1.
Selection of ﬁxing nodes for FETI-DP method in 3D
Selection of ﬁxing nodes for FETI-DP method in 3D
Jaroslav Broˇ , Jaroslav Kruis
z
Department of Mechanics
Faculty of Civil Engineering
Czech Technical University in Prague
The Seventh International Conference on Engineering
Computational Technology
Valencia, Spain
16 September 2010
2.
Selection of ﬁxing nodes for FETI-DP method in 3D
Contents
Contents
1 FETI-DP Method
Short Introduction
Fixing nodes
Algorithm for Selection of Fixing Nodes in 3D
2 Numerical Tests
Boxwise partitioning - Dam
Stripwise partitioning - Bridge
3 Conclusions
3.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Short Introduction
FETI-DP Method
Short Introduction
FETI-DP (Finite Element Tearing and Interconnecting
Dual-Primal) method was published by C. Farhat et all
Non-overlapping domain decomposition method
Based on the combination of the original FETI method and the
Schur complement method
Unknowns in the problem are splitted into three categories
1 ﬁxing unknowns
2 remaining interface unknowns
3 internal unknowns
4.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Short Introduction
Continuity Conditions
Continuity conditions are enforced
by Lagrange multipliers deﬁned on remaining interface
unknowns
directly on ﬁxing unknowns.
5.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Short Introduction
Coarse problem is obtained after elimination of internal and
remaining interface unknowns.
Coarse Problem
FIrr FIrf λ dr
= (1)
FTrf
I −K∗ ff uf −f∗
f
Coarse Problems - Lagrange Multipliers
−1 −1
FIrr + FIrf K∗ FTrf λ = dr − FIrf K∗ f∗
ff I ff f (2)
Solving of Reduced Coarse Problem
Matrix of system of equations of coarse problem is symmetric and
positive deﬁnite → solving by conjugate gradient method.
6.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Fixing nodes
Deﬁnition of Fixing Nodes in Original Farhat’s Article
D1 Cross-points - the nodes belonging to more than two
subdomains
D2 The set of nodes located at the beginning and end of
each edge of each subdomain
D2
Ω2 Ω4
D1
D2 D2
Ω1 Ω3
D2
7.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Fixing nodes
Importance of Proper Selection of Fixing Nodes
Non-singularity of subdomain matrix Ks
Non-singularity of matrix of coarse problem
FIrr FIrf
FTrf
I −K∗ ff
Suitable selected ﬁxing nodes ⇒ relatively small condition
number κ of matrix of coarse problem ⇒ speed of convergence
of solution of coarse problem
Theoretically, all interface nodes can be selected as ﬁxing nodes
→ Schur complement method
8.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Fixing nodes
Difﬁculties with Choice of Fixing Nodes
Deﬁnition 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 of
graphs
Problem with interpretation of deﬁnition in case of meshes
splitted by mesh decomposer
Limited data - only mesh of ﬁnite elements
9.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Algorithm for Selection of Fixing Nodes in 3D
Deﬁnition of Graphs
Decomposed cube domain into
8 subdomains Boundary Graph B
10.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Algorithm for Selection of Fixing Nodes in 3D
Deﬁnition of Graphs
Boundary Curve Graph C
Boundary Surface Graph P
11.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Algorithm for Selection of Fixing Nodes in 3D
Deﬁnition of Minimal Number Algorithm
The Deﬁnition of the Basic Fixing
Node for Three-Dimensional Mesh
Let be v vertex of the graph
v ∈ V (C ) and let be dC (v) the
degree of vertex v. If
dC (v) = 1 or dC (v) > 2 (3)
then the vertex v is called ﬁxing
node.
Selected ﬁxing nodes
12.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Algorithm for Selection of Fixing Nodes in 3D
Geometrical Conditions
Following geometrical condition
are checked
distance between two ﬁxing
nodes
angle between two ﬁxing
nodes
area of triangle among ﬁxing
nodes
13.
Selection of ﬁxing nodes for FETI-DP method in 3D
FETI-DP Method
Algorithm for Selection of Fixing Nodes in 3D
Extended Algorithm
Additional ﬁxing nodes can be chosen with the help of
boundary curve subgraphs Cj
boundary surface subgraphs Pj
combination of above possibilities
14.
Selection of ﬁxing nodes for FETI-DP method in 3D
Numerical Tests
Boxwise partitioning - Dam
Dam
Linear elasticity problem
Tetrahedral elements with 3
DOFs
METIS partitionining into 10
subdomains
Boxwise partitioning
15.
Selection of ﬁxing nodes for FETI-DP method in 3D
Numerical Tests
Boxwise partitioning - Dam
Dam - obtained results
160
minimal
155 curve - centroid
curve - each n-th
150 curve random
Number of iterations in coarse problem
145 surface random
surface centroid
140 comb - centroid+centroid
135
130
125
120
115
110
105
100
95
90
85
80
75
70
0 25 50 75 100 125 150 175 200 225 250 275 300
Number of fixing nodes in whole problem
16.
Selection of ﬁxing nodes for FETI-DP method in 3D
Numerical Tests
Boxwise partitioning - Dam
Dam - obtained results
1140
minimal
curve - centroid
1130 curve - each n-th
curve random
surface random
1120 surface centroid
comb - centroid+centroid
Time of whole solution [s]
1110
1100
1090
1080
1070
1060
1050
1040
0 25 50 75 100 125 150 175 200 225 250 275 300
Number of fixing nodes in whole problem
17.
Selection of ﬁxing nodes for FETI-DP method in 3D
Numerical Tests
Stripwise partitioning - Bridge
Bridge
Linear elasticity problem
Hexahedral elements with 3 DOFs
METIS partitionining into 10 subdomains
Stripwise partitioning
18.
Selection of ﬁxing nodes for FETI-DP method in 3D
Numerical Tests
Stripwise partitioning - Bridge
Bridge - obtained results
400
minimal
375 surface n-th ring
surface centroid
350 surface random
Number of iterations in coarse problem
surface max ring
325 surface max triangle
300
275
250
225
200
175
150
125
100
75
50
25
0
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Number of fixing nodes in whole problem
19.
Selection of ﬁxing nodes for FETI-DP method in 3D
Numerical Tests
Stripwise partitioning - Bridge
Bridge - obtained results
250
minimal
245 surface n-th ring
240 surface centroid
surface random
235 surface max ring
230 surface max triangle
225
Time of whole solution [s]
220
215
210
205
200
195
190
185
180
175
170
165
160
155
150
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Number of fixing nodes in whole problem
20.
Selection of ﬁxing nodes for FETI-DP method in 3D
Conclusions
Conclusions
Algorithm for selection of ﬁxing nodes for 3D meshes was
developed
Following behaviour was observed
The increasing of the number of ﬁxing nodes leads to the
decreasing of the number of iterations in the coarse problem
The large number of ﬁxing nodes leads to the prolongation of the
whole time of solution
Addition of a few further nodes is the best solution
Further work can be an automatic choice of optimal number of
ﬁxing nodes
21.
Selection of ﬁxing nodes for FETI-DP method in 3D
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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