Stationary Incompressible Viscous Flow Analysis by a Domain Decomposition Method

Stationary Incompressible
Viscous Flow Analysis by a
Domain Decomposition Method
Hiroshi Kanayama, Daisuke
Tagami
and Masatsugu Chiba
（ Kyushu University ）

Contents
 Introduction
 Formulations
 Iterative Domain Decomposition Method
for Stationary Flow Problems
 Numerical Examples
　　　　 1 million DOF cavity flow, DDM v.s.
FEM
A concrete example
 Conclusions

Objectives
 In finite element analysis for stationary
flow problems, our objectives are to
analyze large scale (10-100 million
DOF) problems.
Why Iterative DDM ?
HDDM is effective.
Ex. Structural analysis （ 100 million DOF:
1999 R.Shioya and G.Yagawa ）

Formulations
 Stationary Navier-Stokes Equations
 Weak Form
 Newton Method
 Finite Element Approximation
 Stabilized Finite Element Method
 Domain Decomposition Method


















∂
∂
+
∂
∂
=
+−=
=⋅∇
=⋅∇−∇⋅
Ω
.
2
1
)(
),(2),(
,0
,),()(
:
i
j
j
i
ij
ijijij
x
u
x
u
D
Dpp
p
u
uu
u
fuuu
νδσ
σ
jn ϕ=Γ u:D
.deltasKronecker':],m[viscositykinematic:
,]mforce[body:],mpressure[:,]smvelocity[:
2
222
ijs
ssp
δν
fu
Stationary Navier-Stokes Eqs.
0),(: =⋅Γ nu pN σ

Weak Form
( ) ( ) thatsuch,Find QVp ×∈ ϕu
( ) ( ) ( ) ( ) ( )vfuvvuvuu ,,,,,, =+++ qbpbsa ( ) QVq ×∈∀∀
,for v
( )( )
( ) { }
( ).
),0(,on;
,
2
31
Ω≡
≡Γ=∈≡
Ω≡
LQ
VVXV
HX
Dϕϕ vv
( ) ( )[ ]
( )
( ) ∫
∑∫
∫
Ω
Ω
Ω
⋅∇−=
=
∇⋅=
uu
vuvu
vuwvwu
qqb
DDs
a
d
ml
lmlm
:,
)()(2:,
:,,
,
ν

Newton Method
( ) ( ) thatsuch,Find QVpkk
×∈ ϕu
( ) ( ) ( )
( ) ( ) ( ) ( )vuuvfuv
vuvuuvuu
,,,,,
,,,,,
11
11
−−
−−
+=++
++
kkkk
kkkkk
aqbpb
saa
( ) QVq ×∈∀∀
,for v
( ) ( ) ( ) ( ) ( ) ( )vfuvvuvuwvwu ,
~
,,,,,,, =++++ qbpbsaa
( ) fuuf
wu
uu
~11
1
→∇⋅+
→
→
→
−−
−
kk
k
k
k
pp

Finite Element Approximation
0
1
2
3
element.ltetrahedraais,
elements,ltetrahedraofconsisting
ofiontriangulatA
hK
KK
h
ℑ∈
=Ω
Ωℑ
( ) ( ){ }
( ) ( ){ }
( ) { } ).0(,on;
,,|;
,,|;
1
0
3
1
30
hhDhhhh
hKhhh
hKhhh
VVXV
KKPqCQqQ
KKPCXX
≡Γ=∈≡
ℑ∈∈Ω∩∈≡
ℑ∈∈Ω∩∈≡
∀
∀
ϕϕ vv
vv

Stabilized Finite Element Method
( ) ( ) thatsuch,Find hhhh
QVp ×∈ ϕu
( ) ( ) ( ) ( ) ( )
( ) ( )({∑ℑ∈
∇+∇⋅+∇⋅+
++++
hK
hhhhhK
hhhhhhhhhhhh
p
qbpbsaa
,
,,,,,,,
uwwu
uvvuvuwvwu
τ
( ) ( ) ) ( ) }KhhKKhhhhh q vuvwwv ⋅∇⋅∇+∇−∇⋅+∇⋅ ,δ
( ) hhhh QVq ×∈∀∀
,for v








=
∞
ν
τ
24
,
2
min
2
K
h
K
K
hh
w 







= ∞
∞
Kh
Kh
K
h
h
w
w
λ
ν
λ
δ ,
12
min
22
KhK ofdiameterais
Stabilized Parameter
( ) ( ) ( )( )∑ℑ∈
∇−∇⋅+∇⋅+=
hK
KhhhhhKh
qvwwvfvf ,
~
,
~
τ
1=λ

Domain Decomposition Method






=












b
i
b
i
bbbi
ibii
f
f
a
a
KK
KK
[ ]{ } { }faK =
12γ
2Ω∂
1Ω∂
DΓ
1Ω 2Ω
Stabilized Finite Element Method
Decomposition
i ： corresponding to Inner DOF
b ： corresponding to Interface

Inner DOF
Solver by Skyline Method
bibiiii aKfaK −=
Interface DOF
iiibibbibiibibb fKKfaKKKK 11
)( −−
−=−
Solver by BiCGSTAB or GPBiCG
χ=λS

BiCGSTAB for the Interface Problem
end
,
),(
),(
,
,
,
),(
),(
,
,
),(
),(
),(
begin
:dountil,,for
.set,,guessinitialanis
)()(
)()(
)(
)(
)(
)()()()(
)()()()()()(
)()(
)()(
)(
)()()()(
)()(
)()(
)(
)()()()()()(
)()(
)()()()(
k
k
k
k
k
kkkk
kkkkkk
kk
kk
k
kkkk
k
k
k
kkkkkk
k
gg
gg
Sttg
tw
StSt
tSt
Swgt
Swg
gg
Swwgw
Errggk
Sλg
0
10
1
1
0
0
1111
0
1000
10
0
+
+
+
−−−−
−
⋅=
−=
−−=
=
−=
=
−+=
≤=
=−=
ς
ρ
γ
ς
ςρλλ
ς
ρ
ρ
ςγ
γχλ












=




















t
b
i
t
b
i
btbbbi
itibii
f
f
f
a
a
a
E
KKK
KKK
00
,χ=bSa




−−−=
−=
−−
−
,)(
),(
11
1
titiibibtiiibib
ibiibibb
aKKKKfKKf
KKKKS
χ
Equations on the interface

[ ]{ } ;)0()0(
i
T
tbiitibii faaaKKK =
[ ]{ } ;)0()0()0()0(
b
T
tbibtbbbi faaaKKKwg −==
BiCGSTAB
(1) Initialization
(a) Set 　．
(b) Solve ．

(c) Solve ．
)0(
ba
)0(
ia
)0()0(
,wg

(2) Iteration
(a) Solve ．

(b) Solve ．

(c) Compute ．

;
;
),(
),(
)()()()(
)()0(
)()0(
)(
kkkk
k
k
k
Swgt
Swg
gg
ρ
ρ
−=
=
[ ]{ } ;00)()(
=
Tkk
itibii wvKKK
[ ]{ } ;0)()()( Tkk
btbbbi
k
wvKKKSw =
)(k
v
)(k
Sw
)()(
, kk
tρ

;
;
;
),(
),(
)()()()1(
)()()()()()1(
)()(
)()(
)(
kkkk
kkkkk
b
k
b
kk
kk
k
Sttg
twaa
StSt
tSt
ς
ςρ
ς
−=
−−=
=
+
+
[ ]{ } ;00)()(
=
Tkk
itibii tvKKK
[ ]{ } ;0)()()( Tkk
btbbbi
k
tvKKKSt =
(d) Solve ．

(e) Solve ．

(f) Compute ．

)(k
v
)(k
St
)1()1()(
,, ++ kk
b
k
gaς

(g) Convergence check for ． If converged
, If not converged go to (h) ．
(h) Compue ．

(3) Construction of solution ．
　 (a) Solve ．
);(
;
),(
),(
)()()()()1()1(
)()0(
)1()0(
)(
)(
)(
kkkkkk
k
k
k
k
k
Swwgw
gg
gg
ςγ
ς
ρ
γ
−+=
⋅=
++
+
[ ]{ } ;**
i
T
tbiitibii faaaKKK =
)1( +k
g
)1(* +
= k
bb aa
)1()(
, +kk
wγ
*
ia

Preconditioned BiCGSTAB for the Interface Problem
end
,
),(
),(
,
,
,
),(
),(
,
,
),(
),(
),(
begin
:dountil,,for
,set,,guessinitialanis
)()(
)()(
)(
)(
)(
)()()()(
)()()()()()(
)()(
)()(
)(
)()()()(
)()(
)()(
)(
)()()()()()(
)()(
)()()()(
k
k
k
k
k
kkkk
kkkkkk
kk
kk
k
kkkk
k
k
k
kkkkkk
k
gg
gg
tSMtg
tMwM
tSMtSM
ttSM
wSMgt
wSMg
gg
wSMwgw
Errggk
Sλg
0
10
11
111
11
1
1
10
0
11111
0
1000
10
0
+
−+
−−+
−−
−
−
−
−−−−−
−
⋅=
−=
−−=
=
−=
=
−+=
≤=
=−=
ς
ρ
γ
ς
ςρλλ
ς
ρ
ρ
ςγ
γχλ


GPBiCG for the Interface Problem (1/2)
),
),(
),(
then,if(
,
),)(,(),)(,(
),)(,(),)(,(
,
),)(,(),)(,(
),)(,(),)(,(
,
,
,
),(
),(
),(
begin
:dountil,,for
,,0set,χ,guessinitialanisλ
)(
)()(
)()(
)(
)()()()()()()()(
)()()()()()()()(
)(
)()()()()()()()(
)()()()()()()()(
)(
)()()()(
)()()()()()()(
)()(
)()(
)(
)()()()()(
)()(
)()()()()()(
00
10
0
11
0
0
111
0
111000
===
−
−
=
−
−
=
−=
+−−=
=
−+=
≤=
===−=
−−
−−−
−−−
k
kk
kk
k
kkkkkkkk
kkkkkkkk
k
kkkkkkkk
kkkkkkkk
k
kkkk
kkkkkkk
k
k
k
kkkkk
k
StSt
tSt
k
yStStyyyStSt
tStStytyStSt
yStStyyyStSt
ySttytStyy
Spgt
Spwgty
Spg
gg
upgp
Errggk
wtSλg
ης
η
ς
α
αα
α
β
β


GPBiCG for the Interface Problem (2/2)
end
,
,
),(
),(
,
,λλ
,
),(
)()()()(
)()(
)()(
)(
)(
)(
)()()()()()(
)()()()()(
)()()()()()()(
)()()()()()()()(
kkkk
k
k
k
k
k
kkkkkk
kkkkk
kkkkkkk
kkkkkkkk
SpStw
gg
gg
Stytg
zp
uzgz
ugtSpu
β
ς
α
β
ςη
α
αης
βης
+=
⋅=
−−=
−−=
−+=
+−+=
+
+
+
−
−−−
0
10
1
1
1
111

[ ]{ } ;)()(
i
T
tbiitibii faaaKKK =00
[ ]{ }
;
;
)()(
)()()(
00
000
gp
faaaKKKg b
T
tbibtbbbi
=
−=
GPBiCG
(1) Initialization
(a) Set ．
(b) Solve ．

(c) Set and ．
)(0
ba
)(0
ia
)(0
g )(0
p

(2) Iteration
(a) Solve ．

(b) Set ．

(c) Compute ．

[ ]{ } ;)()(
00 =
Tkk
itibii pvKKK
[ ]{ } ;)()()( Tkk
btbbbi
k
pvKKKSp 0=
)(k
v
)(k
Sp
)()()(
,, kkk
tyα
,
,
,
),(
),(
)()()()(
)()()()()()()(
)()(
)()(
)(
kkkk
kkkkkkk
k
k
k
Spgt
Spwgty
Spg
gg
α
αα
α
−=
+−−=
=
−− 11
0
0

[ ]{ } ;00)()(
=
Tkk
itibii tvKKK
[ ]{ } ;0)()()( Tkk
btbbbi
k
tvKKKSt =
(d) Solve ．

(e) Set ．

(f) Compute ．

)(k
v
)(k
St
)()(
, kk
ης
),
),(
),(
then,if(
,
),)(,(),)(,(
),)(,(),)(,(
,
),)(,(),)(,(
),)(,(),)(,(
)(
)()(
)()(
)(
)()()()()()()()(
)()()()()()()()(
)(
)()()()()()()()(
)()()()()()()()(
)(
00 ===
−
−
=
−
−
=
k
kk
kk
k
kkkkkkkk
kkkkkkkk
k
kkkkkkkk
kkkkkkkk
k
StSt
tSt
k
yStStyyyStSt
tStStytyStSt
yStStyyyStSt
ySttytStyy
ης
η
ς

(g) Compute ．
)()()(
,, 1+kkk
gzu
,
,
),(
)()()()()()(
)()()()()()()(
)()()()()()()()(
kkkkkk
kkkkkkk
kkkkkkkk
Stytg
uzgz
ugtSpu
ςη
αης
βης
−−=
−+=
+−+=
+
−
−−−
1
1
111

(h) Convergence check for ． If converged
If not converged go to (h) ．
(i) Compute ．

(3) Construction of solution ．
　 (a) Solve ．
[ ]{ } ;**
i
T
tbiitibii faaaKKK =
)1( +k
g
)()()(
,, 11 ++ kkk
pwβ
*
ia
),(
,
,
),(
),(
)()()()()(
)()()()(
)()(
)()(
)(
)(
)(
kkkkk
kkkk
k
k
k
k
k
upgp
SpStw
gg
gg
−+=
+=
⋅=
++
+
β
β
ς
α
β
11
0
10
,λλ )()()()()(* kkkkk
b zpa −−== +
α1

Preconditioned GPBiCG for the Interface Problem (1/2)
),
),(
),(
then,if(
,
),)(,(),)(,(
),)(,(),)(,(
,
),)(,(),)(,(
),)(,(),)(,(
,
,
,
,
),(
),(
),(
begin
:dountil,,for
,,0set,χλ,guessinitialanisλ
)(
)()(
)()(
)(
)()()()()()()()(
)()()()()()()()(
)(
)()()()()()()()(
)()()()()()()()(
)(
)()()()(
)()()()(
)()()()()()()(
)()(
)()(
)(
)()()()()(
)()(
)()()()()()(
00
10
0
11
1
1111
1111
1111
11
111
11
0
0
1111
0
111000
===
−
−
=
−
−
=
−=
−=
+−−=
=
−+=
≤=
===−=
−−
−
−−−−
−−−−
−−−−
−−
−−−
−−
−−−−
−−−
k
kk
kk
k
kkkkkkkk
kkkkkkkk
k
kkkkkkkk
kkkkkkkk
k
kkkk
kkkk
kkkkkkk
k
k
k
kkkkk
k
tSMtSM
ttSM
k
ytSMtSMyyytSMtSM
ttSMtSMytytSMtSM
ytSMtSMyyytSMtSM
ytSMtyttSMyy
SpMgMtM
Spgt
Spwgty
Spg
gg
upgMp
Errggk
wtSg
ης
η
ς
α
α
αα
α
β
β


Preconditioned GPBiCG for the Interface Problem (2/2)
end
,
,
),(
),(
,
,
,
),(
)()()()(
)()(
)()(
)(
)(
)(
)()()()()()(
)()()()()(
)()()()()()()(
)()()()()()()()(
kkkk
k
k
k
k
k
kkkkkk
kkkkk
kkkkkkk
kkkkkkkk
SptSMw
gg
gg
tSMytg
zpλλ
uzgMz
ugMtMSpMu
β
ς
α
β
ςη
α
αης
βης
+=
⋅=
−−=
−−=
−+=
+−+=
−
+
−+
+
−−
−−−−−−
1
0
10
11
1
11
111111

One More Analysis of Subdomains
Output Results
Read Data
Analyze Analyze Analyze
Converged?
Change B.C.
Yes
No
System Flowchart
Skyline Method
BiCGSTAB Method
Newton Method

Whole domain Parts
Subdomains
HDDM Parents
only
ParentDisk
Part_1
Part_n
Part_2

AdvTetMesh
AdvBCtool
AdvsFlow
AdvMetis
AdvVisual
AdvCAD
AdvTriPatch
Commercial
CAD ・・・
Configure
・・・ Patch
・・・ Mesh
・・・ Boundary
Cond.
・・・ DD (-difn 4)
・・・ Flow
Analysis
・・・
Visualization
Adventure System

( )
( )










=→===
=→





=
==
==
=→=
.05.0,5.0,5.0
:FEM
,0,0,0
,0
,1,0
,1,0
,0,0,1,1
321
3
22
11
3
pxxx
x
xx
xx
x
D
D
ϕ
ϕ
1.0
1.0
0.0
1.0
1x
3x
2x
Boundary Conditions
Numerical Examples
(The Cavity Flow Problem ）

(8 parts, 8*125 subdomains)
Total DOF ： 1,000,188
Interface DOF ： 384,817
About 1,000 DOF/ subdomain
8 processors for parents
Domain Decomposition

Precond. ： Diagonal Scaling （ Abs. ）
6-1.0E
)0()0()()(
<
∞∞
gg
nk
Criterion ：
1.00E- 07
1.00E- 06
1.00E- 05
1.00E- 04
1.00E- 03
1.00E- 02
1.00E- 01
1.00E+00
1.00E+01
1.00E+02
0 500 1000 1500 2000 2500 3000
反復回数
相対残差
1回目 2回目 3回目 4回目 5回目
一回の反復が約 5 秒弱
Convergence of BiCGSTAB

収束履歴（ Newton 法）
Initial Value ： Sol.of Stokes
Criterion ： 4E0.1)0()()1(
−<−
∞∞
+
uuu nn
Iteration counts of
Newton method
1.00E- 06
1.00E- 05
1.00E- 04
1.00E- 03
1.00E- 02
1.00E- 01
1.00E+00
0 1 2 3 4 5
反復回数
相対変化量
(Nonlinear Convergence ）

Velocity Vectors
Visualization of AVS
Pressure Contour
x2 ＝ 0.5

1.00E- 07
1.00E- 06
1.00E- 05
1.00E- 04
1.00E- 03
1.00E- 02
1.00E- 01
1.00E+00
1.00E+01
0 100 200 300 400 500 600
反復回数
相対残差
1回目 2回目 3回目 4回目 5回目
Precod. ： Diagonal Scaling （ with sign ）
6-1.0E
)0()0()()(
<
∞∞
gg
nk
Criterion ：
一回の反復が約 5.5 秒弱
Convergence of GPBiCG

Initial ： Sol. of Stokes
−<−
∞∞
+
uuu nn
Iteretion counts of
Newton method
1.00E- 06
1.00E- 05
1.00E- 04
1.00E- 03
1.00E- 02
1.00E- 01
1.00E+00
0 1 2 3 4 5
反復回数
相対変化量

x1 component of the velocity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
- 0.3 - 0.1 0.1 0.3 0.5 0.7 0.9
BiCGSTAB GP- BiCG

9 hours (BiCGSTAB)→ 　 1 hour 40 min.(GPBiCG)
GPBiCG is a liitle faster than BiCGSTAB for small problems
High Reynolds number problems are not solved.
Strong preconditioners may be required.

（ 2 parts ， 2*75 subdomains, 800 DOF/subdomain≒
Total DOF ： 119,164
Interface DOF ： 42,417
Domain Decomposition

Initial ： Sol. of Stokes
−<−
∞∞
+
uuu nn
Iteration counts of
Newton method
1.00E- 06
1.00E- 05
1.00E- 04
1.00E- 03
1.00E- 02
1.00E- 01
1.00E+00
0 1 2 3 4 5
反復回数
相対変化量
HDDM FEM

ＦＥＭ
Velocity vectors and pressure at x2 ＝ 0.5
ＨＤＤＭ

x1-velocity component
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
- 0.3 - 0.1 0.1 0.3 0.5 0.7 0.9
x 1方向の流速
x3座標
Ghia 1000188dof(HDDM) 119164dof(HDDM) 119164dof(FEM)

DDM （ 1 ）
　　　 No. of Subdomins 　 64
　　　 No. of Nodes 　 9261
　　　 No. of DOF 　 37044
　　　 No. of Interface DOF 　 11718
DDM （ 2 ）
　　　 No. of Subdomains 　 125
　　　 No. of Nodes 　 9261
　　　 No. of DOF 　 37044
　　　 No. of Interface DOF 　 14800
Computinal Conditions

he Vector Diagram and the Pressure Contour-Line on x2 ＝ 0.5 （ Re:1
DDM(1) DDM(2) FEM

Comparison of the
Velocity （ Re:100 ）
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
- 0.2 0 0.2 0.4 0.6 0.8 1
x1 c ompone nt of ve loc ity
x3
DDM(1) DDM(2) FEM

Relative Residual History
of Newton Method
（ Re:100 ）
1.00E- 09
1.00E- 08
1.00E- 07
1.00E- 06
1.00E- 05
1.00E- 04
1.00E- 03
0 1 2 3 4
No. of Ite rations
RelativeResidual
DDM(1) DDM(2) FEM

DDM(1) DDM(2) FEM
Vector Diagram and the Pressure Contour-Line on x2 ＝ 0.5 （ Re:100

Comparison of the
Velocity （ Re:1000 ）
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
- 0.2 0 0.2 0.4 0.6 0.8 1
x1 c ompone nt of ve loc ity
x3
DDM(1) DDM(2) FEM

Relative Residual History
of Newton Method
（ Re:1000 ）
1.00E- 08
1.00E- 07
1.00E- 06
1.00E- 05
1.00E- 04
1.00E- 03
0 1 2 3 4 5 6
No. of Ite rations
RelativeResidual
DDM(1) DDM(2) FEM

A subway station model
Constant flows
the natural boundary condition

Computational Conditions
)100(30Alpha21264
49
)cos](/[1.0
)]([0.1)]([24
Re
664,943,12
916,235,3
133,873,18
2
hourstimenalcomputatio
ityviskinematicsm
velocitysmlengthm
DOF
Nodes
ElementsofNumber
≈×•
≈
×
=•
•
•
•

Convergence Criteria
5)0(
2
)0()(
2
)(
100.1 −
×≤gg
nk
Convergence of Newton method
4)1()()1(
100.1 −
∞
+
∞
+
×≤− nnn
aaa
Convergence of the interface problem
with GPBiCG method
Initial values of the interface problem
with GPBiCG method
( )



=
=
0
0,0,0
pλ
λu
The solution of the previous step
• 0 step of Newton method
• other steps of Newton method

Nonlinear Convergence
(Newton Method)

Visualization by AVS
(Velocity)

Visualization by AVS
(Pressure)

Conclusion
　
　
Future Works
A HDDM computing system for
stationary Navier-Stokes problems
has been developed and applied to
1- 10 million problems successfully.
More larger scale analysis based on
strong preconditioners and applications to high
Reynolds number problems and coupled problems

Stationary Incompressible Viscous Flow Analysis by a Domain Decomposition Method

Recommended

Recommended

More Related Content

What's hot

What's hot (19)

Similar to Stationary Incompressible Viscous Flow Analysis by a Domain Decomposition Method

Similar to Stationary Incompressible Viscous Flow Analysis by a Domain Decomposition Method (20)

More from ADVENTURE Project

More from ADVENTURE Project (20)

Recently uploaded

Recently uploaded (20)

Stationary Incompressible Viscous Flow Analysis by a Domain Decomposition Method