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Computer Methods in Applied Mechanics and Engineering manuscript No.

(will be inserted by the editor)

ILS-MPM: an implicit level-set-based material point method for frictional1

particulate contact mechanics of deformable particles2

Chuanqi Liu ·WaiChing Sun3

4

Received: May 21, 2020/ Accepted: date5

Abstract

Finite element simulations of frictional multi-body contact problems via conformal meshes can be

6

challenging and computationally demanding. To render geometrical features, unstructured meshes must

7

be used and this unavoidably increases the degrees of freedom and therefore makes the construction of

8

slave/master pairs cumbersome. In this work, we introduce an implicit material point method designed

9

to bypass meshing of bodies by employing level set functions to represent boundaries at structured grids.

10

This implicit function representation provides an elegant means to link an unbiased intermediate reference

11

surface with the true boundaries by closest point projection as shown in [Leichner et al.,2019]. We then

12

enforce the contact constraints by a penalty method where the Coulomb friction law is implemented as an

13

elastoplastic constitutive model such that a return mapping algorithm can be used to provide constitutive

14

updates for both the stick and slip states. To evolve the geometry of the contacts properly, the Hamilton-

15

Jacobi equation is solved incrementally such that the level set and material points are both updated according

16

to the deformation ﬁeld. To improve the accuracy and regularity of the numerical integration of the material

17

point method, a moving least square method is used to project numerical values of the material points back

18

to the standard locations for Gaussian-Legendre quadrature. Several benchmarks are used to verify the

19

proposed model. Comparisons with discrete element simulations are made to analyze the importance of

20

stress ﬁelds on predicting the macroscopic responses of granular assemblies.21

Keywords Frictional contact; penalty method; level set; improved material point method22

1 Introduction23

Modeling frictional contacts in assemblies of deformable bodies has been an important subject of interest

24

for centuries [Coulomb,1776,Oden and Pires,1983,Laursen,1993,Mitchell et al.,2005,De Lorenzis et al.,

25

2011]. Despite the signiﬁcant progress on numerical simulations of granular materials over the last several

26

decades, capturing the macroscopic path-dependent responses of the assemblies evolving multiple contacts

27

remains a challenging task, especially for contacts involving multiple deformable bodies of non-convex and

28

non-smooth geometries (e.g. sand, coal, slit, snow, powder). Nevertheless, understanding the grain-scale

29

particulate frictional interactions is critical to analyze and predict macroscopic behaviors of particulate

30

materials [Sun et al.,2013,Hurley et al.,2014,Wang and Sun,2019a,b].31

A popular way to handle the computational challenges of the multi-particle contact problems is to

32

simplify the nature of the contact by allowing overlapping of particles and then approximate the quasi-static

33

responses with an explicit time integration scheme with an artiﬁcial damping [Cundall,1988,Wang and Sun,

34

2016,Liu et al.,2016a]. These treatments are implemented in the discrete (distinct) element method (DEM)

35

[Cundall,1988] and many of its extensions that incorporate the effect of shape either explicitly through the

36

Corresponding author: WaiChing Sun

Assistant Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University , 614 SW Mudd, Mail

Code: 4709, New York, NY 10027 Tel.: 212-854-3143, Fax: 212-854-6267, E-mail: wsun@columbia.edu

2 Chuanqi Liu, WaiChing Sun

more sophisticated representation of shapes (e.g. level set and potential) or implicitly through more complex

37

phenomenological laws with different degrees of success. While these DEM models may gain efﬁciency

38

by bypassing the predictions of particle deformation and reduce the total number of degrees of freedom

39

(DOFs), the trade-off of this convenience is the less realistic prediction on the evolution of contact force and

40

moment due to the issues below:41

1. The local stress and deformation state of individual particles remain unknown since force is generated42

from overlapping particles or through rigid contacts [Rougier et al.,2004].43

2.

The steady-state of the particulate system must be obtained via relaxation or ad hoc damping [Liu et al.,

44

2016b,Rojek and O˜

nate,2007,Modenese et al.,2012].45

3.

Non-spherical, especially the non-convex particles, with complex geometries are costly represented via

46

implicit functions [Houlsby,2009,Cho et al.,2006,Andrade et al.,2012].47

4.

The evolution of contact geometry due to particle deformation is either neglected or greatly simpliﬁed

48

[Janko et al.,2018]. Even for the cases where the stick-slip does not involve any history variables at the49

grain scale, history-dependence at the macroscopic scales can be manifested due to the re-arrangement

50

of the particles. Both the local geometry and the deformation of the particles at the contact may affect

51

how stresses distribute inside the particles and on the contacts. Neglecting both effects may lead to

52

different resultant force and moment for each contact pair and that in return may lead to different grain

53

connectivity and make the resultant force chain less realistic.54

5.

Any behavior of contacts between two particles can only manifest and be stored in one particle-contact-

55

pair, i.e., an edge of the connectivity graph (cf. Satake [1993], Kuhn et al. [2015]), but not the particle

56

surfaces. As contacts may form and eliminate at different sub-domains of particles during an event,

57

storing one set of internal variables for each particle pair may lead to unrealistic predictions due to the

58

lack of the appropriate data structure to store history of contacts properly.59

To capture the interactions of particles of non-spherical shapes, an implicit function representation

60

approach (cf. Houlsby [2009], Andrade et al. [2012]) is often used. First proposed in Houlsby [2009], this

61

numerical algorithm uses level set, or signed distance function to identify the location of the particle

62

boundaries. Contacts and the overlapping distance of the particles are then computed via a constrained

63

optimization algorithm. While these improved models are useful to locate the contact location, the only

64

kinematic information stored remains the relative displacement of the overlapped particles in each contact

65

pair. As such, the only kinetic data it can generate is the force and moment exerted at the contact pair.

66

This approach is therefore limited to the applications to the few cases where the geometrical nature of the

67

contacts, the evolution of the stress, and the resultant nature of the history-dependence (e.g. frictional wear

68

vs. bulk plasticity of particles) are negligible.69

An alternative but computationally more demanding approach is to explicitly model the deformation of

70

particles and capture the evolution of contact in a ﬁnite element (FE) framework. In fact, FE algorithms for

71

frictional contact have been extensively studied over the past few decades. By designating master/slave (or

72

mortar/non-mortar) pairs, contact constraints are imposed via node-to-segment [Wriggers and Zavarise,

73

2004,Zavarise and De Lorenzis,2009], or segment-to-segment (latest mortar method) [Puso and Laursen,

74

2004,Tur et al.,2009,Zimmerman and Ateshian,2018]. These biased approaches are confronted with

75

difﬁculties especially in the case of multi-body contact where it is impractical to priori nominate a master

76

surface and a slave one [Chouly et al.,2018]. Some unbiased formulations for contact were proposed to

77

avoid these difﬁculties, for instance, Sauer and De Lorenzis [2015], Mlika et al. [2017]. Another important

78

ingredient to complete the contact formulation is the method chosen to enforce contact constraints. While a

79

penalty method may simply enforce the constraints by adding penalty or virtual power in the formulation

80

[Belytschko et al.,2013], the tuning of the penalty parameter must be carried out to strike a delicate

81

balance between preventing overlapping and avoiding ill-conditioned tangential stiffness matrix for the

82

implicit algorithm [Khoei and Nikbakht,2007,Liu and Borja,2010]. The Lagrange multiplier method

83

introduces extra unknown(s) to enforce the constraint exactly and therefore does not need tuning. However,

84

introducing an additional governing equation may lead to a saddle-point problem that is more difﬁcult

85

to solve [Tur et al.,2009,Fortin et al.,2013]. Based on the penalty and Lagrange multiplier methods, some

86

variations of strategies can be developed. For instance, one may add extra terms to the total energy to

87

improve the stabilization of calculation and conditioning of the matrix. This may lead to the augmented

88

Lagrangian method (which introduces an additional functional to modify the effective stiffness) or the

89

implicit level-set-based material point method 3

perturbed Lagrangian method (which introduces an additional functional to enable static condensation for

90

the Lagrange multiplier) [Belytschko et al.,2013,Tur et al.,2015]. The challenge with Lagrange multiplier

91

methods for this class of problems concerns the construction of a stable Lagrange multiplier space to

92

satisfy the inf-sup condition, which often requires the interpolation space for the Lagrange multiplier

93

ﬁeld to be coarsened with respect to the underlying mesh [Ji and Dolbow,2004,Kim et al.,2007,B

´

echet

94

et al.,2009,Sun and Mota,2014,Sun et al.,2017]. Nitsche’s method was originally proposed in Nitsche

95

[1971] to enforce a Dirichlet boundary condition weakly. As a result of the pioneering work of Hansbo and

96

Hansbo [2002], Nitsche’s method has become popular for a wide class of contact problems. This popularity

97

could be attributed to the elegant treatment that eliminates both the outer augmentation loop as well as

98

additional unknowns, i.e., the Lagrange multiplier, and therefore also eliminates the need for an inf-sup

99

stable mixed-ﬁeld discretization [Annavarapu et al.,2014,Mlika et al.,2017,Chouly et al.,2018].100

Although the FM method can be applied to simulate contact problems, the generation of conformal

101

meshes for ﬁnite element simulations remains a time-consuming process. For objects of complex geometries,

102

conformal meshes may lead to a local reﬁnement that signiﬁcantly increases the computational time but

103

contributes little insight to the state of the assemblies. To overcome this issue, the extended ﬁnite element

104

method (X-FEM) [Mo

¨

es et al.,1999,Dolbow et al.,2001,Liu et al.,2019a] or the generalized ﬁnite element

105

method (G-FEM) [Melenk and Babu

ˇ

ska,1996,Simone et al.,2006] can be used to capture the interface

106

geometry without a conformal unstructured mesh. By selecting the enrichment function to span the solution

107

space, this method can handle calculations with discontinuities provided that the speciﬁc nodal points are

108

enriched via the addition of extra DOFs [Mousavi and Sukumar,2010]. The strategy of implementing phase

109

ﬁeld (PF) method to deal with frictional contact problems is similar to the enriched methods, except that the

110

strong discontinuity is represented by a regularized indicator function [Fei and Choo,2019].111

Another strategy to bypass the need for conformal meshing is to use the immersed boundary (IB)

112

elements. Using a Cartesian structured grid consisting of regular hexahedral elements, we represent the

113

boundaries by partitioning the elements across the interfaces. Any integration involving the cut element is

114

then only performed in the interior portion of the element [Zhang et al.,2004]. Previous work, such as Tur

115

et al. [2015], R

¨

uberg et al. [2016], has shown that it is possible to leverage the IB elements to reduce the total

116

number of DOFs in the contact problems. Leichner et al. [2019] develop a contact algorithm for voxel-based

117

meshes enriched by level sets to handle the frictionless contacts.118

The material point method (MPM), originally developed to deal with large deformation problems

119

[Sulsky et al.,1994,1995,Zhou et al.,1999,Bardenhagen and Kober,2004], is another approach that may

120

solve contact problems without a conformal mesh. In the MPM, all physical information is carried by

121

a set of material points. These points are connected with a background grid that is regularly composed

122

with structured cells. After solving the governing equations, the nodal solutions of the background mesh

123

are updated and the movements of nodes are projected to the material points via the interpolation basis

124

functions. The background mesh is then reset for the next incremental step. The implicit MPM assembles

125

the global matrices in a similar fashion to the FE method counterpart where the material points work

126

as integration points [Sulsky and Kaul,2004]. However, since the locations of the material points can be

127

arbitrary, the accuracy of the volume integration can be low and/or ﬂuctuating in the temporal-spatial

128

domain. Sulsky and Gong [2016] implemented an integration algorithm that employs ﬁxed integration

129

points in the MPM. The major idea is to project the data stored in the scattered material points back to the

130

standard integration points via the moving least squares (MLS) method. The projected data is then used

131

to compute integrals. This method is convenient for handling issues, such as hourglass mode or lockings.

132

However, we need to pay attention to this competing mechanism since the accuracy would reduce due to

133

the projection.134

To handle contact problems, the MPM provides a convenient means by aligning the normal direction

135

of the contact surface with the gradient of mass or volume [Bardenhagen et al.,2000,2001]. However, the

136

resolution of the interface depends on both the size of the background mesh and the number of material

137

points. It is also difﬁcult to cope with quasi-static contacts since contact is detected via existing relative

138

velocities between two contacting bodies [Liu et al.,2018]. As an alternative, we can implicitly represent

139

the boundaries or interfaces via level set functions deﬁned on an Eulerian grid [Osher et al.,2004]. Zhang

140

et al. [2017] enrich the background mesh of the MPM with a level set function to capture the evolution of

141

boundaries. Moreover, the level set function is a versatile tool to help handle contacts since it provides an

142

easy means to compute the normal directions of the contact surfaces [Chi et al.,2015]. The reproducing

143

4 Chuanqi Liu, WaiChing Sun

kernel particle method, as a meshfree method, also can handle contact problems within the framework of

144

kernel functions [Sherburn et al.,2015,Hillman et al.,2014,Chen et al.,2017].145

In this work, our new contribution is two-fold. First, we introduce the level set representation of

146

boundaries for material point contact mechanics problems. As the implicit MPM allows one to keep track of

147

the evolving interface without requiring small time step, this treatment provides a simple way to capture

148

the interplay between the evolving geometry of the contact surfaces and the macroscopic frictional response

149

of the materials. Second, by introducing the level set representation of interfaces, the new implicit level set

150

material point method, which we referred to as ILS-MPM herein, is able to handle both the slip and stick

151

states in the same manner while resolving the mesh-distortion and potential spurious oscillation through a

152

projection technique inspired by the MLS approach used in Bardenhagen et al. [2001].153

The rest of the paper is organized as follows. Section 2 states the problem setting giving the expression of

154

total energy and the enforced constraints for frictional contact problems. Section 3 shows the formulations

155

related to the level set, such as the evolution equation, the extension of physical ﬁelds, and the gap function in

156

the context of the level set. Section 4 presents the spatial discretization scheme and the temporal integration

157

algorithm, focusing on the algorithm of the MPM. Section 5 gives some veriﬁcations and numerical examples

158

to show the effectiveness of the proposed method to simulate granular materials. Section 6 concludes the

159

paper by the major remarks.160

2 Problem setting161

In this section, we review the total energy functional and the constraints required to constitute the boundary

162

value problems for simulating contacts. The energy contributions arising from the surface traction obtained

163

by the penalty method and the analogy of the tangential constraint to an elastoplastic constitutive model

164

are brieﬂy reviewed for completeness.165

2.1 Total energy in a Lagrange multiplier formulation166

We consider contact problems involving ﬁnite solid bodies

Ω(i)

with boundaries of

∂Ω(i)

, where

i∈ I :=167

{1, ·· · ,NΩ}index the solids. There exists a ﬁnite bounding box Bcontaining all Ω(i), and we may write:168

[

i∈I

Ω(i)=:Ω⊂ B ⊂ Rd, (1)

where

d

is the number of spatial dimensions. Similar to

Ω

, we can deﬁne

∂Ω⊂Rd−1

as a union of

∂Ω(i)

, and

169

assume

∂Ω=Γd∪Γn∪Γc

with

Γd∩Γn=∅

,

Γd∩Γc=∅

and

Γn∩Γc=∅

, where

Γd

,

Γn

, and

Γc

represent

170

the Dirichlet, Neumann and contact boundary surfaces, respectively. We denote

u=ˆu

on

Γd

and

σ·n=ˆ

t171

on

Γn

, where

u

is the displacement,

σ

is the stress tensor,

n

is the normal direction of the surface,

ˆu

is the

172

prescribed displacement, and ˆ

tis the prescribed traction. Fig. 1shows a schematic of contact problems.173

The weak form of contact problems can be derived from the minimization of the total energy

W

with

174

certain constraints [Tur et al.,2009]. In the Lagrange multiplier formulation, the total energy for the contact

175

problems is deﬁned as the sum of potential energy πp(u)and the contact contributions:176

W(u,λn,λt):=πp(u) + πn(u,λn) + πt(u,λt), (2)

where

λn

and

λt

are the Lagrange multipliers along normal and tangential directions of the contact surface,

177

respectively,

πn

and

πt

are contact contributions arising from normal and tangential tractions, respectively.

178

Prior to considering the speciﬁc expressions of energy contributions, we discuss the constraints should be

179

fulﬁlled for the contacts, which are called Karush-Kuhn-Tucker (KKT) conditions [Laursen,2013]:180

(1) normal direction181

gn(u)≥0, τn(u)≤0, τn(u)gn(u) = 0 on Γc, (3)

(2) tangential direction182

|τt(u)| ≤ µ|τn(u)|on Γc, (4)

implicit level-set-based material point method 5

(1)

(2)

(3)

(1)

ˆ

u

(1)

u

(1)

c

(1)

c

(1)

c

2

ˆ

t

(2)

c

(2)

c

(3)

c

(3)

c

(3)

c

Fig. 1: Schematic of contact problems

and183

gt(u) = γτt(u), where (γ=0, if |τt(u)|<µ|τn(u)|

γ≥0, if |τt(u)|=µ|τn(u)|, (5)

where

gn

is the normal gap function,

τn=τ·n

and

τt=τ·t

are normal and tangential components of the

184

contact traction, respectively,

τ

is the traction vector applied on the contact surface,

n

and

t

are normal and

185

tangential directions, respectively,

gt

is the tangential slip,

µ

is the Coulomb coefﬁcient of friction and

γ

is a

186

nonnegative coefﬁcient. We brieﬂy interpret the KKT conditions as follows.187

1.

For condition (3), the ﬁrst term describes the non-penetration condition for any contact, the second term

188

ensures that only inward contact forces act over the contact area, and the third term ensures that if there

189

is contact (contact pressure is non-zero) the global gap is zero.190

2.

Condition (4) requires that the magnitude of the tangential stress not exceed the coefﬁcient of friction

191

times the contact pressure. Condition (5) represents two important physical ideas associated with the

192

Coulomb law, ﬁrst, that the tangential slip

gt(u)

be identically zero when the tangential stress is less than

193

the Coulomb limit, and second, that any tangential slip that does occur be collinear with the frictional

194

stress exerted by the sliding point on the opposing surface.195

2.2 Contact contributions obtained by the penalty method196

As shown in (2), two extra DOFs, i.e.,

λn

and

λt

, are introduced in the Lagrangian formulation. We thus

197

need to choose suitable functional spaces to ensure the inf-sup condition for the multiple ﬁelds [B

´

echet

198

et al.,2009], which complicates the computation. The penalty method can be derived from a perturbed

199

Lagrangian method, which is implemented here for simplicity. By introducing a regularization on the

200

tangential constraint, we can introduce a uniﬁed procedure to update the tangential states for both stick and

201

slip conditions, in a manner similar to the branched responses of an elastoplastic problem. More details on

202

the methods enforcing constraints of contacts can be found in Wriggers and Zavarise [2004], Laursen [2013].

203

For clarity, we consider only the contact between

Ω(i)

and

Ω(j)

and show the subscripts when it is necessary.

204

The normal contribution πnis ﬁrst deﬁned depending on a perturbed Lagrangian method as :205

πn(u,λn) = ZΓc

gn(u)λndγ−en

2ZΓc

λ2

ndγ, (6)

6 Chuanqi Liu, WaiChing Sun

where enis a coefﬁcient. For the saddle point problem, it then follows:206

∂πn(u,λn)

∂λn

=ZΓc

[gn(u)−enλn]dγ=0. (7)

We thus can assume:207

τn(u) = λn(u) = 1

en

gn(u). (8)

Substituting (8) into (6), we ﬁnal obtain the normal contribution without introducing extra DOFs as:208

πn(u) = 1

2enZΓc

(gn(u))2dγ, (9)

which is identical to the penalty method.209

In the tangential direction, (4) and (5) represent unregularized constraints implying that the relationship

210

between

gt

and

τt

is a strong discontinuous function, which can cause zig-zagging in practice. As shown in

211

Fig. 2, following Laursen [2013], we additively decompose the tangential slip into an ”elastic” or recoverable

212

part and a non-recoverable plastic part, such that:213

gt(u) = ge

t(u) + gp

t(u). (10)

Therefore, we regularize the tangential constraints to a weak discontinuous function avoiding zig-zagging.

214

Since the Coulomb friction law behaves analogously to an elastoplastic material model, we can consider the

215

physical ideas reﬂected by (4) and (5), writing a related constitutive law as:216

φ(τt,τn):=|τt(u)|−µ|τn(u)| ≤ 0,

˙

gt=˙

γτt(u)

|τt(u)|,˙

γ≥0, ˙

γφ =0,

˙

γ˙

φ=0(if φ=0),

(11)

where

φ

is termed the slip function and is a direct analogue of the yield function in theories of plasticity. The

217

second line in (11) express the colinearity of slip displacement

gt

and frictional stress

τt

in rate form, and

˙

γ218

represents the slip rate. The third line in (11) is the persistency condition to ensure that if elastic unloading

219

begins to occur while a point is still on the yield surface, the plastic strain rate will be zero. The numerical

220

beneﬁt of the analogy to elastoplasticity is that it enables an appeal to the return mapping strategies leading

221

a uniﬁed form to compute the energy contribution of tangential traction for all conditions of stick and slip.

222

Unregularized

Regularized

1

Fig. 2: Schematic depiction of the penalty regularized Coulomb friction law.

implicit level-set-based material point method 7

In analogy to the derivations of (8) and (9), we thus can obtain the tangential traction and the corre-

223

sponding contribution to the energy as:224

τt(u) = λt(u) = 1

et

(gt(u)−gp

t(u)), (12)

and225

πt(u) = 1

2etZΓc

(gt(u)−gp

t(u))2dγ, (13)

where

et

is a coefﬁcient representing the elastic stiffness, as shown in Fig. 2. Again, (12) and (13) are identical

226

to the expressions shown in the penalty method. The above treatment of the Coulomb friction law is called

227

the penalty regularization [Laursen,2013]. Substituting (9) and (13) into (2), we can derive the total energy

228

without Lagrange multipliers. The main problems need to be solved now are how to deﬁne the gap functions

229

and what is the contact surface. We introduce the level set method to answer these questions in the following

230

section.231

3 Level set method for boundary tracking232

In this section, we ﬁrst present the deﬁnition of level set and the corresponding governing equation. We

233

then demonstrate how we adopt the approach introduced by Leichner et al. [2019] for frictionless contacts234

to frictional contact problems within the framework of the MPM method. The core is to construct a potential

235

contact reference without a priori nomination of a master surface and the slave counterpart. We then deﬁne

236

the gap functions in the context of the level set.237

3.1 Representation of boundaries via level set238

A level set

Φ(x)

is expressed in the Eulerian coordinates to implicitly represent boundary surfaces. In a

239

bounded domain B, the level set reads,240

Φ:B → R,Φ(x)

<0x∈Ω

=0x∈∂Ω

>0x∈ B/¯

Ω

, (14)

where

¯

Ω=Ω∪∂Ω

constitutes the interior body and boundary, and

B/¯

Ω

represents the exterior domain.

241

The outward-pointing unit normal on the boundary surface is easily obtained by:242

n(x) = ∇Φ(x)

||∇Φ(x)||,x∈∂Ω, (15)

where

|| ·||

is the norm operator and

∇

is the gradient operator. The evolution of level set is governed by

243

the convection equation:244

Φt+v·∇Φ=0, (16)

where the

t

subscript denotes a temporal partial derivative in the time variable

t

and

v

is the velocity vector.

245

Osher et al. [2004] shows that (16) is valid for

∀x∈ B

when the

v

is smooth without sharp gradients, which

246

is assumed in this work. For quasi-static problems, we can set

v=u

by assuming the pseudo time interval

247

between two loading steps is unit. Since we only can compute

u(x)

for

x∈Ω

, an extension algorithm of

248

u(x)

to

B/¯

Ω

is needed to evolve the level set. The widely employed algorithm for the ﬁeld extrapolation of

249

a scalar ﬁeld

f(x)

is to carry out a constant normal extrapolation by solving the partial differential equation

250

(PDE) to steady state:251

ft+n·∇ f=0. (17)

In practice, this procedure can be carried out using a fast marching method (FMM), in which the ﬁeld values

252

are set by using

n·∇ f=

0. Details of the algorithm of the FMM can be found in Osher et al. [2004], Chopp

253

[2012]. Once we achieve

u(x)

for

∀x∈ B

, we need a stable algorithm to solve (16) and extensive efforts

254

8 Chuanqi Liu, WaiChing Sun

have been made for such an algorithm in the past decades. We here implement the Hamilton-Jacobi weight

255

essentially non-oscillatory (HJ WENO) scheme [Jiang and Peng,2000] to discretize the spatial terms to ﬁfth-

256

order accuracy and total variation diminishing (TVD) Runge-Kutta (RK) methods [Shu and Osher,1988] to

257

increase the accuracy of temporal discretization. A review of level set methods and some recent applications

258

can be found in Gibou et al. [2018]. Note that Aslam [2004] proposed an algorithm for higher-order ﬁeld

259

extrapolation and Rycroft and Gibou [2012] came up with an algorithm to update the level set without

260

solving

(16)

based on the high order ﬁeld extrapolation and the FMM. However, for convenience, we here

261

adopt an open-source code ”Level Set Method Library (LSMLIB)” [Chu and Prodanovic,2008] to track

262

boundaries.263

3.2 Construction of the candidate contact surfaces264

An unbiased candidate contact surface can be conveniently inferred from the level set as illustrated in

265

Chi et al. [2015], Leichner et al. [2019]. The term ”unbiased” refers to the fact that the contact is set at an

266

intermediate contact reference between the contacting bodies without a priori designation of master and

267

slave contact surfaces. For completeness, we provide a brief account on the procedure that constructs the

268

contact reference. As an instance, we still consider the contact between

Ω(i)

and

Ω(j)

and the level sets are

269

deﬁned by the signed distance functions denoting as

Φ(i)(x)

and

Φ(j)(x)

, respectively. The procedure to

270

generate a contact reference is as follows and is illustrated in Fig. 3.271

1. We ﬁrst deﬁne the minimum level set ﬁeld as:272

Φ(ij)

min(x) = (Φ(i)(x)if Φ(i)(x)<Φ(j)(x)

Φ(j)(x)otherwise . (18)

2. The region indicting the contact is speciﬁed with three conditions:273

Ω(ij)

c:=nx|Φ(i)(x)>0∩Φ(j)(x)>0∩reinitialize(Φ(ij)

min(x)−e) + e<0o, (19)

where ”reinitialize” refers to the procedure that converts the level set into a signed distance function (cf.

274

Osher and Fedkiw [2003], Li et al. [2005], Sun et al. [2011]). eis a shift parameter.275

3.

We further construct an intermediate level set

Φ(ij)

int (x)

whose zero-isocontour surface

Γ(ij)

int

is used as a

276

reference for contact. The intermediate level set is deﬁned by277

Φ(ij)

int (x) = Φ(i)(x)−Φ(j)(x)

2, (20)

which indicates that Φ(i)(x) = Φ(j)(x)for x∈Γ(ij)

int (i.e., Φ(ij)

int (x) = 0).278

4.

Then, the following portion of the intermediate surface offers an alternative to the master and slave

279

surfaces:280

Γ(ij)

c:=Γ(ij)

int ∪Ω(ij)

c. (21)

Note that

Γ(ij)

c

, as a ﬁctitious interface aligned with neither surfaces, is an initial guess for a contact surface

281

between

Ω(i)

and

Ω(j)

. In the previous conﬁguration,

Γ(ij)

c

acts as a reference surface which supports the

282

active set strategy [Laursen,2013]. However, in the updated deformed conﬁguration,

Γ(ij)

c

,

Γ(i)

c

and

Γ(j)

c

283

coincide when a new equilibrium state is achieved, where

Γ(i)

c

and

Γ(j)

c

are the contact surfaces for

Ω(i)

and

284

Ω(j), respectively.285

implicit level-set-based material point method 9

Fig. 3: Procedure to generate a contact reference. The contours for the distance functions

Φ(i)

and

Φ(j)

are

omitted and we start from the scalar ﬁeld Φ(ij)

min(x)deﬁned in (18).

3.3 Determination of the gap function286

In order to enforce the contact constraints, such as non-penetration, we require a gap function that indicates

287

the distance of boundaries of potentially contacting bodies. For the contact reference surface

Γ(ij)

c

(shown

288

in Fig. 4a), any point on this surface

x∈Γ(ij)

c

(shown in Fig. 4b) can be mapped onto the true boundary

289

of the body

∂Ω(i)

via a closest point projection

P(i)(x):Γ(ij)

c→∂Ω(i)

, such that

Φ(i)(P(i)(x)) =

0 and

290

(P(i)(x)−x)×∇Φ(i)(P(i)(x)) = 0

(cf. Rycroft and Gibou [2012]). This projection can be achieved by the

291

following iterative algorithm by setting x(0)∈Γ(ij)

c[Chopp,2001,Rycroft and Gibou,2012]:292

δ1=−Φ(i)(x(k))∇Φ(i)(x(k))

∇Φ(i)(x(k))·∇Φ(i)(x(k)),

x(k+1/2)=x(k)+δ1,

δ2= (x(0)−x(k))−(x(0)−x(k))· ∇Φ(i)(x(k))

∇Φ(i)(x(k))·∇Φ(i)(x(k))∇Φ(i)(x(k)),

x(k+1)=x(k+1/2)+δ2,

(22)

where P(i)(x) = x(k), k →∞.293

Considering the displacement (shown in Fig. 4c), we can deﬁne a gap vector for two contacting bodies294

with regard to x∈Γ(ij)

cin the current conﬁguration, as:295

g(u(i)(P(i)(x)),u(j)(P(j)(x))) :=P(j)(x)− P(i)(x) + u(j)(P(j)(x)) −u(i)(P(i)(x)), (23)

where

u(i):Ω(i)→Rd

is the displacement ﬁeld of the

i

-th body. As shown in Fig. 4b, the normal direction of

296

the contact at

x∈Γ(ij)

c

is deﬁned as the gradient of

Φ(ij)

int

via (15) and denoted to

n

. The tangential direction

t297

can therefore be easily determined. We split the normal gap into two parts, i.e.,298

gn(u(i)(P(i)(x)),u(j)(P(j)(x))) = gn0(x) + j(ij)

int (u(x)), (24)

where the ﬁrst part

gn0(x)

represents the normal gap in the undeformed conﬁguration (independent of the

displacement ﬁeld) and j(i j)

int (u(x)) is the normal jump (a functional of displacement), i.e.,

gn0(x) = (P(j)(x)− P(i)(x)) ·n, (25)

j(ij)

int (u(x)) = (u(j)(P(j)(x)) −u(i)(P(i)(x))) ·n. (26)

10 Chuanqi Liu, WaiChing Sun

()ij

c

()

int

ij

()i

()j

()ij

c

(a) (b) (c)

deformation

Fig. 4: Contact detecting: (a) tight proximity zone, (b) normal and tangential directions and closest point

projections, and (c) standard gap vector.

In contrast to gn, the tangential slip gtis the relative displacement along the tangential direction as:299

gt(u(i)(P(i)(x)),u(j)(P(j)(x))) = (u(j)(P(j)(x)) −u(i)(P(i)(x))) ·t. (27)

For clarity, we also deﬁne a shear jump s(ij)

int (u(x)) in analogy to j(i j)

int (u(x)) as:300

s(ij)

int (u(x)) :=gt(u(i)(P(i)(x)),u(j)(P(j)(x))) −gp

t(u(i)(P(i)(x)),u(j)(P(j)(x))). (28)

where gp

tis the non-recoverable plastic part as shown in Eq. (10).301

3.4 Active set strategy302

The integration points along

Γ(ij)

c

are contact monitoring points. Since we analogize the Coulomb friction

303

law to an elastoplastic constitutive law, it is not required to distinguish the tangential state, i.e., stick or slip

304

explicitly. We only rely on the points with negative normal traction

τn

to update the constitutive responses

305

properly. The set of active points for contact between Ω(i)and Ω(j)is denoted as:306

A(ij):={x∈Γ(i j)

c|τ(ij)

n(x)<0}. (29)

where Γ(ij)

cis deﬁned in (21) and τ(ij)

n(x)is deﬁned at (8).307

4 Variations, discretization and integration308

We have derived the expression of total energy and gap functions as shown above. In this section, we

309

ﬁrst show the procedure to derive the weak form of frictional contact problems via the ﬁrst variation of

310

the energy. We second present the spatial discretization and updating procedure for an individual body

311

and then discuss the contact contributions to the residual. The computation of the tangent matrix by the

312

numerical ﬁnite-differencing method is also given. We summarize the computational procedure at the end

313

of this section.314

implicit level-set-based material point method 11

4.1 Variations and weak form315

The ﬁrst variation of πp(u)is316

δπp(u,w) = a(u,w) + f(w)(30)

where

w

is the variation of displacement, and the bilinear forms

a(·

,

·)

and linear operator

f(·)

are deﬁned

317

as:318

a(u,w) = ZΩ

σ(u):ε(w)dxf(w):=ZΩb·wdx+ZΓn

ˆ

t·wdγ, (31)

where εis strain tensor and bis body force.319

Considering (24) and (28), the ﬁrst variations of (9) and (13) are:320

δπn(u,w) = d

dεπn(u+εw)|ε=0=ZΓc

j(ij)

int (w)τn(u)dγ, (32)

and321

δπt(u,w) = ZΓc

s(ij)

int (w)τt(u)dγ, (33)

respectively. Combining (30), (32) and (33), the weak form for the contact problems is: ﬁnd

u∈H1(Ω)

, such

322

that323

r(u,w) = a(u,w) + ZΓc

j(ij)

int (w)τn(u)dγ+ZΓc

s(ij)

int (w)τt(u)dγ+f(w) = 0, ∀w∈H1

0(Ω). (34)

Note that the admissible

u

must satisfy the Dirichlet boundary condition. In this work, we directly impose

324

boundary conditions on the nodes of the background mesh of the MPM or specify a rigid-body motion

325

to apply loadings. For more discussions on the imposition of Dirichlet boundary conditions for meshless

326

methods, we refer to Fern´

andez-M´

endez and Huerta [2004] and Liu and Sun [2019].327

4.2

Individual body contribution to the residual: the material point method implementing the MLS method

328

We now consider the discretization and update of

Ω(i)

with the boundary of

Γ(i)

in the ILS-MPM framework.

329

For simplicity, variables in this subsection omit the superscript

(i)

except for speciﬁcation. We denote to

xp

,

330

xI

, and

xq

as the positions of particles (material points), nodes of background mesh, and quadrature points,

331

respectively and provide some highlights on key ingredients and treatments to implement the ILS-MPM for

332

completeness. More comprehensive reviews on the theory and implementation of the MPM can be found in

333

Sulsky et al. [1994,1995], Zhou et al. [1999], Bardenhagen and Kober [2004].334

1.

Setting of DOFs in the background mesh. The DOFs are set at the nodes of a subset of the background

335

mesh. In the standard MPM, all nodes belonging to those cells that contain at least one material point

336

are activated. In the proposed model, we activate the nodes based on the status of cells (i.e. inside the

337

boundary, cut by the boundary, or outside the boundary). The cells completely separated from a body

338

have no contribution to the residuals, therefore we do not activate the nodes of these cells. As shown in

339

Fig. 5, the unknowns are set at the nodes belonging to the elements inside of

Ω(i)

or cut by the boundary

340

Γ(i)

, e.g. the triangle labels in Fig. 5. Note that we here only show the DOFs setting for one body and

341

other bodies are independently treated in the same manner.342

2.

Locations of quadrature points. The locations of quadrature points are generated according to the

343

location of boundary for accurate integration, rather than using material points as quadrature points

344

as employed in the standard implicit MPM method. For the element inside of

Ω(i)

, the locations of

345

the integration points are identical to the typical Gauss’s quadrature scheme. For the element cut by

346

the boundary, we partition the element into a set of sub-triangles in terms of the location of

Γ(i)

(the

347

boundary is assumed to be a straight line in the element) and the integration points are at the centers

348

of the sub-triangles. This integration scheme based on the element-partitioning is developed from the

349

X-FEM [Sukumar et al.,2000]. Recently, an integration scheme without element-partitioning is also

350

developed, see Chin et al. [2015,2017], Liu et al. [2019b]. The quadrature points are labeled as squares in

351

Fig. 5.352

12 Chuanqi Liu, WaiChing Sun

Undeformed configuration

Deformed configuration

Evolution of boundary

(LSM) Evolution of material points

(MPM)

Reconstruct physical field

(MLS)

Background mesh deforms

Background mesh resets

Note: dofs are set at nodes of background mesh

Quadrature point, xq

Material point, xP

Nodes of background mesh, xI

Fig. 5: Spatial discretization and information update.

3.

Initialization and update of material points. At the beginning of the computation, the material points

353

share the same positions with the quadrature points (as shown in the undeformed conﬁguration in Fig.

354

5). After deformation, the material points do not coincide with the quadrature points that are determined

355

by the locations of boundaries. We then project stresses stored at the material points to the integration

356

points using the MLS method:357

σ(xq) = MLS(σ(xp)), (35)

where

MLS

is the operator minimizing the weighted least squares measure biased towards the region

358

around the point at which the reconstructed value is requested. The details of this projection can be

359

found in Gong [2015], Sulsky and Gong [2016], Liu et al. [2019b]. The residual arising from volume

360

integration at the node Iis,361

rI(u) = −

nI

∑

q=1Zσ(xq)· ∇ NI(xq)dx+fext

I, (36)

where nIis the number of quadrature points inﬂuencing the node I,∇NI(xq)is the spatial gradient at

xq

of the shape function

NI

, and

fext

I

is the external force applied at the node

I

. As mentioned above, the

unknowns are set at the nodes of the background mesh and therefore we compute displacements at the

nodes, i.e.

u(xI)

, eliminating the total residual (contact contributions would be discussed at the next

implicit level-set-based material point method 13

subsection). Once obtain

u(xI)

, we then compute the displacement, stress tensor, and other variables of

the material points, such as

u(xp) =

np

∑

I=1

NI(xp)u(xI), (37)

σ(xp) = C:(

np

∑

I=1∇NI(xp)u(xI))sym, (38)

where

np

is the number of nodes inﬂuencing the particle

p

,

C

is the forth-order elastic matrix, and

362

(·)sym

means the symmetric part. The update process is illustrated in the vertical middle inset in Fig.

363

5. In the next time step, the background mesh is reset, which means the background mesh is identical

364

(as a Eulerian grid) at the beginning of each time step. We mention that deformation within two

365

consecutive incremental steps is assumed inﬁnitesimal for simplicity. The MPM within the framework of

366

ﬁnite-deformation can be found in Charlton et al. [2017], Liu et al. [2019b], Coombs and Augarde [2020].

367

4.

Update of boundary Once we obtain the displacement at the activated nodes, we extend the information

368

to the whole domain according to (17) and then update the level set by evolving (16).369

The update procedure is illustrated in Fig. 5. For convenience, we incorporate the open-source numerical

370

analysis and data processing library (ALGLIB) [Bochkanov,2019] into our code for the interpolations, where

371

the k-d tree is adopted for the efﬁcient neighbor searching.372

4.3 Contact contributions to the residual373

From (34), the residual contributed from the surface integral is termed as contact residual and deﬁned as:374

rc(u,w):=ZΓc

JwK(ij)

int τ(u)dγ=ZΓc

j(ij)

int (w)τn(u)dγ+ZΓc

s(ij)

int (w)τt(u)dγ, (39)

where

JwK(ij)

int =nj(ij)

int (w) + ts(ij)

int (w)

is the displacement jump and

τ(u) = nτn(u) + tτt(u)

is the traction

375

along the contact force. As shown in Fig. 6, the two projected points

P(i)(x)

and

P(j)(x)

of the surface

376

integration point

x∈Γc

are within elements

Ω(i),e

and

Ω(j),e

, respectively. We construct a generalized nodal

377

displacement vector

{u}={u(i),e

,

u(j),e}

and a shape function matrix

[N]=[−N(i)(P(i)(x))

,

N(j)(P(j)(x))]

,

378

where

u(i),e

and

u(j),e

are nodal displacement of elements

Ω(i),e

and

Ω(j),e

, respectively, and

N(i)(P(i)(x))379

and

N(j)(P(j)(x))

are nodal shape functions at the projected points. We then implement a radial extension

380

of displacement [Osher and Fedkiw,2006] leading to

u(i)(x) = u(i)(P(i)(x))

and

u(j)(x) = u(j)(P(j)(x))

.

381

Therefore, the virtual displacement jump can be expressed as:382

JwK(ij)

int = [N]T{w}, (40)

where the vector

{w}

is the general virtual displacement vector deﬁned in a same way of

{u}

. Due to the

383

arbitrariness of w, the contribution of xto the residual in a matrix form is:384

rc=ZΓc

[N]Thnxtx

nytyinτn

τtodγ. (41)

We now discuss the calculation of the traction along the contact surface. Again, the traction is updated by

385

applying the classical return mapping algorithm over the incremental form of (11). We here modify the

386

algorithm of Annavarapu et al. [2014], which is shown in Algorithm 1.387

14 Chuanqi Liu, WaiChing Sun

Fig. 6: Computation of tangent matrix for surface integrations

Algorithm 1

Update tractions on the contact surface at the

(k+1)

-th iteration given converged results at

k-th iteration

for all active Gauss-points on Γcdo

Compute normal gap g(k+1)

nas (24)

Compute tangential gap g(k+1)

tas (27)

Compute trial normal traction τtri,(k+1)

nas (8)

Compute trial tangential traction τtri,(k+1)

t=1

et(g(k+1)

t−gp,(k)

t)

Compute trial yield function φtri,(k+1)=φ(τtri,(k+1)

t) = |τtri,(k+1)

t|−µ|τtri,(k+1)

n|

if φtri,(k+1)≤0then

Trial state is true state: τ(k+1)

n=τtri,(k+1)

nand τ(k+1)

t=τtri,(k+1)

t

else

Normal direction - no yielding: τ(k+1)

n=τtri,(k+1)

n

Tangential direction - return-mapping algorithm

∆γ=etφtri,(k+1)

τ(k+1)

t=τtri,(k+1)

t−∆γ

et

τtri,(k+1)

t

|τtri,(k+1)

t|

gp,(k+1)

t=gp,(k)

t+∆γτtri,(k+1)

t

|τtri,(k+1)

t|

4.4 Convergence scheme388

In this work, we consider linear elastic materials, so the tangent matrix for the term in (34) involving volume

389

integration, i.e. (36), is conventional. We here only show the tangent for the terms involving integrals along

390

Γc

, i.e. (41). Note that the tangential traction would be affected by the normal traction due to the Coulomb

391

friction law. Rather than deriving theoretical tangent matrix, we here adopt the ﬁnite-differencing of the

392

residual to compute the Jacobian determinant matrix (tangent matrix) for simplicity [Prevost and Sukumar,

393

2016].394

In general, the entry

JAB

of the Jacobian determinant matrix

J

is approximated by ﬁnite-differencing as:

395

JAB =∂rA

∂uB≈rA(uB+¯

h)−rA(uB)

¯

h, (42)

implicit level-set-based material point method 15

where

A

and

B

are global equation numbers corresponding to the unknown displacement. The perturbation

396

of the degrees of freedom ¯

his a small parameter chosen as:397

h=3

√eM|uB|,¯

h=max(3

√eM,h), (43)

where

eM

is machine precision. In practice, it is hard to directly compute

rA(uB+¯

h)

. In our code, we

398

loop activated points along

Γc

to assemble the tangent matrix. For instance, we consider the case that the

399

background grid consists of structured quadrilateral cells for 2-dimensional problems, as shown in Fig. 6.

400

Therefore, the integration point

x

along

Γc

inﬂuences eight nodes with two DOFs for each node. For the

401

point

x

, we assign a pointer vector containing 16 pointers,

∗ue={∗ue

b}(b=

1,

·· ·

, 16

)

, pointing the global

402

unknowns

uB

with

ue

b=

&

uB

, where subscript

e

representing elemental variables,

∗

meaning pointers, and &

403

assigning the address. The map between elemental DOFs and global DOFs is denoted as

M(b)7→ B

, which

404

is one to one correspondence. The perturbation displacement vector is denoted as

∆ua= [

0,

·· ·

,

¯

h

,

·· ·

, 0

]T

405

with the nonzero entry holding

a

-th position. The

a

-th column of the local tangent matrix corresponding to

406

the integration point xis computed by407

Je

c(:, a) = rc(∗ue+∆ua)−rc(∗ue)

¯

h, (44)

where

rc

is computed via (41). Looping

a=

1

·· ·

16, we can compute the contribution to the tangent matrix

408

arising from the contact,

Je

c

. We then can obtain the global Jacobian determinant matrix via assembling the

409

local matrix point-by-point via the map M(b)7→ B.410

4.5 Calculation procedure411

The calculation procedure is detailed as follows.412

1. Choose a proper spatial region Bcontaining all bodies.413

2. Initialize the level set for each body Φ(i)(x),∀i∈ I.414

3.

Generate a set of material points for each body according to its boundary

Γ(i)

and set unknowns

415

regarding to the boundary conditions, as shown in Fig. 5.416

4. Determine the potential contact pairs and set the reference contact surface Γcas (21).417

5.

Implement the Newton-Raphson iteration scheme to solve the displacement vanishing the residuals

418

shown in (34) by repeating the following steps (krepresenting the k-th iteration).419

(a) Compute the tractions for all integration points xpon the Γc, according to Algorithm 1.420

(b)

If

τn(xp)≤

0 at

xp

, compute the contribution to the contact residual according to (41) and its

421

contribution to Jacobian matrix according to (44).422

(c)

Considering the stiffness matrix coming from the traditional volume integration, assemble the global

423

tangent matrix J(k).424

(d) Compute the global residual r(k).425

(e) Update the displacement by u(k+1)=u(k)+δu(k), where δu(k)= (Jk)−1r(k).426

6. Update the locations and stresses of the materials points, as shown in Fig. 5.427

7. Extend the displacement of each body to the whole region Baccording to the steady state of (17).428

8. Evolve the level set Φ(i)(x),∀i∈ I, according to (16).429

9. Set the integration points and unknowns according to the updated level sets and boundary conditions.430

10. Project the information of the scattered material points to the integration points, according to (35).431

11. Repeat from step 4.432

One last remaining problem is the setting of the penalty parameters

en

in (8) and

et

in (12). Following

433

the appoarch from Kikuchi and Oden [1988], which is also adopted in Leichner et al. [2019], we set the

434

parameters in relation to the average Young’s modulus ¯

Eand mesh size h:435

en=et=e0

h

¯

E. (45)

We hereafter set

e0=

1 for all cases in terms of accuracy and convergence rate. Furthermore, this factor is

436

kept as a constant during the iterations.437

16 Chuanqi Liu, WaiChing Sun

5 Numerical Examples438

In this section, we provide numerical examples to verify the model and demonstrate its capacities. We

439

ﬁrst consider the cases containing a single contact to verify the implementation via comparisons of our

440

results and results in the literature or analytical solutions. The contacts involving multiple bodies with

441

simple and complicated shapes are then simulated to demonstrate the efﬁciency of the proposed model to442

capture complex contacts. The last two examples are with multiple loading steps considering the evolution

443

of boundaries, and one is used to validate and the other to show the perspectives of our model in terms of

444

linking particle physical states and macroscopic responses of the assemblies.445

5.1 Veriﬁcation Problems446

5.1.1 Two contacting blocks447

To verify our implementation, we ﬁrst simulate the deformation of two contacting rectangular blocks

448

discretized by structured cells, as discussed in Tur et al. [2009]. As shown in Fig. 7, a vertical displacement

449

uy=−

1.6

×

10

−6

m is imposed on the top boundary of the upper body and stress distributions

px=450

4

×

10

11y(

0.01

−y)

Pa and

py=

10

12y(

0.01

−y)

Pa are applied on the left and right sides of the lower

451

body. The origin of the coordinate system locates at the lower-left corner of the lower body. Other boundary

452

conditions and geometries are shown in Fig. 7. A linear elastic material is assumed with plane strain

453

and Young’s modulus

E=

100GPa and Poisson’s ratio

ν=

0.3 for the two bodies. Coulomb model with

454

coefﬁcient friction of

µ=

1.0 is implemented to distinguish stick and slip regions. Contact reference is

455

exactly set along the line of y=0.01 m and the mesh size is set as h=0.0002 m.456

Body1

Body 2

0.01

0.01

0.01

x

y

uy

Fig. 7: Model and boundary conditions

Fig. 8shows the comparison of our result for the distribution of

σx

on the distorted domain (ampliﬁed

457

by a factor of 500) and the result shown in Tur et al. [2009]. Fig. 9shows the contact tractions along the

458

implicit level-set-based material point method 17

contact surface. As can be seen, the contact area is split into a central stick zone and two slip zones with

459

opposite slip directions, which also coincides with the observation in Tur et al. [2009]. Unlike setting two

460

loops to vanish the residuals and to ensure the correct shear states [Tur et al.,2009], we here do not need

461

to distinguish the shear states since the Coulomb friction model is treated as an elastoplastic model and

462

a typical return-mapping algorithm is implemented to converge the simulation. The convergence for this

463

example is achieved within eight iterations.464

(a) (b)

Fig. 8: Comparison of stress component σxx: (a) results in Tur et al. [2009] and (b) our results.

−1

−0.5

0

0.5

1

0 0.002 0.004 0.006 0.008 0.01

τn,τt(107Pa)

x

τn

τt

Fig. 9: Contact stresses: τnis the normal traction and τtis the tangential traction.

18 Chuanqi Liu, WaiChing Sun

5.1.2 Two contacting trapezoids465

In this example, we consider two bodies contacting each other along an inclined plane as proposed in

466

Annavarapu et al. [2014]. As shown in Fig. 10a, the equation of the inclined plane is

y−

0.2

x−

0.4586

=

0

467

leading a slope with

tan θ=

0.2. Vertical displacement of

uy=−

10

−3

m is imposed on the top boundary

468

of the upper body. Other boundary conditions and geometries can be found in Fig. 10a. Young’s modulus

469

and Poisson’s ratio are

E=

1GPa and

ν=

0.3, respectively, for the two bodies. Fig. 10b shows the subsets

470

of background mesh with unknowns for different bodies and the red dots denote the contact monitoring

471

points on the contact reference.472

Body 1

1

1

uy

Body 1

Body 2

x

y

(a) (b)

Body 2

Fig. 10: Two contacting bodies with an inclined contact plane (a) loading and boundary conditions and

(b) subsets of the background mesh with unknowns for different bodies (red dots denoting the contact

monitoring points on the potential contact surface).

The friction coefﬁcient is chosen as

µ=

0.19 and

µ=

0.21 for two separate computations. The problem

473

serves as a good benchmark for ﬁctional sliding problems since we can predict slipping behavior when

474

the friction coefﬁcient

µ<tan θ

and a stick state otherwise. Fig. 11 shows the distributions of horizontal

475

displacements for different friction coefﬁcients. As expected, slipping occurs when the friction coefﬁcient is

476

less than the tangent of the inclination of the surface while it is larger, we see a stick state. We repeat that the

477

cells used for computations are squares as shown in Fig. 10b and the subcells (triangles) partitioning the

478

squares cut by the contact surface are only used for volume integrals and post-processing as shown in Fig.

479

11. Hereafter, all results containing cells are plotted following this treatment except for some speciﬁcations.

480

5.1.3 Hertzian contact481

The third problem to validate our model is the cylinder on plane Hertzian contact. It has a closed analytical

482

solution for the contact stresses and so is used as a comparison with the numerical results. Fig. 12a shows

483

the boundary conditions and geometries. The radius of the cylinder is 10 mm and the height of each

484

body is 4 mm. Young’s modulus is 10 GPa and Poisson’s ratio is 0.3 for both bodies. A displacement with

485

u= (

0.03,

−

0.04

)

mm is applied on the top surface of the cylinder part. Fig. 12b shows the meshes used in

486

the computations and the contact reference. The mesh size is h=0.1 mm.487

Fig. 13 plots the distribution of vertical stress. To compare the numerical and analytical results, we ﬁrst

488

compute the reaction forces acting on the top boundary, as

f= (−

570.73, 1617.64

)

kN. The distributions of

489

tangential and normal tractions obtained from simulations are compared with the analytical solutions as

490

implicit level-set-based material point method 19

(a) (b)

Fig. 11: Horizontal displacement contours for different friction coefﬁcient (a) friction angle is less than

incline angle with slip expected and (b) friction angle is greater than incline angle with stick expected.

x

y

ux

uy

Body 1

Body 2

8

4

R=10

(a) (b)

4

Fig. 12: Hertz contact problem (a) geometry and boundary conditions (b) meshes and intermediate contact

reference.

shown in Fig. 14. We can see a stick zone existing between two slip zones and distributions of tractions are

491

almost identical.492

5.2 Frictional contacts involving multiple bodies493

5.2.1 Symmetric problems for nine-discs under an isotropic compression494

We here further verify our model for symmetrical problems. As shown in Fig. 15a, nine discs with the

495

same radius of 1.7mm are conﬁned by four deformable plates. The thickness of the conﬁning plates is 1mm

496

and the length 9.8mm. Isotropic compression is imposed by moving the plates towards the center with a

497

displacement of

d=

10

−4

mm. Note that even we only ﬁx the conﬁning plates without any displacement,

498

the discs are compressed since there are initial gaps between bodies. Fig. 15b shows the meshes used for

499

20 Chuanqi Liu, WaiChing Sun

Fig. 13: Distribution of vertical stress.

−1500

−1000

−500

0

500

1000

−0.8−0.4 0 0.4 0.8

τn,τt(MPa)

x(mm)

τ(num)

n

τ(num)

t

τ(ana)

n

τ(ana)

t

Fig. 14: Comparison of numerical results and analytical results for tractions along the contact surface

(superscripts ”num” denoting numerical results and ”ana” representing analytical results).

the computations. The material is also assumed as linear elastic with Young’s modulus

E=

100 GPa and

500

Poisson’s ratio ν=0.3. The friction coefﬁcient is set as 0.5.501

The computation rapidly converged within four iterations due to the high symmetry of the problem. Fig.

502

16 shows the distributions of the maximum principal stress and the shear stress. The left inset details the

503

meshes for two contacting bodies, where we can ﬁnd that high stress concentrations occur at the contacting

504

points. In a summary, all stresses are symmetrically distributed consistent with expectations.505

5.2.2 Brazilian test for a grain obtained from Micro-CT images506

In this example, we simulate the Brazilian test for a grain whose geometry is inferred from micro-CT

507

scanning images. Since we only consider two-dimensional problems in this paper, the geometry of the grain

508

is a cross section of a real Hostun grain reconstructed based on a binarized 3D image [Gupta et al.,2019],

509

which is also used in Liu and Sun [2019]. To increase the difﬁculty and test the robustness of the algorithm,

510

we set up the conﬁguration as shown in Fig. 17, where two potential contact points locate on the lower plate.

511

The width and length of the conﬁning plates are 10 mm and 65 mm, respectively. The displacement of the

512

top boundary of the upper plate is ﬁxed as

u= (

0,

−

0.1

)

mm, and

−u= (

0, 0.1

)

mm for the bottom boundary

513

of the lower plate. Again, the materials for these three deformable bodies are identical and assumed as linear

514

elastic with Young’s modulus 10 GPa and Poisson’s ratio 0.3. The friction coefﬁcient is also set as 0.5. Fig. 17b

515

implicit level-set-based material point method 21

9.8

d

d

d

d

9.8

1

1

R=1.7

1

1

(a) (b)

Fig. 15: Problem for nine particles: (a) geometry and boundary conditions and (b) meshes and closest

projections from contact reference to boundaries.

(a) (b)

Fig. 16: Distributions of stresses (a) the maximum principal stress and (b) shear stress.

shows contact references and the meshes used in the computation with the mesh size

h=

1 mm. The inset

516

of Fig. 17 illustrates the enlargement of the closest projections from the surface integration points on the

517

contact reference surface to the boundaries of bodies. Note that the contact reference is the zero-isocontour

518

surface of the intermediate level set. This reference is a priori, however, the activated integration points on

519

the reference are according to the active set strategy discussed in subsection 3.4.520

We perform the simulations twice for different mesh sizes, i.e.

h=

1 mm and

h=

0.5 mm, to examine

521

the sensitivity of results to the mesh size. Fig. 18 compares the distributions of vertical stresses for these two

522

mesh sizes with the same legend. The left inset of Fig. 18 shows the enlarged contours at the left bottom

523

contact points for different meshes. The contour patterns are consistent for various mesh sizes though the

524

maximum values are a bit different since the stress at the contacting point is highly concentrated. From this

525

example, we can see the advantage of the proposed method over the DEM where we cannot simultaneously

526

handle two contact points within one contact.527

5.2.3 Four real grains under an isotropic compression528

We deepen the last example to the case of four grains under an isotropic compression. The grains hold

529

the same shape but with various sizes and locations via rotation and translation. Fig. 19 shows the initial

530

22 Chuanqi Liu, WaiChing Sun

65

u=(0, -uy)

10

Body1

Body2

Body3

10

-u

1

2

(12)

int

(a) (b)

Fig. 17: Brazilian test for real sand based on Micro-CT scanning images: (a) geometry and boundary

conditions and (b) meshes and contact references (inset: local enlargement of closest projections).

Coarse mesh with h = 1mm Fine mesh with h = 0.5mm

h = 1mm

h = 0.5mm

Fig. 18: Distributions of vertical stresses for different mesh sizes in the same legend.

conﬁguration. Four deformable plates move toward the center with

d=

10

−3

mm. The material parameters

531

are identical to the last example. The mesh size is set as h=0.1 mm.532

Fig. 20 plots the distribution of the maximum principal stress and enlargement for the vicinity of a

533

speciﬁc contact point. We can observe an obvious shift between two stress concentration regions due to

534

slipping.535

5.3 Contacts with incrementally increasing loads considering the evolution of the level set536

In the following examples, we incrementally increasing the loading to consider the evolution of level set

537

and material points. The ﬁrst example is to show the effectiveness of the proposed method to consider large

538

deformation due to the essence of the MPM. The second example, we explore the capability of the proposed

539

method to simulate an assembly of granular materials with more accurate information than the DEM.540

implicit level-set-based material point method 23

8

d

d

d

d

8

(13)

1

1

Fig. 19: Geometry and boundary conditions for the model of four grains under an isotropic compression.

Fig. 20: Distribution of the maximum principal stress.

5.3.1 Brazilian test for a disc541

As shown in Fig. 21a, a deformable disc is vertically compressed by two rigid rectangles. The low rectangle

542

is ﬁxed and a constant vertical displacement

∆uy=

0.02 mm is applied on the top rectangle in each time

543

step. The radius of the disc is

R=

10 mm. Other geometries can be found in Fig. 21a. The material for the

544

disc is assumed as linear elastic with Young’s modulus

E=

100 GPa and Poisson’s ratio

ν=

0.3. The friction

545

coefﬁcient is also set as 0.5. Note that the movements of the rectangles are prescribed as rigid bodies. Fig.

546

21b shows the contact references and the meshes for the initial conﬁguration with mesh size

h=

0.2 mm.

547

We simulate 50 steps.548

Since the displacement of the top rigid rectangle is not equivalent to the penetration

d

used in the Hertz

549

theory, we validate our model by comparing the curves of contact radius vs. contact force, for the analytical

550

and numerical solutions. According to the Hertz contact theory [Barber,1992], we have551

2a=rF(1−ν2)R

πE, (46)

where

a

is the radius of the contact surface and

F

is the vertical contact force. For the top contact surface, we

552

here approximate

a

by the distance between the farthest activated integration point on the contact surface

553

24 Chuanqi Liu, WaiChing Sun

Rigid body

R=10

14

2

Deformable body

Rigid body fixed

2

14

y

u

(a) (b)

Fig. 21: Brazilian test for a disc: (a) geometry and boundary conditions and (b) meshes and contact references.

and the center of the contact for simplicity. Note that this measurement would introduce a small error within

554

one mesh size since the integration point cannot exactly locate on the boundary of the contact surface. Fig.

555

22 compares the relationship of

a−F

for numerical and analytical solutions. The tendencies are identical

556

although there are some discrepancies due to the inaccurate measurement of a.557

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

F(109N)

a(mm)

Anal. approx.

Num. solution

Fig. 22: Comparison of the relationships between the contact radius and the contact force for the numerical

and analytical solutions.

Fig. 23 shows the evolution of the vertical stress of the deformable disc at different loading steps. The

558

top row of Fig. 23 illustrates the locations of the material points. It should be noted that the material points

559

carrying all physical information do not coincide with the integration points for the volume integration. The

560

elements used for the volume integration (see the inset of Fig. 23) are regenerated according to the updated

561

level set at the beginning of each time step.562

implicit level-set-based material point method 25

n =1 n =10 n =30 n =50

Fig. 23: Evolution of the vertical stress of the deformable disc (top row shows the locations of material

points)

5.3.2 Fifteen non-sphere particles563

In this example, we examine the capacities of the proposed model to predict the evolution of stress distri-

564

bution at the subscale level and the resultant macroscopic responses of granular assembles. We compare

565

the local stress ﬁeld with the force chain obtained from a two-dimensional DEM simulation to examine

566

how the lack of deformability and the information on the local stress distribution affects the conﬁguration

567

of the grain contacts and the resultant macoroscopic constitutive responses. We study a grain assembly

568

consisting of multiple non-sphere particles, which was previously studied using a version of DEM called

569

granular element method in Andrade et al. [2012]. As shown in Fig. 24a, ﬁfteen deformable bodies are

570

conﬁned by four rigid plates with a uniform thickness of 20 mm. The Young’s modulus is

E=

100 GPa,

571

the Poisson’s ratio is

ν=

0.2, and the friction angle is 15

◦

. Note that, since we only recover the geometry

572

of each particle from the images available in Andrade et al. [2012], there could be slight discrepancy on

573

the particle shapes Furthermore, if there exists an isolated particle without any overlap with neighboring

574

particles in the redrawn ﬁgure, we will confront convergence issues due to the implicit algorithm. To avoid

575

this problem, we generate the initial conﬁguration as follows. First, we enlarge the grains in the redrawn

576

ﬁgure by setting the level set of the boundaries as

Φ=

0.5

h

(rather zero), where

h

is the mesh size, to ensure

577

that there are overlaps between bodies for convergence. We then ﬁx the plates and conduct a simulation to

578

achieve the equilibrium state. The deformed conﬁguration is set as the initial conﬁguration and stresses and

579

displacements of the deformed particles are reset to zero. Fig. 24b shows this procedure. We also show the

580

deformation of one particle in Fig. 24, where we emphasize that the deformation of particles may cause a

581

convex particle becoming non-convex locally. This is particularly likely to happen on contacts of particles of

582

different sizes or when the contact surface area is small for a given force. When a particle of non-convex

583

shape is in contact of another particle, there could be more than one non-connected contact areas form and

584

hence the data structure employed by the classical DEM may not work well without special treatments (also

585

see Fig. 20).586

In this example, we ﬁrst isotropically compress the sample by moving the plates toward the center with

587

an incremental displacement

∆d=

0.1mm. Fig. 25 shows the evolution of the normal force and the ratio of

588

tangential force and normal force within 30 loading steps. The normal and tangential forces are computed

589

by a summation of contact forces for the contacts involving the top plate, which is approximately equivalent

590

to the forces acting on the cutting plane as denoted by the red line in Fig. We observe that the normal force

591

linearly increase with the loading, but the ratio of the tangential force with the normal force reaches a steady

592

state after several steps due to the deformation of the bodies (see Figure 24).593

Fig. 26(a-b) compare the force chains obtained from the proposed method and the DEM [Andrade et al.,

594

2012]. We can see that the general networks of force chains are almost identical, but with some discrepancies.

595

Fig. 26(e) shows details of two contacting bodies.596

26 Chuanqi Liu, WaiChing Sun

250

20

20

300

Enlarged particles

(containing overlaps) Deformed particles (being reset to

generate the initial configuration)

Deformation of one

particle

(amplified factor 5)

(a) (b)

Unreformed

configuration

Fig. 24: Model containing 15 particles: (a) conﬁguration published in Andrade et al. [2012] (the red line

represents the cut surface to compute tractions) and (b) the process to generate the initial conﬁguration.

1.5

2

2.5

3

3.5

0 0.511.522.5 3 0.08

0.1

0.12

0.14

0.16

0.18

0.2

Fn(107N)

Ft/Fn

d(mm)

Fn

Ft/Fn

Fig. 25: Loading curves for the isotropic compressing phase

In the shear phase, we apply a shear load on the sample by rotating the plates around their vertices,

597

following the treatment in Andrade et al. [2012]. Note that this load is not a pure shear load due to the

598

volume change caused by the imposition of boundary conditions. Nevertheless, this does not affect our

599

purpose, which is to compare the results between the DEM and the proposed model. Fig. 26(c-d) shows

600

this comparison. Since the particles are extensively compressed, the stresses at the regions far away the

601

contacting points are also very large, which cannot be reﬂected by the DEM.602

To explore the versatility of the proposed method, we conduct a simple shear of the sample after the

603

compression phase by ﬁxing the top and bottom plates and rotating the lateral plates with an incremental

604

rotation angle of 0.3

/

180 rad. We simulate 30 steps for the simple shear test. For the isotropic compression

605

and the pure shearing, to some extent, the structures of the particles are stable resulting in continuous

606

loading curves as shown in Fig. 25. However, for the simple shear, we frequently observe rearrangements of

607

particles and reconstructions of the force chains. Fig. 27 shows the distribution of the maximum principal

608

stress at the 25-th step. The left insets represent the evolutions of the stresses and boundaries of a speciﬁc

609

implicit level-set-based material point method 27

(a) (b)

(c) (d)

(e)

Fig. 26: Comparison of the results obtained from the proposed method and the DEM [Andrade et al.,2012]:

(a) force chains obtained by the DEM for the compression, (b) distribution of the maximum stress obtained

by the proposed method for the compression, (c) force chains obtained by the DEM for the pure shearing,

(d) distribution of the maximum stress obtained by the proposed method for the pure shearing, and (e)

details of the contact between two particles.

particle. We can see that the stresses vary with the shear. The evolutions of normal and tangential forces are

610

plotted in Fig. 28. Since the number of particles is only ﬁfteen, the normal contact force cannot maintain

611

constant for the conserve-volume simple shear test. Also, as discussed before, the curves are more ﬂuctuated

612

than in the case of isotropic compression. We conducted another simulation with a different friction angle of

613

30◦for comparison. As shown in Fig. 28, the shear force increases with the shear for large friction angle.614

In conclusion, the proposed model is capable of generating the macroscopic responses of the assembly

615

of the non-spheres as well as the local stress and strain ﬁelds inside each particle. The latter information

616

is crucial for predicting hysteresis effect due to damage, fracture and fragmentation of particles. The

617

incorporation of the stress and energy ﬂux to predict fracture of grains and the corresponding macroscopic

618

responses of granular assembles will be considered in future studies.619

5.4 Conclusions620

In this work, we model the multi-body frictional contacts using level sets in voxel meshes. The boundaries

621

of the bodies are implicitly represented by level sets and an unbiased contact reference is constructed to

622

deﬁne the gap functions. The Coulomb friction law is analogized to an elastoplastic constitutive law to

623

implement the regular return-mapping algorithm to unify the treatment for slip and stick states. We store

624

the information of bodies by a set of material points and the level set variable is stored at the background

625

28 Chuanqi Liu, WaiChing Sun

n=5 n=10

n=15

Shear increase

Fig. 27: Distribution of the maximum principal stress for the simple shear loading. The left subﬁgures are

these distributions for a speciﬁc particle and its boundaries at different steps.

1.8

2.1

2.4

2.7

3

3.3

3.6

0 0.01 0.02 0.03 0.04 0.05

Fn(107N)

γ

ϕ= 15o

ϕ= 30o

0.3

0.6

0.9

1.2

1.5

0 0.01 0.02 0.03 0.04 0.05

Ft(107N)

γ

ϕ= 15o

ϕ= 30o

Fig. 28: Evolutions of normal and tangential forces for the simple shear test with different friction angles.

mesh. After each step, both the material points and level sets are updated. The information of material

626

points is reconstructed to the traditional Gauss’s quadrature points of the elements discretized the updated

627

boundaries. Examples are given to show the veriﬁcations and effectiveness of the proposed method to handle

628

contacts involving multiple bodies and multiple loading steps. In this work, we bypass the generations

629

of conformal meshes and constructing cumbersome master-slave contact pairs. By using the voxel-based

630

meshes, the number of DOFs dramatically reduces. Compared to the DEM, this method provides more

631

information for frictional contact problems. Since we know the stress ﬁeld, in the future, we will consider

632

the damage or crush for the particles. Thus, the macroscopic loading curves would be more accurate. We

633

aim to establish a bridge between real grain images and macroscopic constitutive laws.634

implicit level-set-based material point method 29

5.5 Acknowledgments635

The authors are supported by by the Dynamic Materials and Interactions Program from the Air Force

636

Ofﬁce of Scientiﬁc Research under grant contracts FA9550-17-1-0169 and FA9550-19-1-0318, and the NSF

637

CAREER grant from Mechanics of Materials and Structures program at National Science Foundation under

638

grant contracts CMMI-1846875. These supports are gratefully acknowledged. The views and conclusions

639

contained in this document are those of the authors, and should not be interpreted as representing the

640

ofﬁcial policies, either expressed or implied, of the sponsors, including the Army Research Laboratory

641

or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for

642

Government purposes notwithstanding any copyright notation herein. The views and conclusions contained

643

in this document are those of the authors, and should not be interpreted as representing the ofﬁcial

644

policies, either expressed or implied, of the sponsors, including the Army Research Laboratory or the U.S.

645

Government. The U.S. Government is authorized to reproduce and distribute reprints for government

646

purposes notwithstanding any copyright notation herein.647

30 Chuanqi Liu, WaiChing Sun

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