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Smoothed Particle Hydrodynamics (SPH)
Huamin Wang
Fluid Representation• Fluid is represented using a set of particles.• Particle i has position xi, velocity vi, and mass mi.
Animation Example
• Using 10 million particles– http://www.youtube.com/watch?v=Q4_quqtGtM
A
Smoothed Particle Interpolation• Suppose each particle j has a physical quantity Aj.• The quantity can be: velocity, pressure, density, temperature….• How to estimate the quantity at a new location Xi?• A simple solution:
Ai?
A0
A4
A3
A2
A1
A5
A6
Aismooth = Aj
j∑ For |Xi – Xj| < r
Smoothed Particle Interpolation• Instead of assuming a particle is volumeless, let us assume
each one has a volume Vi.• So a better solution is:
Ai?
A0
A4
A3
A2
A1
A5
A6
Aismooth = Vj Aj
j∑ For |Xi – Xj| < r
Smoothed Particle Interpolation• One problem of this solution:
• Not smooth! (red 7 -> green 9!)
i
Aismooth = Vj Aj
j∑ For |Xi – Xj| < r
Smoothed Particle Interpolation
• Final solution:• Wij is called smoothing kernel.• When |Xi – Xj| is large, Wij is small. • When |Xi – Xj| is small, Wij is large.
i
For |Xi – Xj| < rAismooth = Vj AjWij
j∑
Particle Volume• But how do we get the volume of particle i?
ρi = Vjρ jWijj∑
m0
m4
m3
m2
m1
m5
m6
ρi = mjWijj∑
Vi =mi
ρi Vi =mi
mjWijj∑
Final Interpolation Solution• So the actual solution is:
Vj =mj
mkWjkk∑Ai
smooth = Vj AjWijj∑
Aismooth =
mj
mkWjkk∑
AjWijj∑
The advantage of This Interpolation
• We can easily compute its derivatives:– Gradient
– Laplacian
Aismooth = Vj AjWij
j∑ ∇Ai
smooth = Vj Aj∇Wijj∑
Aismooth = Vj AjWij
j∑ ∆Ai
smooth = Vj Aj∆Wijj∑
A Smoothing Kernel Example
Wij =3
2πh3
23− q2 +
12
q3 (0 ≤ q < 1)
16
(2 − q)3 (1 ≤ q < 2)
0 (2 ≤ q)
q =xi − x j
h
h is called smoothing length
Kernel Derivatives• Gradient at particle i (a vector)
∇iWij =
∂Wij
∂xi
∂Wij
∂yi
∂Wij
∂zi
=∂Wij
∂q∇iq =
∂Wij
∂qxi − x j
xi − x j h q =xi − x j
h
∂Wij
∂q=
32πh3
−2q + 32
q2 (0 ≤ q < 1)
−12
(2 − q)2 (1 ≤ q < 2)
0 (2 ≤ q)
Wij =3
2πh3
23− q2 +
12
q3 (0 ≤ q < 1)
16
(2 − q)3 (1 ≤ q < 2)
0 (2 ≤ q)
Kernel Derivatives
• Laplacian at particle i (a scalar)
∇iWij =∂2Wij
∂xi2 +
∂2Wij
∂yi2 +
∂2Wij
∂zi2 =
∂2Wij
∂q2
1h2 +
∂Wij
∂q2h q =
xi − x j
h
∂Wij
∂q=
32πh3
−2q + 32
q2 (0 ≤ q < 1)
−12
(2 − q)2 (1 ≤ q < 2)
0 (2 ≤ q)
∂2Wij
∂q2 =3
2πh3
−2 + 3q (0 ≤ q < 1)2 − q (1 ≤ q < 2)0 (2 ≤ q)
Fluid Dynamics• We model fluid dynamics by applying three
forces on particle i.– Gravity– Fluid Pressure– Fluid Viscosity
i
Gravity Force• Gravity Force is:
i
Figravity = mig
Pressure Force• Pressure is related to the density
– First compute the density of Particle i:
– Convert it into pressure (some empirical function):
High Pressure Low pressure
ρi = mjj∑ Wij
Pi = k ρi
ρconstant
7
−1
Pressure Force• Pressure force depends on the difference of
pressure:
i
High Pressure
High Pressure
High Pressure
High Pressure
High Pressure
No pressure force!
i
High Pressure
High Pressure
LowPressure
LowPressure
High Pressure
Pressure force!
Pressure Force• Mathematically, the difference of pressure =>
Gradient of pressure.
• To compute this pressure gradient, we assume that the pressure is also smoothly represented:
• So:
Fipressure = −Vi∇iP
smooth
Pismooth = VjPjWij
j∑
Fipressure = −Vi VjPj∇iWij
j∑
Viscosity Force• Viscosity effect means: particles should move together
in the same velocity.• In other words, minimize the difference between the
particle velocity and the velocities of its neighbors.
Before Ultimately
Viscosity Force• Mathematically, it means:
• To compute this Laplacian, we assume that the velocity is also smoothly represented:
• So:
Fivis cos ity = −νmi∆iv
smooth
v ismooth = Vjv jWij
j∑
Fivis cos ity = −νmi Vjv j∆iWij
j∑
Algorithm• For every particle i
– Compute its neighborhood set– Using the neighborhood, compute:
• Force = 0• Force + = The gravity force• Force + = The pressure force• Force + = The viscosity force
– Update vi = vi + t * Force / mi;– Update xi = xi + t *vi; What is the bottleneck of the
performance here?
Exhaustive Neighborhood Search• Search over every particle pair? O(N2)• 10M particles means: 100 Trillion pairs…
i
Solution: Spatial Partition– Separate the space into cells– Each cell stores the particles in it– To find the neighborhood of i, just look at the surrounding
cells
i
Spatial Partition
– What if particles are not uniformly distributed?– Solution: Octree, Binary Spatial Partitioning tree…
i
Fluid Display• Need to reconstruct the water surface from
particles!
Ongoing Research
• How to make the simulation more efficient?
• How to make fluids incompressible?
• When simulating water, only use water particles, no air particles. So particles are sparse on the water-air boundary. How to avoid artifacts there?
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