(APR) applied to SPH method : Stability, accuracy and CPU analysis

Adaptive Particle Renement (APR) applied to SPH
method : Stability, accuracy and CPU analysis
Laurent Chiron
LHEEA Lab - Ecole Centrale Nantes
1er juillet 2015
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
1 / 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
2 / 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
3 / 25
Sph-ow scopes
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
4 / 25
SPH background
Conservation law
∂t U + ∂x f (U ) = 0
∂x f (U ) →

Finite dierences



Finite volumes
Discontinuous Galerkin



SPH
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
5 / 25
SPH background
Conservation law
∂t U + ∂x f (U ) = 0
∂x f (U ) →

Finite dierences



Finite volumes
Discontinuous Galerkin



SPH
• How to calculate the term ∂x f (U ) in SPH ?
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
5 / 25
SPH background
Conservation law
∂t U + ∂x f (U ) = 0
∂x f (U ) →

Finite dierences



Finite volumes
Discontinuous Galerkin



SPH
• How to calculate the term ∂x f (U ) in SPH ?
Convolution product properties
f ∗δ =
R
R f (y)δ(x
− y)dy = f
∇f ∗ g = f ∗ ∇g
δ is approached by a kernel function W
∇fi ≈ ∇f ∗ W = f ∗ ∇W =
P
wj fj ∇Wij
j
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
5 / 25
SPH background
Benets
• Meshless (particles)
• Lagrangian method
Weacknesses
• Low spatial order
• No convergence theorem
"Convergence" conditions
h
• ∆x
→∞
and
h→0
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
6 / 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
7 / 25
Particle renement theory
Some parameters
• Spacing parameter ∈ [0, 1]
• Radius ratio α ∈]0, 1]
• Switch parameter γ ∈ [0, 1]
∆x
•
•
•
•
Rm
∆xm
•
•
•
•
•
Laurent Chiron
Renement
Process
• • • m•
•
•
• • • •
• •R • •
• m •
• • • •
• • • •
•
•
• • • •
• •
•
• •
• •
•
•αRm•
• •
•
• •
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
8 / 25
Particle renement theory
Switch parameter properties
• γ = 1 → particle is turned on (active particle)
• γ = 0 → particle is turned o (passive particle)
γ
1
γm
γf
Rened area
0
Buer zone
Laurent Chiron
Modied SPH operator
Buer zone
• h∇f ii =
P
fj ωj ∇Wij γj
j
er juillet: 2015
Adaptive Particle Renement (APR) applied to SPH 1method
Stability, accuracy
9 / 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 10
accuracy
/ 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 11
accuracy
/ 25
Arbitrary Lagrangian Eulerian (ALE) process
d~xi
= ~v0i ,
dt
(1)
X
dωi
= ωi
ωj (~v0j − ~v0i )∇Wij ,
dt
j
(2)
X
dωi ρi
= −ωi
ωj 2ρE (~vE − ~v0 (~xij ))∇Wij ,
dt
j
(3)
X
dωi ρi~vi
= −ωi
ωj 2 ρE ~vE ⊗ (~vE − ~v0 (~xij )) + PE I¯ ∇Wij + ωi ρi~g ,
dt
j
(4)

 ~vi (Lagrangian)
3 possibilities : ~v0i = 0 (Eulerian)

~vi + ~δvi (ALE)
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 12
accuracy
/ 25
ALE process : benets on a dambreak test case
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 13
accuracy
/ 25
Why using such an approach ?
Pressure instabilities at the ne/coarse interface
• Particles move in a Lagrangian way (Streamlines)
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 14
accuracy
/ 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 15
accuracy
/ 25
The APR concept
Prolongation
φd = φm + R D h (φ)m .(xd − xm )
Quantity
Before renement
Mass
mm
After renement
D
X
mj
j
Kinetic energy
Linear momentum
•
•
Laurent Chiron
D
X
mj u~j .u~j
j
D
X
mj u~j
j
Angular momentum
•
1
2
mm u~m
•
•
•
1
m ~
u .u~
2 m m m
x~m × mm u~m
D
X
x~j × mj u~j
j
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 16
accuracy
/ 25
The APR concept
Prolongation
φd = φm + R D h (φ)m .(xd − xm )
Without prolongation
Laurent Chiron
With prolongation
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 16
accuracy
/ 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 17
accuracy
/ 25
The utility of well distributed tasks between processors
•
Neighbors number
•
Particle radius
Poor distribution of particles between processors
• Particles inside/near the buer zone have 2 to 4 more neighbors than
other particles
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 18
accuracy
/ 25
New algorithm eciency
Load balancing
τ=
min λ(k)
max λ(k)
•
where λ(k) is the number of interactions of processor k
Former algorithm (cutting into an
equal number of particles)
•
New algorithm (cutting into an
equal number of interactions)
At least of 30% CPU gain
On this test case, τ = 0.8 with the new algorithm (against τ = 0.54 before)
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 19
accuracy
/ 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 20
accuracy
/ 25
Plan
1 SPH background
2 Particle Renement theory
3 Particle renement improvements
Arbitrary Lagrangian Eulerian (ALE) process
The APR concept
Parallelization
4 Some results
Kleefsman dambreak
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 21
accuracy
/ 25
Kleefsman dambreak : pressure eld analysis
0.68 m 0.6 m
0.45 m
Rened area
0.55 m
1.22 m
•
Laurent Chiron
1.17 m
0.16 m
0.67 m
Coarse
• APR
• Fine
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 22
accuracy
/ 25
Kleefsman dambreak : pressure eld analysis
0.68 m 0.6 m
0.45 m
Rened area
0.55 m
1.22 m
•
Coarse
Laurent Chiron
1.17 m
•
APR
0.16 m
0.67 m
•
Fine
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 22
accuracy
/ 25
Kleefsman dambreak : Comparison of pressure sensors
•
Laurent Chiron
Sensor P1
•
Sensor P1 (Zoom)
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 23
accuracy
/ 25
Kleefsman dambreak : Comparison of pressure sensors
•
Sensor P3
•
Kinetic Energy
Simulation performed CPU Time (seconds)
Laurent Chiron
Prot
Fine
21 000
-
APR
9000
57.2%
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 23
accuracy
/ 25
Thank you for your attention !
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 24
accuracy
/ 25
References
D.A. Barcarolo, G. Oger, and D. Le Touzé.
Adaptive particle renement and derenement applied to Smoothed Particle
Hydrodynamics method.
Journal of computer physics, 273 :640657, 2014.
J.Feldman and J.Bonet.
Dynamic renement and boundary contact forces in SPH with applications in uid ow
problems.
Int. J. Numer. Meth. Eng, 72 :295324, 2007.
S. Kitsionas and A. Whitworth.
Development of SPH variable resolution using dynamic particle coalescing and splitting.
Monthly Notices of the Royal Astronomical Society, 330 :129136, 2002.
K.M.T. Kleefsman, G. Fekken, A.E.P. Veldman, B. Iwanowski, and B. Buchner.
A volume-of-uid based simulation method for wave impact problems.
Journal of Computational Physics, 206 :363393, 2005.
Y. R. Lopez, D. Roose, and C. R. Morfa.
Dynamic particle renement in SPH : application to free surface ow and non-cohesive soil
simulations.
Comput Mech, 51 :731741, 2013.
S. Marrone, A. Colagrossi, D. Le Touzé, and G. Graziani.
Fast free-surface detection and level-set function denition in SPH solvers.
Journal of Computational Physics, 229 :36523663, 2010.
Laurent Chiron
Adaptive Particle Renement (APR) applied to SPH1er
method
juillet :2015
Stability, 25
accuracy
/ 25