Calculus and NASA Michael Bloem February 15, 2008 Calculus Field Trip Presentation 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  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 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  FoilSim 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

Airfoil Design: Computing Lift

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

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)  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)

3D Visualization of Air Traffic

Air Traffic Flow Models LagrangianEulerian Keep track of each planeKeep track of the number of planes in different areas

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

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

Optimization with the Aggregate Flow Model

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

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)  Newton’s Method –Find where derivative = 0 –Iteration: –Why?  Works well on a computer  Works well on big problems (many variables)

Picture for Newton’s Method

Newton’s Method

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

Example of Constrained Optimization  Constrained optimization problem?  Is it convex?  Try our condition  Constrained optimization problem?  Is it convex?  Try our condition

Example of Constrained Optimization (continued)

Optimal Traffic Flow Management

Conclusions  NASA uses calculus a lot because calculus helps solve real problems

Websites  Altair Lunar Lander Altair Lunar Lander  CFD at Ames CFD at Ames  FoilSim FoilSim  Aviation Systems Division Aviation Systems Division  Altair Lunar Lander Altair Lunar Lander  CFD at Ames CFD at Ames  FoilSim FoilSim  Aviation Systems Division Aviation Systems Division