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

http://youtube.com/v/cI_B9iqsB9E

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