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

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

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

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 …

FIRST APPLICATION

AERODYNAMIC DESIGN

Free-Jet Test Concept

WIND TUNNEL

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.

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

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

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

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)

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

Design of Optimal Test Forebody

DEVELOPED A NEW MATHEMATICAL METHOD

“CONTINUOUS SENSITIVITY EQUATION METHOD”

HOW WELL DID WE DO ???

HOW DID WE DO IT?

NEXT APPLICATION

NANO-TECHNOLOGY(THE ULTRA SMALL)

Control of Thin Film Growth

Ei = .1 eV Ei = 5.0 eV

“VARIABLE ENERGY ION SOURCE”

OR

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)

Control of Thin Film Growth

h(t,x,y )q =

d

:

Sensitivity of h(t,x,y,,,,, d ) to - h(t,x,y,,,,, d )

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)

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.

Numerical Solutions

WHAT ABOUT THE CONTROL PROBLEM?

In Real Life …

NEED FEEDBACK CONTROL

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

LQG Feedback Control

LQG Feedback Control

“ABSTRACT” MATHEMATICS MADE THE DIFFERENCE

NEXT APPLICATION

LARGE SPACE STRUCTURES(THE ULTRA LARGE)

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

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

Inflatable Assembled Structures

UV Curing Thermosets Thermoplastics Elastic Memory Stem Aluminum

Temperature, ºC

Psi, Pa

Inflatable/RigidizableAnd Assembled

Structures

Inflatable Truss Structures

Deploy and assemble into large structures

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

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

NEXT APPLICATION

DESIGN OF JET ENGINES

Design of Injection Scram Jets

q1U

q2U

j

q3j

Design/Control Variables

Slip LIne

Air

H2

U

UJ

j

H2

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

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

H2O Mass Fraction Sensitivity

Slip line shifts down

Sensitivity to q3 = j Converges 15 OM in 4 iterations

USED

“CONTINUOUS SENSITIVITY EQUATION METHOD”

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

OTHER SENSE™ APPLICATIONS

SENSITIVITIES FOR 3DSHAPE OPTIMIZATION

WITH …

COMPLEX GEOMETRIES

NEXT APPLICATION

SYSTEM BIOLOGY/EPIDEMICS

Epidemic Models

Susceptible Infected

Removed ASSUME A WELL MIXEDUNIFORM POPULATION

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)

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

TRANSIMS - EpiSIMSC. Barrett - Los Alamos R. Laubenbacher - VBI

Ω(t)

10 years for transportation model Clearly a “fake” cloud …

Dynamic Pathogen & Migration

MODELS? ID? SENSITIVITY? COMPUTATIONAL TOOLS?WHAT ARE THE (SOME) PROBLEMS?

“SEIJR” PDE Equations

DIFFUSION CONVECTION

VERYCOMPLEXSYSTEMSOF PDEs

COMMONLINK BETWEEN

ALL THE PROBLEMS

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

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

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 …

THEEND