Calculus and NASA

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Calculus and NASA

• Michael Bloem
• February 15, 2008
• Calculus Field Trip Presentation

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Outline

• NASAs (many!) uses of calculus
• Space
• Airfoil design
• My use of calculus at NASA
• Optimization for air traffic management

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Airfoil Design

• FoilSim
• Pressure is change in force per area
• On a wing, the lift is the difference between the
  forces acting on the bottom and top of the wing

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Airfoil Design Computing Lift
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Airfoil Design Computing Lift

• FoilSim says pressure 7731 lb
• How could I improve my estimate?

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Traffic Flow Management

• Planning of air traffic to avoid exceeding
  airport and airspace capacity, and effective use
  of available capacity
• Cost of Delay to airlines in 2005 5.9 Billion
  (Air Transportation Association Estimate)

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3D Visualization of Air Traffic
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Air Traffic Flow Models
Lagrangian
Eulerian
Keep track of each plane
Keep track of the number of planes in different
areas
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Aggregate Flow Model
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Optimization with the Aggregate Flow Model

• Minimize quadratic cost on the difference
  between the scheduled and actual arrivals and
  departures
• Subject to
• Follow system dynamics equations
• Do not have more cumulative arrivals or
  departures than scheduled
• Count of aircraft in each center stays below a
  time-varying maximum
• Cumulative arrivals and departures are
  non-decreasing

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Optimization with the Aggregate Flow Model
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How do we optimize?

• Consider a simple case with one variable
• Check convexity
• Set derivative 0

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Another way to optimize?

• Newtons Method
• Find where derivative 0
• Iteration
• Why?
• Works well on a computer
• Works well on big problems (many variables)

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Picture for Newtons Method
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Newtons Method
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Constrained Optimization

• What if we have bounds on x?
• Optimality condition for a convex function and a
  convex constraint set

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Example of Constrained Optimization

• Constrained optimization problem?
• Is it convex?
• Try our condition

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Example of Constrained Optimization (continued)
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Optimal Traffic Flow Management
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Conclusions

• NASA uses calculus a lot because calculus helps
  solve real problems

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Websites

• Altair Lunar Lander
• CFD at Ames
• FoilSim
• Aviation Systems Division