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ILS-MPM: An implicit level-set-based material point method for frictional particulate contact mechanics of deformable particles

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Finite element simulations of frictional multi-body contact problems via conformal meshes can be challenging and computationally demanding. To render geometrical features, unstructured meshes must be used and this unavoidably increases the degrees of freedom and therefore makes the construction of slave/master pairs cumbersome. In this work, we introduce an implicit material point method designed to bypass meshing of bodies by employing level set functions to represent boundaries at structured grids. This implicit function representation provides an elegant means to link an unbiased intermediate reference surface with the true boundaries by closest point projection as shown in [Leichner et al., 2019]. We then enforce the contact constraints by a penalty method where the Coulomb friction law is implemented as an elastoplastic constitutive model such that a return mapping algorithm can be used to provide constitutive updates for both the stick and slip states. To evolve the geometry of the contacts properly, the Hamilton-Jacobi equation is solved incrementally such that the level set and material points are both updated according to the deformation field. To improve the accuracy and regularity of the numerical integration of the material point method, a moving least square method is used to project numerical values of the material points back to the standard locations for the Gaussian-Legendre quadrature. Several benchmarks are used to verify the proposed model. Comparisons with discrete element simulations are made to analyze the importance of stress fields in predicting the macroscopic responses of granular assemblies.
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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
(i)
with boundaries of
(i)
, where
i∈ I :=167
{1, ·· · ,N}index the solids. There exists a finite bounding box Bcontaining all (i), and we may write:168
[
i∈I
(i)=:⊂ B ⊂ Rd, (1)
where
d
is the number of spatial dimensions. Similar to
, we can define
Rd1
as a union of
(i)
, and
169
assume
=ΓdΓnΓc
with
ΓdΓn=
,
ΓdΓc=
and
ΓnΓc=
, 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
Fig. 1: Schematic of contact problems
and183
gt(u) = γτt(u), where (γ=0, if |τt(u)|<µ|τn(u)|
γ0, 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
(i)
and
(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)λndγen
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)|,˙
γ0, ˙
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
=0x
>0x∈ B/¯
, (14)
where
¯
=
constitutes the interior body and boundary, and
B/¯
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, (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
, an extension algorithm of
248
u(x)
to
B/¯
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
(i)
and
(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
(ij)
c:=nx|Φ(i)(x)>0Φ(j)(x)>0reinitialize(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 xΓ(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 (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
(i)
and
(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
(i)
and
284
(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
xΓ(ij)
c
(shown in Fig. 4b) can be mapped onto the true boundary
289
of the body
(i)
via a closest point projection
P(i)(x):Γ(ij)
c(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 xΓ(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):(i)Rd
is the displacement field of the
i
-th body. As shown in Fig. 4b, the normal direction of
296
the contact at
xΓ(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 (i)and (j)is denoted as:306
A(ij):={xΓ(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) = Z
σ(u):ε(w)dxf(w):=Zb·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
uH1()
, 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, wH1
0(). (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
(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
(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
(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=1NI(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
xΓc
are within elements
(i),e
and
(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
(i),e
and
(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)
tgp,(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)
tγ
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
uBrA(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)7B.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
0.80.4 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(1ν2)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
aF
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
30for 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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... For each packing lattice, similar to the procedure adopted in [40], we prepare an initial configuration by stacking disks with slightly larger radii 5.01 mm and relaxing the configuration with the four boundaries being held fixed. We apply this protocol to generate a very small overlap between two disks. ...
... where J k c (:, m) means the m-th column of J k c , and m represents the global indexing of a node of a solid body along either the x or the y degree of freedom (in 2D). e m,k is a unit vector with all entries being zero except the m-th one which has a value of h with h being a small value that takes the following form similar to [40]: ...
... We take this role-interchanging step as being beneficial to minimize the potential bias [35,45,46] that can arise in the computed contact forces. Alternatively, one can perform contact computations on a so-called neutral contact surface [47,48,40] without computing contact forces twice. Finally, our implementation is outlined by Algorithm 1 (see Appendix) where we leverage the sparse representations and solvers available in the open-source library Eigen [49] to efficiently store K and J k c , and to solve for U a,k . ...
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Under external perturbations, inter-particle forces in disordered granular media are well known to form a heterogeneous distribution with filamentary patterns. Better understanding these forces and the distribution is important for predicting the collective behavior of granular media, the media second only to water as the most manipulated material in global industry. However, studies in this regard so far have been largely confined to granular media exhibiting only geometric heterogeneity, leaving the dimension of mechanical heterogeneity a rather uncharted area. Here, through a FEM contact mechanics model, we show that a heterogeneous inter-particle force distribution can also emerge from the dimension of mechanical heterogeneity alone. Specifically, we numerically study inter-particle forces in packing of mechanically heterogeneous disks arranged over either a square or a hexagonal lattice and under quasi-static isotropic compression. Our results show that, a hexagonal packing exhibit a more heterogeneous inter-particle force distribution than a square packing does. For both packing lattices, preliminary analysis shows the consistent coexistence of outliers (i.e., softer disks sustaining larger forces while stiffer disks sustaining smaller forces) in comparison to their homogeneous counterparts, which implies the existence of nonlocal effect. Further analysis on the portion of outliers and on spatial contact force correlations suggest that the hexagonal packing shows more pronounced nonlocal effect over the square packing under small mechanical heterogeneity. However, such trend is reversed when assemblies becomes more mechanically heterogeneous. Lastly, we confirm that, in the absence of particle reorganization events, contact friction merely plays the role of packing stabilization while its variation has little effect on inter-particle forces and their distribution.
... On the contrary, with BFEMP, MPM particles can freely move around outside the FEM mesh. To verify the accuracy of the contact model, BFEMP is studied on the Brazilian disk test, which is a special case of the plane Hertzian contact problem [86,87]. The Brazilian disk test can be used for tensile strength testing, which involves a 2D elastic disk squeezed between two rigid objects. ...
... The Hertzian model requires to measure the contact radius a. Following [87], we use half of the horizontal range of the particles within the contact distance around the bottom FEM plate to approximate it. Here we test both linear elasticity and neo-Hookean elasticity. ...
Preprint
Full-text available
This paper introduces BFEMP, a new approach for monolithically coupling the Material Point Method (MPM) with the Finite Element Method (FEM) through barrier energy-based particle-mesh frictional contact using a variational time-stepping formulation. The fully implicit time integration of the coupled system is recast into a barrier-augmented unconstrained nonlinear optimization problem. A modified line-search Newton's method is adopted to strictly prevent material points from penetrating the FEM domain, ensuring convergence and feasibility regardless of the time step size or the mesh resolutions. The proposed coupling scheme also reduces to a new approach for imposing separable frictional kinematic boundaries for MPM when all nodal displacements in the FEM domain are prescribed with Dirichlet boundary conditions. Compared to standard implicit time integration, the extra algorithmic components associated with the contact treatment only depend on simple point-segment (or point-triangle in 3D) geometric queries which robustly handle arbitrary FEM mesh boundaries represented with codimension-1 simplices. Experiments and analyses are performed to demonstrate the robustness and accuracy of the proposed method.
... With they researching on these problems, how to deal with the deformable boundaries of the solid model in Euler grids has been a topic of great interest for centuries (e.g.,[6-10]). Prof. Liu Chuanqi and his colleagues [11] have done a thorough research on this problem ,they proposed an implicit material point method designed to bypass meshing of bodies by employing level set functions to represent the boundaries at Euler grids. This implicit function representation provides an elegant mean to link an unbiased intermediate reference surface with the true boundaries by closest point projection as shown in Leichner et al. (2019). ...