Computing Curvature Using Amorphous Computing

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Computing Curvature Using Amorphous Computing

• Omari Carter-Thorpe
• August 5th, 1998
• Research Science Institute
• Mentor Radhika Nagpal

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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

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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

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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))

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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
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Constructing a Tree

• Source processor finds neighbors
• Neighbors find neighbors not in tree.
• Store parent and child

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Sources of Error

• Variation in density
• Maximum communication distance lt r.
• Kleinrock Formula

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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

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Simulation Results Computing Area

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

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Simulation Results Computing Radius

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

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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