The Deflation Accelerated Schwarz Method for CFD

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The Deflation Accelerated Schwarz Methodfor CFD

• C. Vuik
• Delft University of Technology
• c.vuik_at_ewi.tudelft.nl
• http//ta.twi.tudelft.nl/users/vuik/

J. Verkaik, B.D. Paarhuis, A. Twerda TNO Science
and Industry
ICCS congres, Atlanta, USA May 23, 2005
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Contents

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

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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

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Problem description
Incompressible Navier-Stokes equations
Discretisation Finite Volume Method on
colocated grid
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Problem description
SIMPLE method
pressure- correction system ( )
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Schwarz domain decomposition
Minimal overlap
Additive Schwarz
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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

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Deflation basic idea
Problem convergence Schwarz method deteriorates
for increasing number of
subdomains
Solution remove smallest eigenvalues that
slow down the Schwarz method
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Deflation deflation vectors

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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
??
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GCR Krylov acceleration
Objective efficient solution to
Additive Schwarz
Property slow convergence Krylov
acceleration
GCR Krylov method

• 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

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Numerical experiments
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Numerical experiments
Buoyancy-driven cavity flow
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Numerical experiments
Buoyancy-driven cavity flow inner iterations
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Numerical experiments
Buoyancy-driven cavity flow outer iterations
without deflation
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Numerical experiments
Buoyancy-driven cavity flow outer iterations
with deflation
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Numerical experiments
Buoyancy-driven cavity flow outer iterations
varying inner iterations
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Numerical experiments
Glass tank model
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Numerical experiments
Glass tank model inner iterations
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Numerical experiments
Glass tank model outer iterations without
deflation
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Numerical experiments
Glass tank model outer iterations with deflation
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Numerical experiments
Glass tank model outer iterations varying inner
iterations
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Numerical experiments
Heat conductivity flow
h30 Wm-2K-1
T303K
K 1.0 Wm-1K-1
K 100 Wm-1K-1
Q0 Wm-2
Q0 Wm-2
K 0.01 Wm-1K-1
T1773K
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Numerical experiments
Heat conductivity flow inner iterations
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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