Seminary of numerical analysis 2010

Selection strategy for fixing nodes in FETI-DP method

Selection strategy for fixing nodes in FETI-DPmethod

Jaroslav Brož1, Jaroslav Kruis

Katedra mechanikyFakulta stavebníCVUT v Praze

Seminár numerické analýzy18. leden - 22. leden 2010

Zámek Nové Hrady

[email protected]


Outline

Outline

1 FETI-DP Method

2 Algorithm for Fixing Node Selection in 2D

3 Numerical Tests of Algorithm for 2D



6 Conclusions and Future Works


FETI-DP Method

Outline

1 FETI-DP MethodIntroductionCoarse ProblemFixing Nodes







FETI-DP Method

Introduction

FETI-DP MethodIntroduction

One of non-overlapping domain decomposition methods

Method was published by prof. Farhat and his collaborators inthe article: Farhat, C., Lesoinne, M., LeTallec, P., Pierson, K. &Rixen, D. (2001): FETI-DP A dual-primal unified FETImethod-part I: Faster alternative to the two-level FETImethod. International Journal for Numerical Methods inEngineering, Vol. 50, 1523–1544.

Method was developed due to problems with singulars matrix inoriginal FETI Method


FETI-DP Method

Introduction

FETI-DP MethodIntroduction

Unknowns are divided into two groups - interior unknowns andinterface unknowns among subdomains

Continuity conditions are ensured by Lagrange multipliers andfixing nodes

Interior unknowns are eliminated and a coarse problem areobtained


FETI-DP Method

Coarse Problem

Coarse Problem

(−S[cc] F[cr]

F[rc] F[rr]

)(d[c]

λ

)=

(−sg

). (1)

whered[c] vector includes DOF defined on fixing nodes.λ vector includes Lagrange multipliers.S[cc], F[cr], F[rc], F[rr] are blocks of matrix of coarse problem.

d[c] =−(

S[cc])−1 (

−s−F[cr]λ

). (2)

(F[rr] +F[rc]

(S[cc]

)−1F[cr]

)λ = g−F[rc]

(S[cc]

)−1s. (3)


FETI-DP Method

Fixing Nodes

Definition of Fixing Nodes

Simple definition in the case of a regular mesh

x

y

1

2

3

4

5


FETI-DP Method

Fixing Nodes

Definition of Fixing Nodes

Problem with definition of fixing nodes in the case of non-regularmeshes which are decomposed by a mesh decomposer (e.g.METIS, http://glaros.dtc.umn.edu/gkhome/views/metis).

Minimal number of fixing nodes due to the nonsingular matrix ofsubdomainsTheoretically the number of fixing node = the number of allnodes on boundaries


Algorithm for Fixing Node Selection in 2D

Outline










Definition of Nodal MultiplicityNodal multiplicity - the number of subdomains which belongs to node

Definition of Fixing NodesNode with node multiplicity > 2→ fixing node

Node with node multiplicity = 2 and only with one neighborwith node multiplicity = 2→ fixing node.

x

y




Definition of Boundary CurvesBoundary curve connect boundary nodes between two fixing nodes.

Fixing nodes can be added into:Centroid of boundary curveEach n-th member of the boundary curveEach n-th end of the part of the boundary curve“Integral points” of the boundary curveRandom position

x

y


Numerical Tests of Algorithm for 2D

Outline









Numerical testsIrregular Domain - Slope

NS NN NE NN-SUB NE-SUB NDOF-SUB4 105182 208840 26448 52210 528464 186577 371124 46847 92781 936279 105182 208840 11834 23204 236479 186577 371124 20923 41236 41816



SlopeResults of Tests - The Number of Iterations with Respect to the Number of Fixing Nodes



SlopeResults of Tests - Time of Condensation with Respect to the Number of Fixing Nodes



SlopeResults of Tests - Total Time of the Solution with Respect to the Number of Fixing Nodes



Outline










Definition of Edges and SufracesEdge - defined by boundary nodes which belongs to more than twosubdomainsSurface - defined by boundary nodes which belongs to exactly twosubdomains




Definition of Fixing nodesnode with maximal nodal multiplicity→ fixing node

end of edge→ fixing node

Definition of Boundary CurvesBoundary curve→ edge between two fixing nodes



Algorithm for Fixing Node Selection in 3DNext Step - Under Developement

Definition of Boundary SurfaceBoundary surface - created by boundary nodes with nodal multiplicityequal two



Outline









Numerical testsRegular Domain - Cube

NS NN NE NN-SUB NE-SUB NDOF-SUB8 29791 27000 4096 3375 119048 68921 64000 9261 8000 27121



Cube - 27000 elementsResults of Tests - The Number of Iterations with Respect to the Number of Fixing Nodes



Cube - 27000 elementsResults of Tests - Time of Condensation with Respect to the Number of Fixing Nodes



Cube - 27000 elementsResults of Tests - Total Time of the Solution with Respect to the Number of Fixing Nodes



Cube - 64000 elementsResults of Tests - The Number of Iterations with Respect to the Number of Fixing Nodes



Cube - 64000 elementsResults of Tests - Time of Condensation with Respect to the Number of Fixing Nodes



Cube - 64000 elementsResults of Tests - Total Time of the Solution with Respect to the Number of Fixing Nodes


Conclusions and Future Works

Outline










The algorithm for selection of fixing nodes for arbitrary 2D meshhas been developed

Increasing of the number of the fixing nodes leads to decreasingof the number of iterations in coarse problem and its time of thesolution

Big number of fixing nodes leads to prolongation of the wholetime of the solution

Developing of the algorithm for the selection of fixing nodes forregular 3D mesh

Optimization of the algorithm


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.

Seminary of numerical analysis 2010

Education

Transcript of Seminary of numerical analysis 2010