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From Nano-Technologyto Large Space Structures
orHow Mathematical Research is
Becoming the Enabling Science From the
Ultra Small to the Ultra Large
John A. Burns
Center for Optimal Design And Control
Interdisciplinary Center for Applied MathematicsVirginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0531
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GOALS
1. TO DESCRIBE SOME OF OUR RECENT (EXCITING) PROJECTS WHERE MATHEMATICAL RESEARCH HAS MADE A BIG DIFFERENCE
2. TO TRY TO EXPLAIN THE FOLLOWING …
MATHEMATICS IS THE ENABLING SCIENCEFOR MANY OF THE GREAT BREAKTHROUGHS
IN MODERN SCIENCE AND TECHNOLOGY
3. TO CONVINCE EVERYONE THAT …
I HAVE THE BEST JOB IN THE WORLD
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Joint Effort Virginia Tech
J. Borggaard, J. Burns, E. Cliff, T. Herdman,T. Iliescu, D. Inman, B. King, E. Sachs
J. Singler, E. Vugrin Texas Tech
D. Gilliam, V. Shubov George Mason University
L. ZietsmanOTHERS ...
D. Rubio (U. Buenos Aires)J. Myatt (AFRL)A. Godfrey (AeroSoft, Inc.)M. Eppard (Aerosoft, Inc.)K. Belvin (NASA) ….
FUNDING FROMAFOSR
DARPA
NASA
FBI
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Key Points
A GOOD THEORY CAN LEADTO GREAT ALGORITHMS
MATHEMATICS IS OFTEN THE ENABLING TECHNOLOGY
BIG TECHNOLOGICAL ADVANCES HAVE COME BECAUSE WE HAVE
GENERATEDNEW MATHEMATICS Differentiation of functions with respect to shapes
Integration of set-valued functions Control of infinite dimensional systems …
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FIRST APPLICATION
AERODYNAMIC DESIGN
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Free-Jet Test Concept
WIND TUNNEL
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Design of Wind Tunnel Facility
This problem is based on a research effort that started with a joint project between the Air Force's Arnold Engineering Design Center (AEDC) and ICAM at Virginia Tech. The goal of the initial project was to help develop a practical computational algorithm for designing test facilities needed in the free-jet test program. At the start of the project, the main bottleneck was the time required to compute cost function gradients used in an optimization loop. Researchers at ICAM attacked this problem by using the appropriate variational equations to guide the development of efficient computational algorithms this initial idea has since been refined and has now evolved into a practical methodology known as the Sensitivity Equation Method (SEM) for optimal design.
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Design of Wind Tunnel Facility
For the example here we discuss a 2D version of the problem. The green sheet represents a cut through the engine reference plane and leads to the following problem.
Real forebody test shapes have been determined by expensive cut-and-try methods.
Goal is to use computational - optimization tools to automate this process
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Design of Optimal Forebody
INFLOWOUTFLOW
TEST CELL WALL
CENTERLINE FOREBODY
S
DATA GENERATED AT Mach # = 2.0 AND LONG FOREBODY
INFLOWOUTFLOW
TEST CELL WALL
CENTERLINE
SHORT FOREBODY
S
FOREBODY RESTRICTED TO 1/2 LENGTHMATCH
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Long and Short Forebody
direction- yin momentum - energy, - direction-x in momentum - density -
)y,x(n)y,x(E)y,x(m,)y,x(
LONG FOREBODY
SHORTFOREBODY
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Design of Optimal Test Forebody
Data Optimal DesignInitial Design
direction- yin momentum - energy, - direction-x in momentum - density -
)y,x(n)y,x(E)y,x(m,)y,x(
Momentum in x-direction - m(x,y)
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Design of Optimal Test Forebody
OPTIMIZATION LOOPS (TRUST REGION METHOD)
INITIAL ITR # 1 ITR # 5ITR # 2 ITR # 12
THE “SENSITIVITY EQUATION METHOD” WAS100 TIMES FASTER
THAN PREVIOUS “STATE OF THE ART” METHODS
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Design of Optimal Test Forebody
DEVELOPED A NEW MATHEMATICAL METHOD
“CONTINUOUS SENSITIVITY EQUATION METHOD”
HOW WELL DID WE DO ???
HOW DID WE DO IT?
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NEXT APPLICATION
NANO-TECHNOLOGY(THE ULTRA SMALL)
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Control of Thin Film Growth
Ei = .1 eV Ei = 5.0 eV
“VARIABLE ENERGY ION SOURCE”
OR
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Control of Thin Film Growth
Optimized ion beam processing through Modulated Energy Deposition • Low energy for initial monolayers
• Moderate energy for intermediate layers
• High energy to flatten film surface
Successful proof-of-concept experiments using Modulated Energy Deposition approach (Honeywell)Successful proof-of-concept experiments using Modulated Energy Deposition approach (Honeywell)
Cambridge Hydrodynamics, SC Solutions, Colorado, Oak Ridge National Lab
Atomistic Model-Based Design of GMR Processes. Virginia(PI: H. Wadley)
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Control of Thin Film Growth
h(t,x,y )q =
d
:
Sensitivity of h(t,x,y,,,,, d ) to - h(t,x,y,,,,, d )
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Control of Thin Film Growth
Phenomenological models (Ortiz, Repetteo, Si, Zangwill, … 1990s)
)l/)y,x,t(h(fV
),y,x,t(F)y,x,t(hD
)y,x,t(h))y,x,t(h()y,x,t(ht
]1[4
22
q
p)q/z(e)z,q,p(f
MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT
Generalized Transition Function (Stein, VA TECH)
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Models (Ortiz, Repetteo, Si)Raistrick, I. And Hawley, M., Scanning Tunneling and Atomic Force Microscope Studiesof Thin Sputtered Films of YBa2Cu3O7 , Interfaces in High Tc Superconducting Systems, Shinde, S. L. and Rudman, D. A. (eds.), 1993, 28-70.
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Numerical Solutions
WHAT ABOUT THE CONTROL PROBLEM?
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In Real Life …
NEED FEEDBACK CONTROL
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Infinite Dimensional Theory
uopt(t)=
d
:
dy (t,x,y)dxhΩ
y),f(x)(tuopt
h (t,x,y)
OBSERVER
y=C[h(t,x,y)]
SensorInformation
COMPUTATIONAL PROBLEM THE FUNCTIONAL GAIN
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LQG Feedback Control
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LQG Feedback Control
“ABSTRACT” MATHEMATICS MADE THE DIFFERENCE
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NEXT APPLICATION
LARGE SPACE STRUCTURES(THE ULTRA LARGE)
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Control of Large Space Structures
NIA
Active ShapeAnd Vibration
Control
SkilledR&D
Workforce
Inflatable/RigidizableAnd Assembled
Structures
VT- ICAM Modeling
VT- ICAMNASA LaRC
FUNDING FROM DARPA and NASA
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Control of Large Space Structures
Solar Array Flight experiment had unexpected thermal deformation
Early satellites lost because of thermal instabilities
Hubble had large thermal excitations (later fixed)
All of these where not modeled and hence unpredicted
Photos courtesy of W. K. Belvin, NASA Langley
shadesunlight
AVOID THESE PROBLEMS IN FUTURE SPACE STRUCTURES
NEW APPLICATIONS REQUIRE STRUCTURES > 100 m2
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Inflatable Assembled Structures
UV Curing Thermosets Thermoplastics Elastic Memory Stem Aluminum
Temperature, ºC
Psi, Pa
Inflatable/RigidizableAnd Assembled
Structures
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Inflatable Truss Structures
Deploy and assemble into large structures
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New Mathematical Theory
SENSOR
(MFCTM)Flexible Actuators
2
2
2 2 3( , ) [ ( , ) ( , )] ( ) ( )
2 2 2 y t x EI y t x y t x b x u t
t x x x t
INFINITE DIMENSIONAL OPTIMAL CONTROL THEORY IMPLIES
2''( ) EI ( ) ( , ) ( ) ( , )1 220 0
L Loptu t k x y t x dx k x y t x dxtx
VERY PRACTICAL INFORMATION
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New Mathematical Models
2
2 2
02 2 3( , ) [ ( , ) ( ) ( , ) ]
2 2y t x EI y t x s y t s x ds
t x x x t
Including Thermal Effects Changes Everything
02
2 3( , ) ( , ) ( , ) ( , )
2t x t x y t x f t x
t x x t
( , )x t x ADD THERMAL
EQUATIONS ( ) ( )b x u t
MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT
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NEXT APPLICATION
DESIGN OF JET ENGINES
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Design of Injection Scram Jets
q1U
q2U
j
q3j
Design/Control Variables
Slip LIne
Air
H2
U
UJ
j
H2
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Design of Injection Scram Jets
Objective: Prioritization of Design / Control Variables
Free-stream & Design Variables Free-stream: N2 / O2 mixture
M = 3, T = 800 K Injectant: H2
M = 1.7, T = 291 K Momentum ratio = 1.7
Slip LIne
Air
H2
Virginia TechGene Cliff
&AeroSoft, Inc.
Andy GodfreyMark Eppard
q1U
q2U
j
q3j
Design/Control Variables
U
UJ
j
SHAPE
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N2 and H2O Contours
Wedge Angle: 15 deg Shock Angle: 32 deg Flow Solver GASP™ Marching
– 2nd Order Upwind– 3rd Order
Converges 70 planes 3 OM in 60-70 Iters/plane Grid Sizes:
– Zone 1: 41 x 57 x 2– Zone 2: 31 x 81 x 2
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H2O Mass Fraction Sensitivity
Slip line shifts down
Sensitivity to q3 = j Converges 15 OM in 4 iterations
USED
“CONTINUOUS SENSITIVITY EQUATION METHOD”
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Mathematics Impacts “Practically”
UNDERSTANDING THE PROPER MATHEMATICAL FRAMEWORK CAN BE EXPLOITED TO PRODUCE BETTER SCIENTIFIC COMPUTING TOOLS
A REAL JET ENGINE WITH 20 DESIGN VARIABLES PREVIOUS ENGINEERING DESIGN METHODOLOGY
REQUIRED 8400 CPU HRS ~ 1 YEAR USING A HYBRID SEM DEVELOPED AT VA TECH AS
IMPLEMENTED BY AEROSOFT IN SENSE™ REDUCED THE DESIGN CYCLE TIME FROM ...
8400 CPU HRS ~ 1 YEAR TO 480 CPU HRS ~ 3 WEEKS
NEW MATHEMATICS WAS THE ENABLING TECHNOLOGY
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OTHER SENSE™ APPLICATIONS
SENSITIVITIES FOR 3DSHAPE OPTIMIZATION
WITH …
COMPLEX GEOMETRIES
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NEXT APPLICATION
SYSTEM BIOLOGY/EPIDEMICS
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Epidemic Models
Susceptible Infected
Removed ASSUME A WELL MIXEDUNIFORM POPULATION
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Epidemic Models (SARS) SEIJR: Susceptibles – Exposed - Infected - Removed
)()()(
)()()(
222
111
tPtSrtSdt
d
tPtSrtSdt
d
)()()(
)()()()(
)()()()(
21
2
1
tJtItRdt
d
tJtItJdt
d
tItkEtIdt
d
)()()()()()( 2211 tkEtPtSrtPtSrtEdt
d
)(/))()()(()( tNtlJtqEtItP
Model of SARS Outbreak in Canada
byChowell, Fenimore, Castillo-Garsow & Castillo-Chavez (J. Theo. Bio.)
MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT
EXTENSION OF CLASSICALSIR Models
(Kermak – McKendrick, 1927)
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Other Problems Cancer
Cell Growth Vascularization Capillary Formulation
– Reaction diffusion– Moving boundary problems
Heart Models Nerve Membranes Blood flows
– FitzHugh-Nagumo– Navier-Stokes
Enzyme Kinetics Biochemistry Cell Growth
– Michaelis-Menton– Extensions …
J. D. Murray,Mathematical Biology: I and II,Springer, 2002 (2003).
Reference
FAR OUT PROBLEMS
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TRANSIMS - EpiSIMSC. Barrett - Los Alamos R. Laubenbacher - VBI
Ω(t)
10 years for transportation model Clearly a “fake” cloud …
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Dynamic Pathogen & Migration
MODELS? ID? SENSITIVITY? COMPUTATIONAL TOOLS?WHAT ARE THE (SOME) PROBLEMS?
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“SEIJR” PDE Equations
DIFFUSION CONVECTION
VERYCOMPLEXSYSTEMSOF PDEs
COMMONLINK BETWEEN
ALL THE PROBLEMS
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Common Link
WIND TUNNEL EQUATIONS: q = ( M0, q1, q2)
T)]E(x,y,),n(x,y,),m(x,y,),(x,y,)(x,y,U [
qq q q q
(x,y) in (q)0,,21
))y,(xU(Fy
))y,(xU(Fx
),UG(
qqq
)l/)y,x,t(h(fV
),y,x,t(F)y,x,t(hD
)y,x,t(h))y,x,t(h()y,x,t(ht
]1[4
22
NANO-FILM EQUATIONS: q = (, , , , d )
q
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Common Link
LARGE STRUCTURE EQUATIONS: q = q(x)
)( ),(
),(
),(
)2
3
2
2
2
2
2( tuxty
txxty
xEI
xyty
t
q(x)
JET ENGINE EQUATIONS: q = (U , Uj , j)
T)]E(x,y,),n(x,y,),m(x,y,),(x,y,)(x,y,U [
qq q q q
(x,y) in (q)0,,21
))y,(xU(Fy
))y,(xU(Fx
),UG(
qqq
TONOHHON
yxyxyxyxyxyx ]),(),,(),,(),,(),,([ ),(2222
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Remarks
MATH COMBINED WITH COMPUTATIONAL SCIENCE WILL BE THE KEY TO FUTURE
TECHNOLOGY BREAKTHROUGHSCOMPUTATIONS MUST BE DONE RIGHT
LOTS OF APPLICATIONS OPPORTUNITIES FOR MATHEMATICS TO LEAD
THE WAY TO NEW SOLUTIONS = JOB SECURITY FOR APPLIED MATHEMATICIANS
NEW MATHEMATICS NEED TO BE DEVELOPED FOR MODERN PROBLEMS IN PHYSICS, CHEMISTRY, BIOLOGY … ENGINEERING, FLUID & STRUCTURAL DYNAMICS, NANO-
SCIENCE …
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THEEND