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# Conference ECT 2010

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### Conference ECT 2010

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.