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The Deflation Accelerated Schwarz Method for CFD

The Deflation Accelerated Schwarz Method for CFD. J. Verkaik, B.D. Paarhuis, A. Twerda TNO Science and Industry. C. Vuik Delft University of Technology c.vuik@ewi.tudelft.nl http://ta.twi.tudelft.nl/users/vuik/. ICCS congres, Atlanta, USA May 23, 2005. Contents. Problem description

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The Deflation Accelerated Schwarz Method for CFD

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  1. The Deflation Accelerated Schwarz Methodfor CFD J. Verkaik, B.D. Paarhuis, A. Twerda TNO Science and Industry C. Vuik Delft University of Technology c.vuik@ewi.tudelft.nl http://ta.twi.tudelft.nl/users/vuik/ ICCS congres, Atlanta, USA May 23, 2005

  2. Contents • Problem description • Schwarz domain decomposition • Deflation • GCR Krylov subspace acceleration • Numerical experiments • Conclusions

  3. Problem description GTM-X: • CFD package • TNO Science and Industry, The Netherlands • simulation of glass melting furnaces • incompressible Navier-Stokes equations, energy equation • sophisticated physical models related to glass melting

  4. Problem description Incompressible Navier-Stokes equations: Discretisation: Finite Volume Method on “colocated” grid

  5. Problem description SIMPLE method: pressure- correction system ( )

  6. Schwarz domain decomposition Minimal overlap: Additive Schwarz:

  7. Schwarz domain decomposition GTM-X: • inaccurate solution to subdomain problems: 1 iteration SIP, SPTDMA or CG method • complex geometries • parallel computing • local grid refinement at subdomain level • solving different equations for different subdomains

  8. Deflation: basic idea Problem: convergence Schwarz method deteriorates for increasing number of subdomains Solution: “remove” smallest eigenvalues that slow down the Schwarz method

  9. Deflation: deflation vectors +

  10. Deflation: Neumann problem Property deflation method: systems with have to be solved by a direct method Problem: pressure-correction matrix is singular: has eigenvector for eigenvalue 0 singular Solution:adjust non-singular 

  11. GCR Krylov acceleration Objective: efficient solution to Additive Schwarz: • for general matrices (also singular) • approximates in Krylov space such that is minimal • Gram-Schmidt orthonormalisation for search directions • optimisation of work and memory usage of Gram-Schmidt: restarting and truncating Property: slow convergence Krylov acceleration GCR Krylov method:

  12. Numerical experiments

  13. Numerical experiments Buoyancy-driven cavity flow

  14. Numerical experiments Buoyancy-driven cavity flow: inner iterations

  15. Numerical experiments Buoyancy-driven cavity flow: outer iterations without deflation

  16. Numerical experiments Buoyancy-driven cavity flow: outer iterations with deflation

  17. Numerical experiments Buoyancy-driven cavity flow: outer iterations varying inner iterations

  18. Numerical experiments Glass tank model

  19. Numerical experiments Glass tank model: inner iterations

  20. Numerical experiments Glass tank model: outer iterations without deflation

  21. Numerical experiments Glass tank model: outer iterations with deflation

  22. Numerical experiments Glass tank model: outer iterations varying inner iterations

  23. Numerical experiments Heat conductivity flow h=30 Wm-2K-1 T=303K K = 1.0 Wm-1K-1 K = 100 Wm-1K-1 Q=0 Wm-2 Q=0 Wm-2 K = 0.01 Wm-1K-1 T=1773K

  24. Numerical experiments Heat conductivity flow: inner iterations

  25. Conclusions • using linear deflation vectors seems most efficient • a large jump in the initial residual norm can be observed • higher convergence rates are obtained and wall-clock time can be gained • implementation in existing software packages can be done with relatively low effort • deflation can be applied to a wide range of problems

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