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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

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Stationary Incompressible Viscous Flow Analysis by a Domain Decomposition Method

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  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 0 2 1 3

  9. Stabilized Finite Element Method Stabilized Parameter

  10. Domain Decomposition Method Stabilized Finite Element Method Decomposition i：corresponding to Inner DOF b：corresponding to Interface DOF

  11. Interface DOF Solver by BiCGSTAB or GPBiCG Inner DOF Solver by Skyline Method

  12. BiCGSTAB for the Interface Problem

  13. 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

  19. GPBiCG for the Interface Problem (1/2)

  20. GPBiCG for the Interface Problem (2/2)

  21. Equations on the interface

  22. GPBiCG (1) Initialization (a) Set． (b) Solve ． (c) Set and ．

  23. (2)     Iteration (a) Solve． (b) Set ． (c) Compute ．

  24. (d) Solve． (e) Set ． (f) Compute ．

  25. (g) Compute ．

  26. (h) Convergence check for ．If converged • If not converged go to (h)． • (i) Compute ． • (3)Construction of solution． • 　(a) Solve ．

  27. Preconditioned GPBiCG for the Interface Problem (1/2)

  28. Preconditioned GPBiCG for the Interface Problem (2/2)

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

  30. Disk Parent Part_1 Part_n Part_2 HDDM Parents only Whole domain Parts Subdomains

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

  32. 1.0 1.0 0.0 1.0 Numerical Examples (The Cavity Flow Problem） Boundary Conditions

  33. Domain Decomposition (8 parts, 8*125 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： 一回の反復が約5秒弱

  35. (Nonlinear Convergence ） Iteration counts of Newton method Initial Value：Sol.of Stokes Criterion： 収束履歴（Newton法）

  36. Visualization of AVS x2＝0.5 Velocity Vectors Pressure Contour

  37. Convergence of GPBiCG Precod.：Diagonal Scaling（with sign） Criterion： 一回の反復が約5.5秒弱

  38. 収束履歴（Newton法） Iteretion counts of Newton method Initial：Sol. of Stokes Criterion：

  39. x1 component of the velocity

  40. 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.

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

  42. 収束履歴（Newton法） Iteration counts of Newton method Initial：Sol. of Stokes Criterion：

  43. Velocity vectors and pressure at x2＝0.5 ＦＥＭ ＨＤＤＭ

  44. x1-velocity component

  45. 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 x2＝0.5（Re:100） DDM(1) DDM(2) FEM

  48. Comparison of the Velocity（Re:100）

  49. Relative Residual History of Newton Method （Re:100）

  50. The Vector Diagram and the Pressure Contour-Line on x2＝0.5（Re:1000） DDM(1) DDM(2) FEM

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