Percolation Properties of
Triangles with Variable Aspect Ratios
Gabriela Calinao CorreaUniversity of Massachusetts at Amherst
Department of Electrical & Computer Engineering
Dr. Alan FeinermanUniversity of Illinois at Chicago
Department of Electrical & Computer Engineering
Outline• Percolation Overview• Methods & Analysis• Matlab• AutoCAD• Data• Conclusions• Acknowledgements & Citations
Percolation Theory OverviewPercolation: study of random networks
& their nature of connectivity
Percolation Threshold ( pc ):Below pc no giant connected component existsAbove pc a giant connected component exists
Methods & Analysis
DataPercolatedAluminum
Mylar
CO2 LASER
AutoCAD
MATLAB
ResistanceMeasured
pc
Matlab
Define Parameters
Percolation Area
• Location in AutoCAD file• Dimensions
Values• Kerf• Node
Size
Triangle• Aspect Ratio• Area
Location• Pseudorandom• Within
Boundary
Orientation
• Pseudorandom Angle
• Rotation Matrix
Generate Triangles
AutoCAD
• CAD Commands
• Define Shapes
DetermineArea Loss
Binary Matrix
• 1 - indicates Node removed
• 0 - indicates untouched Node
Threshold
• Label Clusters• Find if Path
Exists
Percent
• Area Left• Percolated
Logic for Triangle Generation
if intersection exists
if i in box
edge on box
add 1st intersect
add point i
if i+1 in box
add point i+1
if intersection DNE
add 1st intersect
two points in boxYes No
add point i+2 add 2nd intersect
if triangle on corner
add corner point
Draw Whole Triangle
Draw Whole Triangle
i+1
i
i+2
intersection point
AutoCAD
• Loads file made by Matlab• Defines Laser Settings• Creates Parts
o Guideso Acrylic Fasteners
Aluminum Coated Mylar Film
Paper
Sandblasted Aluminum Base
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 thresholdnc is the hole density per unit
volume at pc
b/a is the aspect ratio of an ellipse
x = Aspect Ratiof(x) = Percolation Thresholdf(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)
Conclusions• Generated Data from
o Matlabo Experimental Setup
• Exponential Relationshipo Triangular relationship ~ Elliptical
relationship• Additional Experiments
o Refine Datao Percolate over Larger Area
• Further Geometric Exploration:o Double Triangles
e.g. equilateral triangle to equilateral rhombus
• Nils Burglund• B. Xia and M. Thorpe, Physical Review A 38,
2650 (1988).
• EEC-NSF Grant # 1062943• Dr. Takoudis, Dr. Jursich, & the REU staff• Tatjana Dankovic, Prateek Gupta, Michael Walsh• The University of Illinois at Chicago
Citations
Acknowledgements