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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 [9–13] have been

successfully applied in challenging problems including multiphase ﬂows, fracture, contact, adaptivity, free-surface

ﬂows, soil-ﬂuid mixture, explosives, and granular media [14–22].

Although MPM has been demonstrated eﬀective on a wide range of materials, many application scenarios favor

other discretization schemes due to considerations in eﬃciency, 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 [24–26], the Finite Diﬀerence 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 suﬀers from unique challenges in aspects

such as stability, accuracy, boundary condition enforcement, and numerical fracture [29–31]. 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 stiﬀ

nonlinear problems. Implicit MPM/GIMP has been explored by many researchers [29,30,49–52]. 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) [57–59] was recently

proposed for nonlinear elastodynamics with linear kernel FEM, which has also shown to be eﬀective 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 stiﬀness 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 Ωtwith x∈Ωtand time t∈[0,∞) are given by [66]

Dρ

Dt +ρ∇x·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 X∈Ω0from the material space to x∈Ωtin 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

X∈Ω0. Here we can pull back the momentum equation to the material space. Denoting the ﬁrst Piola-Kirchhoﬀ

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 Kirchhoﬀstress 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(FTF−I)−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

J≤0. 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) : Ω0→R3, the corresponding Lagrangian weak form of Equation (4) is

ZΩ0

Qα(X,t)R0Aα(X,t)dX=Z∂Ω0

QαTαdS (X)−ZΩ0

Qα,βPαβ dX+ZΩ0

Qα(X,t)R0gαdX,(8)

where Tα=PαβNβ(with N(X) being the material space normal) is the traction ﬁeld at the domain boundary ∂Ω0, 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 Ωt. We can either push forward Equation (8) or directly integrate Equation (2)

to reach

ZΩt

qα(x,t)ρ(x,t)ai(x,t)dx=Z∂Ωt

qαtαds(x)−ZΩt

qα,βσαβ dx+ZΩt

qα(x,t)ρ(x,t)gαdx,(9)

where q(x,t)=Q(Φ−1(x,t),t) is the push forward of Q,i.e., an Eulerian trial function, a(x,t)=A(Φ−1(x,t),t)=

Dv

Dt (x,t)=∂v

∂t+v· ∇v, and t=σnis the traction at ∂Ωtwith n(x) being the outward pointing normal.

4

2.2. Incremental Variational Form

To enable the development of eﬃcient 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)=ZΩ0 1

2

R0

β∆t2kφn+1k2+2αΨ(Fn+1)!dX−ZΩ0

R0¯

Bn+1·φn+1dX−2αZ∂Ω0

T·φn+1dS (X),(10)

where

¯

Bn+1=2αg+1

β∆t2 φn+ ∆t∂φ

∂t(X,tn)+α(1−2β)∆t2∂2φ

∂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 (1−γ)∂2φ

∂t2(X,tn)+γ∂2φ

∂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+1−1

4∇X·Pn+1=1

4R0g+R0

∆t2(φn+ ∆tVn+1

4∆t2An).(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 ΩF(Section 3.1), while the

Eulerian form is discretized on the MPM domain ΩM(Section 3.2). We assume

Ω0

F∩Ω0

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=ΩF∪ΩM.

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 (Ω0

Fand Ω0

M) on the left are mapped via φFand φMto the world space

(Ωn+1

Fand Ωn+1

M) on the right, with Ωt

F∩Ωt

M=∅for all t∈[0,∞). With Updated Lagrangian kinematics, MPM treats Ωn

Mas the “intermediate”

material space and focuses on studying the deformation from Ωn

Mto Ωn+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:

Ωt

F∩Ωt

M=∅,∀t∈[0,∞) (17)

between the two domains. Note that the time is discretized with t=t0,t1,t2,...,tn. We use Ωnto denote Ωtnfor

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),∀XM∈Ω0

M,XF∈Ω0

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 Ωt

Mand Ωt

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 eﬀects.

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+α∆t2((1 −2β)an+2βan+1)),(21)

and it is equivalent to ﬁrst solving the optimization problem

min

x: 1

2kx−ˆxnk2

M+2αβ∆t2Ψ(x)!(22)

to get ˜xn+1, and then explicitly calculating ˜vn+1. Here ˆxn=xn+vn∆t+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)B−1

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 Ωn

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 Aﬃne 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

i−xn

p)(xn

i−xn

p)T

mn

ivn

i=X

p

wip mn

p(vn

p+Bp(Dp)−1(xn

i−xn

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

i−xn

p+˜

xn+1

i−xn+1

p)T

+(xn

i−xn

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

i−vn

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 ∂ΩMand ∂ΩFto be

Z∂Ω0

M

b( min

xf∈∂Ωt

F

dPP(xm,xf),ˆ

d)dXm,(26)

where dPP(xm,xf)=kxm−xfkis the point-point distance function, and

b(d,ˆ

d)=

−κ(d

ˆ

d−1)2ln ( d

ˆ

d) 0 <d<ˆ

d

0d≥ˆ

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 stiﬀness [57,59], which scales the magnitude

of contact forces at a certain distance.

Intuitively, minxf∈∂Ωt

FdPP(xm,xf) is the distance between a material point xm∈∂Ωt

Mand surface ∂Ωt

F, and Equation

(26) can be viewed as an integration of the point(xm)-surface(∂Ωt

F) contact energy density over surface ∂Ω0

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

d→0, the approximation error between the contact energy density function band the real contact phenomenon

described in Equation (20) decreases, which also makes ∂ΩMand ∂ΩFinterchangeable 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 diﬀerent 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!T∂B

∂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

q/π in 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

eﬃciently 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

k∈ˆ

B

(ηk−1)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

e∈ˆ

B

(ηe−1)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

2kx−ˆxnk2

M+2αβ∆t2(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×(d−1) 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)) ∈Rd−1provides the local relative sliding displacement in the frame orthogonal to the distance gradient. The

corresponding sliding velocity is then vk=uk/∆t∈Rd−1.

Maximizing dissipation rate subject to the Coulomb constraint deﬁnes friction forces variationally [79,80]

Fk(x)=Tk(x) argmin

β∈Rd−1

βTvks.t. kβk ≤ µλk,(34)

where λkis the contact force magnitude and µis the local friction coeﬃcient. 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 eﬃcient 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

v∆t2+2y

v∆t,y∈[0,∆tv)

1,y≥∆tv,

(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

p−xn

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 diﬀerent 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

2kx−ˆxnk2

M+2αβ∆t2(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

M←particleToGrid(xn

P,vn

P,MP)Table 1,2, and 3

3: ˜xn+1

F←xn

F, ˜xn+1

M←xn

Mfor initial guess

4: j←0

5: Fj,λj,Tj←computeFrictionOperator( ˜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: j←j+1

9: Fj,λj,Tj←computeFrictionOperator( ˜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

P←gridToParticle( ˜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: H←computeProxyMatrix(x)applying projected Newton [81]

6: p← −H−1∇E(x)solved using CHOLMOD [82]

7: τ←initStepSize(x)line search ﬁltering [77]

8: do Armijo line search [83]

9: x←xprev +τp

10: τ←τ/2

11: while E(x)>Eprev

12: Eprev ←E(x), xprev ←x

13: while kpk∞/∆t> dSection 5

14: ˜xn+1←x, ˜vn+1←vn+1

∆t(Mn)−1((γ−1)∇E(xn)−γ∇E(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 stiﬀness in Pa). Here κalso aﬀects 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

stiﬀness of the objects in the simulation can provide eﬃcient 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 stiﬀness are set to ˆ

d=10−2mand κ=107Pa respectively. To minimize

numerical dissipation, we use the Newmark time integrator with time step size ∆t=2×10−4s. 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=10−6m/s. We also compare the APIC and FLIP transfer schemes. Their diﬀerences 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 stiﬀness is set to ˆ

d=10−3mand κ=104Pa respectively. The

implicit Euler time integration with time step h=10−3sis used. The Newton tolerance is set to d=10−4m/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

1−ν2

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 Coeﬃcient. (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.

eﬀect, we artiﬁcially decrease the density of the material, which is set to ρ=100kg/m2. The contact active distance

is ˆ

d=10−4mand the contact stiﬀness 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=10−2sand

the Newton tolerance is d=10−8m/s. Friction coeﬃcient 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 F−a

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 Coeﬃcient

To verify the accuracy of BFEMP’s friction model, an experiment with a stiﬀMPM box resting or sliding on a

ﬁxed FEM slope (or BFEMP’s friction-controllable boundary condition) with a certain friction coeﬃcient 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 θ−µcos θ), θ ≥tan θ,

0, θ < tan θ, (40)

where µis the friction coeﬃcient 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 coeﬃcient 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 Coeﬃcient. At all friction coeﬃcients, 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 diﬀerent friction

coeﬃcients 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 coeﬃcient µ=0, 0.1, 0.1999,

and 0.2 have been tested, all with contact active distance ˆ

d=0.001m, contact stiﬀness κ=106Pa, static friction

velocity threshold v=10−5m/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=10−8m/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 coeﬃcient), 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=10−9m/s. The contact stiﬀness 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 diﬀerence 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 diﬀerent 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 Diﬀerent Friction Coeﬃcients

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, diﬀerent buckling patterns of the inner semi-

circular ring under diﬀerent friction coeﬃcients 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 10−3mabove 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 stiﬀness are set to ˆ

d=10−3mand

κ=105Pa. The static friction velocity threshold is set to v=10−5m/s. Implicit Euler time integrator with time step

size h=10−2sand Newton tolerance d=10−4m/sare used.

We vary the friction coeﬃcient 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 diﬀerence 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 Diﬀerent Friction Coeﬃcients. The experiment setup is illustrated on the left. Under diﬀerent friction

coeﬃcients, the buckling appears at diﬀerent 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 diﬀerent friction coeﬃcients, it is

expected to observe diﬀerent 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 10−2mbelow

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 stiﬀness are set to ˆ

d=10−2mand κ=107Pa. For settings with frictions, the static friction

velocity threshold is set to v=10−3m/s. The simulation uses the implicit Euler time integrator with the time step

size h=10−2s. The Newton tolerance is set to d=10−3m/s.

With diﬀerent friction coeﬃcients, the slipping between the shell bottom and the top center of the cube happens

after diﬀerent 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 diﬀerent friction coeﬃcients 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 diﬀerent 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 diﬀerent time points under diﬀerent friction coeﬃcients.

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