simulating complex surface flow by smoothed particle hydrodynamics & moving particle...
TRANSCRIPT
Simulating complex surface flow by Smoothed Particle Hydrodynamics
& Moving Particle Semi-implicit methods
Benlong Wang Kai Gong Hua Liu
Shanghai Jiaotong University
Contents
• Introduction • SPH & MPS methods• Parallel strategy and approaches
– SPH:– MPS:
• Numerical results– 2D dam breaking– 2D wedge entry– 3D cavity flow– 3D dam breaking
Modeling free surface flows
• Multiphase flows:
MAC, VOF, LevelSet etc.
• ALE
• Meshless methods
& particle methods
SPH & MPS LBM
Kernel function
• Properties:– Narrow support
–
– decreases monotonously as increase
– h->0, Dirac delta function
r( ; )W r h
( '; ) 1W r r h dV
( ) ( ') ( '; )
( ') ( ' ; )
f r f r W r r h dV
f r W r r h dV
-3 -2 -1 0 1 2 3
0
h
dx
a b ab bb
f f W V
( )r
W
expression of derivatives
'( ) '( ') ( '; )
( ') ( '; ) ( ') '( '; )
f r f r W r r h dV
f r W r r h nds f r W r r h dV
' 'a b ab bb
f f W V-3 -2 -1 0 1 2 3
0
hWW’
( ) ( )b bb
f x dV f x V
Integral Summation
Trapeze like quadrature formula
h
dx 1.3 ~ 1.5
3.0 2h 130+ (2D)
0
( ') '( '; ) ( ') '( ' ; )f r W r r h dV f r W r r h dV
Correction and Consistance——advanced topic …
' 'a b ab bb
f f W V
' ( ) 'a b a ab bb
f f f W V
-3 -2 -1 0 1 2 3
0 f const ' 0af
( ) ( )a a a a a a b b ab b a b ab b b b a ab bb b b
f f f f W V f W V f f W V
f ax by c
Lists of kernel function
Cubic spline 2h
Quartic spline 2.5h
Fifth order B-spline 3h
Truncated Gaussian
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
h
w
2 3
22
3 31 0 1
2 410 1
( ) (2 ) 1 27 4
0 2
s s s
W s s sh
s
2 2
2 2
2 2 22
2
exp exp1
( ) 3
exp2
sh h
W s hh
hh
Hydrodynamics governing equations
2
( )( )2 1( )a b a b a ab b
b a b a abb ba b ba b
p p r r W mm v v W
tr r
20
02
d
d
2 ( )( )( )
a aa
a
b a b a b a aba b a ab b
b ba b a b a b
v pg v
t
m v v r r Wg p p W m
r r
1 aa
a
vp
t
MPS: projection method: Pressure Poisson Equation
SPH: weakly compressible method: State Equation1
0
1aap
Ma < 0.1d
( )da
a a b a b a abb
v m v v Wt
d
da
a
xv
t
0av
Link-List neighbour search
back ground mesh (L X L)
L=2h, 3h, support distance
L
SPH: the most time consuming part ~90%
MPS: generally less than PPE solver
Boundary Condition
• Sym: ghost particles,
• Free surface, p0
Identify the surface particle: 95% const. density
0 p p
gx y
'v v 'n nv v
ba b ab
b b
mW
Large Scale Computation(a few millions particles)
share memory architecture(NEC SX8: 8 nodes, 128G RAM)(Dell T5400: 2 Quad cores Xeon 16G RAM)
• SPH– Particle lists partition, NOT domain partition
• MPS – parallel ICCG method
Black-box Parallel Sparse Matrix Solver
SPH MethodLagrangian Method
Large deformation
Continue changing domain
Complex domain structure
Why not Domain decomposition ?
So, Black-box solver
give me a matrix, I will solve it in parallel…
PPE solver : ICCG method
• Precondition ILU(0)• Forward and backward substitutions• Inner products• Matrix-vector products• Vector updates
Parallel
Ax bSparse symmetric positive definite matrix
Direct solver or Iterative solver
Coloring• COLOR: Unit of independent sets.• Any two adjacent nodes have different colors. Elements
grouped in the same “color” are independent from each other, thus parallel/vector operation is possible.
• Many colors provide faster convergence, but shorter vector length.
Main Idea of the Coloring
Algebraic Multi-Color OrderingThe number of the colors has a lower boundary
the max bandwidth of the sparse matrix
Any two adjacent nodes have different colors
2hT. Iwashita & M. Shimasaki
2002 IEEE Trans. Magn.
The connection info could be obtained from the distribution of non-zeros in the sparse matrix
bcsstk14 n=1806,nnz=63454
MC=50
MC=180
Parallelized ICCG with AMC
Ax b
1,1 1,2 1,
2,1 2,2 2,
,1 ,2 ,
nc
nc
nc nc nc nc
C C C
C C C
C C C
1
2
nc
x
x
x
A TLDL
1
2,1 2
,1 ,2nc nc nc
D
L D
L L D
Ly r1
1,
1
ic
ic ic ic ic k kk
y D r L r
Forward and backward substitutions: parallelized in each color
SPH Parallel Strategy: OpenMP
MPS Parallel Strategy: OpenMP
Almost linear speedup
Numerical Results
• 2D dam breaking• 2D wedge water entry• 3D cavity flow• 3D dam breaking
Dambreaking Test
Surge front location
Water entry of a wedge
4.5M particles Speed up around 7Dell T5400 2 Xeon Quadcores
t(s)
Fy(
N)
0 0.005 0.01 0.015 0.02 0.0250
2000
4000
6000
8000
SPH ResultsAnalysisExperimentOger' s results
t(s)
v(m
/s)
0 0.005 0.01 0.015 0.02 0.0254.5
5
5.5
6
6.5
SPH ResultsExperiment
3D Cavity Flow: Re=400
Yang Jaw-Yen et al. 1998 J. Comput. Phys. 146:464-487
45 X 45 X 45 nodes
h/dx=1.5
3D Dambreaking Tests
Kleefsman, K.M.T. et al 2005J. Comput. Phys. 206:363-393
0.0 0.5 1.0 1.5 2.00.0
0.1
0.2
0.3
0.4
0.5
0.6 H4 H3 H2 H1
Wa
ter
Le
ve
l (m
)
Time (s)
MARIN Exp. Results
SPH Results
Conclusions
• 2D code is developed for both SPH and MPS methods
• 3D code is developed for complex free surface flows
• Computation costs of SPH is generally cheaper than MPS method
• Good agreements are obtained, a promising method for complex free surface flows.