Solving Quasi-Static Equations with the Material-Point Method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2010; 00:1–20Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme

Solving Quasi-Static Equations with the Material-Point Method

J. Sanchez1, H. Schreyer1, D. Sulsky2 ∗, P. Wallstedt3

1Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 871312Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131

3Caterpillar Inc., Peoria, IL 61629

SUMMARY

The material-point method (MPM) models continua by following a set of unconnected material pointsthrough out the deformation of a body. This set of points provides a Lagrangian description of the materialand geometry. Information from the material points is projected onto a background grid where equations ofmotion are solved. The grid solution is then used to update the material points. This paper describes how touse this method to solve quasi-static problems. The resulting discrete equations are a coupled set of nonlinearequations that are then solved with a Jacobian-free, Newton-Krylov algorithm. The technique is illustratedby examining two problems. The first problem simulates a compact tension test and includes a model ofmaterial failure. The second problem computes effective, macroscopic properties of a polycrystalline thinfilm. Copyright c© 2010 John Wiley & Sons, Ltd.

Received . . .

KEY WORDS: material-point method (MPM), quasistatic, compact tension, effective moduli,polycrystal

1. INTRODUCTION

The Lagrangian finite element method is often the method of choice for solving problems in solidmechanics. The Lagrangian formulation is particularly suited to applications with complicatedconstitutive behavior where the stress response is history dependent and must be integratedalong material-point trajectories. However, reliance of the method on a Lagrangian mesh is stillproblematic when modeling large deformations since considerable loss of accuracy is possiblewhen elements in the mesh become distorted. In fluid mechanics, fluid flow almost always makesa Lagrangian approach impractical and the method of choice is often an Eulerian finite differencemethod. In an Eulerian frame, one must discretize nonlinear advection terms in the equations ofmotion which introduces complications. Moreover, Eulerian methods diffuse edges and contactdiscontinuities are difficult to treat accurately. The material-point method (MPM) was developedto solve problems in continuum mechanics by combining the best properties of Lagrangian andEulerian methods.

In MPM, two descriptions of the continuum are employed. A Lagrangian description is givenby a set of material points identified in the original configuration and tracked throughout thedeformation history. The geometry of the body is represented by this unconnected set of points,and its distortion is not tied to a mesh. Complicated geometry is relatively easy to model sincethe domain is discretized by filling regions with material points. This construction is much simpler

∗Correspondence to: Department of Mathematics and Statistics, MSC01 1115, 1 University of New Mexico,Albuquerque, NM 87131

Copyright c© 2010 John Wiley & Sons, Ltd.Prepared using nmeauth.cls [Version: 2010/05/13 v3.00]

2 SANCHEZ, ET AL.

than constructing a body-fitted mesh. Constitutive equations are solved at each material point, somaterial models with history dependence are implemented just as easily as in a Lagrangian finiteelement method. A background computational grid is used to calculate the interactions among theLagrangian points through the solution of the momentum equation. During a computational cycle,information is projected from the material points to a background grid; equations of motion aresolved on the grid in an updated Lagrangian frame; the grid solution is then used to move andupdate material-point properties; finally, the old grid is discarded and a new grid is constructed forthe next time step.

The first implementations of MPM were serial and solved dynamic equations using explicittime integration [20, 23, 22]. In [21] and [5] the dynamic equations were made implicit intime. A nice feature of MPM is that the stability properties of the grid solution govern overallstability of the method [9] and we can exploit standard mesh-based solvers when solving theequations on the background grid. Guilkey and Weiss [5] used the trapezoidal rule in time with ahyperelastic constitutive model for two-dimensional problems. The nonlinear equations were solvedusing Newton’s method with the Jacobian matrix explicitly formed for the problem. Kaul [21]examined solutions of implicit dynamic equations using Jacobian-free Newton-Krylov methods forhypoelastic models in two space dimensions. The Jacobian-free approach is particularly attractivefor complicated constitutive models since the tangent stiffness is not required. Love and Sulsky[8, 9] extended these ideas to hyperelastic models and elastic-plastic models. These two-dimensionalapplications were also implemented in parallel using calls to PETSc libraries [1]. More recently, [3]used MPM for quasi-static, large strain problems in two and three space dimensions. They used15-noded prismatic and 10-noded tetrahedral elements with quadratic interpolation. Their analysesincluded modifying the integration in MPM and explicitly forming and decomposing the elasticstiffness matrix, with an iteration to impose all material nonlinearity.

This work extends the Jacobian-free approach to a formulation for quasi-static problems.In Section 2 we describe the MPM algorithm. Although [3] shows that higher order elementformulations are possible, the applications in this work are implemented using standard 4-nodedquadrilateral elements in two dimensions and 8-noded brick elements in three dimensions, withoutmodifying the quadrature. Section 2.1 introduces our notation and summarizes the continuumequations to be solved. Section 2.3 presents the MPM computational cycle for quasi-static problems.The details surrounding implementation of the Jacobian-free Newton-Krylov solver are reservedfor the Appendix. Although we have performed several verification tests of the code, we highlighttwo new applications of the method in Sections 3 and 4. First, in Section 3, we examine failureof a compact tension specimen. The specimen is concrete and is modeled by a decohesive failuremodel for quasi-brittle materials. The constitutive model is sufficiently complex that this examplehighlights the efficacy of the Jacobian-free approach compared with finding the tangent stiffness.Moreover, MPM has some advantages in modeling material failure because the material pointsare not logically connected and are free to separate as the material fails without having to deleteelements, insert special elements, or otherwise break element connectivity. The second example,discussed in Section 4, computes effective material properties of a polycrystalline material. Thegeometry of the crystals within the specimen are reminiscent of what is typically obtained usingelectrodeposition during the fabrication of MEMS devices. This example highlights the ability ofMPM to model materials in contact and the trivial meshing of the polycrystalline sample enabledby the representation of crystals as collections of material points.

2. THE QUASI-STATIC MPM ALGORITHM

The governing equations of a quasi-static deformation of a solid medium are presented. Thedeformation process is assumed to be isothermal, so the continuum model consists of forceequilibrium, boundary conditions and a constitutive model, as discussed in Section 2.1. The weakform of the equilibrium equation is also presented. The discrete quasi-static MPM equations given inSection 2.2 are derived from their continuous counterpart. Section 2.3 then discusses the numericalimplementation of the discrete quasi-static MPM equations.

Copyright c© 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2010)Prepared using nmeauth.cls DOI: 10.1002/nme

SOLVING QUASI-STATIC EQUATIONS WITH THE MATERIAL-POINT METHOD 3

2.1. Continuum Equations

Consider two configurations of a deformable body. The initial or reference configuration of thebody is a set of points Ω0 ⊂ R3. The position vector of a material point in the initial configuration isdenoted by X ∈ Ω0. The body deforms relative to its initial state and this deformed state, representedby the set of points Ω ⊂ R3, is referred to as the current or spatial configuration of the body. Theposition vector of a material point in the current configuration is denoted by x ∈ Ω. A one-to-onemapping is assumed to exist between the material point positions in the current and reference statessuch that x = x(X) = X−1(x) and X = X(x) = x−1(X). The displacement of a material point asa function of the current state is defined as u(x) = x−X(x). The local form of the equilibriumequation is obtained by setting the sum of internal and external forces in Ω equal to zero. Theresulting governing equation for equilibrium is

∇ · σ + b = 0 (2.1)

where σ(x) is the symmetric Cauchy stress tensor and b(x) is the body force per unit volume. Theboundary is divided into two disjoint sets of points such that ∂Ω = ∂Ωu + ∂Ωt. The displacement(or essential) boundary conditions are applied on ∂Ωu and the traction (or natural) boundaryconditions are applied to ∂Ωt as follows

u(x) = g(x) ∀x ∈ ∂Ωu τ (x) = t(x) ∀x ∈ ∂Ωt, (2.2)

where g(x) is the given prescribed boundary displacement on ∂Ωu and t(x) is the prescribed tractionon ∂Ωt. The traction is denoted by τ = σ · n, where n is the unit outward normal to the boundaryin the current configuration.

The measure of deformation between the initial and deformed configurations is the strain. Thesmall strain approximation, denoted by ε, is used for some applications and is computed as thesymmetric part of the displacement gradient

ε =12[∇u + (∇u)T

]. (2.3)

The constitutive relationship, denoted symbolically as σ = σ(ε), is necessary to form a completeset of equations for obtaining the displacement field solution, u(x). The weak (variational) form ofequilibrium in equation (2.1) is obtained using the usual arguments. The final result is

−∫

Ω

∇w : σ dv +∫

∂Ωt

w · τ da+∫

Ω

w · b dv = 0 (2.4)

where the test function, w, is an admissible variation of the solution.

2.2. The Discrete Quasi-Static MPM Equations

MPM involves two discretizations. Space is discretized by a set of grid cells or elements defined bya set of Nn grid node positions xiNn

i=1. This grid covers Ω, but does not necessarily conform to itsboundary. The solid body is discretized by a set of Np material points xp

Np

i=1 ∈ Ω. The subscriptsi and p are used to denote quantities associated with grid nodes and material points, respectively.The initial position of the material point is Xp = X(xp). Each point Xp in the initial configurationrepresents a volume of material Ω0

p such that Ω0 = ∪Np

p=1Ω0p.

The discrete set of MPM equilibrium equations is formed from the weak form of equilibrium inequation (2.1). A simple one-point quadrature rule is used over each volume Ωp associated with thepth material point

−Np∑p=1

Ωpσ(xp) : ∇w(xp) +∫

∂Ωt

w · τ (x) dA+Np∑p=1

Ωpw(xp) · b(xp) = 0. (2.5)

The volume Ωp in the current configuration is related to the initial volume according to Ωp =J(Xp)Ω0

p, where J(X) = det F(X) is the determinant of the deformation gradient, F(X) =


4 SANCHEZ, ET AL.

∂x/∂X. We use the notation Fp to denote the numerical approximation of the deformation gradientevaluated at the material point position, Fp ≈ F(Xp). Similarly, Jp = det Fp is the determinant ofthe deformation gradient evaluated at the material point position.

The variation in equation (2.5) is approximated with standard finite element nodal basis functions,Ni(x), i = 1, 2, . . . , Nn which satisfy the partition of unity property such that

∑Nn

i=1Ni(x) = 1. Inthis work, we build the nodal basis functions from tensor products of one-dimensional piecewiselinear hat functions on a quadrilateral mesh. The finite element approximation of the displacementfield variation is

w(x) =Nn∑i=1

wiNi(x). (2.6)

The test function approximation in equation (2.6) is substituted into equation (2.5). Aftersimplification, the final result is

−Nn∑i=1

wi ·Np∑p=1

Ωpσp · ∇Ni(xp) +Nn∑i=1

wi ·∫

∂Ωt

τ (x)Ni(x) da+Nn∑i=1

wi ·Np∑p=1

Ωpb(xp)Ni(xp) = 0.

(2.7)The components of wi are arbitrary except for those points where components of displacementare prescribed. With the understanding that the constraints on the displacement field given by theboundary conditions are enforced, the discrete MPM equations reduce to the following

−Np∑p=1

Ωpσp · ∇Ni(xp) +∫

∂Ωt

τ (x)Ni(x) da+Np∑p=1

Ωpb(xp)Ni(xp) = 0. (2.8)

The nodal value of the internal forces is defined to be

f inti = −

Np∑p=1

Ωpσp · ∇Ni(xp). (2.9)

The external forces at the nodes are defined to be

f exti = bi + τ i, (2.10)

where

bi =Np∑p=1

ΩpbpNi(xp) (2.11)

and

τ i =∫

∂Ωt

τ (x)Ni(x) da. (2.12)

Equations (2.8)-(2.12) are combined to form a simplified expression for the discrete MPMequilibrium equations. The result is

Ri(u) = f inti + f ext

i = 0, i = 1, . . . , Nn. (2.13)

where the functional dependence on the nodal displacement, u, is emphasized through the use ofthe notation for the residual, Ri(u) in equation (2.13). Note that in the vector u, only the degreesof freedom associated with unconstrained values of the displacement are considered unknown. Theobjective of the implicit method for quasi-static problems is to find values of u that make the residualzero, i.e. solve equation (2.13).



2.3. Numerical Implementation of Quasi-static MPM

The numerical solution of the quasi-static problem is typically obtained in steps by incrementallyapplying displacement boundary conditions, external forces or both in order to obtain incrementsin displacement, ∆u. The discrete quasi-static equations are solved for each load step denoted by k(k = 1, 2, . . . ,K). Quantities associated with a discrete load step are denoted with the superscript k.Each discrete load step corresponds to a discrete configuration, Ωk, of the solid body. Note that thegrid nodes are assumed to move with the deforming body during a load step so xk+1

i = xki + ∆ui.

Since the grid points move with the deformation, the nodal basis functions are invariant over the step,Ni(xk+1

p ) = Ni(xkp). For displacement control, a boundary displacement increment, ∆gk, is applied

on ∂Ωku for each step such that

∑Kk=1 ∆gk = g. In general, load control is achieved by incrementing

the external force on the nodes by ∆f ext,k for each load step such that∑K

k=1 ∆f ext,k = f ext. Theloading is accomplished by incrementing the corresponding body force bi for xi ∈ Ωk and surfacetractions τ i for xi ∈ ∂Ωk

t appropriately on the corresponding grid nodes.Consider an intermediate equilibrium state of loading at step k for which the material point

quantities, xkp , Fk

p , Jkp , Ωk

p , εkp and σk

p are known. The known external force at the grid nodes isf ext,ki =

∑kj=1 ∆f ext,j

i for predetermined load increments, ∆f ext,ji (j = 1, 2, . . . k). The equilibrium

state at the (k + 1)st load step is obtained once the nodal displacement increment solution, ∆ui, isfound which satisfies the following discrete MPM equation for the incremental quasi-static problem

Ri(∆u) = f int,k+1i + f ext,k+1

i = 0, i = 1, 2, . . . , Nn (2.14)

where the external force at the nodes for the (k + 1)st loading step is computed to be

f ext,k+1i = f ext,k

i + ∆f ext,k+1i (2.15)

and the following displacement boundary conditions are satisfied

∆ui = ∆gk+1(xi) ∀xi ∈ ∂Ωku. (2.16)

To be more specific about the solution to equation (2.14), the goal is to find the incrementaldisplacement, ∆u(x), so that the residual, Ri(∆u) is zero at the nodes. To see the connectionbetween the incremental displacement and the residual, evaluate the residual in the following steps.The finite element approximation of the incremental displacement is written using nodal basisfunctions as follows

∆u(x) =Nn∑i=1

∆uiNi(x). (2.17)

The material point strain increment is computed from equations (2.3) and (2.17) to be

∆εp =12

Nn∑i=1

[∇Ni(xk

p)⊗∆ui +(∇Ni(xk

p)⊗∆ui

)T ]. (2.18)

The updated strain isεk+1

p = εkp + ∆εp. (2.19)

Next, the material point stress increment is obtained by evaluation of the constitutive model.Symbolically, the stress increment computation is

∆σp = ∆σp(∆εp), (2.20)

and the updated stress isσk+1

p = σkp + ∆σp. (2.21)

Other constitutive models might be functions of the deformation gradient which, if needed, isupdated using a multiplicative incremental update

Fk+1p = Fk

p

(I +∇∆u(xk+1

p ))

= Fkp

(I +

Nn∑i=1

∇Ni(xkp)⊗∆ui

), (2.22)


6 SANCHEZ, ET AL.

where I is the second order identity tensor. If the full deformation gradient is not needed, theJacobian can be updated directly

Jk+1p = (1 + trace(∆εp)) Jk

p . (2.23)

The new volume isΩk+1

p = Jk+1p Ωk

p. (2.24)

The nodal internal forces at the (k + 1)st loading step are computed from equation (2.9) in terms ofthe corresponding material point stress σk+1

p ,

f int,k+1i = −

Np∑p=1

Ωk+1p σk+1

p · ∇Ni(xkp). (2.25)

These internal forces then should balance the external forces so that equation (2.14) holds forunconstrained nodes and the given boundary values are used at constrained nodes. Note thatarbitrarily general constitutive models can be used. In the general case, equations (2.20-2.21) shouldbe interpreted as invoking a constitutive routine for each material point to obtain the stress from thestrain increment according to a general constitutive model.

In general, the function Ri = Ri(∆u) in equation (2.14) is a non-linear function of ∆u. Thesolution, ∆u, to R(∆u) = 0 is obtained using the implicit solution method described in AppendixA in which equations (2.14) - (2.25) are solved iteratively with a Jacobian-free, Newton-Krylovmethod to obtain the new equilibrium state at the (k + 1)st load step. Once the displacementincrement is known, the material point position, xp, and displacement, up, are updated as follows

xk+1p = xk

p +Nn∑i=1

∆uiNi(xkp) (2.26)

uk+1p = uk

p +Nn∑i=1

∆uiNi(xkp). (2.27)

Information is retained on the material points only. After each load step the grid nodes can beredefined as needed. Typically (and in this work), the spatial mesh of grid cells is returned to itsundeformed state at the end of the load step.

3. FAILURE OF A COMPACT-TENSION SPECIMEN

In this section, the quasi-static implementation of MPM is applied to study failure of a concretecompact tension specimen (CTS). In the following, a brief description is given of the experimentalsetup and results. In the numerical simulations, failure is predicted through the use of an elastic-decohesive constitutive model, also described below. The model is sufficiently complicated tomake it inconvenient to compute the tangent stiffness; thus, the Jacobian-free approach to theNewton-Krylov solver is particularly advantageous. Moreover, the representation of material failurethrough the constitutive equation allows a procedure to detect initiation of failure, to determine theorientation of crack planes and to evolve cracks, all integrated through one model. As will be seen,the material points are free to separate as the sample is deformed and a crack propagates throughthe specimen.

3.1. Description of Experiments

The experiments were performed by Wittman, et al. [25] in order to study the effect of samplesize on the measured mode I fracture energy of concrete, denoted as Gf . Work of fracture testsare designed to measure Gf for geological materials such as concrete [14]. Figure 1 displays anillustration of a CTS work of fracture experiment. The compact tension sample is characterized by



the ligament length, b. The appendages of the specimen are separated by a notch. A material testingmachine pulls the appendages apart in the horizontal direction using displacement control of theload point position, δ. The applied horizontal force P , which is continuously measured, increasesup to a peak value. As the sample fails, roughly along the dominant crack path (Figure 1a), thevalue of P decreases gradually to zero. The area under the measured P vs. δ response (Figure 1b)is the total energy to fail the sample. The value of the failure energy divided by the ligament area(ligament length b times specimen depth d) is the measured value of Gf for the material.

(a) CTS sample (b) experimental data

Figure 1. Illustration of CTS experiment

3.2. The Quasi-Brittle Failure Constitutive Model

A discrete constitutive model for quasi-brittle material failure is applied to the CTS problem, forwhich the failure process initiates on a discrete surface, defined by the unit normal vector n andreferred to as the failure surface. The determination of n depends on the local stress σ at a pointin the deformable body. The traction on a potential failure surface defined by n is τ = σ · n.The failure process is governed by a softening relationship, τ = τ ( [[u]] ), which is applied onthe failure surface, where [[u]] is the jump in the displacement field across the failure surface.Only mode I type failures are considered for this problem for which the normal componentsof traction τn = n · σ · n = τ · n and displacement jump [[un]] = [[u]] · n are related through asoftening function, τn = τn([[un]]).

The chosen material failure model, developed by Schreyer [18] for quasi-brittle materials isassociated with a failure function, Fn = Fn(τ , [[u]]), that governs the initiation and evolutionof a discrete failure surface (i.e. crack). Since the model is intended to predict failure for manystress states including mixed-modes in triaxial compression, the form of Fn(τ , [[u]]) is complex.However, application to the CTS problem only requires consideration of mode I failure. In this caseit can be shown [17] that Fn(τ , [[u]]) reduces to the following form:

Fn =τnτnf

+[[un]]u0− 1 (3.1)

In Equation (3.1) τnf is the ultimate tensile strength and u0 is the critical crack opening definedas the value of [[un]] for which τn = 0. Initiation of failure occurs on a surface defined by n thatmaximizes Fn such that Fn(τ · n , 0) = 0. It is easily shown that this failure initiation criterioncorresponds to the Rankine criterion for which n coincides with the direction of maximum principalstress. Subsequent softening, for which [[u]] > 0 and Fn(τ , [[u]]) = 0, is governed by the linearsoftening relationship, τn = τnf (1− [[u]]n/u0). SinceGf is taken to be the area under the softeningcurve, the critical crack opening is computed to be u0 = 2Gf/τnf .

The failure state is completely defined by σ and [[u]]. The failure model is formulated as adirect analogy to rate independent inelasticity models [19] for which isotropic linear elastic material


8 SANCHEZ, ET AL.

behavior is assumed up to a yield point. The failure function, Fn (σ, [[u]]), plays the role of a yieldfunction and the displacement discontinuity, [[u]], is treated as an internal variable that evolves withcontinued loading. The material failure model is implemented into MPM using a multiple smearedcrack representation of failure [2, 15]. Material failure is represented through a continuum inelasticstrain contribution, denoted as εdc, and referred to as de-cohesion strain.

The smeared crack representation of the quasi-brittle failure model is summarized in rate form asfollows:

ε = εe + εdc (additive strain composition)

σ = C : (ε− εdc) (isotropic linear elasticity)

τ = σ · n (traction rate)

[[u]] = ω∂Fn

∂τ, (ω > 0) (evolution of displacement jump)

εdc =1

2Lc([[u]]⊗ n + n⊗ [[u]]) (de-cohesion strain)

Fn (σ, [[u]]) = 0 (consistency condition)

(3.2)

In Equation (3.2) ε and εe are the total strain rate and the elastic strain rate respectively, C is thefourth order isotropic elasticity tensor and Lc is the smearing length. The quantity, Lc, is computedas the ratio of the material-point volume to the crack area, across the material point. Details of thefailure model implementation into MPM can be found in [17]. Previous applications of this failuremodeling approach in MPM include dynamic spalling [24] and dynamic motion of sea ice [13].

3.3. Numerical Results for the CTS Simulation

The concrete CTS failure tests performed by Wittman, et al., [25] on specimens of size b = 15cm, b = 30 cm and b = 60 cm are simulated using quasi-static MPM with a smeared materialfailure representation. Newton’s method is combined with GMRES to find equilibrium after eachloading increment. The depth of the sample is d = 12 cm. Nominal un-reinforced concrete materialparameter values used for Young’s modulus, Poisson’s ratio and ultimate tensile strength areE = 34.9 GPa, ν = 0.18 and τnf = 2.77 MPa, respectively. The measured size-dependent valuesof Gf , taken from [25], are Gf = 122.7 J/m2, Gf = 160.6 J/m2 and Gf = 163.6 J/m2 for b = 15cm, b = 30 cm and b = 60 cm, respectively.

Figure 2 displays a sample two-dimensional discretization of the CTS problem in MPM. Auniform grid of four node squares with one material point per cell is used. Loading is appliedwith displacement boundary conditions located at the top and center of each appendage. The leftappendage is constrained in the x and y directions while a constant displacement increment isapplied to the right appendage in the positive x direction. Material failure is only permitted onsurfaces aligned with grid cell edges in the y direction in order to preclude the development ofnon-physical stress build up commonly found in smeared failure representations of failure in finiteelement analysis [15] and MPM [17]. The notch width is taken to be one element side length, h. Theanalysis is performed subject to the following restrictions: (i) the measured size-dependent fractureenergy is used for each geometry, (ii) cracking is allowed only in the elements containing the centerline of the specimen, and (iii) only one crack with normal in the x-direction is permitted within anelement.

The size of the displacement increment is is 1× 10−3 mm, resulting in a total number of steps of1000, 2000 and 3000 for b = 15 cm, b = 30 cm and b = 60 cm, respectively. The error tolerance forboth the linear and nonlinear iterations is 1× 10−3. The number of nonlinear iterations per loadingstep ranged from 1 to 6 and the maximum number of nonlinear iterations was 450. Comparison ofsimulated and experimental results for load versus deflection are displayed in Figure 3. Numerical



Figure 2. MPM discretization of CTS

results for b = 15 cm, b = 30 cm and b = 60 cm correspond to grid resolutions of h = 1 cm, h = 2cm and h = 4 cm, respectively. In general, the computed global responses of the samples agree wellwith experimental observations. The initial linear responses and peak loads match well. The globalsoftening of the samples are close to that of the test data near the peak, but deviate with continuedloading, predicting a steeper softening response. This difference is most likely due to the chosenlinear form of the softening function. It has been demonstrated that the use of different softeningfunctions produces very different global softening responses for the same problem [15]. However,the simple linear softening function provides an adequate approximation of the failure of the CTSsamples, and demonstrates the numerical method.

The progression of the simulated CTS failure is visualized in plots of[[un]]/u0, σxx/τnf andσyy/τnf at three times during the loading scenario in Figs. 4a, 4b, and 4c, for the specimenwith b = 15 cm. The top figures represent the initiation of failure, the second set denotes partialfailure, and the final set complete failure. Normalized crack opening values of [[un]]/u0 = 0,0 < [[un]]/u0 < 1 and [[un]]/u0 = 1 correspond to virgin, partially failed and completely failedmaterial, respectively. Initially, a tensile stress concentration forms at the notch of the undamagedspecimen (Fig. 4a). Mode I failure initiates when σxx/τnf = 1 . As failure progresses, the stressconcentration remains at the tip of the traveling crack opening along the CTS ligament as indicatedby the plots of σxx/τnf for partial failure (Fig. 4b) and complete failure (Fig. 4c).

As indicated previously, these results are based on the use of size-dependent values for fractureenergy, a parameter normally considered to be a material constant. One possible explanation for thesize-dependence is the observation that the normal component of stress, σyy, as shown in Fig. 4b,evolves to such an extent that the state of stress at the crack tip is one of equal biaxial tension. Oncethis biaxial state has developed, the crack may extend in any direction, at least theoretically andfor a short distance. The physical manifestation may be that for larger specimens a crack begins tomeander and becomes jagged. A second consideration not in the present analysis is the possibilityof the formation of microcracks in a zone beyond the single center-line of elements, as indicated bythe lateral extent of the maximum values of both σxx and σyy in Figs. 4b and 4c. The combinationof jaggedness of the primary crack and the existence of microcracks may account for the apparentsize-effect of the fracture energy. A preliminary analysis along these lines but with cracks restrictedto being parallel to the element sides, i.e., crack normals only in the x-and y-directions, has beenperformed by Sanchez [17]. A single value for fracture energy is able to provide the trend shown byexperimental data. However, a complete analysis requires the inclusion of multiple cracks at anglesto the element sides, a feature not currently included.


10 SANCHEZ, ET AL.

Figure 3. CTS failure simulation using quasi-static MPM superimposed onto experimental data obtained byWittman, et al., [25]



[[un]]/u0 σxx/τnf σyy/τnf

(a) Initiation of failure

(b) Partial failure

(c) Complete failure

Figure 4. The first column is the normalized crack opening, [[un]]/u0; the second and third columns arenormalized stress, σxx/τnf and σyy/τnf for the CTS failure simulation.


12 SANCHEZ, ET AL.

4. EFFECTIVE PROPERTIES OF POLYCRYSTALS

Polycrystalline Ni films electrodeposited in the fabrication of RF MEMS switches studied inthe PRISM Center at Purdue University have been found to have a preferred orientation of thecrystallites that causes their in-plane Young’s modulus to differ from the bulk isotropic value. Themodulus plays a role in the lifetime and reliability of the switches. For example, larger stiffnessof the bridge in these devices increases the pull-in voltage necessary to activate the switch. Higherpull-in voltages accelerate charging of the electrode’s dielectric coating, a key failure mechanism inthe device [4]. Hence, an accurate estimate of the true modulus is critical to the analysis and designof these MEMS switches. Moreover, the PRISM Center has a focus on quantifying uncertaintyin the predictions of performance of the devices. Accordingly, we are interested in estimatingthe probability distribution of material properties for use in downstream simulations aimed atpredictions of performance, lifetime or reliability.

MPM provides a means to obtain this distribution of properties. Given properties of the individualNi grains comprising the polycrystalline structure, we construct an aggregate of crystals or grainsrepresenting a sample cut from the Ni film bridge making up the RF MEMS switch. Grainorientations are assigned to the polydisperse aggregate of grains based on texture measuredexperimentally using X-ray diffraction on real devices [4]. The aggregate is then tested numericallyto calculate the elastic compliance tensor, from which the in-plane modulus can be estimated. Inthis paper we describe the implementation details of the numerical method and discuss the resultsfor one realization of material properties of the crystals, grain geometry and orientation distribution.Ultimately, many realizations are required to fill out a complete picture of the probability distributionfunction, which will be reported elsewhere.

4.1. Description of Experimental Data

There are several methods for experimentally measuring texture to determine the OrientationDistribution Function (ODF). The details of efficient and accurate methods used at the PRISMCenter can be found in [4]. Briefly, crystal orientation is measured experimentally using 2D X-raydiffraction (XRD). From the XRD data, texture analyses are then carried out using the Rietveldrefinement software MAUD [10] to calculate inverse pole figures. These pole figures are thenimported into MTEX [6, 11] for further analysis.

The ODF is denoted by f(g). This function describes the volume fraction in the sample ofcrystallites with a given crystallographic orientation, g, with respect to the sample coordinate axes.As is common, we describe orientations using Euler angles with Bunge’s convention, so that anorientation is defined by three angles, g = (φ1,Φ, φ2). If the sample has orthorhombic symmetryand the crystals are of cubic symmetry, as in our case, the Euler space reduces to

0 ≤ φ1,Φ, φ2 ≤ 90.

The ODF, f(g), satisfies the property

∆VV

=

∫∆ω

f(g)dg∫ω0f(g)dg

, (4.1)

where ω0 is the total volume of Euler space, ∆ω is a region of Euler’s space, V is the total samplevolume and ∆V the portion of the sample that contains crystals with orientations in ∆ω. The left andright terms of (4.1) are the physical volume fraction and volume fraction due to crystal orientations,respectively. The element of volume in Euler space is dg = sin Φ dφ1 dΦ dφ2. In this work, we usea discrete ODF described by values given on a grid defined in Euler space with angles taken in stepsof 5.

4.2. Grain Geometry

We imagine cutting a small cube of approximately 1000 grains from the MEMS device underconsideration. Electrodeposition tends to produce columnar grains with the grain axis perpendicular



to the plane of the film. Thus, we represent the microstructure geometry of this sample with perfectlycolumnar grains based on a Voronoi diagram construction. Columnar grains are obtained bydistributing generating points for the Voronoi diagram in a single plane, constructing a tessellationof the plane to represent grain cross-sections, and then extending the grain geometry to the thirddimension to make columns. Figure 5 shows the 1000-grained microstructure used in the simulation.Each grain is represented by a collection of material points. Colors are used to indicate separategrains in the microstructure.

Figure 5. Columnar grain geometry based on a Voronoi construction. Grains are colored to distinguish them.

4.3. Grain Orientations

Assume a model microstructure is given, for example one generated using the Voronoi constructionof the last section. The microstructure has grains of unequal volume. We follow the algorithmsuggested by Melchoir and Delannay [12] to assign orientations to the microstructure. First, thesample is divided into Nu elementary volumes each of equal volume Vu. Each grain contains manyelementary volumes.

Next, the Euler space is divided into N cubic cells Ci, i = 1, 2, . . . , N , with a side length of 5.The volume fraction of each cell in the orientation space is given by

fi =

∫Cif(g)dg∫

ω0f(g)dg

, i = 1, 2, . . . , N.

A cumulative distribution function of volume fractions is defined in the following way

F (M) =M∑i=1

fi, 1 ≤M ≤ N.

By construction, we have 0 < F (M) ≤ 1. Each integer M is identified with a value F (M) and anorientation gM at the center of CM . This establishes a correspondence between the numbers F (i)and an orientation gi, i = 1, . . . , N .

The texture discretization algorithm or the numerical method that assigns orientations to crystalsof a polycrystalline microstructure is summarized in the following two steps.

1. Sample Nu uniformly distributed random numbers between zero and one. Each randomnumber, x, will satisfy F (i) ≤ x < F (i+ 1) for some i, 1 ≤ i ≤ N ; and thus, can be identifiedwith the corresponding orientation gi. After identifying the number x with an orientation gi

for i = 1, . . . , Nu, a list of Nu orientations is obtained.2. Select the largest grain of the microstructure. Represent this grain by the indexG, and callNG

the number of elementary volumes that are contained in the grain. From the list of orientations


14 SANCHEZ, ET AL.

obtained in 1, select NG of them so that the disorientation between them is less than a giventhreshold value. Following [12] we choose 7 for the threshold value. Compute the meanorientation of the NG orientations. The resulting mean orientation is the orientation assignedto grain G. Delete this grain and the NG orientations used for grain G from the list. Repeatthis step until all grains are assigned an orientation.

It is possible for this algorithm to fail if we cannot find NG orientations all within the thresholddisorientation. Melchoir and Delannay [12] suggest a variation of this algorithm that prevents thisfailure. However, we did not encounter this case in our assignment of texture to our numericalsample.

The accuracy of the above algorithm is measured with the misfit

Misfit =

∫ω0

(fexp(g)− freconstructed(g)

)2dg∫

ω0

(fexp(g)

)2dg

,

where fexp stands for the experimental ODF and freconstructed is the ODF reconstructed from theorientations generated by the above algorithm. We calculate freconstructed with the MTEX software.The input parameters for MTEX are the crystal symmetry, sample symmetry and the volumefractions of both, physical and orientation space. As a rule of thumb, misfits less than 15% areconsidered acceptable numerical representations of experimental crystallographic texture. Using50,000 elementary volumes, the misfit for the orientations assigned to the geometry in Fig. 5 is10.44%.

4.4. Numerical Calculation of Effective Material Properties

Given the description of the geometry and the grain orientations, we are now ready to determine theeffective material properties of the sample using numerical simulations in MPM. Each grain in thegeometry is filled with material points and the grain’s orientation with respect to the sample axes isstored using Euler angles. Hooke’s law is evaluated at each material point, taking into account thecrystal orientations. The strain increment for each material point comes from the grid solution, thecomponents are stored in the sample coordinate system. The strain increment is rotated from thesample coordinates to the material axes associated with the grain. The constitutive model is appliedin the material frame to obtain the stress increment. The stress components in the material frame arethen rotated to obtain the stress in the sample coordinates.

Voigt introduced a convenient notation where the components of the symmetric second orderstress, σ, and strain tensor, ε, are put into vectors of length six, and the components of the fourthorder compliance and stiffness tensors are put in a six-by-six matrix. In Voigt’s notation, Hooke’slaw for a cubic crystal is

σ11

σ22

σ33

σ23

σ13

σ12

=

C11 C12 C12 0 0 0C12 C11 C12 0 0 0C12 C12 C11 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C44

ε11

ε22

ε33

2ε23

2ε13

2ε12

. (4.2)

The components are given with respect to material axes fixed in the crystal. Each grain is nickelwith single-crystal elastic moduli given by C11 = 249 GPa, C12 = 155 GPa, and C44 = 114 GPa.



Our goal is to compute the components of C, the effective stiffness tensor for the sample, whereσ = C : ε is Hooke’s law relating the effective stress, σ, to the effective strain, ε. In Voigt’s notation

σ11

σ22

σ33

σ23

σ13

σ12

=

C11 C12 C13 C14 C15 C16

C12 C22 C23 C24 C25 C26

C13 C23 C33 C34 C35 C36

C14 C24 C34 C44 C45 C46

C15 C25 C35 C45 C55 C56

C16 C26 C36 C46 C56 C66

ε11

ε22

ε33

2ε23

2ε13

2ε12

. (4.3)

We do not assume a priori that the sample has any symmetry and therefore C potentially has 21independent moduli. We will compute all entries of C with tests designed to isolate each columnof C. The tests are shown schematically in Fig. 6. A one micron cube of material is confined onfour sides. The confined sides have boundary conditions that allow the material to slide tangentially,but have zero normal displacement. The other two opposing sides are given prescribed equal andopposite displacements of magnitude 0.002 m in order to deform the cube under uniaxial strain if theprescribed displacement is normal to the face, or under simple shear if the prescribed displacementis tangential to the face. The displacement is applied over two load steps, for a total effectivestrain of 0.4%. The quasi-static solver is used with the momentum formulation and the Newton-GMRES solution algorithm. The error tolerance on the Newton iteration is 10−6 and the GMRESerror tolerance is set to 10−8.

x

yz

Figure 6. Six tests applied to the polycrystal, each test determines a row of C. The top row illustrates thethree uniaxial strain tests and the bottom row illustrates the three simple shear tests.


16 SANCHEZ, ET AL.

For each of the six tests, the total force is calculated over each face of the cube. The force is thenaveraged over each of the three pairs of opposing faces. The resulting three net forces per unit areaprovide the effective stress components for each test. Since each test has only one nonzero straincomponent, the stress divided by the size of the strain component (0.004), gives the column of C inEq. 4.3 corresponding to the nonzero strain component. The result of these computations is

C =

286.3 130.2 142.5 0.3 0.8 8.9130.1 286.3 142.5 1.3 0.3 −0.7142.5 142.5 273.9 0.8 4.5 0.07−0.04 0.09 −0.04 122.9 0.12 −1.1−0.1 −0.03 0.14 −0.2 120.1 0.3

0.2 −0.2 −0.03 −0.30 0.6 113.5

∼

286 130 143 0 0 0130 286 143 0 0 0143 143 274 0 0 0

0 0 0 123 0 00 0 0 0 120 00 0 0 0 0 114

(4.4)

where the entries are in GPa. The resulting in-plane Young’s modulus is Ein−plane = 197.4 GPa.Texture plays an important role in the prediction of the in-plane Young’s modulus. Table I

summarizes values of the effective in-plane Young’s modulus approximated using a Hill average,assuming perfect fiber texture, the experimentally determined texture and uniform (random) texture[4]. Our computed value, Ein−plane = 197.4 GPa, is close to the Hill average based on using theexperimentally measured texture; and, as expected falls within the bounds provided by the Reussand Voigt averages. There is no theoretical basis, but it is often found that the Hill average is closeto experimentally measured values. Our computed value, using experimentally measured texture, is14.5% larger than would be obtained assuming perfect fiber texture, and 5.9% smaller than the valueof Young’s modulus for bulk Ni. The different assumptions about texture in the sample significantlyimpacts the prediction of in-plane Youngs modulus, emphasizing the importance of knowing theODF for the sample in order to predict its effective material properties.

Table I. The in-plane Young’s modulus of a simulated perfect 〈001〉 fiber texture sample, the modulus forbatch #1 of RF MEMS switches calculated from experimentally measured texture and the modulus for a

uniform texture sample [4].

Perfect 〈001〉 RF MEMS UniformTexture Switches Texture

(batch #1)(GPa) (GPa) (GPa)

ERin−plane 163.3 178.8± 1.1 192.4

EHillin−plane 172.4 194.7± 1.3 209.7

EVin−plane 182.7 210.3± 1.5 225.7



5. CONCLUSIONS

This paper has examined the use of MPM to solve quasi-static continuum mechanics problems.The Jacobian-free, Newton-Krylov approach works well for complicated constitutive models, suchas the elastic-decohesive model of Section 3, since the tangent stiffness is not required. Moreover,the material points are not logically connected and are free to separate as the material fails. Thus,MPM models failure without having to delete elements, insert special elements, or otherwise breakelement connectivity. As applied to failure of concrete in a compact tension test, load-deflectioncurves for three differently sized specimens agree well with experimental measurements.

In Section 4, the quasi-static formulation is used to determine effective material properties of athin polycrystalline Ni film. MPM is advantageous in modeling the polycrystalline geometry sincecomplicated meshing is not required - each crystal is simply represented by a collection of materialpoints. The crystal orientation is a property assigned to the material points, and the constitutiveequations are updated for each material point. Using an assignment of crystal orientations consistentwith experimental measurements, we are able to predict the macroscopic, effective elastic propertiesby performing six simple, virtual mechanical tests. The resulting effective properties predict thatthese thin films, composed of cubic crystals, are macroscopically orthotropic materials. For the in-plane Young’s modulus we compute a value close to the Hill average for one realization of geometryand grain orientations. We expect other realizations will produce a similar value, and together yielda probability distribution for the modulus that is sharply peaked about the Hill average.

ACKNOWLEDGEMENTS

This work is partially supported by NNSA Center for the Prediction of Reliability, Integrity, andSurvivability of Microsystems, and Department of Energy under Award Number DE-FC52-08NA28617and by the National Science Foundation under grants ARC-0621173 and ARC-1023667.

REFERENCES

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2. Z.P. Bazant and B.H. Oh. Crack band theory for fracture of concrete. RILEM Materials and Structures, 16:155–177, 1983.

3. L. Beuth, Z. Wieckowski, and P.A. Vermeer. Solution of quasi-static large-strain problems by the material pointmethod. Int. J. Numer. Anal. Meth. Geomech., 35:1451–1465, 2011.

4. Patrick R. Cantwell, Hojim Kim, Mattew M. Schneider, Hao-Han Hsu, Dmitrios Peroulis, Eric. A. Stach, andAlejandro Strachan. Estimating the in-plane young’s modulus of polycrystalline films in mems. J. MEMS, pages1–10, 2012.

5. J. E. Guilkey and J. A. Weiss. Implicit time integration for the material point method: Quantitative and algorithmiccomparisons with the finite element method. International Journal for Numerical Methods in Engineering,57:1323–1338, 2003.

6. R. Hielscher and H. Schaeben. Journal of Applied Crystallography, 41:1024, 2008.7. C.T. Kelley. Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA, 1995.8. E. Love and Deborah L. Sulsky. An energy-consistent material-point method for dynamic finite deformation

plasticity. International Journal for Numerical Methods in Engineering, 65:1608–1638, 2005.9. E. Love and Deborah L. Sulsky. An unconditionally stable, energymomentum consistent implementation of the

material-point method. Computer Methods in Applied Mechanics and Engineering, 195:3903–3925, 2006.10. L. Lutterotti, M. Bortolotti, G. Ischia, and I. Lonardelli. Zeitschriftfur Kristallographie Supplement, 26:125, 2007.11. D. Mainprice, R. Hielscher, and H. Schaeben. Calculating anisotropic physical properties from texture data using

the mtex open source package. Deformation Mechanism, Rheology & Tectonics: Microstructures, Mechanics &Anisotropy - The Martin Casey Volume, Geological Society of London, Special Publications, 2011.

12. Maxime A. Melchior and Laurent Delannay. A texture discretization technique adapted to polycrystallineaggregates with non-uniform grain size. Computational Materials Science, 37:557–564, October 2006.

13. K. Peterson. Modeling arctic sea ice using the material-point method and a elastic-decohesive rheology. PhDthesis, University of New Mexico, Albuquerque, New Mexico, U.S.A, 2008.

14. P.E. Petersson. Fracture energy of concrete; method of determination. Cement and Concrete Research, 10:78–89,1980.

15. J.G. Rots. Smeared Computational modeling of concrete fracture. PhD thesis, Delft University of Technology,Delft, The Netherlands, 1988.

16. Y. Saad and M.H. Schultz. Gmres: A generalized minimal residual algorithm for solving nonsymmetric linearsystems. SIAM J. Sci. Stat. Comput., 7:856–869, 1986.


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17. J.J. Sanchez. A Critical Evaluation of Computational Fracture Using a Smeared-crack Approach in MPM. PhDthesis, University of New Mexico, 2010.

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19. J.C. Simo and T.J.R Hughes. Computational Inelasticity. Springer-Verlag New York, Inc., 1998.20. D. Sulsky, Z. Chen, and H.L. Schreyer. A particle method for history dependent materials. Computer Methods in

Applied Mechanics and Engineering, 118:179–196, 1994.21. D. Sulsky and A. Kaul. Implicit dynamics in the material-point method. Computer Methods in Applied Mechanics

and Engineering, 193:1137–1170, 2004.22. D. Sulsky and H.L. Schreyer. Axisymmetric form of the material point method with applications to upsetting and

taylor impact problems. Computer Methods in Applied Mechanics and Engineering, 139:409–429, 1996.23. D. Sulsky, S. Zhou, and H.L. Schreyer. Application of a particle-in-cell method to solid mechanics. Computer

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Mechanics, 35:107–115, 1990.

A. IMPLICIT SOLUTION OF NON-LINEAR, QUASI-STATIC, MPM EQUATIONS

The solution of the non-linear system of discrete MPM equations describing equilibrium follows from thesolution methods used for implicit dynamics in MPM [21]. Equation (2.14) represents the set of discreteincremental MPM equations of equilibrium for the (k + 1)st load step. The non-linearity is due to non-linear constitutive models that relate stress to strain and the changing geometry. Strain is updated from thedisplacement increment solution (equation (2.3)) and the internal forces at the nodes are computed fromthe stresses (equation (2.9)). As a result the quantity Ri is generally a non-linear function of the nodaldisplacement increment ∆u. To make the notation less cumbersome, we consider the solution for one loadstep, and from this point forward, the superscript, k, denoting the load step is dropped from the equations.We also consider Equation (2.14) restated as a vector equation

R(∆u) = 0, (A.1)

where the vector length is the number of nodal degrees of freedom. The problem is now to find nodal valuesof the unknown quantity ∆u that satisfy equation (A.1). Newton’s method is employed to find a solution.In this approach, if ∆u0 is an approximate solution to equation (A.1), an improvement, ∆u1, is found bysolving the linearized system of equations through ∆u0, namely

J(∆u0)s = −R(∆u0), (A.2)

where J is the Jacobian matrix of R and s = ∆u1 −∆u0. (The Jacobian matrix, denoted here with a boldfont, should not be confused with the scalar, J , which is the determinant of the deformation gradient.) Thesolution, s, to (A.2) is the increment in the solution. Thus, ∆u1 = ∆u0 + s is hopefully a better solution,and the process can be repeated by linearizing about ∆u1. This iterative procedure is Newton’s method forsolving (A.1). The complete algorithm is summarized next.

A.1. Newton Algorithm

The subscript n ≥ 0 refers to the iteration counter within the Newton iteration that solves equation (A.1)for the displacement increment ∆u, associated with the current load step. Thus, we generate a sequence∆u0,∆u1, . . . that we hope converges to the desired solution ∆u to within the desired accuracy. Newton’smethod consists of the following:

1. Initialize the Newton iteration. Choose an error tolerance, γ. Set the counter n = 0 and make an initialguess for the solution ∆u0. Since we are solving for an increment in displacement, a reasonable initialguess is zero, except at the nodes controlled by a prescribed displacement, which should be assignedthe known displacement increment.

2. Evaluate R(∆un). If ‖R(∆un)‖ ≤ γmax(1, ‖R(∆u0‖) then the desired solution has been obtainedand the iteration is terminated. Otherwise continue. (Note that this stopping criterion combines anabsolute error criterion when the initial residual is small, with a relative error criterion otherwise.)

3. Compute J(∆un).4. Solve the linear system J(∆un)s = −R(∆un) for s. The components of s corresponding to nodes

constrained by the boundary conditions are set to zero.5. Update the independent variable ∆un+1 = ∆un + s, increment the Newton iteration counter n←

n+ 1, and return to step 2. Note that each iterate ∆un satisfies the prescribed, known displacementincrement on the boundary.



Step 3 can be difficult since explicit computation of the matrix J(∆un) requires computation of thematerial tangent tensor. The tangent tensor is not readily available for complicated constitutive models.However, if the linear system in Step 4 is solved iteratively with a Krylov space method (such as theconjugate gradient method or GMRES), then such methods only require computation of the matrix-vectorproduct, J(∆un)s, rather than the matrix J(∆un) itself. Moreover, the left-hand side of the linear equationin Step 4 can be interpreted as the directional derivative of the function R at ∆un in the direction s. Thisdirectional derivative can be approximated using a finite difference. Thus, for a Newton-Krylov method,Step 3 can be omitted and an approximate directional derivative can be used in Step 4. This approach iscalled a Jacobian-free Newton-Krylov method since the matrix J is never explicitly formed.

The directional derivative is approximated using a finite difference of values of R as follows

J(∆un)s ≈ DhR(∆un, s) =R(∆un + hs)−R(∆un)

h, (A.3)

where h is a difference approximation parameter that determines the step length in the direction of s. Kelley[7] suggests choosing

h =

8><>:0 s = 0√εprec‖∆un‖/‖s‖ s 6= 0,∆un 6= 0√εprec‖s‖ s 6= 0,∆un = 0

(A.4)

where εprec is a parameter that measures errors in the evaluation of R. If the evaluation errors are mainly dueto roundoff errors, then √εprec ≈ 10−7 for 64-bit floating-point numbers. In practice, εprec is often adjusted,sometimes by trial and error, for a specific application.

Note that since R(∆un) is already computed in Step 2, computing Dh requires one additional evaluationof the function R, at ∆un + hs. We use GMRES to solve for s in Step 4. GMRES is an iterative method thatgenerates a sequence of vectors which are used to form the solution. The GMRES algorithm is summarizedin the next section. To use GMRES, we need to evaluate R for an arbitrary vector of nodal values. Let θrepresent such an arbitrary vector. The evaluation of R(θ) is summarized in the following steps. Quantitiescomputed for the current load step during the solver iterations are temporary quantities that are overwrittenat every iteration and are denoted below by a tilde. Quantities without a tilde are known values from theprevious load step and quantities with the superscript L denote values at the current load step.

1. Apply the kinematic boundary conditions to the argument θ on applicable grid nodes as inequation (2.16),

θi = ∆gL(xi) ∀xi ∈ ∂Ωu. (A.5)

2. Compute the material-point strain increment as in equation (2.18)

∆εp =1

2

NnXi=1

h∇Ni(xp)⊗∆θi + (∇Ni(xp)⊗∆θi)

Ti. (A.6)

3. Update the strainεp = εp + ∆εp. (A.7)

4. Evaluate the constitutive model to compute the material-point stress increment, ∆σp, and then updatethe stress σp = σp + ∆σp.

5. Evaluate the material-point volume as in equations (2.23)-(2.24)

Jp = (1 + trace(∆εp)) Jp, (A.8)

Ωp = JpΩp. (A.9)

6. Compute the nodal internal force vector as in equation (2.9)

f inti = −

NpXp=1

Ωpσp · ∇Ni(xp). (A.10)

7. Compute the nodal external force vector as in equation (2.10).8. Evaluate R as in equation (2.13).


20 SANCHEZ, ET AL.

A.2. The GMRES Algorithm in MPM

The iterative GMRES algorithm for obtaining a solution to a linear system of equations is presented for thespecific case of the linearized equations encountered in Step 4 of Newton’s method applied to the MPMequations. A complete description of the GMRES algorithm can be found in [16].

1. Choose an appropriate tolerance, ε, and the maximum number of GMRES iterations kmax. Set k = 1.Set b = −R(∆un) except for components associated with the prescribed boundary displacementwhich are set to zero. Initialize and store the first orthonormal basis vector, q1 = b/‖b‖ and setβ = ‖b‖e1. Compute the first residual error to be β1 = ‖b‖.

2. Check the residual error. If βk < ε or k ≥ kmax then terminate the procedure and proceed to Step 3.Otherwise continue.

(a) Perform k steps of Arnoldi iteration as follows:i. Compute qk+1 = DhR(∆un,qk). Set qk+1 to zero at nodes constrained by displacement

boundary conditions.ii. For j = 1, 2, . . . , k, compute Hjk = qT

j qk+1 and overwrite qk+1 ← qk+1 −Hjkqj

iii. Compute Hk+1k = ‖qk+1‖. Overwrite and store qk+1 ← qk+1/Hk+1k.(b) Compute and store the components of the Givens rotation matrices

cj =Hjjq

H2jj +H2

j+1j

, sj =Hj+1jq

H2jj +H2

j+1j

(c) For j = 1, 2, . . . , k, overwrite the components of the kth column of Hessenberg matrix Hk.HjkHj+1k

ff←»cj sj−sj cj

–HjkHj+1k

ff(d) Overwrite the kth and (k + 1)st components of the vector β as follows:

βkβk+1

ff←»cj sj−sj cj

–βkβk+1

ff(e) Increase the GMRES iteration count k ← k + 1(f) Return to Step 2 to check the error tolerance and the number of iterations

3. Solve the upper triangular system Hky = β using back substitution.(a) Compute yk = βk/Hkk(b) For j = 1, 2, . . . , k + 1, perform the following computations:

i. Compute yk−j = βk−jii. For i = (k − j + 1), . . . , k overwrite yk−j ← yk−j −Hk−jiyi

iii. Overwrite yk−j ← yk−j/Hk−jk−j

4. Compute the solution s =Pk

j=1 yjqj .


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Solving Quasi-Static Equations with the Material-Point Method