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Multi-species simulation of porous sand and water mixtures

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We present a multi-species model for the simulation of gravity driven landslides and debris flows with porous sand and water interactions. We use continuum mixture theory to describe individual phases where each species individually obeys conservation of mass and momentum and they are coupled through a momentum exchange term. Water is modeled as a weakly compressible fluid and sand is modeled with an elastoplastic law whose cohesion varies with water saturation. We use a two-grid Material Point Method to discretize the governing equations. The momentum exchange term in the mixture theory is relatively stiff and we use semi-implicit time stepping to avoid associated small time steps. Our semi-implicit treatment is explicit in plasticity and preserves symmetry of force linearizations. We develop a novel regularization of the elastic part of the sand constitutive model that better mimics plasticity during the implicit solve to prevent numerical cohesion artifacts that would otherwise have occurred. Lastly, we develop an improved return mapping for sand plasticity that prevents volume gain artifacts in the traditional Drucker-Prager model.
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Multi-species simulation of porous sand and water mixtures
ANDRE PRADHANA TAMPUBOLON, University of California, Los Angeles
THEODORE GAST, University of California, Los Angeles and Jixie Eects
GERGELY KLÁR, DreamWorks Animation
CHUYUAN FU, University of California, Los Angeles
JOSEPH TERAN, University of California, Los Angeles and Jixie Eects
CHENFANFU JIANG, University of Pennsylvania and Jixie Eects
KEN MUSETH, DreamWorks Animation
Fig. 1. Levee breach. Water pours in from a reservoir and slowly erodes a dam. As water seeps into the sand, its cohesivity decreases. When it eventually
breaks, the landslide creates interesting dynamics in the debris flow.
We present a multi-species model for the simulation of gravity driven land-
slides and debris ows with porous sand and water interactions. We use
continuum mixture theory to describe individual phases where each species
individually obeys conservation of mass and momentum and they are cou-
pled through a momentum exchange term. Water is modeled as a weakly
compressible uid and sand is modeled with an elastoplastic law whose
cohesion varies with water saturation. We use a two-grid Material Point
Method to discretize the governing equations. The momentum exchange
term in the mixture theory is relatively sti and we use semi-implicit time
stepping to avoid associated small time steps. Our semi-implicit treatment is
explicit in plasticity and preserves symmetry of force linearizations. We de-
velop a novel regularization of the elastic part of the sand constitutive model
that better mimics plasticity during the implicit solve to prevent numerical
cohesion artifacts that would otherwise have occurred. Lastly, we develop
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DOI: http://dx.doi.org/10.1145/3072959.3073623
an improved return mapping for sand plasticity that prevents volume gain
artifacts in the traditional Drucker-Prager model.
CCS Concepts: Computing methodologies Physical simulation;
Additional Key Words and Phrases: MPM, elastoplasticity, porous media
ACM Reference format:
Andre Pradhana Tampubolon, Theodore Gast, Gergely Klár, Chuyuan Fu,
Joseph Teran, Chenfanfu Jiang, and Ken Museth. 2017. Multi-species simula-
tion of porous sand and water mixtures. ACM Trans. Graph. 36, 4, Article 1
(July 2017), 11 pages.
DOI: http://dx.doi.org/10.1145/3072959.3073623
1 INTRODUCTION
While wet sand is both ubiquitous and literally child’s play, simulat-
ing the underlying interaction of water and sand certainly is not. In
fact, this type of visual eect is rarely seen in feature movie produc-
tions, typically because it is considered too complex and dicult
to achieve with existing particle-based simulation techniques. This
is surprising given that water simulations, and to a lesser extent
also sand simulations, are routinely undertaken in VFX. It is the
complex nature of the changing material behaviors resulting from
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
1:2 Pradhana Tampubolon, A. et al
the mixing of sand and water that complicates simulation. However,
we argue that this is precisely what makes the animation of wet
sand so visually intriguing and hence desirable to master.
Most practitioners in the VFX industry would agree that anima-
tion of water is a mature area of research, as evident by its abundance
and the fact that several excellent commercial solutions exist today,
c.f. Houdini, Maya, and RealFlow just to mention a few. Conversely,
animation of sand is considered challenging by most, and is still a
subject of signicant research. Although the Material Point Method
(MPM) [Sulsky et al
.
1995] has been demonstrated to produce en-
couraging results for sand simulations, commercial solutions tend
to use simpler particle based techniques, like Position Based Dy-
namics (PBD), which is easier to implement, but less well founded
in continuum theory. However, mixtures of water and sand present
a completely new set of simulation challenges for which, to the
best of our knowledge, there is still no established best practice in
VFX, let alone o-the-shelf commercial solutions. While there are
many successful examples of wet sand animation, they tend to be
based on art-directed constrained simulations or even procedural
particle systems. Given the success of simulating sand with MPM it
is natural to explore a similar approach for wet sand. Such a unied
MPM description of the mixing of water and sand is exactly what
motivated our work.
In this paper, we develop a method for multi-species MPM to allow
interactions between sand and water for gravity driven landslides
and debris ows. We resolve wet sand transitions from cohesive
rigid grains to owing slurries as water saturation increases. We
use a semi-implicit time stepping scheme to deal with sti terms in
the multi-species momentum exchange. The semi-implicit approach
in [Stomakhin et al
.
2013] produces articial cohesive eects for
sand simulations. We introduce a unilateral modication to the
elastic energy function that better mimics the eect of plasticity
to prevent these cohesive artifacts. Furthermore, we develop an
improved return mapping for Drucker-Prager plasticity that reduces
the volume gain observed with lower grid resolutions. In our multi-
species treatment, we use the two grid scheme proposed by Bandara
et al. [2015]. Each material point is associated with a set of grid nodes
and the interaction between the two species is via an interaction
term proportionate to relative velocity between the phases. This
term is sti for practical porosities, and it is impractical to do explicit
simulation since it would require a time-step restriction on the
order of
t
10
6
10
5
. Our semi-implicit treatment allows
us to use a time step on the order of
t=
10
3
. Furthermore,
using our modication to the elastic energy function and lagged
plasticity approach, the system to be solved remains symmetric. We
summarize our contributions as:
A unilateral elastic constitutive model modication frame-
work that removes numerical cohesion with semi-implicit
time stepping and plasticity.
An improved return mapping algorithm for Drucker-Prager
plasticity with reduced volume gain artifacts.
A semi-implicit two grid MPM discretization of sti inter-
action terms in multi-species continuum equations.
A model for wet sand cohesion based on water saturation.
2 PREVIOUS WORK
Probably the earliest work on water and sand in computer graphics
is by Peachey [1986]. Rungjiratananon et al. simulate sand-water
interaction in real-time using a hybrid Smoothed Particles Hydro-
dynamics (SPH) and Discrete-Element Method (DEM) approach
[Rungjiratananon et al
.
2008]. Lenaerts and Dutre [2009] also couple
water with porous granular materials using SPH. Notably, these
approaches capture a wider range of porous phenomena than that
considered in our approach. While we focus on gravity driven land-
slides and debris ows, their approaches more accurately capture
surface tension driven eects like capillary action drawing water
into dry sand. They can also handle landslides and debris ows,
but they do so with SPH where we develop an MPM approach that
naturally allows for implicit time stepping and high resolution simu-
lation. Other graphics approaches have shown the ecacy of hybrid
Lagrangian/Eulerian approaches like FLIP and MPM, including sand
[Daviet and Bertails-Descoubes 2016; Klár et al
.
2016; Narain et al
.
2010; Zhu and Bridson 2005] and various other elastoplastic materi-
als [Jiang et al
.
2015; Ram et al
.
2015; Stomakhin et al
.
2013, 2014;
Yue et al
.
2015]. Unilateral incompressibility is an eective assump-
tion for granular materials [Alduán and Otaduy 2011; Daviet and
Bertails-Descoubes 2016; Ihmsen et al. 2013; Narain et al. 2010].
Our approach is the rst MPM technique in graphics that consid-
ers multi-species modeling for porous sand/water. However, mixture
theory and multi-species simulations have been used for a wide
range of eects in computer graphics. Nielsen and Osterby [2013]
simulate spray and mist with a two-continua mixture model. Takashi
et al. [2003] use the Cubic Interpolation Propagation method to cou-
ple spray, water and foam continua. Losasso et al. [2008] and Yang et
al. [2014] also represent spray and dense water with multiple phases.
Similar multi-species interaction ideas have been used for bubbles
in incompressible ow [Mihalef et al
.
2009; Ren et al
.
2015; Song
et al
.
2005; Thürey et al
.
2007]. Liu et al. [2008] use two continua to
simulate mixtures of air and dust. Multi-species approaches have
been used for miscible and immiscible uids [Bao et al
.
2010; He
et al. 2015; Kang et al. 2010; Ren et al. 2014; Yang et al. 2015].
Various researchers in engineering have shown the ecacy of sim-
ulating water and soil interactions with the MPM. Abe et al. [2014]
solve coupled hydromechanical problems of uid-saturated soil sub-
jected to large deformation with a two grid MPM algorithm based on
Biot’s mixture theory. Bandara et al. use a single grid MPM method
for saturated and unsaturated soils that undergo large deformations
in [2016] and two grid MPM to represent soil skeleton and pore
water layers in [Bandara and Soga 2015]. Jassim et al. [2013] also
develop a two grid MPM approach for soil mechanics problems.
Mast et al. [Mast et al
.
2014] use MPM to simulate large deforma-
tion, gravity-driven landslides of porous soil. Mackenzie-Helnwein
et al. [2010] examine the multi-species momentum exchange terms
for problems with liquefaction, landslides, and sedimentation with
two grid MPM.
3 MATHEMATICAL BACKGROUND
We model sand and water as a multi-species continuum using mix-
ture theory [Atkin and Craine 1976; Borja 2006]. With this approach,
each species is given distinct material properties, and their motion is
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
Multi-species simulation of porous sand and water mixtures 1:3
derived from distinct velocity elds. This kinematic assumption al-
lows sand and water to occupy the same points in space at the same
time to create the mixture. We use superscript
s
to represent sand
quantities and superscript
w
to represent water quantities. With this
convention, the primary state is dened in terms of mass density
ρα(x,t)
and material velocity
vα(x,t)
, where
α=s,w
. The momen-
tum density of phase
α
is given by
ραvα
. The total mass density of
the mixture is the sum
ρ=ρs+ρw
and total momentum is the sum
ρv=ρsvs+ρwvw
. This denes the velocity
v=1
ρ(ρsvs+ρwvw)
of the mixture as the mass-averaged velocity of the constituents.
Each species obeys the following conservation of mass with re-
spect to its own motion
Dαρα
Dt +ρα∇ · vα=0.(1)
Here the operator
Dαf
Dt =f
t+vα· ∇f
is the material (or total)
derivative of function
f
with respect to the motion of either the sand
(
α=s
) or water (
α=w
) phases. By summing over
α
in Equation 1,
we obtain the standard conservation of mass
Dρ
Dt +ρ∇ · v=0,(2)
by noting that
ρDf
Dt =PαραDαf
Dt
where
Df
Dt =f
t+v· ∇f
is the
material derivative of function
f
with respect to the motion of the
mixture. Each species also obeys conservation of linear momentum
as
ραDαvα
Dt =∇ · σα+pα+ραg,α=s,w,(3)
where
pα
represents the transfer of momentum due to the relative
motion of the constituents,
σα
is the partial stress tensor associated
with species
α
, and
g
is the gravitational acceleration. Because
pα
represent the exchange of momenta between species, the sum
Pαpα=0
must be zero to not aect the total linear momentum of
the mixture. Indeed with this constraint we can show by summing
over αin Equation 3
ρDv
Dt =∇ · σ+ρg,(4)
where we have summed over
α
and introduced
σ=Pασα
, which
denotes the Cauchy stress in the mixture expressed as the sum of
the partial stresses in each species. In other words, with this notion
of the Cauchy stress, conservation of linear momentum for the
individual species implies conservation of linear momentum for the
mixture.
3.1 Sand elastoplasticity
We dene the constitutive behavior of sand with elastoplasticity
model as in Klár et al. [2016]. We modify it slightly to include
cohesive stresses. The amount of cohesion varies with the saturation
level of water in the mixture. We assume that the sand partial stress
σs
is dened in terms of the hyperelastic potential energy density
ψsas
σs=1
det(Fs)
ψs
F(Fs,E)Fs,E.(5)
Here,
Fs
is the deformation gradient of the sand motion, which
evolves as
Ds
Dt Fs=vsFs
. The appearance of the
1
det(Fs)
and
Fs,E
terms arise because we write the potential energy density in terms
of the deformation gradient. With this convention, its derivative
gives rise to the rst Piola-Kirchho stress and these terms are need
to transform it into the Cauchy stress. As typically done in nite
strain elastoplasticity [Bonet and Wood 2008], it is decomposed into
Fs=Fs,EFs,P
to dene a plastic ow. For sand,
Fs,E
represents
the remembered compression and shearing, while
Fs,P
represents
the forgotten sliding and separation. We use the Drucker-Prager
[Drucker and Prager 1952] plastic ow and yield condition to de-
termine the evolution of the elastic (
Fs,E
) and plastic (
Fs,P
) parts
of the deformation gradient. The Drucker-Prager yield condition
is dened from the constraint that the shear stress should be no
larger than the compressive normal stress in all directions. This
expresses a mechanical interaction that is consistent with Coulomb
friction. While dry sand is modeled eectively with this assumption,
it precludes the eects of cohesion. However, cohesive eects can
be modeled by modifying the elastic stress yield condition to be
cFtr(σs)+σstr(σs)
dIFcC,(6)
where
d=
2
,
3is the spatial dimension,
cC
0increases with
the amount of cohesion in the material and
cF
0increases with
amount of friction between grains. E.g. for dry sand in Klár et
al. [2016] they used
cC=
0. A positive
cC
shifts the yield surface
along the hydrostatic axis (the line where
tr(σs)=
0), which allows
the material to exhibit stress under tension and thus cohere to itself.
Here we model cohesivity as a function of water saturation in the
sand. This is naturally measured in terms of the volume fraction of
water in the mixture ϕw=ρw
ρso that cC=cC(ϕw).
Inequality 6 is referred to as the plastic yield condition, if it is
satised, there will be no further plastic deformation. The boundary
of the region in stress space dened by
cFtr(σs)+σstr(σs)
dIF=
cC(ϕw)
is called the yield surface. For states of stress on the yield
surface, plastic ow will commence when a perfectly elastic assump-
tion would drive the stress out of the region. The plasticity functions
as a means of satisfying this inequality constraint.
3.1.1 Unilateral hyperelasticity. Here we describe the elastic part
of the constitutive behavior for the sand phase. This is largely
the same as in Klár et al. [2016] where the elastic potential en-
ergy density is dened in terms of the logarithmic strain
ϵ
as
ψ(Fs)=˜
ψ(ϵ)=µtr(ϵ2)+λ
2tr(ϵ)
where
Fs=UΣV
is the singular de-
composition of
F
and
ϵ=log(Σ)
. However, we provide some novel
modications that allow for more ecient implicit time integra-
tion. We note that with the plastic yield condition from Inequality 6,
much of the energy landscape has no eect since it produces stresses
outside of the yield surface. To mitigate this we modify the energy
density such that it smoothly transitions to zero in these regions.
This has no eect on the continuous behavior of the governing equa-
tions, but it improves the performance of our semi-implicit time
stepping. Essentially, our modication modies the elastic behavior
outside the yield surface to better resemble the eects of plasticity.
Our approach is similar to, and indeed inspired by the unilat-
eral approaches in [Alduán and Otaduy 2011; Daviet and Bertails-
Descoubes 2016; Ihmsen et al
.
2013; Narain et al
.
2010]. We thus refer
to our modied energy as the unilateral energy function
˜
ψU(ϵ)
. We
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
1:4 Pradhana Tampubolon, A. et al
Fig. 2. Two grids. We use separate water grid and sand grids. The blue and red dots denote water and sand particles respectively. In the overlapping region, we
compute the momentum exchange term between the two species. We also compute the force based on individual constitutive model.
Fig. 3. The contour plot of the multiplier in strain space. Region A is where
the multiplier is the identity, region C is where the multiplier is zero, and
region B is where the multiplier transitions from 1to 0in
C2
manner. The
green cone denotes the Drucker-Prager cone in principal strain space.
dene this as the product of the original energy function
˜
ψ(ϵ)
and
a multiplier h(ϵ)
˜
ψU(ϵ)=˜
ψ(ϵ)h(ϵ).(7)
The multiplier serves to (as smoothly as possible) transition the
energy to zero in regions outside yield surface. We dene it to
be symmetric about the hydrostatic axis in the strain space (axis
of equal strain, see Figure 3) since the original energy
˜
ψ
has this
property and we wish to preserve it in
˜
ψU
. To construct it, we
partition strain space into the three regions labeled A, B, and C
in Figure 3. In region A, the value of the multiplier is simply one
and the modied constitutive model is identical to the original one.
Conversely, in region C, the multiplier is set to zero. In region B, the
multiplier smoothly transitions from one to zero while guaranteeing
that we have a continuous second derivative. The boundary of
region A and B denes two envelopes that are symmetric about the
hydrostatic axis. To best preserve the behavior of sand, we want
region A to cover most of the region inside of the Drucker-Prager
cone, illustrated as the green cone in Figure 3.
The construction of this multiplier function is done as a composi-
tion of two functions:
hs
which is a scalar function and
f
which is
a function of the strain ϵ. If we let oto denote the hydrostatic axis
(
o=1
2(1,1)
in 2D or
1
3(1,1,1)
in 3D), then we can compute
u=ϵ·o
to be the component of the strain in the hydrostatic axis
and v=ϵuo. The function fis dened as
f(ϵ)=co
v4
1+|v|3.(8)
The coecient
co
controls the opening of the envelope of region
A and B around the hydrostatic axis. The scalar function
hs
is chosen
so that the multiplier function is twice continuously dierentiable,
and is given by
hs(z)=
1if z<0
0if z>1
110z3+15z46z5otherwise.
(9)
The multiplier function is then dened for some choice of param-
eters a,b,and sC.
h(ϵ)=
1if u+f(ϵ)<a+sC
0if u+f(ϵ)>b+sC
hsu+f(ϵ)asC
baotherwise.
(10)
The parameter
a
and
b
determines the intersection of the hydro-
static axis with the boundary of region A and B respectively. The
parameter
sC
controls a shift of this multiplier region along the
hydrostatic axis.
3.2 Water
We model the water as nearly incompressible [Becker and Teschner
2007] with the partial stress
σw=pwI,pw=k 1
Jwγ1!.(11)
We note that the water pressure is related to a potential
ψw
as
pw=
ψw
Jw(Jw)
, with
ψw(Jw)=k(Jw)(1γ)
1γJw
. This pressure
pw
is designed to stiy penalize the volume change of the water phase,
which is characterized in terms of the determinant of the water
deformation gradient
Jw=det(Fw)
. As with the solid phase, the
uid deformation gradient evolves according to
Dw
Dt Fw=vwFw
.
Intuitively,
Jw
is the ratio of the current to initial local volume of
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
Multi-species simulation of porous sand and water mixtures 1:5
Fig. 4. Sand wedges. A box of sand with various values of cohesion is dropped onto a wedge. The top le simulation is done using an explicit time-stepping
scheme, while all the others are done using a semi-implicit time-stepping scheme.
Fig. 5. Unilateral extension. We demonstrate a 2D sand column collapse
with dierent values of
c0
coeicient in the unilateral modified energy
density function. From top to boom,
c0=
1
.
1
,
2
,
2
.
5
,
3. Here parameters
a
and bfrom Equation 10 are a=0.5, and b=0.
material in the water phase. It evolves as
DwJw
Dt =∇ · vwJw.(12)
Here
k
is the bulk modulus of the water and
γ
is a term that more
stiy penalizes large deviations from incompressibility.
3.3 Momentum exchange
The momentum exchange terms
ps,pw
for water and porous sand
interactions can generally be viewed as a combination of dissipative
and reversible interactions [Borja 2006]. We follow the formulation
of Bandara and Soga [2016] because like them we are concerned pri-
marily with gravity driven ows such as fast catastrophic landslides
and debris ows. Their formulation assumes
ps=cEvwvs+pwϕw,pw=ps(13)
where
cE=n2ρwд
ˆ
k
and
n
is the sand porosity,
ˆ
k
is the sand perme-
ability and
д
is the gravitational acceleration,
ϕw=ρw
ρ
is the water
volume fraction and
pw
is the water pressure. The rst term repre-
sents viscous forces generated by sand particles moving through
the uid. Although it can be conceived from the view of an ideal-
ized particle moving through a Stokes uid, it simply amounts to a
Coulomb-friction-like response [Mackenzie-Helnwein et al
.
2010].
The second term is often called the “buoyancy term" in mixture
theories [Robert and Soga 2013]. It can be conceived from entropy
equilibrium constraints or from the physical consideration that the
pore uid pressure multiplying the porosity is appropriate notion
of reversible pressure, however there is some debate about its ap-
propriateness outside of immiscible mixtures [Drumheller 2000].
Mackenzie-Helnwein et al. [2010] omit the second term and view
the momentum exchange terms as purely dissipative processes. We
also follow this approach for the majority of the examples presented
in the paper, however we include one example demonstrating its
eect in Figure 11. Even with the inclusion of the active term from
[Bandara et al
.
2016], we are only capable of simulating a rather
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
1:6 Pradhana Tampubolon, A. et al
Fig. 6. The levee is going to break. Here we demonstrate the eects of our parameters on levee wall integrity. A simulation with lower drag momentum
exchange coeicient and lower cohesion will fail more easily.
narrow range of porous media phenomena. While this is sucient
for landslides and debris ows, phenomena such as capillary action
drawing water into dry sand is not achievable in our approach.
3.4 Cohesion vs. Saturation
As in Robert and Soga in [2013], we assume that the sand cohesion
varies as a function of water saturation. We measure saturation as
the percentage of water in the mixture which we estimate as the ratio
of the density of the water phase to the total density ϕw=ρw
ρw+ρs.
The cohesion of sand is zero when it is completely dry (
ϕw=
0).
Intuitively, as dry sand becomes saturated with more water, the
cohesivity of the wet sand should increase as wet sand tends to better
hold its shape. Indeed this was observed in the work of Robert and
Soga in [2013]. However, they also observe that this increase only
continues to a maximal value
cmax
C
when the saturation is around
ϕw=
0
.
4. Beyond this point the sand becomes more compliant
to owing and less cohesively elastic. In all of our multispecies
examples, we model water interaction with wet sand that is capable
of holding its shape. We thus set the sand cohesion to be initially
maximal, even in the absence of the water phase. Based on the
observations in Robert and Soga [2013], we then assume that the
cohesion decreases linearly with increasing saturation beyond this
point (with cohesion equal to zero at full saturation ϕw=1).
4 DISCRETIZATION
First, we explain the notation used in the discretization section.
There are two sets of grids: one is associated with sand material
and the other is associated with water. As with the continuous
equations, the superscript
α=s,w
indicates the corresponding
species. Whenever a symbol has a subscript
i
or
j
, this denotes one
degree of freedom in grid node index
i
or
j
. Subscript
p
denotes
attributes that belong to a particle. A symbol that is not followed by
a subscript refers to the whole collection of grid nodes as degrees
of freedom. For example
vs,n+1
refers to all of the sand grid nodes
that are active at step n+1.
We discretize the continuum equation using the Material Point
Method (MPM). As in other approaches in the engineering literature
[Bandara and Soga 2015; Jassim et al
.
2013; Mackenzie-Helnwein
et al
.
2010], we use two sets of grids (Figure 2): one for the solid
particles and the other for the water particles. This is the primary
dierence between our approach and others recently used in the
graphics literature. The overview of the algorithm is as follows:
(1)
Transfer to grids: Transfer the mass and momentum of each
species to its corresponding grid (§4.1).
(2)
Update grids momenta: Solving for the coupled water and
sand grid velocities using semi-implicit backward Euler
(§4.2).
(3)
Update particles: Update all particle state, including the
cohesion based on saturation as well as plasticity return
mappings (§4.3).
4.1 Transfer to grid
We transfer mass and momentum from sand and water particles
to their respective grid using APIC [Jiang et al
.
2015]. For each
species
α
, the particle
p
interacts with the grid node
α
i
with weight
wα,n
ip =N(xα,n
pxα
i)
. The weight is computed using the quadratic
B-spline interpolation kernel. Mass is computed according to
mα,n
i=X
p
wα,n
ip mα
p,(14)
and velocity according to
vα,n
i=1
mα,n
iX
p
wα,n
ip mα
pvα,n
p+Cα,n
pxα,n
ixα,n
p,(15)
where the matrix
Cα,n
p
is an extra matrix stored per particle which
denes an ane velocity eld local to the particle [Jiang et al
.
2015].
Cα,n
p
is initialized with
Cα,0
p=0
and updated at the end of the
previous time step during the grid-to-particle transfer (Equation
29).
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
Multi-species simulation of porous sand and water mixtures 1:7
4.2 Update Grids Momenta
Using MPM, the forces in the sand and water phases are computed
as
fs
i(ˆ
xs)=ψs
ˆ
xs
i
=X
p
V0
p ψU
Fs(FsE
p(ˆ
xs)!(Fs E,n
p)ws,n
ip ,(16)
fw
i(ˆ
xw)=ψw
ˆ
xw
i
=X
p
V0
p ψw
Jw(Jw(ˆ
xw))!Jw,nww,n
ip .(17)
As in [Stomakhin et al
.
2013] we think of
ˆ
xα
i
as the position of
the grid node
i
corresponding to species
α
that has been deformed
from its original position
xα
i
by an amount of
tvα
i
, i.e.
ˆ
xα
i=
ˆ
xα
i(vα,n+1
i)=xα,n
i+tvα,n+1
i
. The discrete momentum balance
to be solved is
ms,n
ivs,n+1
ivs,n
i=tfs
iˆ
xs+ms,n
ig+ds
i(ˆ
x)(18)
mw,n
jvw,n+1
jvw,n
j=tfs
jˆ
xw+mw,n
jg+dw
j(ˆ
x),(19)
where the discrete interaction term is given by
ds
i j (ˆ
x)=cEms
imw
j(vs,n+1
ivw,n+1
j),(20)
dw
ji (ˆ
x)=cEms
imw
j(vs,n+1
ivw,n+1
j),(21)
for some drag coecient cE. Setting
M= Ms,n
Mw,n!,v= vs
vw!,f(ˆ
x(vn+1)) = fs(ˆ
xs)
fw(ˆ
xw)!,
and
D
to be the drag coecient matrix derived from Equations 20
and 21, we can write the coupled system as
(M+tD)vn+1=Mvn+tMд+f(ˆ
x(vn+1)) .(22)
At each time step, we solve this nonlinear system using a few
iterations of a modied Newton’s method. Since the matrix
D
is
symmetric, and both
fs
and
fw
are derived as the negative gradient
of a potential, the whole system is symmetric when linearized (as-
suming the eects of plasticity are ignored in the linearization) and
can be solved using MINRES. We note that we do not include the
eect of plasticity when computing the derivatives of
fs
. Doing oth-
erwise results in non-symmetric sand force derivatives which would
require GMRES. Our omission of these terms in the linearization
of the system is a modication to the standard Newton’s method.
However, it is essential that we use implicit time stepping because
of the sti momentum exchange terms and our lagged plasticity
approach is the key to making this ecient. Notably, this can lead
to cohesion artifacts without the unilateral modication in (§3.1.1)
(see Figure 7).
4.3 Update Particles
Update Jw.
We do not keep track the deformation gradient
Fw
of
water particles, but we keep track of its determinant
Jw
, which is
updated according to the discretization of Equation 12, i.e.
Jw,n+1
p=I+ttr(vw,n+1
p)Jw,n.
We found that in practice, this discretization tends to oer more
stability than the alternative of evolving
Fw
followed by computing
its determinant.
Update Fs.
Because we ignore the eects of plasticity during the
implicit solve for momenta,
ˆ
FsE ,n+1
evolves with the grid during
the grid momentum update as in Stomakhin et al. [2013]
ˆ
FsE ,n+1
p=I+tvs,n+1
pFsE ,n.(23)
ˆ
FsE ,n+1
p
is later processed for plasticity at the end of the time step
to dene FsE ,n+1
p, which we discuss below.
Saturation based cohesion.
We dene the water saturation on
sand particles based on a heuristic. First, we populate a grid whose
domain is the union of the sand grid and water grid domain. We
mark each node that has a non-zero mass for both sand and water
species as
ϕw,n+1
i=
1, otherwise
ϕw,n+1
i=
0. One can think of this
grid as tracking an indicator function of the overlap region between
the sand and water constituents. We then compute the saturation on
the sand particle by interpolating from the grid to the sand particle
according to
ϕw,n+1
p=X
i
wn
s,ip ϕw,n+1
i.(24)
This approximates the saturation as equal to one (maximal) deep in
the interior of the overlap region with a ramp to zero exterior to the
region. In all of our multi-species examples, we assume that the wet
sand is already saturated and has reached its maximum cohesion
level. Hence, any amount of additional water saturation will lower
the cohesion level of the sand. We vary cohesion in a linear fashion
as a function of water saturation as discussed in Section 3.4
cs,n+1
Cp =cs,0
Cp (1ϕw,n+1
p)(25)
where we note that the saturation is always in
(
0
,
1
)
. The approxi-
mation of the saturation in Equation 24 has errors biased towards
full saturation in the interior. This naturally leads to more rapid fail-
ure in the landslides and debris ows we consider in our examples
since the cohesion decreases more rapidly than it should. This is
an extreme simplication to correct behavior dened in Robert and
Soga [2013], but we found that it was eective for simulating these
phenomena.
Projection and volume correction treatment.
We now de-
scribe the plastic projection of
ˆ
FsE ,n+1
to
FsE ,n+1
. We start from the
Drucker-Prager projection,
P
, as implemented by Klár et al. [2016].
P
projects strains outside the yield surface onto the yield surface
according to the ow rule. The yield region has the shape of a cone.
When the material is under expansion, the strain is projected to the
tip. Otherwise it is projected to the side of the cone. However, while
this approach is adequate for projection directions perpendicular to
the hydrostatic axis, steps which project to the tip can induce volume
gain. This occurs when a particle undergoes expansion that induces
a cohesive elastic stress. In this case, stress is projected to the tip,
which is a stress free state. The particle is then in a new rest state
and any motion that would return it to its initial volume would be
penalized elastically. There are existing corrections to this behavior
in the literature, e.g. Dunatunga and Kamrin [2015] model material
as a disconnected stress-free medium under sucient expansion.
To combat this artifact, each sand particle carries an extra scalar
attribute
vs
cp
which tracks changes in the log of the volume gained
during extension. This can be naturally taken into account in the
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
1:8 Pradhana Tampubolon, A. et al
sec/frame Scheme tParticle # x
Dry Sand Wedge 129.65 Explicit 1×1041×1060.003
Wet Sand Wedge 72.65 Implicit 1×1031×1060.003
Dam Breaches 268.99 Implicit 1×1023×1060.01
Piling 11.16 Implicit 1×1034.1×1060.0014
2D Piling 7.26 Explicit 5×1054.7×1040.0025
2D Piling 3.39 Implicit 1×1044.7×1040.0025
2D Piling Unilateral 0.67 Implicit 1×1024.7×1040.0025
2D Hourglass 0.84 Explicit 7.4×1053×1040.0037
2D Hourglass 0.73 Explicit 7.4×1053×1040.0074
Table 1. Performance.
logarithmic-strain-based constitutive model to allow for compres-
sion in the event of prior net expansion. At each time step, we update
vs
cp according to
vs,n+1
cp =vs,n
cp +log(det(Fs E,n+1) ) log(det(ˆ
FsE ,n+1)),(26)
where
vs,0
cp =
0. Finally, when we perform the Drucker-Prager
projection, we update ϵsE,n+1according to
ϵsE,n+1=P*
,ˆ
ϵsE,n+1+
vn
cp
dI+
-,(27)
where Pis the projection operator described above.
This can be interpreted as projecting the elastic strain plus the
volume gain term. Lastly,
FsE ,n+1=Uˆ
ϵsE,n+1V
where
ˆ
FsE ,n+1=
Ueˆ
ΣsE ,n+1
V
and
ˆ
ϵsE,n+1=log(ˆ
ΣsE,n+1)
. See Figures 8 and 9 for
a demonstration of this eect. We note that this projection is par-
ticularly dened for constitutive models written in terms of the
logarithmic strain. For a more general constitutive model, it would
require non-trivial modication.
Update position and velocity.
Velocity is updated according to
vα,n+1
p=X
i
wα,n
ip vα,n+1
i,(28)
the ane velocity matrix is updated according to
Cα,n+1
p=X
i
wn
ip vα,n+1
i(( 4
h2)(xα,n
ixα,n
p)) ,(29)
where
h
is the Eulerian grid spacing. Lastly position is updated
according to
xα,n+1
p=xα,n
p+tvα,n+1
p.(30)
5 IMPLEMENTATION AND RESULTS
We use the sparse grid structure provided by OpenVDB [Museth
2013]. The table of performance for various examples is given by
Table 1. The list of parameters used for these examples are listed in
Table 2.
5.1 Unilateral hyperelasticity and implicit time stepping
Figure 7 demonstrates the eects of our unilateral constitutive model
with semi-implicit time stepping. We note that our unilateral po-
tential removes articial cohesion eects in the simulation of dry
sand. The cohesion in this simulation is zero so the sand should not
stick together. When we use the constitutive model from [Klár et al
.
2016] with a semi-implicit time integration scheme, we see articial
cohesion that gets worse as we increase the time step size. Using
our unilateral elastic energy function removes this articial cohe-
sion. We further demonstrate that our semi-implicit scheme gives
results comparable to the more accurate, but more expensive fully
implicit backward Euler scheme in Figure 10. The importance of
choosing the right unilateral parameters is illustrated by Figure 5. If
the regions
A
and
B
in Figure 3 do not closely t the Drucker-Prager
cone, then the accuracy of the simulation is compromised for large
time steps.
5.2 Volume fix and wet sand
To illustrate our volume x to the Drucker-Prager artifact, we run
a simulation of an hourglass turned three times with explicit sym-
plectic Euler, as shown by Figure 8. The left most gure depicts the
initial state of the sand. For each set of three hourglass gures, we
show the state of the sand after going through the neck the rst
time, after the rst ip, and after the second ip. Figure 9 depicts a
similar artifact in a 3D piling of sand. Without the volume correc-
tion algorithm and without a suciently small grid resolution, we
observe a substantial volume gain artifact.
5.3 Wet sand and dam breach
In Figure 4 we demonstrate how varying cohesion gives rise to dif-
ferent wet sand behaviors. In Figure 6 we demonstrate our approach
with an example that is representative of the types of gravity driven
ows we are interested in with our approach. As water ows into
the wall of a dam, the saturation increases weakening the it. The
cohesion of sand decreases with saturation and the dam eventually
breaks.
We demonstrate the eect of the active component of the momen-
tum exchange terms in Equation 13 using a simulation of a 2D dam
breach, shown in Figure 11. Again water pours in from a reservoir
and slowly erodes a retaining wall. We note that the active term has
only a subtle eect on the bulk dynamics of the motion for these
types of ows. We discretize the active term by adding
Xpw,n
pws,n
ip
mw,n
i
ms,n
i+mw,n
i
to the water drag term in Equation 21. We then dene the solid drag
term to be equal and opposite to the water drag in accordance with
the zero-net-sum nature of the momentum exchange.
6 LIMITATIONS AND FUTURE WORK
Our approach has a number of limitations. The momentum exchange
model we use in the water/sand multispecies examples is rather
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
Multi-species simulation of porous sand and water mixtures 1:9
Fig. 7. From top to boom row: initial, middle, and final configuration of a 2D sand pile. On the top le is the result of explicit time-stepping with max
t=
10
4
. The two columns in the middle are the results of semi-implicit time-stepping with the regular constitutive model and max
t=
10
2
and 10
3
respectively. On the boom right is semi-implicit time stepping with our unilateral energy density function with maximum time step restriction of
t=
10
2
.
Fig. 8. An hourglass is flipped two times. The le most hourglass shaded grey depicts the initial state of sand. Each set-of-three figures depicts the state of the
sand aer going through the neck the first time, and aer flipping the hourglass for the first and second time. In the first two sets, grid
dx =
7
.
4
×
10
3
, while
the last two sets have grid
dx =
3
.
7
×
10
3
. The first and third set are run without any volume correction fix, while the second and last set are run with our
volume fix.
Fig. 9. Sand piling. The top row depicts the result of an explicit simulation with a coarser grid size (
dx =
10
2)
, while the boom row corresponds to a grid
size of 10
3
. The first and third column correspond to a projection step without volume correction, while the second and fourth column uses our volume
correction algorithm.
ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
1:10 Pradhana Tampubolon, A. et al
ρEνFriction angle h0/h1/h2/h3
Castle 2200 3.537 ×1050.3 — 35/0/0.2/10
Friction angle 2200 3.537 ×1050.3 20/25/30/35/40
Hourglass 2200 3.537 ×1050.3 — 35/9/0.3/10
Buttery 2200 3.537 ×1050.3 — 35/9/0.2/10
Buttery close 2200 3.537 ×1050.3 — 35/9/0.2/10
Raking 2200 3.537 ×1050.3 — 35/9/0.2/10
Raking close 2200 3.537 ×1050.3 — 35/9/0.2/10
Pile from spout 2200 3.537 ×1050.3 30
Splash 1582 3.537 ×1060.3 22
Shovel 2200 3.537 ×1050.3 — 35/9/0.2/10
Young’s modulus 2200 103,4,5,60.3 — 35/9/0.3/10
Table 2. Material parameters are provided for all of our 3D simulations.
Friction angle
ϕF
and hardening parameters
h0
,
h1
, and
h3
are listed in
degrees for convenience.
simplied. While adequate for gravity driven ows like landslides
and levee breaches, it is inadequate for capillary driven phenom-
ena like water being drawn in to dry sand. Such phenomena has
been captured by prior approaches like that of Lenaerts and Dutre
[2009]. Furthermore we fail to capture behavior like those in Rungji-
ratananon et al. [2008] where surface tension eects in wetting are
more accurately captured.
Although our approximation to the dependence of sand cohesion
on saturation is useful for facilitating rapid failure of water/sand
mixtures, it is an extreme simplication to the correct behavior
dened in Robert and Soga [2013] and this compromises its accuracy
dramatically. This reduces the applicability of our approach outside
of visually plausible simulation applications.
Large values of the momentum exchange coecient
cE
can lead
to ill-conditioning in the linear systems that arise during implicit
time stepping. We found that these cases required many MINRES
iterations to resolve and lead to excessive run times. This compli-
cated the simulation of slurry materials where the water and sand
remain mixed. In the future we would like to examine appropriate
preconditioners to improve the performance. Also, while we omit
or use a very simplistic buoyancy term for the reversible momenta
exchange in the
pα
equations, we would like to examine the addi-
tion of more accurate terms to produce phenomena like absorbent
sponges interacting with liquids. Lastly, we would like to examine
the suitability of our multiple grid MPM framework for the simu-
lation of more general multi-species interactions like chemically
reacting ow.
ACKNOWLEDGMENTS
The authors thank Qi Guo, Lawrence Lee and Stephanie Wang
for their help with simulations, rendering and video production.
The work is supported by NSF CCF-1422795, ONR (N000141110719,
N000141210834), DOD (W81XWH-15-1-0147), Intel STC-Visual Com-
puting Grant (20112360) as well as a gift from Disney Research.
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ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017.
... The two-layer formulation provides opportunities to solve dynamic flow problems in geomechanics such as internal and external erosions as well as fluid flow failures (seepage failures). The two-layer approach was adopted to study different strategies of modeling dragging interactions; for solid-fluid interaction in animation (Gao, Tampubolon, Jiang, & Sifakis, 2017;Tampubolon et al., 2017); combine with the DDMP (Tran, Solowski, Karstunen, & Korkiala-Tanttu, 2017;Tran, Solowski, Thakur, & Karstunen, 2017;Tran & Sołowski, 2019a, 2019b to model the large consolidation; combine with the GIMP (Liu, Sun, Jin, & Zhou, 2017) and to develop the two-layer approach with a thermodynamic constitutive model (Baumgarten & Kamrin, 2019). For instance, it allows modeling of fluidization of soil particles at the water interface and simulations of internal erosion by transferring a portion of soil particle mass to fluid particle. ...
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Thesis
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Preprint
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This article presents a novel and flexible bubble modelling technique for multi-fluid simulations using a volume fraction representation. By combining the volume fraction data obtained from a primary multi-fluid simulation with simple and efficient secondary bubble simulation, a range of real-world bubble phenomena are captured with a high degree of physical realism, including large bubble deformation, sub-cell bubble motion, bubble stacking over the liquid surface, bubble volume change, dissolving of bubbles, etc. Without any change in the primary multi-fluid simulator, our bubble modelling approach is applicable to any multi-fluid simulator based on the volume fraction representation.
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We simulate sand dynamics using an elastoplastic, continuum assumption. We demonstrate that the Drucker-Prager plastic flow model combined with a Hencky-strain-based hyperelasticity accurately recreates a wide range of visual sand phenomena with moderate computational expense. We use the Material Point Method (MPM) to discretize the governing equations for its natural treatment of contact, topological change and history dependent constitutive relations. The Drucker-Prager model naturally represents the frictional relation between shear and normal stresses through a yield stress criterion. We develop a stress projection algorithm used for enforcing this condition with a non-associative flow rule that works naturally with both implicit and explicit time integration. We demonstrate the efficacy of our approach on examples undergoing large deformation, collisions and topological changes necessary for producing modern visual effects.
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We present a new continuum-based method for the realistic simulation of large-scale free-flowing granular materials. We derive a compact model for the rheology of the material, which accounts for the exact nonsmooth Drucker-Prager yield criterion combined with a varying volume fraction. Thanks to a semi-implicit time-stepping scheme and a careful spatial discretization of our rheology built upon the Material-Point Method, we are able to preserve at each time step the exact coupling between normal and tangential stresses, in a stable way. This contrasts with previous approaches which either regularize or linearize the yield criterion for implicit integration, leading to unrealistic behaviors or visible grid artifacts. Remarkably, our discrete problem turns out to be very similar to the discrete contact problem classically encountered in multibody dynamics, which allows us to leverage robust and efficient nons-mooth solvers from the literature. We validate our method by successfully capturing typical macroscopic features of some classical experiments, such as the discharge of a silo or the collapse of a granular column. Finally, we show that our method can be easily extended to accommodate more complex scenarios including two-way rigid body coupling as well as anisotropic materials.
Conference Paper
The simulation of fluid mixing under the Eulerian framework often suffers from numerical dissipation issues. In this paper, we present a mass-preserving convection scheme that offers direct control on the shape of the interface. The key component of this scheme is a sharpening term built upon the diffusive flux of a user-specified kernel function. To determine the thickness of the ideal interface during fluid mixing, we perform theoretical analysis on a one-dimensional diffusive model using the Fick's law of diffusion. By explicitly controlling the interface thickness using a spatio-temporally varying kernel variable, we can use our scheme to produce realistic fluid mixing effects without numerical dissipation artifacts. We can also use the scheme to control interface changes between two fluids, due to temperature, pressure, or external energy input. This convection scheme is compatible with many advection methods and it has a small computational overhead.
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Introduction Large-scale physical model experiments Behavior of unsaturated sands Numerical modeling of the behavior of unsaturated sands Numerical modeling of the physical model experiments Dimensionless force – H/D relationship for pipelines in unsaturated soils Conclusions Acknowledgments Bibliography
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