Calculus and NASACalculus and NASA

Michael Bloem

February 15, 2008

Calculus Field Trip Presentation

Michael Bloem

February 15, 2008

Calculus Field Trip Presentation

OutlineOutline

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

€

mgm

€

Ft

€

d2y

dt 2=

(Ft − mgm )

m

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.€

yerr

If then

€

yerr > −(1/2γ y )dyerr

dt

⎛

⎝ ⎜

⎞

⎠ ⎟2

+ y⋅

db

€

Fcontrol = −F∗

Airfoil DesignAirfoil 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

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

€

P(A) =dF

dA

€

P(A)dA∫ = F

€

L = FB − FT = PB (A)dA∫ − PT (A)dA∫L = PB (A) − PT (A)[ ]dA∫

Airfoil Design: Computing LiftAirfoil Design: Computing Lift

Airfoil Design: Computing LiftAirfoil 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?

Rectangle Top [psi] Bottom [psi] Height [psi] Width [in2] Pressure [lb]1 14.80 14.10 0.70 3600 25202 14.75 14.22 0.53 3600 19083 14.75 14.35 0.40 3600 14404 14.72 14.60 0.12 3600 432

TOTAL 6300

Traffic Flow ManagementTraffic 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 Traffic3D Visualization of Air Traffic

Air Traffic Flow ModelsAir Traffic Flow Models

Lagrangian Eulerian

Keep track of each plane Keep track of the numberof planes in different areas

Aggregate Flow ModelAggregate Flow Model

Region i

€

x i(k)

Departures from Center i

€

ud ,i(k)

Inflow from Center j

€

β jix j (k)

Outflow to Center j

€

βij x i(k)

Arrivals into Center i

€

ua,i(k)

€

x(k +1) = A(k)x(k) + ud (k) − ua (k)

€

x i(k +1) = x i(k) − β ij x i(k) − ua,i(k) +j=1j≠ i

N

∑ β jix j (k) +j=1j≠ i

N

∑ ud ,i(k)

Optimization with the Aggregate Flow Model

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

Optimization with the Aggregate Flow Model

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

How do we optimize?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:€

minimize f (x) = 2x 4 +10x + 5

€

d2 f (x)

dx 2= 24x 2 > 0

€

df (x)

dx= 8x 3 = −10

x =1.0772

Another way to optimize?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)€

xn +1 = xn −′ f (xn )′ ′ f (xn )

Picture for Newton’s MethodPicture for Newton’s Method

Newton’s MethodNewton’s MethodIteration x df(x)/dt d2f(x)/dt 2

1 25 125010 150002 16.666 37042.5928 6666.133343 11.1091665 10978.1762 2961.925954 7.40273485 3255.38758 1315.21165 4.92755323 967.158718 582.7387396 3.26787514 289.181315 256.296197 2.13956607 88.3550679 109.8658318 1.33535727 29.0494487 42.79629689 0.6565731 12.2643275 10.3461177

10 -0.52883073 8.81684939 6.7118865211 -1.8424479 -40.0351994 81.470742412 -1.35104205 -9.72861447 43.807550913 -1.12896585 -1.51151283 30.589533314 -1.0795531 -0.06519094 27.970437715 -1.0772224 -0.00014064 27.849794116 -1.07721735 -6.5934E-10 27.84953317 -1.07721735 0 27.84953318 -1.07721735 0 27.84953319 -1.07721735 0 27.84953320 -1.07721735 0 27.849533

Constrained OptimizationConstrained 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

€

x is optimal if and only if

df (x)

dx(y − x) ≥ 0

for all y that meet constraints

Example of Constrained Optimization

Example of Constrained Optimization

Constrained optimization problem?

Is it convex? Try our condition

Constrained optimization problem?

Is it convex? Try our condition

€

minimize f (x) = 2x 4 +10x + 5

subject to x ≥ 0

€

x is optimal if and only if

df (x)

dx(y − x) ≥ 0

for all y that meet constraints

€

x is optimal if and only if

(8x 3 +10)(y − x) ≥ 0

for all y ≥ 0

Example of Constrained Optimization (continued)Example of Constrained Optimization (continued)

€

x is optimal if and only if

(8x 3 +10)(y − x) ≥ 0

for all y ≥ 0

€

x is optimal if and only if

(8x 3 +10)y ≥ (8x 3 +10)x

for all y ≥ 0

€

x is optimal if and only if

y ≥ x

for all y ≥ 0

€

x is optimal if and only if

x = 0

Optimal Traffic Flow Management

Optimal Traffic Flow Management

ConclusionsConclusions

NASA uses calculus a lot because calculus helps solve real problems

NASA uses calculus a lot because calculus helps solve real problems

WebsitesWebsites

Altair Lunar Lander CFD at Ames FoilSim Aviation Systems Division

Altair Lunar Lander CFD at Ames FoilSim Aviation Systems Division