point-implicit diffusion operator in a moving particle method ( mppm )
DESCRIPTION
Point-implicit diffusion operator in a moving particle method ( MPPM ). Yao-Hsin Hwang Department of Marine Engineering National Kaohsiung Marine University. Diffusion operator. Time-step constraint for explicit scheme Implicit scheme: Iteration Numerically stable without iteration. - PowerPoint PPT PresentationTRANSCRIPT
1
Point-implicit diffusion operator in a moving particle method (MPPM)
Yao-Hsin Hwang
Department of Marine Engineering
National Kaohsiung Marine University
2
Diffusion operator
Time-step constraint for explicit scheme
Implicit scheme: Iteration Numerically stable without iteration
/2t
3
Features of existing particle methods
SPH, MPS, FVPM…. Material particles (mass point, global mass
conservation) Operators realized with randomly particle
cloud
4
Disadvantages of existing schemes
Inaccurate (inconsistent) operator realization (particle smoothing) —continuity constraint
Assignment of boundary condition Particle management (impractically
clustered or dispersive) Computationally inefficient
5
Essentials of MPPM (1)
Inserted pressure mesh to realize pressure-related operators
Background mesh and length scale Local mass conservation Particle as observation point
6
Essentials of MPPM (2)
No particle management constraint Feasible non-uniform distribution No special boundary treatment Constant and diagonally dominant
coefficient matrix of pressure equation
7
Diffusion Procedure
Equation
Laplacian
, 2
p nb nb p pnb
d d
2D
Dt
2pd O( )
8
Explicit Operator
Scheme
Time-step constraint
n 1 n np p p nb nb
nb
(1 t d ) t d
D pC td 1
9
Implicit Operator
Scheme
Iteration required
n 1 n 1 n 1p p p nb nb
nb
(1 t d ) t d
10
Point-implicit Operator
Operator2 n n n 1
p nb nb p p p pnb
d (1 )d d
p n np p nb nb p p
nb
Dd d (1 )d
Dt
11
Point-implicit Operator
Scheme A
Scheme B
n 1 n n np p nb nb p p
nbD
1( ) t( d d )1 C
DCn 1 n n np p nb nb p p
nbD
1 e( ) t( d d )
C
12
Effective-Time Step
Scheme A
Scheme B
DD,A
D
CC
1 C
DC
D,B
1 eC
13
Stability Limits
Scheme A
Scheme B
D
DD
1C
11
max(1. , 0)C
D
1C n(1 )
0.0 0.2 0.4 0.6 0.8 1.0
CD
0
2
4
6
8
10
scheme A
scheme B
14
Three-Point Analysis(1)
1. Boundary conditions:2. Solution at
3. Results
L(0) , R(2 )
n 1 n nL R( )2
x
D
2 2D
CD
A BD
C (2n 1)n8
ex 3 3n 1
C 1 e, ,
1 C
32( 1)1 e
(2n 1)
15
Three-Point Analysis(2)
Results
CD
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
exact
=0
=0.5
=1.0
scheme A
scheme Ascheme B
scheme B
16
Pure Diffusion(1)
Particle distribution
xi
C C
(i 1) ( 0.5)x
n n
Cn 32, 0.3
yj
C C
( 0.5)( j 1)y
n n
17
Pure Diffusion(2)
0
DC 1.5
exact
-0.9
-0.8
-0.8
-0.7
-0.7-0.7
-0.6
-0.6
-0.6
-0.5
-0.5
-0.5
-0.5
-0.4
-0.4
-0.4
-0.4
-0.4
-0.3
-0.3
-0.3
-0.3
-0.3
-0.2
-0.2
-0.2
-0.2
-0.2
-0.1
-0.1
-0.1
-0.1
-0.1-0.1
0
0
0
0
0
0
0
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.6
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.9
0.9
-0.9
-0.9
-0.8
-0.8
-0.7
-0.7
-0.6-0.6
-0.6
-0.6
-0.5
-0.5
-0.5
-0.5
-0.4
-0.4
-0.4
-0.4
-0.4
-0.3
-0.3
-0.3
-0.3
-0.3
-0.2
-0.2
-0.2
-0.2
-0.2
-0.1
-0.1
-0.1
-0.1
-0.1
0
0
0
0
0
0
0
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.9
0.5
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Pure Diffusion(3)
CD
0 1 2 3 4 5
m
0.0001
0.001
0.01
0.1
1
=0.0 (scheme A)=0.25 (scheme A)=0.5 (scheme A)=0.75 (scheme A)=1.0 (scheme A)=0.0 (scheme B)=0.25 (scheme B)=0.5 (scheme B)=0.75 (scheme B)=1.0 (scheme B)
m
0.0001
0.001
0.01
0.1
1
=0.1 (=0.0)=0.25 (=0.0)=0.5 (=0.0)=0.1 (=0.5)=0.25 (=0.5)=0.5 (=0.5)=0.1 (=1.0)=0.25 (=1.0)=0.5 (=1.0)
CD
0 1 2 3 4 5
m
0.0001
0.001
0.01
0.1
1
=0.1 (=0.0)=0.25 (=0.0)=0.5 (=0.0)=0.1 (=0.5)=0.25 (=0.5)=0.5 (=0.5)=0.1 (=1.0)=0.25 (=1.0)=0.5 (=1.0)
(a) scheme A
(b) scheme B
=1.0
=1.0
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Cavity Flow(1)
time
0 20 40 60 80 100
min
=0=1=
D
nC=10
nC=20
nC=40
-0.05
-0.10
-0.10
-0.10
-0.12
x
0.0 0.2 0.4 0.6 0.8 1.0
v
=0=1=
D
Hwang (FV)
0.0
0.0
0.0
0.5
-0.5
Re 10
20
Cavity Flow Re=10
0 D
21
Cavity Flow(2)
time
0 5 10 15 20-0.2
-0.1
0.0
0.1
0.2
=0=1=
D
Erturk and Dorsum (nC=512)
u(0.5,0.1875)
v(0.925,0.5)
v(0.1875,0.5)
time
0 20 40 60 80 100-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
=0=1=
D
Erturk and Dorsum (nC=512)
u(0.5,0.1875)
v(0.925,0.5)
v(0.1875,0.5)
Re 100 Re 1000
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Cavity Flow(3)
CRe 5000 ( n 160 ) CRe 1000 ( n 60 )
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Cavity Flow(4)
u
y
0.0
0.2
0.4
0.6
0.8
1.0
=0=1=
D
Ghia et al.Erturk et al.
0.0 0.0 0.0-0.5 0.5 1.0
Re=10
0 (n C
=40)
Re=40
0 (n C
=40)
Re=
1000
(nC
=60)
Re=
2500
(nC
=120
)
0.0 0.0
Re=
5000
(nC
=160
)
x
0.0 0.2 0.4 0.6 0.8 1.0
v
=0=1=
D
Ghia et al.Erturk et al.
Re=100 (nC =40)
Re=1000 (nC =60)
Re=400 (nC =40)
-1.0
0.0
0.0
0.0
0.5
-0.5
Re=2500 (nC =120)0.0
0.0
Re=5000 (nC =160)
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Cavity Flow(5)
u
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
y
0.0
0.2
0.4
0.6
0.8
1.0
WCSPH (nC=100), Lee et al.
ISPH (nC=100), Lee et al.
=D (nC=40)
Ghia et al.
u
-0.5 0.0 0.5 1.0
y
0.0
0.2
0.4
0.6
0.8
1.0
WCSPH (nC=240), Lee et al.
ISPH (nC=240), Lee et al.
=D (nC=60)
Ghia et al.Erturk et al.
Re 400 Re 1000
25
Backward-Facing Step Flow(1)
26
Backward-Facing Step Flow(2)
Re 100
Re 200
Re 600
Re 800
27
Backward-Facing Step Flow(3)
y/hT
0.0
0.2
0.4
0.6
0.8
1.0
presentHwang (FV)Armaly et al. (experiment)
0 2.55 3.06 3.57 4.18 4.80 5.41
0 1 2
x/hSTEP
u/um
Re 100
y/hT
0.0
0.2
0.4
0.6
0.8
1.0
presentHwang (FV)Amaly et al. (experiment)
0 2.55 3.06 3.57 4.18 4.80 5.41
0 1 2
x/hSTEP
u/um
Re 389
28
Backward-Facing Step Flow(4)
Re
0 200 400 600 800 1000
Lr/
h ST
EP
0
2
4
6
8
10
12
14
16
Hwang (FV)Armaly et al. (experiment)Erturk (numerical)present
29
Backward-Facing Step Flow(5)
t 307
t 303
t 309
t 301
t 305
30
Backward-Facing Step Flow(6)
x/hStep
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
y/hT
0.0
0.2
0.4
0.6
0.8
1.0
t=301t=307
0 1 2
u/um