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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 - PowerPoint PPT Presentation
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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
.2
1)(
),(2),(
,0
,),()(
:
i
j
j
iij
ijijij
x
u
x
uD
Dpp
p
u
uu
u
fuuu
jn u:D
.delta sKronecker':],m[ viscositykinematic:
,]mforce[body:],mpressure[:,]smvelocity[:2
222
ijs
ssp
fu
Stationary Navier-Stokes Eqs.
0),(: nu pN
Weak Form
thatsuch,Find QVp u
vfuvvuvuu ,,,,,, qbpbsa QVq ,for v
.),0(,on;
,
2
31
LQ
VVXV
HX
D vv
uu
vuvu
vuwvuw
qqb
DDs
a
d
mllmlm
:,
)()(2:,
:,,
,
Newton Method
thatsuch,Find QVp kk u
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 Approximation0
1 2
3
element. ltetrahedraais,
elements, ltetrahedraofconsisting
ofiontriangulatA
hK
KK
h
).0(,on;
,,|;
,,|;
10
31
30
hhDhhhh
hKhhh
hKhhh
VVXV
KKPqCQqQ
KKPCXX
vv
vv
Stabilized Finite Element Method thatsuch,Find
hhhhQVp u
hKhhhhhK
hhhhhhhhhhhh
p
qbpbsaa
,
,,,,,,,
uwwu
uvvuvuwvwu
KhhKKhhhhh q vuvwwv ,
hhhh QVq ,for v
24,
2min
2
K
h
K
K
hh
w
Kh
Kh
Kh
hw
w
,12
min22
KhK ofdiameterais
Stabilized Parameter
hK
KhhhhhKhqvwwvfvf ,
~,
~
1
Domain Decomposition Method
b
i
b
i
bbbi
ibii
f
f
a
a
KK
KK
faK
12
2
1
D
1 2
Stabilized Finite Element Method
Decomposition
i ： corresponding to Inner DOF
b ： corresponding to Interface DOF
Inner DOF
Solver by Skyline Method
bibiiii aKfaK
Interface DOF
iiibibbibiibibb fKKfaKKKK 11 )(
Solver by BiCGSTAB or GPBiCG
χS
BiCGSTAB for the Interface Problem
end
,),(
),(
,
,
,),(
),(
,
,),(
),(
),(
begin
:dountil,,for
.set,,guessinitialanis
)()(
)()(
)(
)()(
)()()()(
)()()()()()(
)()(
)()()(
)()()()(
)()(
)()()(
)()()()()()(
)()(
)()()()(
k
k
k
kk
kkkk
kkkkkk
kk
kkk
kkkk
k
kk
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
ργ
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
,χbSa
,)(
),(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
kk
Swgt
Swg
gg
;00)()( Tkk
itibii wvKKK
;0)()()( Tkkbtbbbi
k wvKKKSw
)(kv
)(kSw
)()( , kk t
;
;
;),(
),(
)()()()1(
)()()()()()1(
)()(
)()()(
kkkk
kkkkkb
kb
kk
kkk
Sttg
twaa
StSt
tSt
;00)()( Tkk
itibii tvKKK
;0)()()( Tkkbtbbbi
k tvKKKSt
(d) Solve ．
(e) Solve ．
(f) Compute ．
)(kv
)(kSt
)1()1()( ,, kkb
k ga
(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
kk
Swwgw
gg
gg
;**i
T
tbiitibii faaaKKK
)1( kg)1(* k
bb aa
)1()( , kk w
*ia
Preconditioned BiCGSTAB for the Interface Problem
end
,),(
),(
,
,
,),(
),(
,
,),(
),(
),(
begin
:dountil,,for
,set,,guessinitialanis
)()(
)()(
)(
)()(
)()()()(
)()()()()()(
)()(
)()()(
)()()()(
)()(
)()()(
)()()()()()(
)()(
)()()()(
k
k
k
kk
kkkk
kkkkkk
kk
kkk
kkkk
k
kk
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
ργ
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
kkk
kkk
kkkkkkkk
kkkkkkkkk
kkkkkkkk
kkkkkkkkk
kkkk
kkkkkkk
k
kk
kkkkk
k
StSt
tStk
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
kk
kkkkkk
kkkkk
kkkkkkk
kkkkkkkk
SpStw
gg
gg
Stytg
zp
uzgz
ugtSpu
β
ς
αβ
ςη
α
αης
βης
0
10
1
1
1
111
t
b
i
t
b
i
btbbbi
itibii
f
f
f
a
a
a
E
KKK
KKK
00
,χbSa
,)(
),(11
1
titiibibtiiibib
ibiibibb
aKKKKfKKf
KKKKS
χ
Equations on the interface
;)()(i
Ttbiitibii faaaKKK 00
;
;
)()(
)()()(
00
000
gp
faaaKKKg bT
tbibtbbbi
GPBiCG(1) Initialization
(a) Set ．
(b) Solve ．
(c) Set and ．
)(0ba
)(0ia
)(0g )(0p
(2) Iteration(a) Solve ．
(b) Set ．
(c) Compute ．
;)()( 00 Tkk
itibii pvKKK
;)()()( Tkkbtbbbi
k pvKKKSp 0
)(kv
)(kSp
)()()( ,, kkk tyα
,
,
,),(
),(
)()()()(
)()()()()()()(
)()(
)()()(
kkkk
kkkkkkk
k
kk
Spgt
Spwgty
Spg
gg
α
αα
α
11
0
0
;00)()( Tkk
itibii tvKKK
;0)()()( Tkkbtbbbi
k tvKKKSt
(d) Solve ．
(e) Set ．
(f) Compute ．
)(kv
)(kSt
)()( , kk ης
) ,),(
),( then , if(
,),)(,(),)(,(
),)(,(),)(,(
,),)(,(),)(,(
),)(,(),)(,(
)()()(
)()()(
)()()()()()()()(
)()()()()()()()()(
)()()()()()()()(
)()()()()()()()()(
00
kkk
kkk
kkkkkkkk
kkkkkkkkk
kkkkkkkk
kkkkkkkkk
StSt
tStk
yStStyyyStSt
tStStytyStSt
yStStyyyStSt
ySttytStyy
ης
η
ς
(g) Compute ．)()()( ,, 1kkk 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( kg
)1()()( ,, kkk pw
*ia
),(
,
,),(
),(
)()()()()(
)()()()(
)()(
)()(
)(
)()(
kkkkk
kkkk
k
k
k
kk
upgp
SpStw
gg
gg
β
β
ς
αβ
11
0
10
,λλ )()()()()(* kkkkkb 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
kkk
kkk
kkkkkkkk
kkkkkkkkk
kkkkkkkk
kkkkkkkkk
kkkk
kkkk
kkkkkkk
k
kk
kkkkk
k
tSMtSM
ttSMk
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
kk
kkkkkk
kkkkk
kkkkkkk
kkkkkkkk
SptSMw
gg
gg
tSMytg
zpλλ
uzgMz
ugMtMSpMu
β
ς
αβ
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
ParentDiskPart_1
Part_n
Part_2
AdvTetMesh
AdvBCtool
AdvsFlow
AdvMetis
AdvVisual
AdvCAD
AdvTriPatch
CommercialCAD ・・・
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
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 ： 1,000,188Interface DOF ： 384,817
About 1,000 DOF/ subdomain 8 processors for parents
Domain Decomposition
Precond. ： Diagonal Scaling （ Abs. ）6-1.0E
)0()0()()(
ggnk
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+000 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()()(
ggnk
Criterion ：
一回の反復が約 5.5 秒弱
Convergence of GPBiCG
Initial ： Sol. of Stokes
Criterion ： 4E0.1)0()()1(
uuu nn
Iteretion counts of Newton method
収束履歴（ Newton 法）
1.00E- 06
1.00E- 05
1.00E- 04
1.00E- 03
1.00E- 02
1.00E- 01
1.00E+000 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 ： 119,164Interface DOF ： 42,417
Domain Decomposition
Initial ： Sol. of Stokes
Criterion ： 4E0.1)0()()1(
uuu nn
Iteration counts of Newton method
収束履歴（ Newton 法）
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+000 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
x1方向の流速
x 3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
DDM(1) DDM(2)
Mesh
The Vector Diagram and the Pressure Contour-Line on x2 ＝ 0.5 （ Re:100 ） 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 component of velocity
x 3
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-030 1 2 3 4
No. of Iterations
Re
lati
ve R
esi
du
al
DDM(1) DDM(2) FEM
DDM(1) DDM(2) FEM
The Vector Diagram and the Pressure Contour-Line on x2 ＝ 0.5 （ Re:1000 ）
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 component of velocity
x 3
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-030 1 2 3 4 5 6
No. of Iterations
Re
lati
ve R
esi
du
al
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)]([24Re
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 ggnk
Convergence of Newton method
4)1()()1( 100.1
nnn aaa
Convergence of the interface problem with GPBiCG method
Initial values of the interface problemwith GPBiCG method
0
0,0,0
pu
The solution of the previous step
• 0 step of Newton method
• other steps of Newton method
Convergence of GPBiCG
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
