Stationary Incompressible Viscous Flow Analysis by a Domain Decomposition Method

1
Stationary Incompressible Viscous Flow Analysis
by a Domain Decomposition Method

• Hiroshi Kanayama, Daisuke Tagami
• and Masatsugu Chiba
• (Kyushu University)

2
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

3
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 )

4
Formulations

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

5
Stationary Navier-Stokes Eqs.
6
Weak Form
7
Newton Method
8
Finite Element Approximation
9
Stabilized Finite Element Method
Stabilized Parameter
10
Domain Decomposition Method
Stabilized Finite Element Method
Decomposition
icorresponding to Inner DOF bcorresponding to
Interface DOF
11
Interface DOF
Solver by BiCGSTAB or GPBiCG
Inner DOF
Solver by Skyline Method
12
BiCGSTAB for the Interface Problem
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Equations on the interface
14
BiCGSTAB (1) Initialization (a) Set
. (b) Solve .     (c)
Solve .    
15
(2)     Iteration (a) Solve
.     (b) Solve .     (c)
Compute .        
16
(d) Solve .     (e) Solve
.     (f) Compute
.        
17

• Convergence check for .If converged
• , If not converged go to (h).
• (h) Compue .
•  
•  
•  
•  
• (3)     Construction of solution.
• (a) Solve .

18
Preconditioned BiCGSTAB for the Interface Problem
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GPBiCG for the Interface Problem (1/2)
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GPBiCG for the Interface Problem (2/2)
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Equations on the interface
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GPBiCG (1) Initialization (a) Set
. (b) Solve .     (c) Set
and .    
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(2)     Iteration (a) Solve
.     (b) Set .     (c) Compute
.        
24
(d) Solve .     (e) Set
.     (f) Compute .        
25
(g) Compute
.
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• (h) Convergence check for .If converged
• If not converged go to (h).
• (i) Compute .
•  
•  
•  
•  
• (3)     Construction of solution.
• (a) Solve .

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Preconditioned GPBiCG for the Interface Problem
(1/2)
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Preconditioned GPBiCG for the Interface Problem
(2/2)
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System Flowchart
Read Data
Skyline Method
Analyze
Analyze
Analyze
Change B.C.
No
BiCGSTAB Method
Yes
One More Analysis of Subdomains
Newton Method
Output Results
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HDDM
Parents only
Whole domain
Parts
Subdomains
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Adventure System
Commercial CAD
AdvCAD
Configure
AdvTriPatch
Patch
AdvTetMesh
Mesh
AdvBCtool
Boundary Cond.
AdvMetis
DD (-difn 4)
AdvsFlow
Flow Analysis
Visualization
AdvVisual
32
Numerical Examples (The Cavity Flow Problem)
Boundary Conditions
33
Domain Decomposition
(8 parts, 8125 subdomains)
About 1,000 DOF/ subdomain 8 processors for
parents
Total DOF 1,000,188 Interface DOF 384,817
34
Convergence of BiCGSTAB
Precond.Diagonal Scaling(Abs.)
Criterion
???????5??
35
(Nonlinear Convergence )
Iteration counts of Newton method
Initial ValueSol.of Stokes
Criterion
????(Newton?)
36
Visualization of AVS
x20.5
Velocity Vectors
Pressure Contour
37
Convergence of GPBiCG
Precod.Diagonal Scaling(with sign)
Criterion
???????5.5??
38
????(Newton?)
Iteretion counts of Newton method
InitialSol. of Stokes
Criterion
39
x1 component of the velocity
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9 hours (BiCGSTAB)? 1 hour 40 min.(GPBiCG) GPBiC
G is a liitle faster than BiCGSTAB for small
problems. High Reynolds number problems are not
solved. Strong preconditioners may be required.
41
Domain Decomposition
(2 parts,275 subdomains, ?800 DOF/subdomain)

Total DOF119,164 Interface DOF 42,417
42
????(Newton?)
Iteration counts of Newton method
InitialSol. of Stokes
Criterion
43
Velocity vectors and pressure at x20.5
FEM
HDDM
44
x1-velocity component
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Computinal Conditions
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
46
Mesh
DDM(2)
DDM(1)
47
The Vector Diagram and the Pressure Contour-Line
on x20.5(Re100)
DDM(1)
DDM(2)
FEM
48
Comparison of the Velocity(Re100)
49
Relative Residual History of Newton
Method (Re100)
50
The Vector Diagram and the Pressure Contour-Line
on x20.5(Re1000)
DDM(1)
DDM(2)
FEM
51
Comparison of the Velocity(Re1000)
52
Relative Residual History of Newton
Method (Re1000)
53
A subway station model
the natural boundary condition
Constant flows
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Computational Conditions
55
Convergence Criteria
Convergence of the interface problem with GPBiCG
method
Initial values of the interface problem with
GPBiCG method

• 0 step of Newton method

• other steps of Newton method

The solution of the previous step
Convergence of Newton method
56
Convergence of GPBiCG
57
Nonlinear Convergence(Newton Method)
58
Visualization by AVS (Velocity)
59
Visualization by AVS (Pressure)
60
Conclusion

A HDDM computing system for stationary
Navier-Stokes problems has been developed and
applied to 1- 10 million problems successfully.
Future Works


More larger scale analysis based on strong
preconditioners and applications to high Reynolds
number problems and coupled problems