Conference ECT 2010

Selection of fixing nodes for FETI-DP method in 3D


Jaroslav Broz, Jaroslav Kruis

Department of MechanicsFaculty of Civil Engineering

Czech Technical University in Prague

The Seventh International Conference on EngineeringComputational Technology

Valencia, Spain16 September 2010


Contents

Contents

1 FETI-DP MethodShort IntroductionFixing nodesAlgorithm for Selection of Fixing Nodes in 3D

2 Numerical TestsBoxwise partitioning - DamStripwise partitioning - Bridge

3 Conclusions


FETI-DP Method

Short Introduction

FETI-DP MethodShort Introduction

FETI-DP (Finite Element Tearing and InterconnectingDual-Primal) method was published by C. Farhat et all

Non-overlapping domain decomposition method

Based on the combination of the original FETI method and theSchur complement methodUnknowns in the problem are splitted into three categories

1 fixing unknowns2 remaining interface unknowns3 internal unknowns


FETI-DP Method

Short Introduction

Continuity Conditions

Continuity conditions are enforced

by Lagrange multipliers defined on remaining interfaceunknowns

directly on fixing unknowns.


FETI-DP Method

Short Introduction

Coarse problem is obtained after elimination of internal andremaining interface unknowns.Coarse Problem

(FIrr FIrf

FTIrf−K∗

ff

) (λuf

)=

(dr

−f∗f

)(1)

Coarse Problems - Lagrange Multipliers

(FIrr + FIrf K

∗−1

ff FTIrf

)λ = dr − FIrf K

∗−1

ff f∗f (2)

Solving of Reduced Coarse ProblemMatrix of system of equations of coarse problem is symmetric andpositive definite→ solving by conjugate gradient method.


FETI-DP Method

Fixing nodes

Definition of Fixing Nodes in Original Farhat’s ArticleD1 Cross-points - the nodes belonging to more than two

subdomains

D2 The set of nodes located at the beginning and end ofeach edge of each subdomain

D1

D2

D2

D2 D2

Ω1

Ω2

Ω3

Ω4


FETI-DP Method

Fixing nodes

Importance of Proper Selection of Fixing NodesNon-singularity of subdomain matrix Ks

Non-singularity of matrix of coarse problem(FIrr FIrf

FTIrf−K∗

ff

)Suitable selected fixing nodes⇒ relatively small conditionnumber κ of matrix of coarse problem⇒ speed of convergenceof solution of coarse problem

Theoretically, all interface nodes can be selected as fixing nodes→ Schur complement method


FETI-DP Method

Fixing nodes

Difficulties with Choice of Fixing NodesDefinition from original article produces a lot of fixing nodes

There is not any suitable software for selection of fixing nodes

There are only tools for automatic domain decomposition(METIS, JOSTLE, CHACO etc.) - based on decomposition ofgraphs

Problem with interpretation of definition in case of meshessplitted by mesh decomposer

Limited data - only mesh of finite elements


FETI-DP Method

Algorithm for Selection of Fixing Nodes in 3D

Definition of Graphs

Decomposed cube domain into8 subdomains Boundary Graph B


FETI-DP Method


Definition of Graphs

Boundary Curve Graph CBoundary Surface Graph P


FETI-DP Method


Definition of Minimal Number Algorithm

The Definition of the Basic FixingNode for Three-Dimensional MeshLet be v vertex of the graphv ∈ V (C ) and let be dC (v) thedegree of vertex v. If

dC (v) = 1 or dC (v) > 2 (3)

then the vertex v is called fixingnode.

Selected fixing nodes


FETI-DP Method


Geometrical Conditions

Following geometrical conditionare checked

distance between two fixingnodes

angle between two fixingnodes

area of triangle among fixingnodes


FETI-DP Method


Extended Algorithm

Additional fixing nodes can be chosen with the help of

boundary curve subgraphs Cj

boundary surface subgraphs Pj

combination of above possibilities


Numerical Tests

Boxwise partitioning - Dam

Dam

Linear elasticity problem

Tetrahedral elements with 3DOFs

METIS partitionining into 10subdomains

Boxwise partitioning


Numerical Tests


Dam - obtained results

70

75

80

85

90

95

100

105

110

115

120

125

130

135

140

145

150

155

160

0 25 50 75 100 125 150 175 200 225 250 275 300

Num

ber

of

iter

atio

ns

in c

oar

se p

roble

m

Number of fixing nodes in whole problem

minimal

curve - centroid

curve - each n-th

curve random

surface random

surface centroid

comb - centroid+centroid


Numerical Tests


Dam - obtained results

1040

1050

1060

1070

1080

1090

1100

1110

1120

1130

1140

0 25 50 75 100 125 150 175 200 225 250 275 300

Tim

e of

whole

solu

tio

n [

s]


minimal

curve - centroid

curve - each n-th

curve random

surface random

surface centroid

comb - centroid+centroid


Numerical Tests

Stripwise partitioning - Bridge

Bridge

Linear elasticity problem

Hexahedral elements with 3 DOFs

METIS partitionining into 10 subdomains

Stripwise partitioning


Numerical Tests


Bridge - obtained results

0

25

50

75

100

125

150

175

200

225

250

275

300

325

350

375

400

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

Num

ber

of

iter

atio

ns

in c

oar

se p

roble

m


minimalsurface n-th ringsurface centroidsurface random

surface max ringsurface max triangle


Numerical Tests


Bridge - obtained results

150

155

160

165

170

175

180

185

190

195

200

205

210

215

220

225

230

235

240

245

250

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

Tim

e of

whole

solu

tio

n [

s]


minimalsurface n-th ringsurface centroidsurface random

surface max ringsurface max triangle


Conclusions

Conclusions

Algorithm for selection of fixing nodes for 3D meshes wasdevelopedFollowing behaviour was observed

The increasing of the number of fixing nodes leads to thedecreasing of the number of iterations in the coarse problemThe large number of fixing nodes leads to the prolongation of thewhole time of solutionAddition of a few further nodes is the best solution

Further work can be an automatic choice of optimal number offixing nodes


Acknowledgement

Acknowledgement

Thank you for your attention.

Financial support for this work was provided by project number103/09/H078 of the Czech Science Foundation. The financial supportis gratefully acknowledged.

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Conference ECT 2010