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The Deflation Accelerated Schwarz Method for CFD

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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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