Seminary of numerical analysis 2010
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Seminary of numerical analysis 2010

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Zamek Nove Hrady

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Seminary of numerical analysis 2010 Seminary of numerical analysis 2010 Presentation Transcript

  • 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 Seminᡠnumerické analýzy r 18. leden - 22. leden 2010 Zámek Nové Hrady 1
  • 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
  • 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
  • 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
  • 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
  • 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)
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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.