Computing Curvature Using Amorphous Computing

Omari Carter-Thorpe

August 5th, 1998

Research Science Institute

Mentor: Radhika Nagpal

Amorphous Computing

• Amorphous Computing
  • Large number of tiny, densely packed, identical processors that communicate locally

• Applications
  • Sensors and Actuators
  • Randomly distributed on a structure
  • Examples:
    • Detecting loads in bridges,
    • Reducing stress in airplane wings

Contributions

• Goal: Self Determine Structure
  • Using Coordinates, Curvature or Edges

• Why: Monitor changes and Affect Structure

• Proposed, simulated and analyzed a method for finding curvature in an amorphous computer

Ideal: Finding Curvature

• Ideal Environment:
  • Surface Area A = 2pR2 [ 1 - cos (h/R)]
  • cos(q) ? 1- q2/2! + q4/4!
  • R = sqrt( ph4 / 12(ph2-A))

Amorphous Computing: Finding Curvature

• Processors communicate within distance r

• Computing area of cap:
  • Cap Area = Number of processors / D

• A cap of arclength h by growing a tree of depth h

r

Constructing a Tree

• Source processor finds neighbors

• Neighbors find neighbors not in tree.

• Store parent and child

Sources of Error

• Variation in density

• Maximum communication distance < r.

• Kleinrock Formula

Simulation Design

• Design
  • Find Radius of Sphere
  • Between 5000 and 44000 processors
  • Spheres of different radii
  • Two densities: 15 and 30 neighbors

• Data
  • Measured Area
  • Measured Area with Kleinrock corrections

PPT Slide

Simulation Results: Computing Area

• Approximation adds no error
• Underestimate Error
• Kleinrock corrects 1 error in area

PPT Slide

Simulation Results: Computing Radius

• Constant percentage of error in area
• Percentage error in radius increases rapidly
• Kleinrock too sensitive to produce radius

Conclusion

• Implemented a technique for finding Curvature using surface Area

• Results:
  • Can compute Area within 1-2% error
  • But radius formula is very sensitive to errors in Area and h, when radius R is large

• Future Work:
  • Less sensitive radius formula
    • New formula using sin