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Percolation Properties of Triangles with Variable Aspect Ratios

Percolation Properties of Triangles with Variable Aspect Ratios. Gabriela Calinao Correa University of Massachusetts at Amherst Department of Electrical & Computer Engineering Dr. Alan Feinerman University of Illinois at Chicago Department of Electrical & Computer Engineering. Outline.

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Percolation Properties of Triangles with Variable Aspect Ratios

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  1. Percolation Properties of Triangles with Variable Aspect Ratios Gabriela Calinao Correa University of Massachusetts at Amherst Department of Electrical & Computer Engineering Dr. Alan Feinerman University of Illinois at Chicago Department of Electrical & Computer Engineering

  2. Outline • Percolation Overview • Methods & Analysis • Matlab • AutoCAD • Data • Conclusions • Acknowledgements & Citations

  3. Percolation Theory Overview Percolation:study of random networks & their nature of connectivity Percolation Threshold ( pc ): Below pc no giant connected component exists Above pc a giant connected component exists

  4. Methods & Analysis MATLAB pc AutoCAD Data CO2 LASER Resistance Measured Percolated Aluminum Mylar

  5. Matlab Define Parameters Values • Kerf • Node Size Percolation Area • Location in AutoCAD file • Dimensions Triangle • Aspect Ratio • Area Generate Triangles Location • Pseudorandom • Within Boundary Orientation • Pseudorandom Angle • Rotation Matrix AutoCAD • CAD Commands • Define Shapes Determine Area Loss Percent • Area Left • Percolated Binary Matrix • 1 - indicates Node removed • 0 - indicates untouched Node Threshold • Label Clusters • Find if Path Exists

  6. Logic for Triangle Generation Draw Whole Triangle if intersection DNE Draw Whole Triangle if intersection exists edge on box if i in box if i+1in box add 1st intersect add 1st intersect add point i+1 add point i i+1 intersection point Yes No two points in box i add point i+2 add 2nd intersect i+2 if triangle on corner add corner point

  7. AutoCAD • Loads file made by Matlab • Defines Laser Settings • Creates Parts • Guides • Acrylic Fasteners

  8. Sandblasted Aluminum Base Aluminum Coated Mylar Film Paper

  9. Exponential Relationship Percolation Properties of Random Ellipses B. Xia and M. Thorpe, Physical Review A 38, 2650 (1988). pc = exp(-πabnc ) pc is the percolation threshold nc is the hole density per unit volume at pc b/a is the aspect ratio of an ellipse x = Aspect Ratio f(x) = Percolation Threshold f(x) = a*exp(b*x) + c*exp(d*x) Coefficients (with 95% confidence bounds): a = 0.3887 ( 0.2579, 0.5195) b = -3.637 (-5.2340, -2.0390) c = 0.4945 ( 0.3562, 0.6328) d = 0.08325 (-0.1832, 0.3497)

  10. Conclusions • Generated Data from • Matlab • Experimental Setup • Exponential Relationship • Triangular relationship ~ Elliptical relationship • Additional Experiments • Refine Data • Percolate over Larger Area • Further Geometric Exploration: • Double Triangles • e.g. equilateral triangle to equilateral rhombus

  11. EEC-NSF Grant # 1062943 Dr. Takoudis, Dr. Jursich, & the REU staff Tatjana Dankovic, Prateek Gupta, Michael Walsh The University of Illinois at Chicago Acknowledgements • Nils Burglund • B. Xia and M. Thorpe, Physical Review A 38, 2650 (1988). Citations

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