Particle-Based non-Newtonian
Fluid Animation for Melting
Objects
Afonso Paiva
Fabiano P. do Carmo
Thomas Lewiner
Geovan Tavares
Matmidia - Departament of Mathematics – PUC-Rio
Fluid for Animation
Melt and Flow
Visually Realistic Computer Animations for Melting Objects
Overview
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Related Works
Governing Equations
Viscoplastic Model
Solid-Liquid Transition
SPH Basics
Implementation
Tracking Free Surface
Results
Related Works
Grid-based
Carlson et al., 2002
Particle-based
Keiser et al., 2005
Smoothed Particle
Hydrodynamics - SPH
Gingold & Monaghan (1977) and Lucy (1977)
Popular in astrophysics
because:
– Resolution automatically
adapts to density
– Easy to combine with NBody algorithms
– Modeling compressible
fluids
– Can be extend to
incompressible fluids
Lagrangian Formulation of
Navier-Stokes Equations
d
  .v
dt
dv
1
1
  p  .S  g
dt


Continuity Equation
Momentum Equation
• PDE → ODEs
• Particle Methods to perform CFD
– SPH, PIC, MAC, MPS, …
Viscoplastic Model
• P. R. S. Mendes, E. S. S. Dutra, J. R. R. Siffert, and M. F. Naccache.
“Gas displacement of viscoplastic liquids in cappilary tubes” Journal of Non-Newtonian Fluid Mechanics, 2005.
• Based on Generalized Newtonian Liquid model
• Stress Tensor:
S    D D
1
2
D
.tr  D 
2
D  v   v 
T
Viscoplastic Model
Viscosity Function
 n1 1 
  D   1  exp   J  1 D   D  
D


• n – power-law index
• J – jump number:
– yield stress
– low shear rate viscosity
– consistency index

Solid-Liquid Transition
Heat Equation
Temperature > Fusion Point
Temperature
Liquid
NS Equations
Jump Number
Viscosity
linearly
Viscosity Function




  D, T   1  exp   J T   1 D   D n1 
1

D
Solid-Liquid Transition
Overview
•
•
•
•
•
•
•
•
Related Works
Governing Equations
Viscoplastic Model
Solid-Liquid Transition
SPH Basics
Implementation
Tracking Free Surface
Results
SPH Basics
Meshless Method
– Particles are moving
interpolation centers
for fluid quantities
– Easy to track fluid free
surface
h
i
j
Quintic Spline
SPH Average Operator
A  xi    A  x j Wh  xi  x j 
n
mj
j 1
j
SPH Gradient
• Scalar
A  xi     Aj  Ai  iWh  xi  x j 
n
mj
j 1
j
• Vector
A  xi     A j  Ai  iWh  xi  x j 
n
mj
j 1
j
• Divergent
.A  x i     A j  Ai  .iWh  x i  x j 
n
mj
j 1
j
SPH Density
SPH Approximation of Continuity Equation
n m
di
j
 i   vi  v j  .iWh  xi -x j 
dt
j 1  j
SPH Momentum Equation
Pressure

1
i
pi
 pi
pj
  m j  2  2

j 1
 i  j
n

iWh  x i -x j 

Equation of state
– we can approximate the incompressible fluid by a quasicompressible fluid (Batchelor, 1974)
pi  c  i  0 
2
dp
c
 10vmax
d
SPH Momentum Equation
Stress Tensor
1
i
n
mj
j 1
i  j
.Si  
Si    Di  Di
S +S  . W  x
i
j
i
h
i
xj
Di  vi   vi
vi    v j  vi  iWh  xi  x j 
n
mj
j 1
j

T
SPH Laplacian
• Traditionally in CG:
 Ti   Tj  Ti   iWh  xi  x j 
n
2
2
j 1
mj
j
• Inspired in SPH Projection Method (Cummins
and Rudman, 1999)
 4 i
 Ti   

j 1  i   j
n
2
x i  x j  .iWh  x i  x j  m j


 T j  Ti 
2
j
xi  x j  

Implementation
Leap-Frog Scheme
Numerical Stability
– Artificial Viscosity
– XSPH
– CFL Condition:

h
h 2 
t  0.1min 
,

2
v

c
6


max 
 max
Tree Search Method
– Complexity: O
 n log n
Rendering the Free Surface
SPH Characteristic Function
1, x  S
  x    Wh  x  x j  
j 1  j
0, x  S
n
mj
Topological Marching Cubes
– T. Lewiner, H. Lopes, A. W. Vieira e G. Tavares. “Efficient
implementation of marching cubes with topological
guarantees” - JGT, 2003
Rendering the Free Surface
Work in Progress
• Phase Transition
→ Multiphase Flow
• Rendering the Free Surface
→ Free Surface Tracking
• Octree Particle Search
→ Pair-List
That’s all Folks
http://www.mat.puc-rio.br/~apneto