇φ as

in uGIMP (see Fig. 4(b) and Fig. 5(b)), the slopesof the left and right regions in Fig. 5(b) are zero,while the slope of the center region is dependenton the smoothing length l and the grid spacing h.Specifically, for these regions, a1 = 0, a3 = 0 anda2 = −2/(hl). When we substitute these parame-ters into Eq. 39, we are left with

E jump =−1h

[Δx− l +2

(1− Δx

l

)δ]. (40)

Here, it is clear that the jump error in this case isO(Δx). If instead, Δx < l and the particle onlyspans one of the uGIMP jumps, the error thentakes the form:

E jump =−1hl

[δ 2 −δΔx+14

Δx2]. (41)

Since l no longer depends on Δx, this is nowO(Δx2).

To test this analysis, we calculate the force on asingle node for a set of particles with constantstress σ . This is the same test performed by Stef-fen, Kirby, and Berzins (2008) when looking ata particle spanning a single jump instead of thetwo-jumps analyzed above. In this case, the inter-nal force on a node is calculated as

f inti = −∑

p∇φip ·σpVp = −σ ∑

p∇φi(xp)Vp. (42)

For a constant stress, internal force should bezero, so any errors are from integrating ∇φi.Fig. 6 shows the errors for various particle spac-ings when the smoothing length l = 1/10. Asexpected, when Δx < l the error converges asO(Δx2). When Δx becomes greater than l, the er-ror tends towards O(Δx).

Here, we have shown errors in the internal forcewhich are either O(Δx) or O(Δx2), depending onthe relationship between the smoothing length land the particle widths Δx. For stability, in ad-dition to the typical CFL constraints one needswhen smooth forces exist, we need to considerfurther time step restrictions when force kicksarise from these integration errors. These timestep restrictions would be similar to those re-quired, as shown by Tran, Kim, and Berzins


10−2

10−1

100

10−3

10−2

10−1

100

101

102

1/PPC

Err

or

UGIMPC

1 x

C2 x2

Figure 6: Errors in internal force vs. particle spac-ing for constant σ = 1, grid spacing h = 1, andsmoothing length l = 1/10. The particles havea global uniform density, however they have alocally non-uniform spacing. Otherwise, super-convergent results are observed. Note that whenparticle spacing is less than the smoothing lengthl, the error converges as second-order. As the par-ticle spacing becomes greater than the smoothinglength, the error tends towards first-order.

(2007), due to force kicks arising from grid cross-ing errors. While a time step of Δt1 may be suf-ficient when we are in the O(Δx2) error region,a smaller Δt2 may be required to control stabilitywhen we are in the O(Δx) error region.

4.3 Impact of Boundary Treatments

In MPM, the union of the particles’ voxels are as-sumed to fill space and define the material of in-terest. However, many calculations are not per-formed directly on the particles, but rather on thebackground grid to which the particle informationis projected. This projection of particle informa-tion leads to a set of “active” basis functions andgrid cells (those which have particles in their sup-port) which, in general, will differ geometricallythan the union of particles’ voxels. This can, anddoes, lead to a further errors in many MPM simu-lations.

In standard MPM with piecewise-linear basisfunctions, the active grid cells are those whichcontain a particle. One could argue that a grid cellwhich contains no particles but still overlaps with

a particle voxel (from a particle in a neighboringcell) should also be active, but is not consideredso in the current MPM framework. In either case,when considering active cells on the grid, theremay be a geometric error of up to h in each direc-tion. When moving to uGIMP, or B-splines, thisgeometric error can become worse since the sup-port of these basis functions are larger. Cubic B-splines, quadratic B-splines, and uGIMP can ex-perience geometric errors of up to 2h, 3h/2 andh+ l/2, respectively. All of these errors are O(h);however it is important to note that these geomet-ric errors are not only a function of how well theobject of interest is aligned with grid cells, butthey are also a function of basis function choice.

Some work has been performed on MPM back-ground grids which more closely represent thematerial of interest. For example, Wallstedt(2008) has worked on an MLS representation ofa material boundary and incorporates this bound-ary into the MPM integration routines. Here, wesidestep part of the issue by developing test prob-lems in Section 5 whose boundaries are perfectlyaligned with the grid boundaries (such as a fixed-fixed elastic bar). Even with these aligned testproblems, geometric errors can still exist since in-formation is projected to extra nodes outside thedomain, as is shown in Fig. 2(b) and Fig. 2(c);information which is still used in standard kine-matic boundary treatments.

To illustrate this geometric error, Fig. 7 showsan example of the density field resulting fromprojecting particles with constant mass (a dis-cretization of a constant density field) to thegrid. The density field is calculated as ρ(x) =∑i ρiφi(x) with ρi = mi/(∑p φipVp). In this ex-ample, the constant density field spans the region[0,1] which is embedded in a grid covering theregion [−0.2,1.2]. Since the deformed configura-tion of the material with respect to the grid is ef-fectively the support of the fields of interest, wecan see from Fig. 7 how implementing bound-ary conditions and modeling contact can presenta challenge when wider basis functions are used.

We postulate that, in general, all of the methodshere can suffer from O(h) geometric errors. Inthe special case of boundary aligned problems,


−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

1

1.5

x

Den

sity

True DensityPiecewise−Linear BasisQuadratic B−spline BasisParticle Locations

Figure 7: Density fields resulting from projecting particle mass to the background grid. The true density fieldis shown, along with density fields calculated with piecewise-linear and quadratic B-spline basis functions.Here we can see that the geometric extent of information projected to the grid not only depends on whichgrid cells contain material, but also on basis function choice.

methods where information is projected to extranodes, such as uGIMP and standard B-splines,will still be affected by this O(h) geometric er-ror when Neumann or Dirichlet boundary con-ditions are applied. These methods should notbe affected by this error when periodic bound-ary conditions are used. Methods which do notrequire the use of extra nodes, such as standardMPM with piecewise-linear basis functions (Fig.2(a)), modified boundary B-splines (Fig. 3), andcpGIMP will not be affected by this geometric er-ror.

It is worth noting that while cpGIMP is imple-mented in the UCF using extra boundary nodes,information is not projected to these extra nodesin well-behaved boundary aligned simulations.This is because the particle p that is closest to theboundary has width lp, and is located at a positionof lp/2 to the inside of the boundary, and the clos-est extra node is at a distance of h + lp/2, whichis the exact location where the extra node’s basisfunction goes to zero (see Fig. 4).

5 Test Problem Development

Code verification has gained renewed importancein recent decades as costly projects rely moreheavily on computer simulations. Full time-dependent test problems with analytical solu-tions are desired so that simulation errors canbe assessed. The Method of Manufactured solu-tions [Schwer (2002); Knupp and Salari (2003);

Banerjee (2006)] begins with an assumed solu-tion to the model equations and analytically deter-mines the external force required to achieve thatsolution. This allows the user to verify the ac-curacy of numerical implementations, understandthe effects of parameter choices within the code,and to find where bugs may exist or improvementscan be made. The critical advantage afforded byMMS is the ability to test codes with boundariesor nonlinearities for which exact solutions willnever be known. It is argued [Knupp and Salari(2003)] that MMS is sufficient to verify a code,not merely necessary.

Since full transient mechanics solutions are of-ten difficult to find in the literature, we will firstpresent an overview of the method of manufac-tured solutions with which we will then developboth 1-D and 3-D test problems.

5.1 Method of Manufactured SolutionsOverview

For this paper we define several non-linear dy-namic manufactured solutions and use them forsubsequent testing. The solutions exercise themathematical and numerical capabilities of thecode and provide reliable test problems for ascer-taining a simulation’s accuracy and stability prop-erties.

Finite Element Method (FEM) texts often presentTotal Lagrange and Updated Lagrange forms ofthe equations of motion. The Total Lagrange form


is written in terms of the reference configurationof the material whereas the Updated Lagrangeform is written in terms of the current configu-ration. Either form can be used successfully in aFEM algorithm, and solutions from Updated andTotal Lagrange formulations are equivalent [Be-lytschko, Liu, and Moran (2000)].

However, within GIMP it is necessary to manu-facture solutions in the Total Lagrange formula-tion so that zero normal stress can be applied tofree surfaces as a boundary condition. This mightat first appear to conflict with the fact that GIMPis always implemented in the Updated Lagrangeform. The equivalence of the two forms and theability to map back and forth between them allowsa manufactured solution in the Total Lagrangeform to be validly compared to a numerical so-lution in the Updated Lagrange form.

The equations of motion are presented in Totaland Updated Lagrangian forms, respectively:

∇P+ρ0b = ρ0a (43)

∇σ +ρb = ρa (44)

whereP 1st Piola-Kirchoff Stress,σ Cauchy Stress,ρ density,b acceleration due to body forces, anda acceleration.

Many complicated constitutive models are usedsuccessfully with GIMP, but for our purposes thesimple neo-Hookean is sufficient to test the non-linear capabilities of the algorithm. The stress isrelated in Total and Updated Lagrangian forms,respectively:

P = λ lnJF−1 + μF−1 (FFT − I

)(45)

σ =λ lnJ

JI+

μJ

(FFT − I

)(46)

whereu displacement,X position in the reference

configuration,F = I+ ∂u

∂X deformation gradient,J = |F| Jacobian,

μ shear modulus, andλ Lamé constant.

The acceleration b due to body forces is used asthe MMS source term. The source term is “man-ufactured” in such a fashion that the equations ofmotion are satisfied for the particular input fields.We select as an ansatz the displacement field, suchas a sine function, and then apply a special bodyforce throughout the object that causes the dis-placement to occur.

5.2 One-Dimensional Periodic Bar

To understand the effect of smoothing length onerrors within MPM, we start by simulating a one-dimensional periodic bar on the domain [0,1].The problem we are considering has an assumedanalytical displacement and resultant deformationgradient of:

u(X , t) = Asin(2πX)cos(Cπt), (47)

F(X , t) = 1+2Aπ cos(2πX)cos(Cπt), (48)

where X is the material position in the referenceconfiguration, A is the maximum deformation per-centage, and C =

√E/ρ0 is the wave speed. The

bar is subjected to a body force of

b(X , t) = C2π2u(X , t)(2F(X , t)−2 +1). (49)

The functions u and F are included in Eq. 49 onlyto simplify notation. The constitutive model isdrawn from Eq. 46 in 1-D with zero Poisson’s ra-tio:

σ =E2

(F − 1

F

). (50)

This constitutive model, when combined with thebody force given by Eq. 49 will lead to the analyt-ical displacement solution in Eq. 47.

While this one-dimensional bar has a periodicsolution, the manufactured solution was chosensuch that the velocity and displacements are bothzero on the boundaries of our simulation domain[0,1]. This allows us to test our simulation withboth Dirichlet and periodic boundary treatmentsof the same problem.

5.3 Axis-Aligned Displacement in a Unit Cube

Displacement in a unit cube is prescribed withnormal components such that the corners and


edges of GIMP particles are coincident andcollinear. This choice allows direct demonstra-tion that GIMP can achieve the same spatial ac-curacy characteristics in multiple dimensions thathave been shown in a single dimension. It is not,however, representative of general material defor-mations usually found in most realistic engineer-ing scenarios.

The displacement field is chosen to be:

u =

⎛⎝ Asin(2πX) sin(Cπt)

Asin(2πY) sin( 23π +Cπt)

Asin(2πZ) sin( 43π +Cπt)

⎞⎠ (51)

where X , Y , and Z are the scalar components ofposition in the reference configuration, t is time,A is the maximum amplitude of displacement,and C =

√E/ρ0 is the wave speed, where E is

Young’s modulus. The factors of two are chosenso that a periodic boundary condition can be usedif desired.

The deformation gradient tensor is found by tak-ing derivatives with respect to position, but for theaxis-aligned problem only the diagonal terms arenon-zero. Therefore:

FXX = 1+2Aπ cos(2πX) sin(Cπt)FYY = 1+2Aπ cos(2πY) sin( 2

3π +Cπt)FZZ = 1+2Aπ cos(2πZ) sin( 4

3π +Cπt)(52)

Acceleration is found by twice differentiating dis-placement Eq. 51 in time. Then substituting stressP into Eq. 43 and solving for the body force b(used as the MMS source term) it is found that:

b = π2

⎛⎜⎜⎜⎝

uX

(4μρ0

−C2 −4 λ(K−1)−μρ0F 2

XX

)uY

(4μρ0

−C2 −4 λ(K−1)−μρ0F2

YY

)uZ

(4μρ0

−C2 −4 λ(K−1)−μρ0F2

ZZ

)⎞⎟⎟⎟⎠ (53)

where K = ln(FXX FYY FZZ) and the subscripts onu and F indicate individual terms of displacementand deformation gradient equations.

6 Results

6.1 One-D Smoothing Length Experiments

We simulated the one-dimensional periodic bardeveloped in Section 5.2 on the domain [0,1] to

understand the effect of smoothing length on er-rors within MPM.

The bar is initially discretized with an even sam-pling of points with initial positions X0

p . Theparticle positions are then adjusted to xp = X0

p +u(X0

p ,0), and deformation gradients set to Fp =F(X0

p ,0). The simulation is run to a final time Tand errors in the particle positions are calculatedas

Error = |xTp −X0

p −u(X0p ,T)|. (54)

This simulation was run with the parameters A =0.02, E = 104, and ρ0 = 1.0 to a final time T =2/C (one full period of oscillation) using uGIMPwith various numbers of particles-per-cell (PPC)and various smoothing lengths. Fig. 8 shows howerrors depend on smoothing length for differentnumbers of PPC when we run at a relatively largetime step corresponding to a CFL number of 0.8.We see that the simulations go unstable when thesmoothing length is close to, or less than the ini-tial width of the particles. The 2 PPC simulationgoes unstable for L < h/2, the 3 PPC simulationgoes unstable for L < h/3, etc.

Fig. 9 shows the effect of smoothing length on thetime-step stability restrictions. Fig. 9(a) showslarger smoothing lengths are stable for a widerrange of CFL values when the problem is dis-cretized with 4 PPC. While the simulation usinga smoothing length of h/8 goes unstable at a CFLnumber of approximately 0.75, the same simula-tion with a smoothing length of h (equivalent toquadratic B-spline basis functions) is stable up toa CFL number of approximately 1.2. Fig. 9(b)shows similar behaviors for 8 PPC. Furthermore,the time-step stability restrictions do not changesignificantly between the 4 PPC and 8 PPC sim-ulations, suggesting that the stability is more de-pendent on the smoothing length than the numberof particles per cell.

6.2 One-D Spatial Convergence Results

To investigate the spatial convergence propertiesof the various MPM methods, we start by simulat-ing the same one-dimensional periodic bar fromSection 5.2, now focusing on the behavior of the


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−6

10−5

10−4

10−3

10−2

10−1

100

Smoothing Length (fraction of h)

RM

S E

rror

8ppc4ppc3ppc2ppc

Figure 8: Stability analysis for uGIMP showingerrors vs. uGIMP smoothing length. The simula-tions were run with a fixed time step correspond-ing to a CFL number of 0.8.

error with respect to grid resolution and how thaterror may differ using different choices of basisfunctions and boundary treatments. The simula-tions were run with the parameters A = 0.05 (5%maximum displacement), E = 104, and ρ0 = 1.0to a final time T = 1/C (one-half period of oscilla-tion). All simulations were run with a time step ofΔt = 4 · 10−6, corresponding to a CFL number ofapproximately 0.2 for 512 grid cells–the highestresolution test case.

Fig. 10 shows results from simulations with boththe standard MPM piecewise linear basis func-tions and quadratic B-splines. With an initial dis-cretization of 4 PPC, we see the standard MPMpiecewise linear basis functions showing no sig-nificant convergence beyond a modest 16 gridcells. Quadratic B-splines show a significantimprovement, demonstrating O(h2) convergence,with an error plateau occurring past 128 grid cells.The 4 PPC quadratic B-spline simulation wasrun with both periodic boundary conditions us-ing standard splines (see Fig. 2(b)) and Dirichletboundary conditions using the modified bound-ary splines (Fig. 3(a)). The results from the twoboundary treatments are nearly identical. As wasshown previously, using smoother basis functionsgreatly improves numerical quadrature errors andstability issues, however nodal integration will al-ways give some quadrature error which can ex-

0.7 0.8 0.9 1 1.1 1.2 1.310

−6

10−5

10−4

10−3

10−2

CFL

RM

S E

rror

L = h/8L = h/4L = h/3L = h/2L = h/1

(a) 4 Particles Per Cell

0.7 0.8 0.9 1 1.1 1.2 1.310

−6

10−5

10−4

10−3

10−2

CFL

RM

S E

rror

L = h/8L = h/4L = h/3L = h/2L = h/1

(b) 8 Particles Per Cell

Figure 9: Examination of stability for uGIMPshowing errors versus CFL number for variouschoices of smoothing length.

plain the error plateaus starting at 128 grid cells.The last set of simulations in Fig. 10 shows thesame quadratic B-spline simulation with Dirichletboundary conditions, except this time the problemhas been discretized using 6 PPC. The extra par-ticles helps lower the quadrature error and lowerthe error plateau.

To further illustrate errors stemming from bound-ary treatments, Fig. 11 shows errors for the sameproblem simulated with B-splines, using threedistinct boundary treatments. Similar to Fig. 10,the Dirichlet boundary conditions with modifiedboundary B-splines and the periodic boundaryconditions with standard B-splines show nearlyidentical results and demonstrate O(h2) conver-gence. Also shown are results for standard


101

102

103

10−6

10−5

10−4

10−3

10−2

10−1

Grid Cells

RM

S E

rror

4PPC, Linear, Dirichlet BCs4PPC, Splines, Periodic BCs4PPC, Splines (MB), Dirchlet BCs6PPC, Quad. Splines (MB), Dirichlet BCs2nd Order

Figure 10: Spatial convergence on a one-dimensional bar manufactured solution problem.

B-splines with Dirichlet boundary conditions.This technique requires handling of the extra or“ghost” boundary nodes as explained in Section3.4.2. As was discussed previously, this can leadto a geometric error on the grid which is O(h)and the results show that the error convergence isin fact reduced to O(h).

101

102

103

10−6

10−5

10−4

10−3

10−2

10−1

Grid Cells

RM

S E

rror

Dirichlet BCs, Ghost NodesDirichlet BCs, Boundary SplinesPeriodic BCs1st Order2nd Order

Figure 11: Spatial convergence on a one-dimensional bar manufactured solution usingquadratic B-splines with various boundary treat-ments.

6.3 Verification with the Method of Manufac-tured Solutions in Multi-D

The full three-dimensional axis-aligned problemfrom Section 5.3 was implemented in the UCF to

both demonstrate the validity of the multi-D man-ufactured solution and show that many of the 1-Dconvergence results from the previous section arealso valid in 3-D. The simulation was run withthe parameters A = 0.05 (5% maximum displace-ment), ρ0 = 1.0, E = 104 and a Poisson’s ratio of0.3. The problem was discretized using 4 PPC ineach dimension (64 total particles per cell). BothB-spline basis functions (with symmetric and pe-riodic boundary conditions) and cpGIMP wereused in the simulations. The study consisted ofgrid resolutions from 8×8×8 cells (32768 parti-cles) up to 64×64×64 cells (16.8 million parti-cles).

The results in Fig. 12 clearly show O(h2) con-vergence with cpGIMP for all grid resolutions inthe study. Using B-spline basis functions withperiodic boundary conditions did nearly as well,with convergence rates trailing off at higher gridresolutions. Similar to the 1-D results, errorswhen using standard B-splines with the extra, or“ghost” boundary nodes (this time with symmet-ric boundary conditions) demonstrate the O(h)convergence we expect due to the geometric er-rors on the grid.

It is not surprising that cpGIMP outperformsother methods, as this problem is well suited forcpGIMP since particles remain axis-aligned andtheir voxels area a true partition of the domain.Fig. 13 is a visualization of a representative 2-D slice of the actual solution, showing the axis-aligned particle voxels and how they partition thedomain. B-splines when using extra boundarynodes performed as expected, demonstrating thesame O(h) error as the 1-D results. There is anobvious benefit to using periodic boundary condi-tions over symmetric boundary conditions for thisproblem since the errors are significantly lower.It is still unclear, however, why the convergencerate for the periodic boundary conditions trails offfrom the O(h2) behavior we would expect fromthe 1-D results. There are a number of possi-bilities, including a more complicated quadratureerror behavior in multi-D, or the buildup of gridcrossing errors (similar to those analyzed by Tran,Kim, and Berzins (2007)), which may be moresignificant in multi-D since many more grid cross-


ing events occur than in 1-D simulations usingsimilar resolutions.

101

102

10−4

10−3

10−2

10−1

Grid Cells (each dimension)

Max

Err

or

B−spline, Symetric BCsB−spline, Periodic BCscpGIMP1st Order2nd Order

Figure 12: Spatial convergence on the three-dimensional axis-aligned manufactured solutionproblem.

Figure 13: Visualization of a representative 2-Dslice of the exact solution (Eq. 51) at time t =.005, showing deformed particle positions (blackdots) and a conceptualization of the axis alignedparticle voxels. The voxels have been roundedand their sizes slightly reduced for visual clarity.

7 Summary

In this paper we have considered some of themany choices one must consider when imple-menting the Material Point Method. Two of thedesign choices which have significant impact onerror properties of the method are which grid basisfunctions to use and how to implement boundaryconditions. We explored and analyzed the numer-ical impact of these algorithmic choices.

A number of basis functions were explored,including: standard piecewise-linear basisfunctions, B-spline basis functions, uniformGIMP (uGIMP), and contiguous particle GIMP(cpGIMP). All these functions were shown tobe connected through a similar constructiontechnique–the convolution of piecewise-constantfunctions of various lengths. Analysis of theuGIMP functions showed an integration, orquadrature error which was second order withrespect to particle spacing when the basis func-tion smoothing length is larger than the particlewidths. When the smoothing length is smallerthan the particle widths, this integration errorbecomes first order. The effects of this rela-tionship between particle widths and smoothinglengths were demonstrated in simulations whereinstabilities occurred when the smoothing lengthwas set smaller than the particle widths.

Boundary condition implementation also had aneffect on the overall errors in the method. The ge-ometric errors present in the grid representationof the deformed material can result in first orderspatial errors when standard kinematic boundaryconditions are applied. These geometric errorsare exacerbated when smoother, and necessarilywider, basis functions are used, such as uGIMP,or B-splines. We were able to eliminate thesefirst order errors when using periodic boundarytreatments. Relaxing the requirement that eachgrid node correspond to a single basis function ledus to a set of modified boundary B-spline basisfunctions which eliminated the geometric errorsfor our problem and allowed second order spa-tial convergence with standard Dirichlet boundaryconditions.


Acknowledgement: This work was supportedby the U.S. Department of Energy through theCenter for the Simulation of Accidental Firesand Explosions (C-SAFE), under grant W-7405-ENG-48. In addition, the authors would like to ac-knowledge helpful discussions with members ofthe Utah MPM Group.

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