Simulation of Fluids
Masters Thesis
Alexander J. Kaminski
N.C.C.A BOURNEMOUTH UNIVERSITY
September 2005
Alexander J. Kamiński
Simulation of Fluids
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Contents
List of Symbols .............................................................................................................. 3
1. Introduction ................................................................................................................ 4
2. Previous Work ........................................................................................................... 4
3. Fluid ........................................................................................................................... 5
3.1 Density ................................................................................................................. 5
3.2 Viscosity .............................................................................................................. 5
3.3 Compressible and Incompressible Flow .............................................................. 6
4. Fluid Dynamics Background ..................................................................................... 7
4.1 Navier-Stokes Equations ...................................................................................... 7
4.1.1 Advection ...................................................................................................... 7
4.1.2 Diffusion ....................................................................................................... 8
4.1.3 Pressure ......................................................................................................... 8
4.1.4 External Forces ............................................................................................. 8
5. Fluid Simulator .......................................................................................................... 9
5.1 MAC Grid ............................................................................................................ 9
5.2 Finite Differences............................................................................................... 10
5.3 MAC Method ..................................................................................................... 11
5.3.1 Simulation Algorithm ................................................................................. 11
5.3.1.1 Calculate Time Step ................................................................................. 11
5.3.1.2 Moving Particles ...................................................................................... 12
5.3.1.3 Identify Empty and Filled Voxels ............................................................ 12
5.3.1.4 Boundary Slip and Continuity Conditions ............................................... 12
5.3.1.4.1 No-Slip Boundary Conditions............................................................... 12
5.3.1.4.2 Slip Boundary Conditions ..................................................................... 13
5.3.1.4.3 Continuity Conditions ........................................................................... 13
5.3.1.5 Solving the Navier-Stokes Equations ...................................................... 13
5.3.2 Pressure Projection Step ............................................................................. 13
5.4 Solving a System of Linear Equations ............................................................... 14
5.4.1 Setting up Matrices ..................................................................................... 14
6. Implementation Issues ............................................................................................. 14
7. Conclusion ............................................................................................................... 17
8. References ................................................................................................................ 18
Alexander J. Kamiński
Simulation of Fluids
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List of Symbols
A
Matrix of size N x N
b
Array of known values
f
body forces as a whole f=(f,g,h)
f
body force in the x direction
g
body force in the y direction
h
body force in the z direction
n
the outward normal of the surface
N
number of fluid filled voxels within the computational
domain
p
pressure
u
velocity u=(u,v,w), in metres per second
u
u component of velocity
u
v
u component of best guess velocity
v
w
v component of best guess velocity
w
x
w component of best guess velocity
v component of velocity
w component of velocity
array of unkown values in the linear system
∆t
Dx
time step
width of the voxel

gradient operator

divergence operator

²
Laplacian opertator



Alexander J. Kamiński
viscosity
density
Simulation of Fluids
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1. Introduction
The modelling of natural phenomena remains a taxing problem in the visual effects
community and attracts a vast amount of research. Natural looking animation of “fluid-like”
behaviour e.g. smoke, liquid and fire is highly desirable and difficult to produce. Arguably the
most demanding of these is to produce is realistic liquid animation. This is a great challenge
in computer graphics due to the complex dynamics of flows. This increasing need for fluid
dynamics in film and computer game productions has made many researchers motivated to
extensively examine computational fluid dynamics (CFD).
This thesis describes the theory and methods behind implementing a stable fluid simulator
that is able to handle varying levels of viscosity. By being able to handle varying levels of
viscosity the simulator should be able to represent fluids from water to honey.
2. Previous Work
Animation of fluids is approached in a number of ways in the computer graphics literature.
The term “fluids” is used to encompass the motion of gases such as air (including simulating
smoke), and liquids such as water.
Several graphics researchers studied the large-scale motion of water in waves [FOU86].
These methods use elevation maps of the terrain underneath the water, and the line of waves
is bent according to the variations in wave speed that the elevation profile induces. The
simulation of breaking waves occurs at a particular sea floor elevation and wave velocity.
The MAC method developed by Harlow and Welch [HAR65] provided a technique that
allowed the Navier-Stokes equations to be applied and this approach was first used by Nick
Foster [FOS96] for use within computer graphics. Foster produced very realistic results while
being very computationally expensive.
Most recently, Smoothed Particle Hydronamics have been used to so that there is no need for
a computational domain for which to compute the Navier-Stokes equations on but rather
momentum equations are directly applied particles that have a mass and interact with each
other.
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3. Fluid
3.1 Density
The density of a fluid, r , is the mass of the fluid per unit volume and depends on both the
temperature and pressure of the material. Table 3.1 lists the densities for common fluids at
constant temperature and pressure values.
Density (kg/m3)
Fluid
Hydrogen
Helium
Air
Carbon Dioxide
Gasoline
Water
Mercury
0.0899
0.1785
1.293
1.977
680
998.2
13600
Table 3.1 – Densities of common Fluids
The gasses are roughly 1000 times denser than the liquids.
3.2 Viscosity
Viscosity, n , is the resistance a fluid has to change in form and can be thought of as internal
friction. It is also perceived as the “thickness”, or resistance to pouring. With this in mind,
water is considered “thin”, having a low viscosity, whereas olive oil would be considered to be
“thick”. Table 3.2 lists some common fluids and their viscosities at a constant room
temperature (20°c).
Fluid
Hydrogen
Air
Water
Whole Milk
Olive Oil
Honey
Peanut Butter
Window Putty
Viscosity (centipoises)
0.0086
0.0182
1.002
2.12
84.0
10,000
250,000
100,000,000
Table 3.2 – Viscosities of common Fluids
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3.3 Compressible and Incompressible Flow
All fluids are to some degree compressible which means that the density of the fluid will
change with applied pressure. However, liquids are extremely hard to compress and in the
computational fluid dynamics world they are usually always held to be incompressible,
whereas gases are easily to compressible.
The density of liquids is therefore assumed to be constant everywhere ( ≡ c), which would
therefore assure incompressibility. A sufficient condition for incompressibility is:
¶r
¶t
+ (u · Ñ )r = 0
(1)
This condition states that flow can have variable density as long as it constant along
streamlines.
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4. Fluid Dynamics Background
4.1 Navier-Stokes Equations
In the following discussion, u represents the velocity field. The p parameter represents the
scalar pressure field and r represents the density of the fluid. The n parameter is used to
represent the fluids kinematic viscosity.
All the current techniques to simulate the motion for a viscous incompressible fluid use the
Navier-Stokes equations (shown below). These two equations are concerned with the
conservation of mass and conservation of momentum within the fluid.
Ñ· u
ut = - (u · Ñ )u + nÑ 2 -
(2)
1
Ñp + f
r
(3)
The first of the equations handles the conservation of mass within the fluid by stating that the
velocity field has zero divergence everywhere. This simply means that in any small region of
fluid, the amount of fluid entering the region is exactly equal to the amount leaving the region.
In reality however, no fluid is ever really incompressible, but incompressibility is assumed for
fluids moving at low speeds because their compression is negligible and the incompressibility
allows for a convenient solution for the pressure.
The second equation describes the conservation of momentum, and has several components.
Reading from left to right, it states that the instantaneous change in velocity of the fluid at a
given position is the sum of four terms: advection, diffusion, pressure, and body forces. The
vector field ut is the time derivative of the fluid velocity.
The Navier-Stokes equations rely heavily on the use of partial differentials, which is
represented by the “del” operator, Ñ . When this operator is applied to a scalar value such as
p a vector is created, and in this case the vector produced is known as the gradient of p
Ñ p = ( px, py, pz )
(4)
When the operator is applied to a vector as it is in the divergence equation (2) a scalar result
is produced.
Ñ · u = u x + v y + wz
(5)
The Laplacian operator, Ñ , is a scalar differential equation and is used as part of the
viscous diffusion equation (EQUATION). The Laplacian operator on a scalar u is:
2
Ñ · (Ñ 2u) = Ñ 2u = uxx + u yy + uzz
(6)
4.1.1 Advection
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The advection term of the Navier-Stokes equation is, shown in equation 8, handles the
transportation of objects, densities and accounts for the direction in which the surrounding
fluid pushes a small region of fluid. This term represents the self-advection of the velocity
field. Fast river water is an advection-dominated flow, and any small amount of water poured
into the river will quickly be swept away with the current.
- (u · Ñ )u
(7)
4.1.2 Diffusion
The momentum diffusion term describes how quickly the fluid damps out variation in the
velocity surrounding a given point. For example, maple syrup flows slowly when poured due
to the higher viscosity, whereas water flows quickly when poured due to the lower viscosity.
The higher the viscosity value, the faster the velocity variations are damped
nÑ 2u
(8)
4.1.3 Pressure
The third term is the pressure gradient, and it describes how a small parcel of fluid is pushed
in a direction from high to low pressure. Since fluids molecules have the ability to freely move
around each other they tend to “squish” and “slosh”. When a force is applied to a fluid, it does
not instantly propagate through the entire volume. Instead, the molecules closer to the source
of the force push on those further away, and pressure builds up. As pressure is a force per
unit area, any pressure in the fluid naturally leads to acceleration. This coincides with
Newton’s second law; F = ma .
1
r
Ñp
(9)
4.1.4 External Forces
The last term in the Navier-Stokes equation (3) defines the acceleration due to the external
forces applied to the fluid. The forces that act upon the fluid may be either classed as ‘body
forces’ or ‘local forces’. Body forces are forces that will affect the fluid as a whole, such as
gravity, whereas the local forces are forces that are only applied to a certain region of the
fluid. An example of this would be the force of a fan blowing air.
Alexander J. Kamiński
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5. Fluid Simulator
In order for a computer to solve the Navier-Stokes equations, they must be broken down into
simple smaller steps. A discrete domain must then be chosen to as a basis for solving these
steps. A discrete computational domain, commonly known as a MAC grid, and a numerical
technique known as finite differences, are used to break the Navier-Stokes equations into
small pieces that the computer can solve. This method is based on a fixed, Eulerian grid of
control volumes.
Foster and Metaxis [FOS96] were the first to attempt to solve the Navier-Stokes equations in
3D computer graphics. In their papers they demonstrate how the Marker-And-Cell (MAC)
approach of Harlow and Welch [HAR65] can be used to animate water and be modified to
control the behaviour of animated fluids. This method demonstrated a new approach and that
representing a liquid is no longer constrained to be a height field.
The approach taken by Foster and Metaxis [FOS96] has continued to be the favourite choice
when attempting to simulate fluids within computer graphics community.
5.1 MAC Grid
The Marker-and-Cell (MAC) grid method was first developed by Harlow and Welch [HAR65]
and is made up of uniformly sized cells or voxels (figure 5.1). Each voxel is a cube with sides
of length D x .
Figure 5.1 A voxel located at i,j,k
There are two types of variable can be stored in one voxel – a scalar value, or a vector value.
The scalar values are stored at the centre of the voxel and are sometimes referred to as the
Alexander J. Kamiński
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cell nodes. The vector values are stored in a staggered grid formation, in which the x
component of the vector is stored at the left face of the cube, the y component is stored at the
bottom face of the cube, and the z component is stored at the back face of the cube. These
face values are sometimes referred to as face nodes. Normally a voxel records two variables:
the pressure and the velocity. The scalar pressure ρ is calculated and stored at the cell node,
and the vector valued velocity u = ( u , v, w ) is stored as separate components at the
appropriate face nodes. A single cell at location i,j,k is shown above in figure 5.1.
The fluid volume is tracked within the grid by a set of marker particles that move with the
velocity field, but have no volume, mass or other properties. The surfaces are simply the
boundaries of the volumes, and in this sense surfaces may appear, merge or disappear.
The grid voxels containing markers are considered occupied by fluid, while those without
markers are empty. A free surface is defined to exist in any grid cell that contains particles
and that also has at least one neighbouring grid cell that is empty. The location and
orientation of the surface within the cell was not part of the original MAC method.
The computational grid that is used with the MAC method uses the dimensions I x J x K, and
the grid is indexed with the integers i Î [0, I - 1] , j Î [0, J - 1] , k Î [0, K - 1] the voxels
positioned at on the outer edges of the grid are considered to be boundary voxels, therefore,
fluid can only exist where 1 £ i £ I - 2,1 £ j £ J - 2 and 1 £ k £ K - 2 .
5.2 Finite Differences
In order to solve the continuous partial differential equations on the discrete computational
domain the finite difference method must be used. Other methods such as finite volumes and
finite elements could be used, but these are more complex and less effective for a regular
grid.
Finite differences use the Taylor series [RAS04] to approximate the derivatives and the
accuracy of approximations is measured in the number of terms that the Taylor series is
expanded to. If the first term is used then the accuracy is first order. The following finite
difference equations are first order accurate.
The finite difference version of the divergence equation (2), of the velocity at the voxel i, j , k
is the scalar:
(Ñ · u)i , j ,k =
ui+ 1, j ,k - ui , j ,k + vi , j+ 1,k - vi , j ,k + wi , j ,k + 1 - wi , j ,k
Dx
(10)
This result is stored at the cell node. The finite difference version of the pressure gradient,
equation 9, at voxel i, j , k is:
æp - pi- 1, j ,k pi , j ,k - pi , j- 1,k pi , j ,k - pi , j ,k- 1 ö
÷
÷
(Ñ p)i , j ,k = çç i , j ,k
,
,
÷
çè
÷
Dx
Dx
Dx
ø
(11)
This is solved for each face node. The finite difference equation in the Navier-Stokes equation
that is second order accurate is the Laplacian operator (equation 12). The finite version of this
for voxel at i, j , k is:
Ñ 2ui , j ,k =
ui- 1, j ,k + ui , j- 1,k + ui , j ,k- 1 - 6ui , j ,k + ui+ 1, j ,k + ui , j+ 1,k + ui, j ,k + 1
Alexander J. Kamiński
D x2
Simulation of Fluids
(12)
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5.3 MAC Method
The MAC method has two major components: the voxels (previously described) and a large
collection of marker particles that are used to carry velocities to previously empty voxels.
Evolution of surfaces is computed by moving the marker particles with locally interpolated
fluid velocities. Special treatments are used to define the fluid properties in newly filled grid
cells and to cancel values in cells that are emptied.
5.3.1 Simulation Algorithm
The method that will be used for the fluid simulation is a combination of the simulator defined
by Nick Foster [FOS96] and the simulator defined by Mark Carlson [CAR04]. Foster
implemented a solid fluid simulator that while being computationally expensive provided very
realistic results. Carlson’s fluid simulator also is computationally expensive but allows for
varying levels of viscosity unlike the simulator defined by Foster. Therefore, a combination of
these designs will be used for the proposed simulator.
One time-step in a fluid dynamic simulation using the MAC method is calculated in several
stages:
1. An appropriate time-step, D t , size is chosen.
2. The marker particles are moved according to the current velocity field, u.
3. Each voxel is marked as being fluid-filled or empty according to whether a given voxel
contains marker particles. The velocities are then set for previously empty voxels that
now contain fluid.
4. Boundary conditions are set for all the solid obstacles and voxel faces that are empty
on one side and have fluid on the other.
5. The velocity, u, values in the fluid-filled voxels are updated based on the NavierStokes equations.
[CAR04]
5.3.1.1 Calculate Time Step
Foster states [FOS96] that to be consistent with a true solution a restriction known as the
Courant-Friedrichs-Lewy (CFL) condition (13) must be enforced. The CFL restriction states
that the time-step must be small enough to make sure each marker particle does not travel
more than one voxel cell at any one time.
Dt
Dx
max(| u |,| v |,| w |)
The Navier-Stokes momentum equation (3) also imposes another restriction on
constant viscosity diffusion equation:
nÑ 2u
Alexander J. Kamiński
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(13)
D t by the
(14)
11
The stability restriction is:
D x2
6n
Dt
(15)
5.3.1.2 Moving Particles
The marker particles in the system are moved forward with an explicit second order RungeKutta technique known as the modified Euler or midpoint method as described by Carlson
[CAR04]. Assume that a particle starts at location x = ( x, y , z ) and is updated in two steps:
step 1: x
n+
1
2
= xn + D tu(xn )
(16)
step 2: x n+ 1 = x n +
1
n+
Dt
[u(x n ) + u(x 2 )]
2
If this term makes a particle move into an empty voxel then the 2 nd step is not taken because
reliable velocities are not available outside the fluid cells. The second order technique is used
as it is the highest order that can be achieved when using linear interpolation.
5.3.1.3 Identify Empty and Filled Voxels
Once the marker particles are moved successfully, they are used to identify which voxels are
fluid-filled. If a previously empty voxel contains a particle the velocities for that voxel must be
set.
The velocities are set by using the marker particles within a face’s region. The region of a
face is defined as size D x centred from the face (figure 5.1).
5.3.1.4 Boundary Slip and Continuity Conditions
The boundary conditions affect how the fluid will react when in contact with the boundary
itself. Without the boundary conditions the fluid would simply flow through walls and obstacles
and cause a simulation failure. The conditions known as ‘no slip’ and ‘free slip’ boundary
condition are defined the same throughout all CFD literature.
5.3.1.4.1 No-Slip Boundary Conditions
At a no-slip boundary, there is no fluid motion relative to the boundary. For a stationary
boundary, the fluid velocity should vanish, i.e.
vn = 0, vt = 0
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(17)
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5.3.1.4.2 Slip Boundary Conditions
At a free-slip boundary, there is no flow through the boundary, but there can be fluid motion
tangential to the boundary:
vn = 0,
Where
¶ vt
= 0
¶n
(18)
¶
denotes the derivative with respect to the normal vn direction.
¶n
5.3.1.4.3 Continuity Conditions
When a voxel contains at least one marker particle it becomes filled. In the case of a filled
voxel neighbouring an empty voxel it becomes a surface cell and the face between the empty
and filled voxel become surface faces. This means that the surface cell contains an outer
layer of marker particles that represent the fluids volume boundary. In a three-dimensional
voxel grid there are 63 possible combinations of surface faces and the velocities for these
surface faces are set using continuity conditions. For information how each of these
combinations is set the reader is referred to Mark Carlson’s thesis [CAR04].
5.3.1.5 Solving the Navier-Stokes Equations
The solving the Navier-Stokes equation is calculated in two major steps while enforcing
incompressibility. The first step is to describe the forward Euler solution to a best guess
velocity,
u.
The equation for the best guess velocity with a constant viscosity is:
u = u + D t[- (u · Ñ )u + nÑ 2u + f ]
(19)
Solving this equation for all the fluid faces (fluid on both sides of the face) within the MAC grid
gives the best guess velocity. There is no pressure term in the best guess velocity as this is
harder to solve.
5.3.2 Pressure Projection Step
Solving the pressure term and combining the result with the best guess velocity will result in
new
the final velocity, u , and will be divergence free so that incompressibility is sustained. The
solving of pressure is known as the pressure projection step and was first used by Chorin
[CAR04]. Since the pressure term (9) was left out of the best guess velocity it must be
accounted for in the final velocity, as shown in equation 20.
u new = u -
Dt
Ñp
r
(20)
To solve the pressure a system of linear equations must be used.
Alexander J. Kamiński
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5.4 Solving a System of Linear Equations
5.4.1 Setting up Matrices
In order to solve the pressure for each of the fluid filled voxels, N , within the MAC grid a
linear system in a matrix form must be built. The notation for this system will is defined as:
Ax = b
(21)
In this linear system, A is an N x N matrix where each row represents an equation; x is a
column vector of size N with an entry for each of the unknowns, and b is a column vector of
size N that represents the known values. A pressure must be found for each of the fluid filled
voxels. To recast equation 21, for use in a linear system:
A º - D x 2Ñ 2
(22)
xº p
(23)
D x 2r Ñ · u
bº
Dt
(24)
The equation 24 for b that solves the each of the known values is straightforward to solve as
all of the velocities have been already set with the best guest and the continuity conditions for
all surface faces. However, the equations 22 and 23 are much harder to solve and require the
use of a conjugate gradient (CG). For further information the reader is referred to
6. Implementation Issues
For the implementation of the proposed simulator C++ and OpenGL will be used to program
and represent the simulator. However, the final rendering of the images will be in Renderman.
A file will be produced that will identify where all the marker particles are at any one frame.
This file will then be used by Jamie Briens and the proposed simulation will be used as a
means of representing his implicit surfaces.
Figure 6.1 shows how the MAC grid is initially setup in the simulator. The blue faces are used
to represent the boundary faces and all other faces are considered to be open and will let fluid
freely pass through the face. The green particles are used to represent the marker particles
within the simulator so that they could be visualised at each step of the simulation.
Alexander J. Kamiński
Simulation of Fluids
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Figure 6.1 MAC Grid represented by the simulator
Figure 6.2 shows the simulation at the first time step. The red faces are used so to represent
the surface faces within the computational domain. Therefore, if a voxel has at least one
surface face then it is considered to be a surface cell and the marker particles it contains
should be used to represent the boundaries of the fluid domain. The velocities for all the red
faces in the simulation use the continuity conditions to solve them.
The pink faces shown in figure 6.2 are used to represent fluid faces, which are faces that
have fluid voxels on both sides of it. The velocities at all fluid faces in the domain are solved
using the Navier-Stokes equations.
Alexander J. Kamiński
Simulation of Fluids
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Figure 6.2: Initial Fluid setup
Figure 6.3 shows the simulation at frame 35. Note how the fluid is sliding correctly up the wall
on the left boundary, whereas, the right hand side boundary is causing errors and the fluid
appears to stick to the boundary.
Figure 6.3: Example of the moving fluid within the Simulator
Alexander J. Kamiński
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7. Conclusion
In conclusion, the simulator is considered to be a success. The fluid motion is realistic and
varying levels of viscosity are able to be simulated from “thicker” fluids such as honey to
“thinner” fluids such as alcohol. Despite the simulator being a success one major known issue
exists. There is a problem with the boundary slip conditions when the fluid reaches the max
bounds of the computational domain in the I, J and K direction. The liquid on contact with
these boundaries will either appear to “stick” to the boundary or simply move through it. This
error has greatly affected the fluid motion when “splashing” back. However, even though this
fault is considered to be a major error, its correction is believed it to be easily solved and has
only been not rectified due to time constraints.
The figure below is a final rendered shot using Jamie Brien’s implicit surfaces on the marker
particles. The initial fluid started off at opposite ends of the domain and was sent towards
each other so that a crashing effect could be demonstrated.
Figure 7.1 Rendered Image of Fluid Motion
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8. References
[HAR65]
[FOS01]
HARLOW, F.H. and WELCH, J.E., "Numerical Calculation of TimeDependent Viscous Incompressible Flow," Phys. Fluids 8, 2182. 1965.
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[FOS96]
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[FOS97]
FOSTER, N. and METAXAS, D. Controlling fluid animation. In Proceedings
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[RAS04]
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[SHA04]
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[SIN03]
SINGH, P., HESLA, T. I., and JOSEPH, D. D., “Distributed Lagrange
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