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Abstract and Figures

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.
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BFEMP: Interpenetration-Free MPM-FEM Coupling with Barrier Contact
Xuan Lia,1,, Yu Fanga,b,1, Minchen Lia, Chenfanfu Jianga,b
aDepartment of Mathematics, University of California, Los Angeles, United States
bDepartment of Computer and Information Science, University of Pennsylvania, United States
Abstract
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 separa-
ble 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 as-
sociated 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.
Keywords: Material Point Method, MPM-FEM Coupling, Implicit Integration, Barrier Method, Frictional Contact
1. Introduction
The Material Point Method (MPM) [1,2] extends the Particle-In-Cell (PIC) [3] and the Fluid Implicit Particle
(FLIP) [4] methods from fluid dynamics to computational solids. In contrast to the commonly used Total Lagrangian
Finite Element Method for elastodynamics [5], MPM utilizes Lagrangian particles to represent continuum materials
and an Eulerian background grid to discretizes the governing equations. Except for recent advancements in Total La-
grangian MPM [6,7], MPM is usually considered to be following an Updated Lagrangian kinematic assumption with
particles tracking historical deformation, strain, stress, and other constitutive variables through evolving them with
velocity fields. The hybrid Lagrangian-Eulerian perspective combined with the Updated Lagrangian kinematics puts
MPM in a very advantageous position in modeling and simulating high-speed, large-deformation, and topologically
changing events [8]. Having gained a lot of attention in the last two decades, MPM and its variants [913] have been
successfully applied in challenging problems including multiphase flows, fracture, contact, adaptivity, free-surface
flows, soil-fluid mixture, explosives, and granular media [1422].
Although MPM has been demonstrated eective on a wide range of materials, many application scenarios favor
other discretization schemes due to considerations in eciency, accuracy, and suitability. Correspondingly, a large
number of engineering applications necessitate hybrid or coupled solvers combining MPM with other discretization
Corresponding Author
Email addresses: xuanli1@math.ucla.edu (Xuan Li), minchen@math.ucla.edu (Minchen Li), cffjiang@math.ucla.edu
(Chenfanfu Jiang)
1These authors contributed equally to this work
choices. For example, MPM has been coupled or hybridized with the Discrete Element Method (DEM) for solid-fluid
interaction [23] and granular media [2426], the Finite Dierence Method (FDM) for multiphase saturated soils [27],
and the Smoothed Particle Hydrodynamics (SPH) for solid mechanics [28].
Even though MPM itself can be derived as a Galerkin Finite Element Method, it is still more common in com-
putational solid mechanics to use the term “FEM” to refer to the standard Total Lagrangian mesh-based FEM dis-
cretization. In this sense, MPM is much less developed than FEM and still suers from unique challenges in aspects
such as stability, accuracy, boundary condition enforcement, and numerical fracture [2931]. Therefore, FEM is often
more suitable for analyzing hyperelastic structures under small or moderate deformations. Resultingly, MPM-FEM
coupling becomes highly desirable in many multi-material simulation tasks or those involving strongly heterogeneous
deformations, for example, the blast event simulations involving vehicles [32]. The coupling between MPM and FEM
has been extensively studied over the last decade. A natural way to hybridize the two schemes is to treat FEM vertices
as MPM particles and embed them into the MPM grid [33,34]. The FEM shell formulation can also be embedded into
the MPM grid [35]. In a similar fashion, EMPFE [36] discretizes the entire domain according to the severity of defor-
mation – small deformation regions with FEM, while large deformation regions with MPM, and then it embeds the
displacement of interface FEM vertices to the MPM grid. Despite its simplicity, additional treatment for eliminating
the hourglass mode is necessary due to simple trilinear MPM kernels for the interface computation. Later on, AFEMP
[37] extends this idea to support the dynamical conversion from severely distorted FEM elements to MPM particles.
These methods handle material interactions through the grid-based MPM contact and inherit common MPM-based
contact characteristics such as the strict nonslip condition between contacting interfaces.
To circumvent these issues, CFEMP [38] was proposed to only use the interface MPM grid for contact detection
while computing frictional contact forces directly based on the contact conditions. CFEMP has been successfully
applied to coupling FEM membranes with MPM solids [39], and also to modeling needle-tissue interactions [40].
However, these grid-based MPM-FEM coupling strategies require the FEM boundary element size to be similar to
the MPM grid spacing. If it is too large, interpenetration can happen, while if it is too small, intrinsic damping will
appear [33], and the time step size for explicit time integration would also be more restricted. Accordingly, Cheon and
Kim [41] proposed to add extra distributed interaction (DI) nodes on the FEM boundary elements to improve contact
detection for large-sized elements.
An improvement to CFEMP that also applies to AFEMP was later proposed to couple FEM with MPM by han-
dling contact primitive pairs between FEM boundary elements and nearby MPM particles [42,43]. A penalty method
is applied for computing the frictional contact forces. Later, Song et al. [44] extended this idea with an iterative con-
tact force computation approach for the simultaneous satisfaction of all contact conditions, together with an improved
local search method to prevent interpenetration issues at the contact crack. Bewick [45] proposed to insert interme-
diate nodes at the interface for 1D impact-resistant design problems, and the coupling forces are calculated by FEM
displacements which are determined by MPM particles.
So far, all the discussed MPM-FEM coupling works are designed for explicit time integration. Imposing frictional
contact between FEM and MPM within implicit time integration is challenging because the associated inequality con-
straints that need to be simultaneously enforced while solving the nonlinear system of equations are also nonlinear
and non-smooth. Aulisa et al. [46] proposed a monolithic coupling method for implicit MPM and FEM through a
conforming interface mesh, while extra care is needed to avoid the sticky artifact in receding contact cases. Larese
et al. [47] discussed implicit MPM-FEM coupling researches in geomechanics. In a soil-structure interaction prob-
lem, the impact forces on the interface are transferred between the soil and the structure solver and iterated until
convergence. A similar method for enforcing nonconforming boundary conditions for MPM [48] was also applied.
However, as pointed out in their work, the method requires smaller time increments for problems with high relative
velocity towards the boundary, as otherwise, the boundary enforcement will be too late, and the incoming material
points may penetrate the nonconforming boundary surfaces.
2
This paper explores MPM-FEM coupling under the assumption of implicit integration in both domains. Compared
to explicit time integration, implicit schemes permit substantially larger time steps with superior stability for sti
nonlinear problems. Implicit MPM/GIMP has been explored by many researchers [29,30,4952]. We refer to these
literature for more discussions about the advantages of implicit time integration. Our work follows the variational
formulation of a wide family of implicit time integrators [53,54], where the displacement evolvement in each time
step is formulated as the stationarity point of a time discretized energy functional. The resulting optimization-based
time integrator has been applied in MPM with Newton-Krylov [55], and more recently, quasi-Newton L-BFGS [56]
solvers.
In this work, integrating both MPM and FEM domains implicitly, we study MPM-FEM coupling based on con-
tact mechanics (thus we do not consider objects with partially mixed discretization choices). A barrier-augmented
variational frictional contact formulation is known as the Incremental Potential Contact (IPC) [5759] was recently
proposed for nonlinear elastodynamics with linear kernel FEM, which has also shown to be eective for codimen-
sional models [60], reduced space dynamics [61,62], and embedded interfaces [63]. It formulates the contact problem
during time stepping as minimizing a potential energy inside the manifold of interpenetration-free displacement tra-
jectories characterized by boundary geometric primitives of elastic structures. Extending this approach, we model
MPM-FEM coupling as jointly finding optimal FEM mesh nodal displacements and MPM grid nodal displacements
under the constraint that MPM particles, with their trajectories embedded in grid nodal displacements, maintain strict
positive distances to the FEM mesh throughout the implicit integration. The resulting method is named barrier FEMP
(BFEMP) because these constraints are enforced using barrier energies. Even though contact conditions are defined
between MPM particles and the FEM mesh, the real displacement unknown variables for the MPM domain which the
implicit time integration solves for are still defined on MPM grids. The MPM particles remain embedded quadrature
points in the MPM grid degrees of freedom. Compared to soft penalty-based methods such as the particle-to-surface
contact algorithm recently proposed by Nakamura et al.[64], BFEMP requires no stiness parameter tuning and guar-
antees strict, hard non-penetration conditions under convergence. Another useful feature of the proposed coupling
scheme is that it enables a new way of imposing irregular, separable, and frictional kinematic boundaries for MPM.
Irregular boundaries for MPM is a recently advanced topic [65]. BFEMP inherently enables it by assigning all nodes
in the FEM domain with prescribed Dirichlet displacements and letting MPM interacts with them.
2. Governing Equations
In this study we focus on elastodynamics based on continuum mechanics. The corresponding governing equations
for a deformed continuum domain twith xtand time t[0,) are given by [66]
Dρ
Dt +ρx·v=0,(1)
ρDv
dt =x·σ+ρg,(2)
where ρ(x,t) is the density, v(x,t) is the velocity, σ(x,t) is the Cauchy stress, and gis the gravitational acceleration
which is assumed to be the only body force. Under the finite strain assumption (as we shall assume throughout this
paper), deformation φ(X,t) maps X0from the material space to xtin the world space: x=φ(X,t). The
deformation gradient is defined as
F=∂φ
X(X,t) (3)
to describe the local deformation.
3
In Lagrangian Finite Element analysis for nonlinear dynamics, it is often preferred to derive the weak form for
X0. Here we can pull back the momentum equation to the material space. Denoting the first Piola-Kirchho
stress P=P(X,t), the Lagrangian momentum equation is then
R0A(X,t)=X·P(X,t)+R0g,(4)
where R0=R(X,0) is the material density at time 0, and A(X,t)=2Φ
t2(X,t) is the Lagrangian acceleration, V(X,t)=
Φ
t(X,t) is the Lagrangian velocity. The Cauchy stress is related to the first Piola Kirchhostress as
σ=det(F)1PFT.(5)
In this paper we are concerned with hyperelasticity. Thus there exists an elastic energy density function ψ(F) such
that
P(F)=∂ψ
F(F).(6)
Without loss of generality, we focus on isotropic materials and adopt a compressible Neo-Hookean constitutive model
with
ψ(F)=µ
2tr(FTFI)µlog(J)+λ
2log(J)2,(7)
where J=det F,µand λare the Lam´
e parameters.
Remark 1.Many hyperelasticity models such as the Neo-Hookean model are only well defined for J>0. Discretely
this will impose a nonlinear strict inequality constraint on the displacements for each quadrature point with a discrete
sample of F. Ignoring these constraints in a nonlinear optimization-based Newton solver will cause floating-point
number failures when a search step tries to evaluate energy or stress quantities at an intermediate configuration with
J0. Instead of requiring a reduction of the time step, we directly enforce this constraint through a line search
filtering strategy (see Section 5).
2.1. Weak Form
Given trial function Q(·,t) : 0R3, the corresponding Lagrangian weak form of Equation (4) is
Z0
Qα(X,t)R0Aα(X,t)dX=Z0
QαTαdS (X)Z0
Qα,βPαβ dX+Z0
Qα(X,t)R0gαdX,(8)
where Tα=PαβNβ(with N(X) being the material space normal) is the traction field at the domain boundary 0, on
which one could presribe traction boundary conditions as needed.
While Total Langrangian FEM typically discretizes Equation (8), MPM usually adopts the Updated Lagrangian
view and consequently discretizes an Eulerian weak form instead [8]. Correspondingly the stress derivatives are
discretized at the current configuration t. We can either push forward Equation (8) or directly integrate Equation (2)
to reach
Zt
qα(x,t)ρ(x,t)ai(x,t)dx=Zt
qαtαds(x)Zt
qα,βσαβ dx+Zt
qα(x,t)ρ(x,t)gαdx,(9)
where q(x,t)=Q(Φ1(x,t),t) is the push forward of Q,i.e., an Eulerian trial function, a(x,t)=A(Φ1(x,t),t)=
Dv
Dt (x,t)=v
t+v· ∇v, and t=σnis the traction at twith n(x) being the outward pointing normal.
4
2.2. Incremental Variational Form
To enable the development of ecient optimization-based time integrators, we follow the variational treatment
of the time-discretized incremental problem [53,67]. Concretely, for a broad family of time discretization schemes
(we will focus on backward Euler and the Newmark-βfamily in this paper), the solution of Equation (8) during an
incremental time step, i.e., the advancement from φn=φ(X,tn) to φn+1=φ(X,tn+1)), for an hyperelastic material is
given by minimizing the following functional:
I(φn+1)=Z0 1
2
R0
βt2kφn+1k2+2αΨ(Fn+1)!dXZ0
R0¯
Bn+1·φn+1dX2αZ0
T·φn+1dS (X),(10)
where
¯
Bn+1=2αg+1
βt2 φn+ ∆t∂φ
t(X,tn)+α(12β)t22φ
t2(X,tn)!(11)
encodes the inertia term and is a constant field in the minimization problem. The velocity update rule is
∂φ
t(X,tn+1)=∂φ
t(X,tn)+ ∆t (1γ)2φ
t2(X,tn)+γ2φ
t2(X,tn+1)!.(12)
Here we have slightly modified the energy proposed by Radovitzky and Ortiz [67] to let it be compatible with back-
ward Euler since their version was explicitly built for Newmark’s algorithm.
Note that in the case of backward Euler (α=1, β=1
2,γ=1), it can be easily shown that the Euler-Lagrangian
equation of functional (10) gives back the time-discretized momentum balance
R0
t2φn+1− ∇X·Pn+1=R0g+R0
t2(φn+ ∆tVn).(13)
Similarly, for Middle-point Newmark (α=1
2,β=1
4,γ=1
2), we would get
R0
t2φn+11
4X·Pn+1=1
4R0g+R0
t2(φn+ ∆tVn+1
4t2An).(14)
Both of them match the results from temporally discretizing the Lagrangian form of Equation (2).
2.3. Coupling
In the proposed framework, the Lagrangian form is discretized on the FEM domain F(Section 3.1), while the
Eulerian form is discretized on the MPM domain M(Section 3.2). We assume
0
F0
M=(15)
and their corresponding deformation map xM=φM(XM,t) and xF=φF(XF,t) are fully governed by their own
variational forms in the absence of any coupling mechanism. In other words, without coupling, evolving the two
domains independently (with two solvers) is equivalent to minimizing
Π(φM, φF)=I(φM)+I(φF) (16)
by directly letting Ω=ΩFM.
5
ϕn
M
ϕn+1
F
Ω0
M
Ω0
F
Ωn+1
M
Ωn+1
F
Material space World space
Ωn
M
Γ0
M
Γn
M
Γn+1
M
Δϕn
M
Γ0
FΓn+1
F
Figure 1: Deformation map. The material space of FEM and MPM domains (0
Fand 0
M) on the left are mapped via φFand φMto the world space
(n+1
Fand n+1
M) on the right, with t
Ft
M=for all t[0,). With Updated Lagrangian kinematics, MPM treats n
Mas the “intermediate”
material space and focuses on studying the deformation from n
Mto n+1
M. Here Γrepresents Dirichlet boundary or nonzero Neumann boundary.
The coupling between FEM and MPM domains are then modeled via imposing the non-interpenetration con-
straints:
t
Ft
M=,t[0,) (17)
between the two domains. Note that the time is discretized with t=t0,t1,t2,...,tn. We use nto denote tnfor
notational simplicity. See Figure 1for an illustration. Thus the final variational MPM-FEM coupling problem can be
described as minimizing Equation (16) under the equality constraint described by Equation (17).
The feasible region described by Equation (17) can be equivalently expressed as
φM(XM,t),φF(XF,t),XM0
M,XF0
F,t[0,).(18)
Moreover, if we define
d(φM, φF,t)=min
XF,XM
kφF(XF,t)φM(XM,t)k(19)
to describe the Euclidean proximity between t
Mand t
F, Equation (18) can be further converted to a strict inequality
constraint
d(φM, φF,t)>0,t[0,).(20)
Note that in Equation (15), we have assumed that the undeformed domains are non-overlapping. Thus the minimiza-
tion problem starts with a strictly feasible solution at t=0.
In Section 4.1 we describe a barrier method that results in a contact pressure for enforcing Equation (15). See
Section 4.3 for extra components on incorporating tangential frictional eects.
6
3. Discretization
The finite element domain is discretized with linear simplex elements (triangles in 2D and tetrahedra in 3D),
and the material point domain is discretized using a collection of material points and an Eulerian background grid
with quadratic B-spline kernels. Both schemes adopt mass lumping and assume zero traction at boundaries unless
otherwise specified in an example. Stacking all nodal positions, velocities, and accelerations from both the FEM
mesh and the MPM grid at time step nas xn=[(xn
F)T,(xn
M)T]T,vn=[(vn
F)T,(vn
M)T]T, and an=[(an
F)T,(an
M)T]Twhere
subscripts F stands for FEM while M for MPM, the unified time integration update rule can be written as
˜vn+1=vn+ ∆t((1 γ)an+γan+1),
˜xn+1=xn+ ∆tvn+αt2((1 2β)an+2βan+1)),(21)
and it is equivalent to first solving the optimization problem
min
x: 1
2kxˆxnk2
M+2αβt2Ψ(x)!(22)
to get ˜xn+1, and then explicitly calculating ˜vn+1. Here ˆxn=xn+vnt+α(1 2β)t2an, and Ψ(x)=PqV0
qψ(Fq(x))
with qbelonging to FEM elements or MPM particles and V0
qthe rest volume of a FEM element or a MPM particle.
For FEM, the time integration is solely performed on the Lagrangian nodal degrees of freedom throughout the
simulation and so ˜xn+1
F=xn+1
Fand ˜vn+1
F=vn+1
F. But for MPM, Equations (21) are only part of the Eulerian time
integration performed on the Eulerian grid before and after particle-grid transfers for xMand vM, so ˜xn+1
M,xn+1
Mand
˜vn+1
M,vn+1
M(see Section 3.2 for details). Note that the minimization problem (22) is equivalent to the discrete form
of the variational time integration in [53,56,68] for hyperelastic problems. xFand xMare coupled through contact
modeling between the two domains (Section 4).
3.1. The Finite Element Domain
For the FEM domain, nodal positions and velocities are stored on mesh vertices and updated directly. The nodal
masses are kept constant, i.e.,mn
i=mi.
Inside any simplex element, the material and world space coordinate of an arbitrary location Xare expressed using
linear interpolation kernels Ni(X) of node Xias
X=X
i
Ni(X)Xiand φ(X,t)=X
i
Ni(X)φ(Xi,t).(23)
Then according to the definition of F(Equation (3)), with Nibeing the linear hat function, the deformation gradient is
piecewise constant. Inside a linear simplex element eit is directly evaluated as a function of x:
Fe(x)=Te(x)B1
e,(24)
where Te(x) is the current triangle basis of element eand Beis the triangle basis of element ein material space.
3.2. The Material Point Domain
For the MPM domain, the nodal positions xn
Mare the uniform Cartesian grid coordinates at each time step. The
grid velocity vn
iand mass mn
iare transferred from particles. The nodal movements are conceptual. ˜xn+1
Mand ˜vn+1
Mwill
be transferred back to particles for advection.
Similar to FEM, each MPM grid node iis associated with a kernel function Ni(x) for the grid to represent the con-
tinuous field. Note that the kernel is defined in terms of xrather than Xbecause the grid is essentially a discretization
7
of n
M– a direct consequence of adopting Updated Lagrangian kinematics. When Niis evaluated at a particle location
xn
q, a shorter notation Ni(xn
q)=wn
iq is from now on used instead. Here Nidirectly takes the current particle location xn
q
as input as opposed to FEM because there is no globally defined reference configuration in MPM and the deformation
is evolved over time steps rather than recomputed using a rest shape. More specifically, the deformation gradient of a
particle qis defined as
Fq(x)=X
i
xi(wn
ip )TFn
q.(25)
In this paper, without loss of generality, we adopt the quadratic B-spline kernel for Ni(x) to avoid MPM’s cell-crossing
instability [69]. Other kernels based on the NURBS [70], Generalized Interpolation Material Point Method (GIMP)
[9], Convective Particle Domain Interpolation (CPDI) [31,71,72], or the Dual Domain Material Point (DDMP)
[73,74] can also be directly used in our framework.
To transfer information between the particles and the grid, we implemented options including the Ane Particle-
In-Cell (APIC) method [34,75] (Table 1), Particle-In-Cell (PIC) method [3] (Table 2), and the Fluid-Implicit Particle
(FLIP) method [4] (Table 3). Note that other transfer schemes such as XPIC [76] can also be applied in our framework
in a straightforward manner.
Particles to grid (APIC) Grid to particles (APIC)
mn
i=X
p
mpwn
ip
Dn
p=X
i
wn
ip (xn
ixn
p)(xn
ixn
p)T
mn
ivn
i=X
p
wip mn
p(vn
p+Bp(Dp)1(xn
ixn
p))
vn+1
p=X
i
˜
vn+1
iwn
ip
xn+1
p=X
i
˜
xn+1
iwn
ip
Bn
p=1
2X
i
wn
ip ˜
vn+1
i(xn
ixn
p+˜
xn+1
ixn+1
p)T
+(xn
ixn
p˜
xn+1
i+xn+1
p)(˜
vn+1
i)T
Fn+1
p=X
i
˜
xn+1
i(wn
ip )TFn
p
Table 1: APIC Particle-Grid Transfer
Particles to grid (PIC) Grid to particles (PIC)
mn
i=X
p
mpwn
ip
mn
ivn
i=X
p
wn
ip mn
pvn
p
xn+1
p=X
i
˜
xn+1
iwn
ip
vn+1
p=X
i
˜
vn+1
iwn
ip
Fn+1
p=X
i
˜
xn+1
i(wn
ip )TFn
p
Table 2: PIC Particle-Grid Transfer
8
Particles to grid (FLIP) Grid to particles (FLIP)
mn
i=X
p
mpwn
ip
mn
ivn
i=X
p
wn
ip mn
pvn
p
xn+1
p=X
i
˜
xn+1
iwn
ip
vn+1
p=vn
p+X
i
wn
ip (˜
vn+1
ivn
i)
Fn+1
p=X
i
˜
xn+1
i(wn
ip )TFn
p
Table 3: FLIP Particle-Grid Transfer
4. The Contact between Domains
4.1. Contact Potential
Recently for Lagrangian FEM, Li et al.[57,59] proposed a consistent variational contact model that smoothly
approximates the nonsmooth contact phenomena with bounded error, and demonstrated its convergence under re-
finement for piecewise linear boundary discretization. Here we customize the FEM contact potential [59] to the
FEM-MPM coupling setting by defining the inter-surface contact potential between surfaces Mand Fto be
Z0
M
b( min
xft
F
dPP(xm,xf),ˆ
d)dXm,(26)
where dPP(xm,xf)=kxmxfkis the point-point distance function, and
b(d,ˆ
d)=
κ(d
ˆ
d1)2ln ( d
ˆ
d) 0 <d<ˆ
d
0dˆ
d(27)
is a smoothly clamped barrier function that serves as the contact energy density with dthe input distance, ˆ
da small
distance threshold below which contact activates, and κin Pa the barrier stiness [57,59], which scales the magnitude
of contact forces at a certain distance.
Intuitively, minxft
FdPP(xm,xf) is the distance between a material point xmt
Mand surface t
F, and Equation
(26) can be viewed as an integration of the point(xm)-surface(t
F) contact energy density over surface 0
M. The
barrier function bsmoothly increases from 0 to infinity as the input distance decreases from ˆ
dto 0, providing arbitrarily
large repulsion to ensure no interpenetration and at the same time bound the contact gap error within ˆ
d(Figure 2).
As ˆ
d0, the approximation error between the contact energy density function band the real contact phenomenon
described in Equation (20) decreases, which also makes Mand Finterchangeable in the limit. Note that the
integration is performed in the material space while the distance is evaluated in the world space.
4.2. Discretization
After applying FEM and MPM discretization schemes, assuming 2D, the contact potential Equation (26) is dis-
cretized to be
B(x)=X
q∈Q
ωqb(min
e∈B dPE (xq,e),ˆ
d),(28)
where Qis the set of all MPM particles, ωqis the integration weight (equivalently, the boundary area) of MPM particle
q,Bis the set of FEM boundary edges, and dPE (xq,e) is the point-edge distance between particle xqand edge e. Note
9
0 0.5 1 1.5
Distance
0
1
2
3
Barrier energy density
Figure 2: The barrier energy density function Equation (27) plotted with dierent ˆ
d.Decreasing ˆ
dasymptotically matches the discontinuous
definition of the contact condition.
that the MPM grid nodal positions ˜xMto be solved and the particle positions xqafter advection using ˜xMare linearly
related through the particle-grid transfer kernel (Figure 3right), and so the contact force on the MPM nodal degrees
of freedom can be calculated by applying the chain rule:
B
xM
=X
q∈Q xq
xM!TB
xq
,(29)
where xq
xM!α,id+β
=δαβwn
iq
with α, β =1,2, ..., dand d=2 or 3 the spatial dimension since xq=Piwn
iqxi.
Ideally, ωqshould be zero for interior particles. It should reveal the proportional surface area of boundary particles.
Since FEM boundary elements are always outside the MPM domain and the barrier function bis only activated at a
small distance, the activation of bcan be applied to conveniently decide whether an MPM particle is at the boundary or
the interior without explicitly identifying the MPM domain boundary in each time step (Figure 3left). Then assuming
a close to uniform particle distribution to be maintained throughout the simulation, ωqcan be set to 2 qV0
qin 2D
and π3V0
q/(4π)2
3in 3D for all particles, which is the area of the largest cross section of a spherical particle q.
The minimization operator in the potential (Equation (28)) helps to compute the point-polyline distance from
point-edge distances. As the barrier function bis monotonically decreasing, the potential can be rewritten as
B(x)=X
q∈Q
ωqmax
e∈B b(dPE (xq,e),ˆ
d).(30)
Due to the existence of a max operator, it is only C0continuous, making the incremental potential challenging to be
eciently minimized by gradient-based optimization methods like Newton’s method. Since the barrier function bis
with local support around each boundary element, and that it maps a majority of the activate distances to tiny potential
values (Figure 4), the maximization of the potential field can be well approximated by summation, with the duplicate
potential around FEM boundary nodes compensated by subtraction as proposed by Li et al. [59]:
B(x)=X
q∈Q
ωq
X
e∈B
b(dPE (xq,e),ˆ
d)X
kˆ
B
(ηk1)b(dPP(xq,xk),ˆ
d)
,(31)
10
̂
d
MPM Grid
MPM Particles
FEM Mesh
Contact Pair
MPM Node
P2G Transfer
Figure 3: Contact constraint pairs. Left: Contact activates on all pairs of MPM particles and FEM boundary elements with distance below ˆ
d.
Right: Contact force is transferred from MPM particles to MPM grids via chain rule.
where ˆ
Bis the set of all FEM boundary nodes and ηkis the number of FEM boundary edges incident to node k. For
closed manifold domains in 2D, ηk=2 for all k.
Similarly, in 3D, the discretized contact potential becomes
B(x)=X
q∈Q
ωq
X
t∈B
b(dPT (xq,t),ˆ
d)X
eˆ
B
(ηe1)b(dPE (xq,e),ˆ
d)+X
xp˜
B
b(dPP(xq,xp),ˆ
d)
,(32)
where Bis now the set of all FEM boundary triangles, ˆ
Bis the set of all edges on the FEM boundary with ηethe number
of FEM boundary triangles incident to edge e,˜
Bthe set of all nodes that are in the interior of the FEM boundary surface
mesh, and dPT (xq,t) the point-triangle distance between particle xqand triangle t. For closed manifold domains in
3D, ηe=2 for all e.
Adding the contact potential into the incremental potential, the minimization problem for time integration is now
fully unconstrained:
min
x:1
2kxˆxnk2
M+2αβt2(Ψ(x)+B(x)).(33)
Since the distance values measured for contacting MPM particle - FEM simplex pairs are all unsigned, Problem (33)
may contain a local optimum at configurations with intersections. To be consistent with the continuous constraint
Equation (20), it is also constrained that the iterates always stay in the feasible region on one side of the barrier
without crossing. This is achieved by applying the interior-point filter line-search algorithm [77] with continuous
collision detection (CCD) [60,78].
4.3. Friction
To model frictional contact, local frictional forces Fkcan be added for every active contact pair k. For each such
pair k, at the current state x, a consistently oriented sliding basis Tk(x)Rdm×(d1) can be constructed, where mis
the total number of colliding nodes and dis the dimension of space, such that uk=Tk(x)T(tvn+ ∆t2((1 γ)an+
11
Barrier energy
Distance
Figure 4: A visual demonstration in 2D of (left) the unsigned distance function to a segmented mesh, and (right) the corresponding barrier function
(27) visualized with an exaggerated ˆ
dparameter.
γan+1)) Rd1provides the local relative sliding displacement in the frame orthogonal to the distance gradient. The
corresponding sliding velocity is then vk=uk/tRd1.
Maximizing dissipation rate subject to the Coulomb constraint defines friction forces variationally [79,80]
Fk(x)=Tk(x) argmin
βRd1
βTvks.t. kβk ≤ µλk,(34)
where λkis the contact force magnitude and µis the local friction coecient. This is equivalent to
Fk(x)=µλkTk(x)f(kukk)s(uk),(35)
with s(uk)=uk
kukkwhen kukk>0, while s(uk) takes any unit vector when kukk=0. The friction magnitude function,
f, is nonsmooth with respect to uksince f(kukk)=1 when kukk>0, and f(kukk)[0,1] when kukk=0. This
nonsmoothness would severely slow and even break convergence of gradient-based optimization.
To enable ecient and stable optimization, the friction-velocity relation in the transition to static friction can be
mollified by replacing fwith a smoothly approximated function. Following Li et al. [57], we use
f1(y)=
y2
2
vt2+2y
vt,y[0,tv)
1,ytv,
(36)
where f0
1(tv)=0 and a velocity magnitude bound v(in units of m/s) below which sliding velocities vkare treated
as static is defined for bounded approximation error (Figure 5).
Note that the velocity used in our friction model on the MPM side is the interpolated grid velocity at particle
quadrature locations, rather than the particle velocity after grid-to-particle transfer. This makes the velocity seen
by frictional forces independent from the choice of the particle-grid transfer scheme. This is important because for
example in FLIP, the particle velocity does not reflect its displacement (vn+1
q,(xn+1
pxn
p)/t) and thus should not be
used to define friction in an implicit solve.
12
-1 -0.5 0 0.5 1
Tangent relative velocity magnitude
-1
-0.5
0
0.5
1
Friction force mollifier
Figure 5: Friction mollifier plotted with dierent v.Decreasing vasymptotically matches the discontinuous Coulomb friction model.
However, challenges remain on incorporating friction into the optimization time integration. A major problem
is that friction is not a conservative force and there is no well-defined potential such that taking the opposite of its
gradient produces the frictional force. Therefore, following Li et al. [57], we fix the friction constraint set Falong
with the normal force magnitude λand the tangent operator Tduring the nonlinear optimization to the last updated
value Fj=F(xj), λj=λ(xj), and Tj=T(xj), which then makes the lagged friction force integrable with the
pseudo-potential
D(x)=X
k∈F j
µλj
kf0(k¯
ukk),(37)
where Fjis the set of all contact pairs with nonzero λj
k,f0
0(y)=f1(y), ¯
uk=(Tj
k)T(tvn+ ∆t2((1 γ)an+γan+1)) and
so we have −∇D(x)=Pk∈F jµλj
kTj
kf1(k¯
ukk)s(¯
uk), which is a semi-implicit discretization of the frictional force with
lagged variables λj
kand Tj
k. Then we can iteratively alternate between the nonlinear optimization with fixed F,λ, and
Tgiven as
min
x:E(x)=1
2kxˆxnk2
M+2αβt2(Ψ(x)+B(x)+D(x)),(38)
and friction update until convergence (Algorithm 1). Although the friction convergence is not guaranteed for arbi-
trarily large time step sizes due to the nonlinearity and asymmetry of the problem, we have confirmed that all our
experiments converge with the practical time step sizes applied (Section 6).
4.4. Irregular Boundaries for MPM
In the BFEMP framework, an experimental setup with a subset or all of the FEM nodes prescribed with Dirichlet
boundary conditions on their displacements can be applied to model irregular boundaries for MPM. This can not only
resolve detailed boundary geometries even when the MPM grid is relatively coarse (Section 6.2), but also provide
accurate and controllable friction on the boundary (Section 6.4).
5. Nonlinear Optimization
The time integration framework of BFEMP for one time step is outlined in Algorithm 1. MPM particle-grid
transfers are performed in the beginning (line 2) and the end (line 12). On the MPM grid and the FEM mesh, the
minimization of incremental potential with lagged friction (line 7) is alternated with the friction update (line 9) until
convergence to the fully implicit friction solution.
Applying the projected Newton’s method [81] for incremental potential minimization (Algorithm 2), we compute
the proxy matrix Hby projecting the local Hessian of every elasticity, barrier, and friction stencil to its closest positive
13
Algorithm 1 BFEMP Time Integration
1: procedure TimeIntegration(xn
F,vn
F,MF,xn
P,vn
P,MP,t)subscript P is for stacked particle variables
2: xn
M,vn
M,Mn
MparticleToGrid(xn
P,vn
P,MP)Table 1,2, and 3
3: ˜xn+1
Fxn
F, ˜xn+1
Mxn
Mfor initial guess
4: j0
5: Fj,λj,TjcomputeFrictionOperator( ˜xn+1
F, ˜xn+1
M)Section 4.3
6: do
7:
˜xn+1
F
˜xn+1
M
,
˜vn+1
F
˜vn+1
M
MinimizeIP(
xn
F
xn
M
,
vn
F
vn
M
,
MF
Mn
M
,t,Fj,λj,Tj,
˜xn+1
F
˜xn+1
M
)Algorithm 2
8: jj+1
9: Fj,λj,TjcomputeFrictionOperator( ˜xn+1
F, ˜xn+1
M)Section 4.3
10: while friction not converged Section 5
11: xn+1
F˜xn+1
F,vn+1
F˜vn+1
F
12: xn+1
P,vn+1
PgridToParticle( ˜xn+1
M, ˜vn+1
M)Table 1,2, and 3
13: return xn+1
F,vn+1
F,xn+1
P,vn+1
P
14: end procedure
Algorithm 2 Line Search Method for Incremental Potential Minimization
1: procedure MinimizeIP(xn,vn,Mn,t,Fj,λj,Tj, ¯x)
2: x¯xinitial guess
3: Eprev E(x), xprev xE(x) defined in (38) also depends on xn,vn,Mn,t,Fj,λj,Tj
4: do
5: HcomputeProxyMatrix(x)applying projected Newton [81]
6: p← −H1E(x)solved using CHOLMOD [82]
7: τinitStepSize(x)line search filtering [77]
8: do Armijo line search [83]
9: xxprev +τp
10: ττ/2
11: while E(x)>Eprev
12: Eprev E(x), xprev x
13: while kpk/t> dSection 5
14: ˜xn+1x, ˜vn+1vn+1
t(Mn)1((γ1)E(xn)γE(x)) Section 3
15: return ˜xn+1, ˜vn+1
16: end procedure
semi-definite form by zeroing out the negative eigenvalues, and then summing them up together with the mass matrix
Mn(line 5). The search direction pis computed by factorizing Hand back-solving it on −∇E(x) using CHOLMOD
[82] (line 6). To obtain global convergence, the backtracking line search that ensures the decrease of energy is applied
(line 8 to 11), starting from a large feasible step size that avoids interpenetration and deformation gradient degeneracy
(line 7). After converging to a local optimum, velocity is updated with the newly obtained position (line 14) and
returned together (line 15). Here we use the infinity norm of the Newton increment (search direction p) in the unit of
velocity (m/s) for the stopping criteria, which provides a 2nd-order approximation on the distance to the true solution.
Similarly, friction convergence in Algorithm 1is also determined this way, but with F,λ, and Tcomputed using the
current x.
Along Newton’s search direction p, we compute the largest step size that will first result in a 0 distance on any
contact pair or a 0 determinant on any deformation gradient. We then set the initial line search step size to be 0.9×of
this critical value. The critical value for 0 distance is computed via continuous collision detection (CCD) [60], and for
14
t = 0.03 s t = 0.05 s t = 0.08 s
t = 0.31 s t = 0.27 s t = 0.23 s t = 0.17 s
Stress
0
2E7
t = 0.13 s
Setup MPM FEM
Figure 6: Colliding rings. The experiment setup and stress wave propagation over time.
0 determinant it is just the smallest positive real root of a polynomial equation [84]. This ensures that interpenetration
or deformation gradient degeneracy could never happen throughout the simulation since the following backtracks
always result in step sizes smaller than the critical value.
The numerical parameters in BFEMP all have physical meanings and directly control the extent of approximation
to the continuous problem. To summarize, we have ˆ
d(contact activation distance in m), v(stick-slip velocity thresh-
old in m/s), d(Newton tolerance in m/s), and physical parameter κ(barrier stiness in Pa). Here κalso aects the
convergence speed of the projected Newton method (Algorithm 2), but the convergence is always guaranteed even-
tually. In our experiments, we observed that setting κseveral orders of magnitude smaller than the average elasticity
stiness of the objects in the simulation can provide ecient convergence.
6. Numerical Simulations
In this section, we provide 6 examples in 2D and 1 example in 3D to verify the contact model and the friction
model in the proposed BFEMP approach. The numerical parameters ˆ
d,v,d, and physical parameter κare all reported
respectively in each experiment. If not mentioned otherwise, all elasticities are with the neo-Hookean model, and all
particle-grid transfer schemes are with APIC. The visualized stresses are all the von Mises stress.
6.1. Momentum and Energy Study
The collision between two elastic rings is simulated to verify the momentum and energy behavior of BFEMP and
to demonstrate the robustness of our framework in handling large deformation. This example is modified from the
MPM-MPM contact version in [85].
The experiment setup is shown in the left-middle subfigure of Figure 6. The two rings are identical except that
the left ring is discretized with MPM, and the right one is discretized with FEM. The inner radius of the ring shape is
3m, and the outer radius of the ring shape is 4m. The Young’s modulus is E=108Pa, the Poisson’s ratio is ν=0.2,
and the density is ρ=1000kg/m2. The MPM ring is discretized by 20098 particles, where the grid spacing is 0.1m.
The FEM ring is discretized by 1830 vertices and 3310 triangles. The gravitational force and the frictional forces are
not included. The two rings are placed 2mapart and then move towards each other with an initial speed of 40m/s.
The contact active distance and the contact stiness are set to ˆ
d=102mand κ=107Pa respectively. To minimize
numerical dissipation, we use the Newmark time integrator with time step size t=2×104s. The Newton tolerance
15
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time [s]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Energy [J]
1e7
Total Energy - FLIP
Total Energy - APIC
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time [s]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Energy [J]
1e7
System Total Energy
FEM Total Energy
MPM Total Energy
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time [s]
750000
500000
250000
0
250000
500000
750000
Momentum [kg m/s]
Total Momemtum
FEM Momemtum
MPM Momemtum
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time [s]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Energy [J]
1e7
Total Energy
FEM Energy
MPM Energy
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time [s]
750000
500000
250000
0
250000
500000
750000
Momentum [kg m/s]
Total Momemtum
FEM Momemtum
MPM Momemtum
APIC
FLIP
Figure 7: Momentum and Energy Behavior. The energy and momentum plot for APIC and FLIP transfer schemes.
is d=106m/s. We also compare the APIC and FLIP transfer schemes. Their dierences in displacement and stress
are small, so only the stress wave propagation with APIC is shown in Figure 6. The energies and momenta over time
are plotted for both APIC and FLIP in Figure 7.
The collision happens between 0.025sand 0.293s. The symmetry of stress patterns is preserved during the colli-
sion. The system’s total momentum is perfectly preserved with both transfer schemes. Part of the energy is lost during
the collision: 8.57% energy is lost with APIC, and 9.67% with FLIP. After the rings are separated, the FEM ring
preserves its energy over time, while the MPM ring gradually loses energy, primarily due to numerical dissipation in
16
BFEMP
MPM
Setup
No-slip ground
Boundary moving path
Figure 8: FEM as Boundary Condition. The friction-free interaction between a sine wave shape boundary and an MPM cube are simulated to
compare BFEMP based slip boundary condition and traditional level set based slip boundary condition. (ux,uy) is the displacement of the sine
wave w.r.t. its initial position. BFEMP based slip boundary condition can guarantee non-penetration and doesn’t have adhesive forces when it is
separating from the object.
the particle-grid transfers.
6.2. FEM as Contact Boundary for MPM
The guaranteed impenetrability between MPM particles and FEM boundaries makes BFEMP a natural strategy
for enforcing kinematic separable boundary conditions in MPM simulations. Here we test the friction-free interaction
between a sine wave shape boundary and an MPM cube. The BFEMP-based boundary condition is compared with
a level-set based slip boundary condition, which enforces a zero normal relative velocity condition at each grid node
inside the sine wave’s level set, i.e., at each time step, for those nodes that are within the level set, their normal
velocities along the level set interface are prescribed, so that the original unconstrained optimization 22 for the time
integration are solved with these equality constraints.
The experiment setup is shown in the left-top subfigure of Figure 8. A 1m×1melastic box with Young’s modulus
E=106Pa, Poisson’s ratio ν=0.2 and ρ=1000kg/m2is placed on a no-slip ground. It is discretized by 21026
particles, with grid spacing 0.02m. A sine wave boundary is placed 0.2mabove the box, whose contour is determined
by y=1
40 cos 2π
0.1x. For BFEMP, the sine wave boundary condition is discretized by a FEM mesh with prescribed
displacements at each time step. While for MPM, it is described by an analytical level set. The sine wave boundary
first moves 0.6mdownwards, then 0.5mto the left, and finally upwards until separation. The moving speed is 1m/s
all the way. The contact active distance and the contact stiness is set to ˆ
d=103mand κ=104Pa respectively. The
implicit Euler time integration with time step h=103sis used. The Newton tolerance is set to d=104m/s.
As shown in Figure 8, the BFEMP-based boundary condition more accurately resolves the complex boundary
geometry without exhibiting any numerical adhesive forces when the boundary is separating from the cube. With the
level-set based slip condition, particles will penetrate the boundary because the boundary condition is only defined
on the MPM grid in a “smeared out” manner. The numerical adhesive force comes from that at each time step, the
17
2.6E9
Stress
0
Figure 9: Brazilian Disk Test. The experiment setup and the compression procedure are shown here. The contact force and contact radius are
illustrated in the middle figure.
grid with slip condition is locked within some plane. On the contrary, with BFEMP, MPM particles can freely move
around outside the FEM mesh.
6.3. Brazilian Disk Test
Figure 10: Brazilian Disk Test. Within the small deformation range, our contact model fits well with Hertzian contact theory. The non-smoothness
of the measured radius from the simulation results can be alleviated as the resolution increases.
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. We use a fixed rigid plate and a moving rigid plate to
simulate the compression procedure. According to the Hertzian contact model, the contact force Fand the contact
radius ahave the following relation:
F=π
4
E
1ν2
a2
R.(39)
The contact force and the contact radius are illustrated in Figure 10.
In this experiment, the radius of the MPM disk is 1m. It is composed of 42920 particles with MPM grid spacing
x=0.025m. The Young’s modulus is E=1010 Pa, and the Poisson’s ratio is ν=0.3. To reduce the inertial
18
Figure 11: Critical Value of Friction Coecient. (a,d) initial configuration with the extra support; (b,c,e) results at t=3sof µ=0, 0.1, and
0.1999, sliding distances all matching analytical solutions; (f) the result at t=3sof µ=0.2, static solution with sliding error bounded by v.
eect, we artificially decrease the density of the material, which is set to ρ=100kg/m2. The contact active distance
is ˆ
d=104mand the contact stiness is κ=104Pa. The two plates are discretized with FEM. Each of them is
composed of four vertices and two triangles. The fixed plate is placed ˆ
dbelow the disk, and the moving plate is placed
ˆ
dabove the disk. The constant velocity 0.1m/sof the moving plate is enforced by prescribing its displacements at
each time step. The simulation is performed with implicit Euler time integration with time step size h=102sand
the Newton tolerance is d=108m/s. Friction coecient µ=1 is used to prevent the disk from slipping.
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. The compression procedure in Figure 9is visualized for the linear elasticity
case. The (a,F) data points within the small deformation range from the two simulations and the analytical Fa
relation from the Hertzian contact model are plotted in Figure 10. The non-smoothness of the measured radius from
the simulation results is due to the inaccurate approximation of the radius athrough a finite number of particles. This
non-smoothness can be alleviated as the resolution increases. Despite that, we observe a qualitative match between
the simulated data and the Hertzian contact theory.
6.4. Critical Value of Friction Coecient
To verify the accuracy of BFEMP’s friction model, an experiment with a stiMPM box resting or sliding on a
fixed FEM slope (or BFEMP’s friction-controllable boundary condition) with a certain friction coecient is created.
When a rigid box is placed on a slope with zero initial velocity, its acceleration has the following analytical form:
ax=
g(sin θµcos θ), θ tan θ,
0, θ < tan θ, (40)
where µis the friction coecient between the box and the slope, gis the gravity acceleration, θ[0, π/4) is the
inclined angle of the slope. Experiments show that BFEMP’s friction model matches analytical solutions on sliding
dynamics and critical value of friction coecient both with bounded and small approximation error.
The initial configuration of this example is obtained by placing the MPM box ˆ
daway from the slope, placing
another fixed plane perpendicular to the slope on the side of the box where it may slide (also ˆ
daway), and then
simulate under gravity (g=5.10m/s2) without friction until the box becomes static (Figure 11a). After obtaining the
19
Figure 12: Critical Value of Friction Coecient. At all friction coecients, including µ=0 (no friction), µ=0.1, µ=0.1999 (99.95% of the
critical value), and µ=0.2 (the critical value), the velocities and accelerations over the releasing period (2sto 3s) are all accurately matching the
analytical solutions.
initial configuration, the slope test simulation is performed without the extra plane and with multiple dierent friction
coecients for each test (Figure 11b,c,d).
Here the MPM box is 0.1m×0.02m, composed of 369 particles (grid dx =0.005) with density ρ=100kg/m2,
Young’s modulus E=4.0×1012 Pa and Poisson’s ratio ν=0.2. Slopes with friction coecient µ=0, 0.1, 0.1999,
and 0.2 have been tested, all with contact active distance ˆ
d=0.001m, contact stiness κ=106Pa, static friction
velocity threshold v=105m/s, and with the lagged normal forces in friction iteratively updated until converging
to a solution with fully-implicit friction. All simulations are using implicit Euler time integration with time step size
h=0.001s, and the Newton tolerance is set to d=108m/s.
With sliding velocity and acceleration of the box’s center of mass plotted over time (Figure 12), they have all
been shown to well match analytical solutions within 0.01% relative errors. Even for µ=0.1999 (99.95% that of the
critical coecient), the sliding behavior can still be accurately captured. For µ=0.2, it is also confirmed that the
acceleration vanishes, and the velocity throughout the simulation is around v, the static friction velocity threshold in
BFEMP’s approximation to provide the static friction force in the same magnitude as dynamic friction.
20
Initial Setup Low-resolution Equilibrium
High-resolution Equilibrium
(a) (b)
(c)
Stress
7E3
0
Figure 13: Convergence under Refinement. (a) Experiment setup. (b) The final equilibrium under low resolution. (c) The final equilibrium under
high resolution. Stress pattern is visualized.
6.5. Convergence under Refinement
To verify the convergence under refinement property of BFEMP, an example with a soft MPM box stacking on a
soft FEM box is created. A series of experiments with increasing resolutions and decreasing contact active distances
are simulated to study the convergence rate under refinement. Results show that BFEMP can achieve a second-order
convergence rate.
The initial configuration is illustrated in Figure 13 (a). The MPM box is with size 2m×1m, Young’s modulus
E=4×104Pa, Poisson’s ratio ν=0.4 and density ρ=103kg/m3. The particles are sampled regularly within each
cell by placing each particle on the center of a sub-cell. The FEM box below is with size 4m×1m, Young’s modulus
E=4×104Pa, Poisson’s ratio ν=0.4 and density ρ=102kg/m2. The minimal edge length of the FEM mesh and the
grid spacing of MPM are with the same value (x) in each experiment. The MPM box initially is placed xabove
the FEM box and then simulated under gravity (g=10m/s2) until no oscillation is observed. To accelerate simulation
to reach its final static state, PIC transfer scheme and implicit Euler time integration with large time step sizes (up to
CFL limit for MPM) are used. The Newton tolerance is set to d=109m/s. The contact stiness is set to κ=106Pa
for all experiments. Figure 13 (b) and Figure 13 (c) show the final equilibria under low resolution and high resolution
respectively.
To examine the convergence rate of displacement to high-resolution results, the example is refined with x=1
N
and ˆ
d=1
N2, where Niterates all positive even numbers smaller than or equal to 20. The reference high-resolution
result is choose as with N=30. The error is defined as the dierence in height of the center of mass of the whole
domain (with both FEM and MPM domains) between each testing resolution and the high-resolution reference. Due
to quadrature error in MPM [2], we also experiment with three dierent particle per cell (PPC) values: 4, 9, and 16.
The three error sequences are plotted in Figure 14. As observed from the plot, a higher PPC value can reduce the
21
Figure 14: Convergence under Refinement. Higher PPC can reduce the noise in the convergence curve at higher resolutions. BFEMP with PPC
=16 achieves a convergence order of 2.75 to high-resolution result. Convergence curves with order 1 and order 2 are also plotted for reference.
noise in the convergence curve. The error sequence with PPC 16 almost falls into line. Under this setting, BFEMP
achieves a convergence order of 2.75.
6.6. Buckling Behaviours under Dierent Friction Coecients
This example tests frictions between two semi-circular rings with large deformation. The two semi-circular rings
are stacked together. As the outer semi-circular ring is compressed, dierent buckling patterns of the inner semi-
circular ring under dierent friction coecients are observed. This example is modified from the version with FEM-
FEM contact in [88].
The experiment setup is shown in Figure 15. The outer semi-circular ring with outer radius 14mand inner radius
12mis discretized by FEM with 2714 vertices and 5067 triangles. The inner semi-circular ring with outer radius
11.99mand inner radius 10mis discretized by MPM with 10261 particles with grid spacing 0.25m. The two semi-
circular rings are both with Young’s modulus E=106Pa, Poisson’s ratio ν=0.3 and density ρ=100kg/m2. One
FEM plate is placed 103mabove the outer semi-circular ring. The displacement of this plate is prescribed to follow a
rigid linear motion with a constant downward velocity 1m/s. Large friction (µ=10) between the plate and the outer
semi-circular ring is activated so that the plate can be viewed as a BFEMP based no-slip boundary condition. The
feet of two semi-circular rings are fixed using the level set-based no-slip boundary condition. Another level set-based
slip boundary condition is added at the bottom middle below the semi-circular rings to prevent the inner semi-circular
rings from colliding into the ground. The contact active distance and the contact stiness are set to ˆ
d=103mand
κ=105Pa. The static friction velocity threshold is set to v=105m/s. Implicit Euler time integrator with time step
size h=102sand Newton tolerance d=104m/sare used.
We vary the friction coecient between the two semi-circular rings from µ=0, µ=0.2 and µ=0.5. The
compression procedure is visualized in Figure 15. In the beginning, there is little dierence between the three settings.
As the FEM plate moves further down, the inner MPM semi ring under the friction-free setting is buckled first as
expected. Friction with µ=0.2 lags the appearance of the buckling. For the large friction case with µ=0.5, no
buckling happens at all.
22
FEM(Rigid)
FEM
MPM
Setup
3E5
Stress
0
No-slip Condition
Slip Condition
Figure 15: Buckling Behaviours under Dierent Friction Coecients. The experiment setup is illustrated on the left. Under dierent friction
coecients, the buckling appears at dierent vertical displacements (uy).
6.7. 3D Twist with Friction
To test BFEMP’s contact and friction model in 3D, a twist test between a FEM spherical shell and an MPM cube
is conducted. The FEM shell is controlled to exert a constant twist speed. Under dierent friction coecients, it is
expected to observe dierent maximal twist angles on the MPM cube. This example is modified from the version with
FEM-FEM contact in [88] as well.
The initial setup is illustrated in Figure 16 (a). The MPM cube with an edge length of 1mis placed 102mbelow
the FEM shell. It is discretized by 90929 particles, where the grid spacing is 0.0625m. The Young’s modulus is
108Pa. The Poisson’s ratio is 0.4. And the density is 100kg/m3. The FEM shell with inner radius 0.45mand outer
radius 0.5mis discretized by 1364 points and 3920 tetrahedra. The Young’s modulus is 1010Pa. The Poisson’s ratio
is 0.4. The density is 104kg/m3. The displacements of the top part of the shell are prescribed to follow a rigid motion
to exert downward compression and constant-speed twist: it first compresses down with a constant speed 0.5m/sfor
1s(Figure 16 (b)) and then rotates around the z-axis with a constant angular velocity π
5for 4.5s. The contact active
distance and the contact stiness are set to ˆ
d=102mand κ=107Pa. For settings with frictions, the static friction
velocity threshold is set to v=103m/s. The simulation uses the implicit Euler time integrator with the time step
size h=102s. The Newton tolerance is set to d=103m/s.
With dierent friction coecients, the slipping between the shell bottom and the top center of the cube happens
after dierent twist angles Rz. The final equilibria when the shell stops twisting are visualized in Figure 16 (c) (d) (e)
(f). The twist angles of the top center part of the cube are plotted in Figure 17. Since the pressure forces are between
triangles and particles, the interface between the shell and the cube is not perfectly smooth. This roughness results in
that the slipping happens when Rz=0.1πin the friction-free settings. The final rotation angle should decrease as the
resolution increases. To verify this, we increase the resolution of the FEM mesh and compare the final equilibria in
the original setting and the higher-resolution setting. The top views are attached in Figure 16, which shows that, with
higher resolution, the final state of the cube is close to the initial state before the twisting. For µ=0.2 and µ=0.5,
the slipping happens when the twists angles are around Rz=0.3πand Rz=0.7πrespectively. With µ=1.0, there is
no slipping between the shell and the cube.
7. Conclusion
In this paper, we proposed a new method for monolithically coupling an MPM domain and a FEM domain for
elastodynamics through frictional contact. By approximating the non-interpenetration constraint with a barrier energy
term and performing time integration using a variational formulation, our method guarantees that no particles will
23
Top views
Low-res
Twist
High-res
Initial Setup
Stress
1E8
0
(a) (b) (c)
(d) (e) (f)
Figure 16: 3D Twist with Friction. (a) Initial setup: The FEM spherical shell is placed ˆ
dabove the MPM cube; (b) Before twist procedure, the
spherical shell is controlled to press down 0.5m; (c, d, e, f) Equilibria under dierent friction coecients when the shell stops rotating. The nonzero
rotation angle with µ=0 is caused by the non-smoothness of the contacting interfaces, which will decrease as the resolution increases (top views).
penetrate into the FEM mesh throughout the simulation. Furthermore, when the displacement of the FEM domain
is fully prescribed, BFEMP reduces to an explicit mesh-based boundary treatment for MPM. Through numerical
experiments validating the energy behavior, robustness, stability, and accuracy, we demonstrated the advantages of
the proposed method.
Limitations and Future Works. For our current formulation, when MPM particles get very close to the FEM boundary,
there is in fact a small portion of overlap between FEM and MPM domains even when there is no interpenetration.
This is because MPM particles represent a region of the domain. Although the overlapping area vanishes under
spatial refinement, it would still be interesting to also consider the size and deformation of the regions when defining
the distance constraints. In addition, it would be meaningful to extend our framework to support cutting of MPM
solids by FEM meshes, where the MPM particles on dierent sides of the FEM mesh should not communicate with
each other even when the FEM mesh is much thinner than the MPM kernel.
24
Figure 17: 3D Twist with Friction. The average twist angle around the z-axis of the top center of the cube. The twist procedure happens between
0.5sand 5s. The slipping between the cube and the shell appears at dierent time points under dierent friction coecients.
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