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Calculus and NASA

Calculus and NASA. Michael Bloem February 15, 2008 Calculus Field Trip Presentation. Outline. NASA’s (many!) uses of calculus Space Airfoil design My use of calculus at NASA Optimization for air traffic management. If then. Airfoil Design. FoilSim

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Calculus and NASA

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  1. Calculus and NASA Michael Bloem February 15, 2008 Calculus Field Trip Presentation

  2. Outline • NASA’s (many!) uses of calculus • Space • Airfoil design • My use of calculus at NASA • Optimization for air traffic management

  3. If then

  4. 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

  5. Airfoil Design: Computing Lift

  6. Airfoil Design: Computing Lift • FoilSim says pressure = 7731 lb • How could I improve my estimate?

  7. 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)

  8. 3D Visualization of Air Traffic

  9. Air Traffic Flow Models Lagrangian Eulerian Keep track of each plane Keep track of the number of planes in different areas

  10. Departures from Center i Outflow to Center j Region i Arrivals into Center i Inflow from Center j Aggregate Flow Model

  11. 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

  12. Optimization with the Aggregate Flow Model

  13. How do we optimize? • Consider a simple case with one variable • Check convexity: • Set derivative = 0:

  14. Another way to optimize? • Newton’s Method • Find where derivative = 0 • Iteration: • Why? • Works well on a computer • Works well on big problems (many variables)

  15. Picture for Newton’s Method

  16. Newton’s Method

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

  18. Example of Constrained Optimization • Constrained optimization problem? • Is it convex? • Try our condition

  19. Example of Constrained Optimization (continued)

  20. Optimal Traffic Flow Management

  21. Conclusions • NASA uses calculus a lot because calculus helps solve real problems

  22. Websites • Altair Lunar Lander • CFD at Ames • FoilSim • Aviation Systems Division

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