1 / 33

Percolation

Percolation. Simulating percolation models Guillermo Amaral Caesar Systems - Argentina. A virtual lab. Percolation deals with…. Propagation of diseases. Propagation of fire. Oil & gas in reservoirs. Gelation & Polymerization. The problem.

samuel-mejia
Download Presentation

Percolation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Percolation Simulating percolation models Guillermo Amaral CaesarSystems - Argentina

  2. Guillermo Amaral

  3. Guillermo Amaral

  4. Guillermo Amaral

  5. A virtual lab Guillermo Amaral

  6. Percolation deals with…

  7. Propagation of diseases Guillermo Amaral

  8. Propagation of fire Guillermo Amaral

  9. Oil & gas in reservoirs Guillermo Amaral

  10. Gelation & Polymerization Guillermo Amaral

  11. The problem

  12. Original problem (Broadbent - Hammersley, 1957) • What is the probability that the water reaches the center of the rock? Guillermo Amaral

  13. The simulation

  14. The mathematical model

  15. The simplest model vϵℤ2 u v u at distance 1 fromv v Open pathfrom u tov u Open clusterfromv e Percolatingcluster v P(e“open”) = p P(e“close”) = 1 - p v Guillermo Amaral

  16. Model types Structure Direction Dimensions Elementbeing open/close 3-D Square Bow-tie p1 p Bond p p2 Hexagonal Kagomé Isotropic Site Anisotropic 2-D n-D… Other… Both… Guillermo Amaral

  17. Phase transition: Critical probability • θ(p) = Pp(a givenvertexbelongsto a percolatingcluster) • θ(p) = 0 si p = 0 • θ(p) = 1 si p = 1 • θ(p) ismonotonically non-decrescent • ThereispcЄ[0, 1] suchthat: • θ(p) = 0 if p < pc • θ(p) > 0 if p > pc • Whenis p = pc? θ(p) 1 pc? p pc 0 1 Guillermo Amaral

  18. Known critical probabilities Guillermo Amaral

  19. Why simulation? • Problems very hard to prove analytically • Square bond model critical probability = 0.5 • Clues for a formal proof • Application to practical cases Guillermo Amaral

  20. Areas of interest • Large-graph representation • Pseudo-random numbers • Graph exploration • Analysis of connected components Guillermo Amaral

  21. Simulation Simulation variables Guillermo Amaral

  22. 2. Generate a “random” configuration • 1. Build the model Simulation process • 3. Search for percolating clusters • 4. Collect results of output variables Guillermo Amaral

  23. The simulator

  24. My experience…

  25. Programming with a solution in mind leads to answers, but modeling the problem also raises new questions Guillermo Amaral

  26. Questions

  27. A case of study

  28. Scope analysis pH x0 • (x0↔v) • (x0↔v’ ) pv v • v = (x, y) • v’ = (y, x) IfpH < pv, P(x0↔v) <P(x0↔v’)? v’ Guillermo Amaral

  29. Scope analysis visualization Mirrorcoloring Scalecoloring > = Guillermo Amaral

  30. Object design

  31. Objects (1) PercolationModel OpenPolicy BondPercolation SitePercolation BondOpenPolicy SiteOpenPolicy LatticeGraph IsotropicPolicy AnisotropicPolicy Lattice SquareVerticalHorizontal … GraphPattern SquareLattice CubicLattice SubgraphPattern NodeBasedPattern AdjacencySolver Square1KVertical1Horizontal Square1Vertical1KHorizontal … PatternAdjacencySolver MatrixAdjacencySolver Guillermo Amaral Caesar

  32. Objects (2) GraphAlgorithm AdjacencyMatrix PSBitMatix PSSparseFloatMatrix QuickUnionFind GraphSearchAlgorithm PSFloatMatrix PSSparseMatrix WeightedQuickUnionFind BreathFirstSearch DepthFirstSearch WQUFPC ModelSampler CriticalRangeFinder NodeScopeAnalizer … ModelEvaluator CompositeSampler ModelHistory VariableWalker UnionFindAnalizer … Guillermo Amaral Caesar

  33. Objects (3) ChartObject ChartAxis XYSerieMarker Chart RangeMark ChartSerie DrawerTool PieChar XYChart NodeLocator XYChartPointLocator ClusterPainter EdgeLocator PSDrawer ChartDrawer CriticalRangeDrawer SquareLatticeGraphDrawer BondPercolationGraphDrawer PieChartDrawer XYChartDrawer SitePercolationGraphDrawer Guillermo Amaral Caesar

More Related