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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.

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Percolation

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  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

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