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ICCS congres, Atlanta, USAMay 23, 2005

The Deflation Accelerated Schwarz Methodfor CFD

C. VuikDelft University of Technology

c.vuik@ewi.tudelft.nlhttp://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 experimentsBuoyancy-driven cavity flow

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Numerical experimentsBuoyancy-driven cavity flow: inner iterations

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Numerical experimentsBuoyancy-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 experimentsGlass tank model

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Numerical experimentsGlass tank model: inner iterations

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Numerical experimentsGlass tank model: outer iterations without deflation

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Numerical experimentsGlass 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