Conference PhD & DLA symposium 2009
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Selection Strategy for Corner Nodes in FETI-DP Method Selection Strategy for Corner Nodes in FETI-DP Method Jaroslav Brož 1 , Jaroslav Kruis Department of Mechanics Faculty of Civil Engineering, Czech Technical University in Prague Czech Republic Fifth PhD & DLA Symposium 19. October - 20. October 2009 Pecs, Hungary 1 [email protected]
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Transcript of Conference PhD & DLA symposium 2009
Selection Strategy for Corner Nodes in FETI-DP Method
Selection Strategy for Corner Nodes in FETI-DPMethod
Jaroslav Brož1, Jaroslav Kruis
Department of MechanicsFaculty of Civil Engineering, Czech Technical University in Prague
Czech Republic
Fifth PhD & DLA Symposium19. October - 20. October 2009
Pecs, Hungary
Selection Strategy for Corner Nodes in FETI-DP Method
Outline
Outline
1 Motivation
2 FETI-DP Method
3 Algorithm for Corner Node Selection in 2D
4 Numerical Tests
5 Conclusions and Future Works
6 Acknowledgement
Selection Strategy for Corner Nodes in FETI-DP Method
Motivation
Motivation
Big demands on computer capacity
Finner mesh in FEM
Complex material models
Large-scale numerical models
Selection Strategy for Corner Nodes in FETI-DP Method
FETI-DP Method
Introduction
FETI-DP MethodIntroduction
Non-overlapping domain decomposition method
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
Selection Strategy for Corner Nodes in FETI-DP 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 andcorner nodes
Interior unknowns are eliminated and coarse problem areobtained
Selection Strategy for Corner Nodes in FETI-DP Method
FETI-DP Method
Coarse Problem
Coarse Problem
(−S[cc] F[cr]
F[rc] F[rr]
)(d[c]
λ
)=
(−sg
). (1)
where
d[c] vector which include DOF defined on coarse nodes.
λ vector which include Lagrange multipliers
S[cc], F[cr], F[rc], F[rr] are blocks of matrix of coarse problem
d[c] =−(
S[cc])(−s−F[cr]
λ
). (2)
(F[rr] +F[rc]
(S[cc]
)−1F[cr]
)λ = g−F[rc]
(S[cc]
)−1s. (3)
Selection Strategy for Corner Nodes in FETI-DP Method
FETI-DP Method
Corner Nodes
Definition of Corner NodesSimple definition in case of regular mesh
x
y
1
2
3
4
5
Problem with definition of corner nodes in case of non-regularmeshes which are decomposed by mesh decomposer (e.g.METIS, http://glaros.dtc.umn.edu/gkhome/views/metis).
Minimal number of corner nodes due to nonsingular matrix ofsubdomainsTheoretically the number of corner node = the number of allnodes on boundaries
Selection Strategy for Corner Nodes in FETI-DP Method
FETI-DP Method
Corner Nodes
Definition of Corner Nodes
Simple definition in case of regular mesh
Problem with definition of corner nodes in case of non-regularmeshes which are decomposed by mesh decomposer (e.g.METIS, http://glaros.dtc.umn.edu/gkhome/views/metis).
Minimal number of corner nodes due to nonsingular matrix ofsubdomains
Theoretically the number of corner node = the number of allnodes on boundaries
Selection Strategy for Corner Nodes in FETI-DP Method
FETI-DP Method
Corner Nodes
Definition of Corner NodesSimple definition in case of regular mesh
Problem with definition of corner nodes in case of non-regularmeshes which are decomposed by mesh decomposer (e.g.METIS, http://glaros.dtc.umn.edu/gkhome/views/metis).
Minimal number of corner nodes due to nonsingular matrix ofsubdomainsTheoretically the number of corner node = the number of allnodes on boundaries
Selection Strategy for Corner Nodes in FETI-DP Method
FETI-DP Method
Corner Nodes
Definition of Corner Nodes
Simple definition in case of regular mesh
Problem with definition of corner nodes in case of non-regularmeshes which are decomposed by mesh decomposer (e.g.METIS, http://glaros.dtc.umn.edu/gkhome/views/metis).
Minimal number of corner nodes due to nonsingular matrix ofsubdomains
Theoretically the number of corner node = the number of allnodes on boundaries
Selection Strategy for Corner Nodes in FETI-DP Method
Algorithm for Corner Node Selection in 2D
Algorithm for Corner Node Selection in 2DThe First Level
The first level based on “nodal multiplicity” (the number of subdo-mains which belongs to node)
Node with node multiplicity > 2→ corner nodeNode with node multiplicity = 2 and only with one neighborwith node multiplicity = 2→ corner node.
x
y
Selection Strategy for Corner Nodes in FETI-DP Method
Algorithm for Corner Node Selection in 2D
Algorithm for Corner Node Selection in 2DThe First Level
The first level based on “nodal multiplicity” (the number of subdo-mains which belongs to node)
Node with node multiplicity > 2→ corner nodeNode with node multiplicity = 2 and only with one neighborwith node multiplicity = 2→ corner node.
x
y
Selection Strategy for Corner Nodes in FETI-DP Method
Algorithm for Corner Node Selection in 2D
Algorithm for Corner Node Selection in 2DThe First Level
The first level based on “nodal multiplicity” (the number of subdo-mains which belongs to node)
Node with node multiplicity > 2→ corner nodeNode with node multiplicity = 2 and only with one neighborwith node multiplicity = 2→ corner node.
x
y
Selection Strategy for Corner Nodes in FETI-DP Method
Algorithm for Corner Node Selection in 2D
Algorithm for Corner Node Selection in 2DThe First Level
The first level based on “nodal multiplicity” (the number of subdo-mains which belongs to node)
Node with node multiplicity > 2→ corner nodeNode with node multiplicity = 2 and only with one neighborwith node multiplicity = 2→ corner node.
x
y
Selection Strategy for Corner Nodes in FETI-DP Method
Algorithm for Corner Node Selection in 2D
Algorithm for Corner Node Selection in 2DThe Second Level
In the second level can be added another corner nodes with help of”boundary curves”. Corner 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 curveRandom position
x
y
Selection Strategy for Corner Nodes in FETI-DP Method
Numerical Tests
Numerical testsNon-regular Domain
Storey
Selection Strategy for Corner Nodes in FETI-DP Method
Numerical Tests
StoreyDecomposition of Domain
Selection Strategy for Corner Nodes in FETI-DP Method
Numerical Tests
StoreyResults of Tests - The Number of Iterations with Respect to the Number of Corner Nodes
Selection Strategy for Corner Nodes in FETI-DP Method
Numerical Tests
StoreyResults of Tests - Time of Condensation with Respect to the Number of Corner Nodes
Selection Strategy for Corner Nodes in FETI-DP Method
Numerical Tests
StoreyResults of Tests - Total Time of the Solution with Respect to the Number of Corner Nodes
Selection Strategy for Corner Nodes in FETI-DP Method
Conclusions and Future Works
Conclusions and Future Works
The algorithm for selection of corner nodes for arbitrary 2Dmesh has been developed
Increasing of the number of the corner nodes leads to decreasingof the number of iterations in coarse problem and its time of thesolution
Big number of corner nodes leads to prolongation of the wholetime of the solution
Optimization of the algorithm
Developing of the algorithm for the selection of corner nodes in3D
Selection Strategy for Corner Nodes in FETI-DP Method
Acknowledgement
Acknowledgement
Thank you for your attention.
Financial support for this work was provided by project number103/07/1455 of the Czech Science Foundation. The financial supportis gratefully acknowledged.
