Transcript of "Seminary of numerical analysis 2010"
1.
Selection strategy for ﬁxing nodes in FETI-DP method
Selection strategy for ﬁxing nodes in FETI-DP
method
Jaroslav Brož1 , Jaroslav Kruis
Katedra mechaniky
Fakulta stavební
ˇ
CVUT v Praze
Semináˇ numerické analýzy
r
18. leden - 22. leden 2010
Zámek Nové Hrady
1
2.
Selection strategy for ﬁxing nodes in FETI-DP method
Outline
Outline
1 FETI-DP Method
2 Algorithm for Fixing Node Selection in 2D
3 Numerical Tests of Algorithm for 2D
4 Algorithm for Fixing Node Selection in 3D
5 Numerical Tests of Algorithm for 3D
6 Conclusions and Future Works
3.
Selection strategy for ﬁxing nodes in FETI-DP method
FETI-DP Method
Outline
1 FETI-DP Method
Introduction
Coarse Problem
Fixing Nodes
2 Algorithm for Fixing Node Selection in 2D
3 Numerical Tests of Algorithm for 2D
4 Algorithm for Fixing Node Selection in 3D
5 Numerical Tests of Algorithm for 3D
6 Conclusions and Future Works
4.
Selection strategy for ﬁxing nodes in FETI-DP method
FETI-DP Method
Introduction
FETI-DP Method
Introduction
One of non-overlapping domain decomposition methods
Method was published by prof. Farhat and his collaborators in
the article: Farhat, C., Lesoinne, M., LeTallec, P., Pierson, K. &
Rixen, D. (2001): FETI-DP A dual-primal uniﬁed FETI
method-part I: Faster alternative to the two-level FETI
method. International Journal for Numerical Methods in
Engineering, Vol. 50, 1523–1544.
Method was developed due to problems with singulars matrix in
original FETI Method
5.
Selection strategy for ﬁxing nodes in FETI-DP method
FETI-DP Method
Introduction
FETI-DP Method
Introduction
Unknowns are divided into two groups - interior unknowns and
interface unknowns among subdomains
Continuity conditions are ensured by Lagrange multipliers and
ﬁxing nodes
Interior unknowns are eliminated and a coarse problem are
obtained
6.
Selection strategy for ﬁxing nodes in FETI-DP method
FETI-DP Method
Coarse Problem
Coarse Problem
−S[cc] F[cr] d[c] −s
= . (1)
F[rc] F[rr] λ g
where
d[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.
−1
d[c] = − S[cc] −s − F[cr] λ . (2)
−1 −1
F[rr] + F[rc] S[cc] F[cr] λ = g − F[rc] S[cc] s. (3)
7.
Selection strategy for ﬁxing nodes in FETI-DP method
FETI-DP Method
Fixing Nodes
Deﬁnition of Fixing Nodes
Simple deﬁnition in the case of a regular mesh
y
4
1 3 5
2
x
8.
Selection strategy for ﬁxing nodes in FETI-DP method
FETI-DP Method
Fixing Nodes
Deﬁnition of Fixing Nodes
Problem with deﬁnition of ﬁxing nodes in the case of non-regular
meshes 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 of
subdomains
Theoretically the number of ﬁxing node = the number of all
nodes on boundaries
9.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 2D
Outline
1 FETI-DP Method
Introduction
Coarse Problem
Fixing Nodes
2 Algorithm for Fixing Node Selection in 2D
3 Numerical Tests of Algorithm for 2D
4 Algorithm for Fixing Node Selection in 3D
5 Numerical Tests of Algorithm for 3D
6 Conclusions and Future Works
10.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 2D
Algorithm for Fixing Node Selection in 2D
Deﬁnition of Nodal Multiplicity
Nodal multiplicity - the number of subdomains which belongs to node
Deﬁnition of Fixing Nodes
Node with node multiplicity > 2 → ﬁxing node
Node with node multiplicity = 2 and only with one neighbor
with node multiplicity = 2 → ﬁxing node.
y
x
11.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 2D
Algorithm for Fixing Node Selection in 2D
Deﬁnition of Nodal Multiplicity
Nodal multiplicity - the number of subdomains which belongs to node
Deﬁnition of Fixing Nodes
Node with node multiplicity > 2 → ﬁxing node
Node with node multiplicity = 2 and only with one neighbor
with node multiplicity = 2 → ﬁxing node.
y
x
12.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 2D
Algorithm for Fixing Node Selection in 2D
Deﬁnition of Nodal Multiplicity
Nodal multiplicity - the number of subdomains which belongs to node
Deﬁnition of Fixing Nodes
Node with node multiplicity > 2 → ﬁxing node
Node with node multiplicity = 2 and only with one neighbor
with node multiplicity = 2 → ﬁxing node.
y
x
13.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 2D
Algorithm for Fixing Node Selection in 2D
Deﬁnition of Nodal Multiplicity
Nodal multiplicity - the number of subdomains which belongs to node
Deﬁnition of Fixing Nodes
Node with node multiplicity > 2 → ﬁxing node
Node with node multiplicity = 2 and only with one neighbor
with node multiplicity = 2 → ﬁxing node.
y
x
14.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 2D
Algorithm for Fixing Node Selection in 2D
Deﬁnition of Boundary Curves
Boundary curve connect boundary nodes between two ﬁxing nodes.
Fixing nodes can be added into:
Centroid of boundary curve
Each n-th member of the boundary curve
Each n-th end of the part of the boundary curve
“Integral points” of the boundary curve
Random position y
x
15.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 2D
Outline
1 FETI-DP Method
Introduction
Coarse Problem
Fixing Nodes
2 Algorithm for Fixing Node Selection in 2D
3 Numerical Tests of Algorithm for 2D
4 Algorithm for Fixing Node Selection in 3D
5 Numerical Tests of Algorithm for 3D
6 Conclusions and Future Works
16.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 2D
Numerical tests
Irregular Domain - Slope
NS NN NE NN-SUB NE-SUB NDOF-SUB
4 105182 208840 26448 52210 52846
4 186577 371124 46847 92781 93627
9 105182 208840 11834 23204 23647
9 186577 371124 20923 41236 41816
17.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 2D
Slope
Results of Tests - The Number of Iterations with Respect to the Number of Fixing Nodes
18.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 2D
Slope
Results of Tests - Time of Condensation with Respect to the Number of Fixing Nodes
19.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 2D
Slope
Results of Tests - Total Time of the Solution with Respect to the Number of Fixing Nodes
20.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 3D
Outline
1 FETI-DP Method
Introduction
Coarse Problem
Fixing Nodes
2 Algorithm for Fixing Node Selection in 2D
3 Numerical Tests of Algorithm for 2D
4 Algorithm for Fixing Node Selection in 3D
5 Numerical Tests of Algorithm for 3D
6 Conclusions and Future Works
21.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 3D
Algorithm for Fixing Node Selection in 3D
Deﬁnition of Edges and Sufraces
Edge - deﬁned by boundary nodes which belongs to more than two
subdomains
Surface - deﬁned by boundary nodes which belongs to exactly two
subdomains
22.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 3D
Algorithm for Fixing Node Selection in 3D
Deﬁnition of Edges and Sufraces
Edge - deﬁned by boundary nodes which belongs to more than two
subdomains
Surface - deﬁned by boundary nodes which belongs to exactly two
subdomains
23.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 3D
Algorithm for Fixing Node Selection in 3D
Deﬁnition of Fixing nodes
node with maximal nodal multiplicity → ﬁxing node
end of edge → ﬁxing node
Deﬁnition of Boundary Curves
Boundary curve → edge between two ﬁxing nodes
24.
Selection strategy for ﬁxing nodes in FETI-DP method
Algorithm for Fixing Node Selection in 3D
Algorithm for Fixing Node Selection in 3D
Next Step - Under Developement
Deﬁnition of Boundary Surface
Boundary surface - created by boundary nodes with nodal multiplicity
equal two
25.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 3D
Outline
1 FETI-DP Method
Introduction
Coarse Problem
Fixing Nodes
2 Algorithm for Fixing Node Selection in 2D
3 Numerical Tests of Algorithm for 2D
4 Algorithm for Fixing Node Selection in 3D
5 Numerical Tests of Algorithm for 3D
6 Conclusions and Future Works
26.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 3D
Numerical tests
Regular Domain - Cube
NS NN NE NN-SUB NE-SUB NDOF-SUB
8 29791 27000 4096 3375 11904
8 68921 64000 9261 8000 27121
27.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 3D
Cube - 27000 elements
Results of Tests - The Number of Iterations with Respect to the Number of Fixing Nodes
28.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 3D
Cube - 27000 elements
Results of Tests - Time of Condensation with Respect to the Number of Fixing Nodes
29.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 3D
Cube - 27000 elements
Results of Tests - Total Time of the Solution with Respect to the Number of Fixing Nodes
30.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 3D
Cube - 64000 elements
Results of Tests - The Number of Iterations with Respect to the Number of Fixing Nodes
31.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 3D
Cube - 64000 elements
Results of Tests - Time of Condensation with Respect to the Number of Fixing Nodes
32.
Selection strategy for ﬁxing nodes in FETI-DP method
Numerical Tests of Algorithm for 3D
Cube - 64000 elements
Results of Tests - Total Time of the Solution with Respect to the Number of Fixing Nodes
33.
Selection strategy for ﬁxing nodes in FETI-DP method
Conclusions and Future Works
Outline
1 FETI-DP Method
Introduction
Coarse Problem
Fixing Nodes
2 Algorithm for Fixing Node Selection in 2D
3 Numerical Tests of Algorithm for 2D
4 Algorithm for Fixing Node Selection in 3D
5 Numerical Tests of Algorithm for 3D
6 Conclusions and Future Works
34.
Selection strategy for ﬁxing nodes in FETI-DP method
Conclusions and Future Works
Conclusions and Future Works
The algorithm for selection of ﬁxing nodes for arbitrary 2D mesh
has been developed
Increasing of the number of the ﬁxing nodes leads to decreasing
of the number of iterations in coarse problem and its time of the
solution
Big number of ﬁxing nodes leads to prolongation of the whole
time of the solution
Developing of the algorithm for the selection of ﬁxing nodes for
regular 3D mesh
Optimization of the algorithm
35.
Selection strategy for ﬁxing nodes in FETI-DP method
Acknowledgement
Acknowledgement
Thank you for your attention.
Financial support for this work was provided by project number
103/09/H078 of the Czech Science Foundation. The ﬁnancial support
is gratefully acknowledged.
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