The document outlines various computational methods for simulating physics and fluids in computer graphics, discussing the mathematical principles underpinning these simulations, such as Newton's laws and fluid dynamics. It addresses the implementation of algorithms for optimal transport and the challenges of achieving precision in simulations. Key concepts include differential calculus, particle systems, and the use of diagrams such as Voronoi for modeling fluid behavior.

1. Mathématiques - Informatique
Ecole Jeunes Chercheurs GDR IM
Géométrie Numérique
Partie I: Transport Optimal (Lundi) B. Lévy
Partie II: Résolution de systèmes linéaires (Jeudi) N. Ray
Bruno Lévy
ALICE Géométrie & Lumière
CENTRE INRIA Nancy Grand-Est
Computer Graphics International 2018 – Bintan – June 11-14
Bruno Lévy - Simulating fluids with the “path bundle” method

2. Part. 1. Introduction
Part. 2. Physics
Part. 3. Fluids
Part. 4. Fluids on computers
Part. 5. To infinity and beyond…
OVERVIEW

3. EMETTEUR - NOM DE LA PRESENTATION
Introduction
00 MOIS 2011 - 3
1

4. 1982

6. 1982
Vol de rêve (Dream Flight)
Produced by Philippe Bergeron, Nadia Magnenat Thalmann, Daniel Thalmann.
Telling stories with computer graphics …

7. 1987
Rendez-vous à Montréal
Directed by Nadia Magnenat Thalmann, Daniel Thalmann.
Telling stories with computer graphics …

8. 1982
Vol de rêve (Dream Flight)
Produced by Philippe Bergeron, Nadia Magnenat Thalmann, Daniel Thalmann.
Telling stories with computer graphics …

9. 1982
Vol de rêve (Dream Flight)
Produced by Philippe Bergeron, Nadia Magnenat Thalmann, Daniel Thalmann.
Telling stories with computer graphics …
Emotions, character animation

10. 1982
Vol de rêve (Dream Flight)
Produced by Philippe Bergeron, Nadia Magnenat Thalmann, Daniel Thalmann.
Telling stories with computer graphics …
Emotions, character animation
Physics

11. How can we simulate physics in a computer ?
2018Telling stories with computer graphics …

12. Physics
2

13. Telling stories…

14. Telling stories…
Tales of matter, energy, time, light, …

15. Which “language” to tell these stories ?
Telling stories…
Tales of matter, energy, time, light, …

16. 1960

17. The connection machine – D. Hillis
R. Feynman,
Playing bongos
1980

18. Isaac Newton
1623-1727
1687
Principia Mathematica
A mathematical language
to talk about physics
Differential calculus
Newton’s principles
(1) Inertia
(2) Fundamental principle
(3) Reciprocal action
1687

19. Isaac Newton
1623-1727
1687
Principia Mathematica
A mathematical language
to talk about physics
Differential calculus
Newton’s principles
(1) Inertia
(2) Fundamental principle
(3) Reciprocal action
x position
1687

20. Isaac Newton
1623-1727
1687
Principia Mathematica
A mathematical language
to talk about physics
Differential calculus
Newton’s principles
(1) Inertia
(2) Fundamental principle
(3) Reciprocal action
x position
x speed
1687

21. Isaac Newton
1623-1727
1687
Principia Mathematica
A mathematical language
to talk about physics
Differential calculus
Newton’s principles
(1) Inertia
(2) Fundamental principle
(3) Reciprocal action
x position
x speed
x acceleration
1687

22. Isaac Newton
1623-1727
1687
Principia Mathematica
A mathematical language
to talk about physics
Differential calculus
Newton’s principles
(1) Inertia
(2) Fundamental principle
(3) Reciprocal action
x position
x speed
x acceleration
F = m x
(2) Fundamental principle of dynamics
Change of speed (acceleration) is proportional to force
Force
Mass
1687

23. Gravity
F = m g = m x
9.81 m / s / s
x = g

24. Gravity
F = m g = m x
9.81 m / s / s
x = g

25. Gravity
F = m g = m x
9.81 m / s / s
x = g

26. Gravity
F = m g = m x
9.81 m / s / s
x = g

27. Gravity
F = m g = m x
9.81 m / s / s
x = g

28. Gravity
F = m g = m x
9.81 m / s / s
x = g

29. Gravity
F = m g = m x
9.81 m / s / s
x = g

30. Gravity
F = m g = m x
9.81 m / s / s
x = g

31. Gravity
F = m g = m x
9.81 m / s / s
x = g

32. Gravity
F = m g = m x
9.81 m / s / s
x = g

33. Gravity
F = m g = m x
9.81 m / s / s
x = g

34. Gravity
F = m g = m x
9.81 m / s / s
x = g

35. Gravity
F = m g = m x
9.81 m / s / s
x = g

36. Gravity
F = m g = m x
9.81 m / s / s
x = g

37. Gravity
F = m g = m x
9.81 m / s / s
x = g

38. Gravity
F = m g = m x
9.81 m / s / s
x = g

39. Gravity
F = m g = m x
9.81 m / s / s
x = g

40. Gravity
F = m g = m x
9.81 m / s / s
x = g

41. Gravity
F = m g = m x
9.81 m / s / s
x = g

42. Gravity
F = m g = m x
9.81 m / s / s
x = g

43. Gravity
F = m g = m x
9.81 m / s / s
x = g

44. Gravity
F = m g = m x
9.81 m / s / s
x = g

45. Gravity
F = m g = m x
9.81 m / s / s
x = g

46. Gravity
F = m g = m x
9.81 m / s / s
x = g

47. Gravity
F = m g = m x
9.81 m / s / s
x = g

48. Gravity
F = m g = m x
9.81 m / s / s
x = gHow to simulate this behavior
on a computer ?
Each second
subtract 9.81 m/s from
the vertical component of speed
move the point along the speed

49. Gravity
F = m g = m x
9.81 m / s / s
x = g
We do computation with a certain precision

50. Gravity
F = m g = m x
9.81 m / s / s
x = g
We do computation with a certain precision
One can increase this precision

51. Gravity
F = m g = m x
9.81 m / s / s
x = g
We do computation with a certain precision
One can increase this precision
Is it possible to invent a langage to talk about
what would happen with infinite precision ?

52. Gravity
F = m g = m x
9.81 m / s / s
x = g
Is it possible to invent a langage to talk about
what would happen with infinite precision ?
Derivatives f’(x)
df(t)
dt
df(x)
dx

53. Gravity
F = m g = m x
9.81 m / s / s
x = g
Is it possible to invent a langage to talk about
what would happen with infinite precision ?
Derivatives f’(x)
df(t)
dt
df(x)
dx
Variation of something
w.r.t time

54. Gravity
F = m g = m x
9.81 m / s / s
x = g
Is it possible to invent a langage to talk about
what would happen with infinite precision ?
Derivatives f’(x)
df(t)
dt
df(x)
dx
Variation of something
w.r.t time
Variation of something
w.r.t. space

55. Gravity
F = m g = m x
9.81 m / s / s
x = g
Is it possible to invent a langage to talk about
what would happen with infinite precision ?
Derivatives – differential calculus
“Ghosts of vanished quantities”
Newton et Leibniz
Derivatives f’(x)
df(t)
dt
df(x)
dx
Variation of something
w.r.t time
Variation of something
w.r.t. space

56. Gravity
F = m g = m x
9.81 m / s / s
x = g
Let’s program it !

57. Fluids
Newton and Euler
3

58. Fluids

59. Fluids“Lagrange” point of view

60. Fluids“Lagrange” point of view
Let’s program it !

61. Fluids“Lagrange” point of view
ρ “nb particles per square”
“Euler” point of view
1757

62. Fluids“Lagrange” point of view
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
1757

63. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
1757

64. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
∂v ∂v dx ∂v dy
∂t ∂x dt ∂y dt
= + +
1757

65. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
∂v ∂v dx ∂v dy
∂t ∂x dt ∂y dt
= + +
∂v ∂v vx ∂v vy
∂t ∂x ∂y
= + +
1757

66. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
∂v ∂v dx ∂v dy
∂t ∂x dt ∂y dt
= + +
∂v ∂v vx ∂v vy
∂t ∂x ∂y
= + +
∂v v. v
∂t
= +
1757

67. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
=
∂v v. v
∂t
+
Q2: incompressible fluids ?
1757

68. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
=
∂v v. v
∂t
+
Q2: incompressible fluids ?
1757

69. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
=
∂v v. v
∂t
+
Q2: incompressible fluids ?
what goes in =
what goes out
1757

70. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
=
∂v v. v
∂t
+
Q2: incompressible fluids ?
what goes in =
what goes out
div(v) = 0
1757

71. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
=
∂v v. v
∂t
+
Q2: incompressible fluids ?
div(v) = 0
Q3: mass conservation ?
1757

72. Fluids
ρ(x,y,t) “nb particles per square”
“Euler” point of view
v(x,y,t) speed of the particle under
“grid point” (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
Q2: incompressible fluids ?
div(v) = 0
Q3: mass conservation ?
d ρ
dt = - div(ρv)
(Continuity equation)
1757
=
∂v v. v
∂t
+

73. Fluids
Newton 2nd law (F = ma)
Incompressibility
Initial condition
Incompressible Euler fluid
1757

74. Fluids
Newton 2nd law (F = ma)
Incompressibility
Initial condition
Incompressible Euler fluid
1757
Q: How to program incompressibility ?

75. Fluids on the computer
4
From Euler to Brenier

79. Fluid element

80. Fluid element
Parameters

81. Fluid element
Parameters
Q1: can we find a parameterization such that
each fluid element has constant volume ?

82. Fluid element
Parameters
Q1: can we find a parameterization such that
each fluid element has constant volume ?
Q2: can we compute the time evolution of the
parameters that mimics the physics ?

83. Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) }
Fluids on the computer: power diagrams

84. Fluids on the computer: power diagrams

85. Power diagram: Pow(xi) = { x | d2(x,xi) – ψi < d2(x,xj) – ψj }
Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) }
Fluids on the computer: power diagrams

86. Fluids on the computer: power diagrams

87. Theorem: (direct consequence of MK duality, Brenier thm
alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ):
Given a point set, there exists a unique weight vector W = [ψ(y1) ψ(y2) … ψ(ym)]
such that the Laguerre cells have the prescribed areas.
Fluids on the computer:
optimal transport

88. Proof: G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj
Is a concave function of the weight vector [ψ(y1) ψ(y2) … ψ(ym)]
Theorem: (direct consequence of MK duality, Brenier thm
alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ):
Given a point set, there exists a unique weight vector W = [ψ(y1) ψ(y2) … ψ(ym)]
such that the Laguerre cells have the prescribed areas.
Fluids on the computer:
optimal transport

89. Proof: G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj
Is a concave function of the weight vector [ψ(y1) ψ(y2) … ψ(ym)]
Theorem: (direct consequence of MK duality, Brenier thm
alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ):
Given a point set, there exists a unique weight vector W = [ψ(y1) ψ(y2) … ψ(ym)]
such that the Laguerre cells have the prescribed areas.
Fluids on the computer:
optimal transport

90. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
The (unknown) weights W = [ψ(y1) ψ(y2) … ψ(ym)]
Fluids on the computer: optimal transport

91. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
T : an arbitrary but fixed assignment.
Fluids on the computer: optimal transport

92. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
T : an arbitrary but fixed assignment.
Fluids on the computer: optimal transport

93. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
T : an arbitrary but fixed assignment.
Fluids on the computer: optimal transport

94. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
T : an arbitrary but fixed assignment.
Fluids on the computer: optimal transport

95. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
W
fT(W)
fT(W)
Fluids on the computer: optimal transport

96. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
W
fT(W)
fT(W)
Fluids on the computer: optimal transport

97. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
fT(W) is linear in W
fT(W)
W
fT(W)
Fixed T
Fluids on the computer: optimal transport

98. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
fT(W) is linear in W
f TW
(W) : defined by Laguerre diagram
fT(W)
fixed W
W
Fluids on the computer: optimal transport

99. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
fT(W) is linear in W
f TW
(W) = minT fT(W)
fT(W)
fixed W
W
Fluids on the computer: optimal transport

100. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
fT(W) is linear in W
f: W → f TW
(W) is concave !!
(because its graph is the lower
enveloppe of linear functions)
fT(W)
W
fT(W)
f TW
(W) = minT fT(W)
Fluids on the computer: optimal transport

101. Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
fT(W) is linear in W
f: W → f TW
(W) is concave
(because its graph is the lower
enveloppe of linear functions)
fT(W)
W
fT(W)
f TW
(W) = minT fT(W)
Consider g(W) = f TW
(W) + ∑vj ψj
Fluids on the computer: optimal transport

102. fT(W)
W
fT(W)
f TW
(W) = minT fT(W)
Consider g(W) = f TW
(W) + ∑vj ψj
vj -∫Lag
ψ
(yj) dμ(x) and g is concave.∂ g / ∂ ψ j =
Idea of the proof
Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x)
fT(W) is linear in W
f: W → f TW
(W) is concave
(because its graph is the lower
enveloppe of linear functions)
Fluids on the computer: optimal transport

103. W
fT(W)
vj =∫Lag
ψ
(yj) dμ(x)
∂ g / ∂ ψ j = 0
Fluids on the computer: optimal transport

104. W
fT(W)
vj =∫Lag
ψ
(yj) dμ(x)
∂ g / ∂ ψ j = 0
Fluids on the computer: optimal transport
Prescribed volume

105. W
fT(W)
vj =∫Lag
ψ
(yj) dμ(x)
∂ g / ∂ ψ j = 0
Fluids on the computer: optimal transport
Prescribed volume
Actual volume

106. Let’s program it !

107. Fluid element
Parameters:
Pow diag. seeds
Q1: can we find a parameterization such that
each fluid element has constant volume ?
R1: use power diagram and compute weights
that maximize F (using Newton optim.)

108. Fluid element
Q2: can we compute the time evolution of the parameters
that mimics the physics ?
Parameters:
Pow diag. seeds
Fluids on the computer: optimal transport

109. Fluid element
Q2: can we compute the time evolution of the parameters
that mimics the physics ?
Parameters:
Pow diag. seeds
Fluids on the computer: optimal transport
F = ma (Newton 2nd law)

110. Fluid element
Q2: can we compute the time evolution of the parameters
that mimics the physics ?
Parameters:
Pow diag. seeds
Fluids on the computer: optimal transport
F = ma (Newton 2nd law)
applies to center of gravity

111. Fluid element
Q2: can we compute the time evolution of the parameters
that mimics the physics ?
Parameters:
Pow diag. seeds
Fluids on the computer: optimal transport
F = ma (Newton 2nd law)
applies to center of gravity
Difficulty: the center of gravity is not a variable/parameter

112. Fluid element
Q2: can we compute the time evolution of the parameters
that mimics the physics ?
Parameters:
Pow diag. seeds
Fluids on the computer: optimal transport
F = ma (Newton 2nd law)
applies to center of gravity
Difficulty: the center of gravity is not a variable/parameter
(1) Apply gravity to the parameters instead

113. Fluid element
Q2: can we compute the time evolution of the parameters
that mimics the physics ?
Parameters:
Pow diag. seeds
Fluids on the computer: optimal transport
F = ma (Newton 2nd law)
applies to center of gravity
Difficulty: the center of gravity is not a variable/parameter
(1) Apply gravity to the parameters instead
(2) Add a spring so that it pulls the center of gravity

114. Fluid element
Q2: can we compute the time evolution of the parameters
that mimics the physics ?
Parameters:
Pow diag. seeds
Fluids on the computer: optimal transport
F = ma (Newton 2nd law)
applies to center of gravity
Difficulty: the center of gravity is not a variable/parameter
(1) Apply gravity to the parameters instead
(2) Add a spring so that it pulls the center of gravity
[Gallout Mérigot]
Converges to sol. of Euler

115. Let’s program it !

116. Moving forward: “path bundle” method
Newton Laws
(1) inertia
(2) F = ma
(3) Fab = -Fba

117. Moving forward: “path bundle” method
∫t1
t2
L(x,x,t) dt
The Least Action Principle
The movement of the system
extremizes the integral :
+ Symmetries
w.r.t. time
w.r.t. space
Newton Laws
(1) inertia
(2) F = ma
(3) Fab = -Fba

118. Moving forward: “path bundle” method
∫t1
t2
L(p,p,t) dt
The Least Action Principle
Extremize
+ Symmetries
w.r.t. time
w.r.t. space
Subject to
Fluid element
Parameters:
Pow diag. seeds

119. Other examples

120. To infinity and beyond
5

121. Simulate the whole universe in a computer ?

122. René Descartes - 1663
Vortices in Ether
… and Voronoi diagrams !

123. Early universe reconstruction.
The data (1) the cosmological microwave background
COBE 1992
To infinity and beyond

124. WMAP 2003
2006
2008
2010
Early universe reconstruction.
The data (1) the cosmological microwave background
To infinity and beyond

125. Early universe reconstruction.
The data (2) redshift acquisition surveys
To infinity and beyond

126. Early universe reconstruction.
The data (2) redshift acquisition surveys
To infinity and beyond

127. Early universe reconstruction.
The data (2) redshift acquisition surveys
To infinity and beyond

128. The millenium simulation project, Max Planck Institute fur Astrophysik
pc/h : parsec (= 3.2 light year)
To infinity and beyond

129. To infinity and beyond
The universal swimming pool

130. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris
Time = Now
Going back in time … 13.7 billion years

131. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris
. . . . .
Going back in time … 13.7 billion years

132. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris
Time = BigBang
(- 13.7 billion Y)
Going back in time … 13.7 billion years

133. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris
“Time-warped” map of the
universe
Going back in time … 13.7 billion years

134. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris
Cosmic Microware Background:
“Fossil light” emitted 380 000 Y after BigBang
and measured now
“Time-warped” map of the
universe
Going back in time … 13.7 billion years

135. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris
Cosmic Microware Background:
“Fossil light” emitted 380 000 Y after BigBang
and measured now
Do they match ?
“Time-warped” map of the
universe
Going back in time … 13.7 billion years

136. Thank you

137. Online resources
All the sourcecode/documentation available from:
http://alice.loria.fr/software/geogram
Demo: www.loria.fr/~levy/GLUP/vorpaview
* L., A numerical algorithm
for semi-discrete L2 OT in 3D,
ESAIM Math. Modeling
and Analysis, 2015
* L. and E. Schwindt,
Notions of OT and how to
implement them on a computer,
Computer and Graphics, 2018.
J. Lévy – 1936-2018 - To you Dad, I miss you so much.
+ My course at SGP
on Optimal Transport
July 08, Paris