23/5/20051 iccs congres, atlanta, usa may 23, 2005 the deflation accelerated schwarz method for cfd...
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ICCS congres, Atlanta, USAMay 23, 2005
The Deflation Accelerated Schwarz Methodfor CFD
C. VuikDelft University of Technology
http://ta.twi.tudelft.nl/users/vuik/
J. Verkaik, B.D. Paarhuis, A. TwerdaTNO Science and Industry
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Contents
• Problem description• Schwarz domain decomposition• Deflation• GCR Krylov subspace acceleration• Numerical experiments• Conclusions
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Problem description
• 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
GTM-X:
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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-correctio
nsystem
( )
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Schwarz domain decomposition
Minimal overlap:
Additive Schwarz:
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• 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
Schwarz domain decomposition
GTM-X:
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Deflation: basic idea
Solution: “remove” smallest eigenvalues that slow down the Schwarz method
Problem: convergence Schwarz method deteriorates for increasing number of subdomains
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Deflation: deflation vectors
+
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Property deflation method: systems with have to be solved by a direct method
Deflation: Neumann problem
singular
Problem: pressure-correction matrix is singular: has eigenvector for eigenvalue 0
Solution: adjust non-singular
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• 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
Additive Schwarz:
Property: slow convergence Krylov acceleration
GCR Krylov acceleration
GCR Krylov method:
Objective: efficient solution to
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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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Buoyancy-driven cavity flow: outer iterations with deflation
Numerical experiments
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Buoyancy-driven cavity flow: outer iterations varying inner iterations
Numerical experiments
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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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Glass tank model: outer iterations varying inner iterations
Numerical experiments
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Heat conductivity flow
Numerical experiments
Q=0 Wm-2Q=0 Wm-2
h=30 Wm-2K-1
T=303K
T=1773K
K = 1.0 Wm-1K-1
K = 0.01 Wm-1K-1
K = 100 Wm-1K-1
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Heat conductivity flow: inner iterations
Numerical experiments
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• 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
Conclusions