Computational Methods for Design

Motivating Applicationsand Introduction to Modeling

John A. BurnsCenter for Optimal Design And Control

Interdisciplinary Center for Applied MathematicsVirginia Polytechnic Institute and State University

Blacksburg, Virginia 24061-0531

A Short Course in Applied Mathematics

2 February 2004 – 7 February 2004

N∞M∞T Series Two Course

Canisius College, Buffalo, NY

Who, What and Why

? WHO MIGHT BE INTERESTED ? STUDENTS IN MATH, ENGINEERING and SCIENCES

? WHAT WILL I TALK ABOUT ? HOW DIFFERENTIAL EQUATIONS ARISE AS

FUNDAMENTAL MODELS IN ALL BRANCHES OF MODERN SCIENCE AND ENGINEERING - MODELING

A SHORT REVIEW/SUMMARY OF THE “BASIC” MATHEMATICS REQUIRED TO UNDERSTAND THE PROBLEMS

A COLLECTION OF CURRENT REAL WORLD APPLICATIONS WHERE NEW MATHEMATICS HAD TO BE DEVELOPED IN ORDER TO SOLVE THESE PROBLEMS

AN INTRODUCTION TO NUMERICAL METHODS NEEDED FOR OPTIMAL DESIGN AND CONTROL OF PHYSICAL AND BIOLOGICAL SYSTEMS

INTRODUCE THE CONTINUOUS SENSITIVITY EQUATION METHODS

Who, What and Why

? WHY DO THIS ? FOR THE STUDENT…

IT IS FUN (AT LEAST IT CAN BE FUN) TO SEE WHY MATHEMATICS IS SO IMPORTANT …

MATHEMATICS IS THE ENABLING SCIENCEFOR MOST OF THE GREAT BREAKTHROUGHS IN

MODERN SCIENCE AND TECHNOLOGY

FOR ME … IT IS FUN (AT LEAST IT CAN BE FUN) I CAN TALK ABOUT THE RESEARCH PROJECTS AT ICAM I CAN TRY TO EXPLAIN WHY …

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

Course Outline Lecture 1 - High Level Description of

Applications Lecture 2 – Some “Simple” Applications Lecture 3 – Elementary Differential Equations Lecture 4 – Introduction to Sensitivities Lecture 5 - Design and Optimization Problems

IF ENOUGH TIME … Modeling and Control of the Growth of Cancer Cells Problems Involving Bioterrorism

General LectureFrom Nano-Technology to Large Space Structures or

How Mathematical Research is Becoming the Enabling Science From the Ultra Small to the Ultra Large

Today’s Topics

Design of Wind Tunnel Test Facilities System Biology: Epidemics and Populations Design and Optimization of Ink Jet Printers Manufacturing Thin Films: Nano-Technology Design of Scram Jets Design and Control of VERY Large Space

Structures

Thing to Remember

A GOOD THEORY CAN LEADTO GREAT ALGORITHMS

MATHEMATICS IS OFTEN THE ENABLING SCIENCE

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

DEVELOPED A NEW MATHEMATICAL METHOD

“CONTINUOUS SENSITIVITY EQUATION METHOD”

HOW WELL DID WE DO ???

HOW DID WE DO IT?

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

NEXT APPLICATION

SYSTEM BIOLOGY/EPIDEMICS

Epidemic Models

Susceptible Infected

Removed ASSUME A WELL MIXEDUNIFORM POPULATION

Epidemic Models SIR Models (Kermak – McKendrick, 1927)

Susceptible – Infected – Recovered/Removed

( ) ( ) ( )d

S t S t I tdt

( ) ( ) ( ) ( )d

I t S t I t I tdt

( ) ( )d

R t I tdt

( ) ( ) ( ) constantS t I t R t N

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

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” Model: Improved

DIFFUSION CONVECTION

HIGHLYCOMPLEX

NEXT APPLICATION

DESIGN OF PRINTERS

Design of Ink Jet Printers

Tektronix Graphics, Printing & Imaging Division (FUNDING - NSF)

Design of Ink Jet Printers

ADJUST THE ACTUATOR

SENSORCONTROL

NEXT APPLICATION

NANO-TECHNOLOGY

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, U. Colorado, Oak Ridge National Lab

Atomistic Model-Based Design of GMR Processes. Virginia(PI: H. Wadley)

Control of Thin Film Growth

MD SIMULATION

MD SIMULATION

Control of Thin Film Growth

h(t,x,y )q =

d

:

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

? Model: The Equations ?

),y,x,t(F)y,x,t(hD

)y,x,t(h))y,x,t(h()y,x,t(ht

4

22

]1[

Phenomenological models (Ortiz, Repetteo, Si, Zangwill, … 1990s)

q

Molecular Dynamic Models (Alder, Wainwright, … 1950s)

),p,,),t(r,t(f)t(rm

)(rf),,p,,,rr(u)(r

r...,r,rr

NN

dtd

Nri

jij,i

jiN

TN

N

i

2

2

21

UU

Position of N - atoms

q

q

N 10 9 ORDINARY

DIFF EQUATIONS

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.

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

u

ze)z()z(f 1Transition Function

• Predicts negative film growth• Parameter identification impossible• Not even necessary in YBCO films!

p)q/z(e)z,q,p(f • Removes negative film growth• Parameters can be tuned• Include more deposition processes

Generalized Transition Function (Stein, VA TECH)

Need “Reasonable” Model

ze)z()z(f 1

Negative Film Height !

Mean Film Height

Mean Film Height

p)q/z(e)z,q,p(f

Mean Height

For YBCO FilmNO TRANSITION FUNCTION

0f

Parameterized Models

• General transition function provides flexibility• However, need to include deposition energy + ...

p)q/z(e)z,q,p(f

MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT

NEXT APPLICATION

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: DETERMINE BEST ANGLE

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

NEXT APPLICATION

LARGE SPACE STRUCTURES

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

Remarks

COMPUTATIONAL MATHEMATICS, SCIENCE AND ENGINEERING WILL BE THE KEY TO FUTURE BREAKTHROUGHS

CMS&E MUST BE DONE RIGHT

LOTS OF APPLICATIONS OPPORTUNITIES FOR MATHEMATICS TO

LEAD THE WAY TO NEW SOLUTIONS = JOB SECURITY FOR APPLIED MATHEMATICIANS NEW MODELS NEED TO BE DEVELOPED

PHYSICS, CHEMISTRY, BIOLOGY … FLUID DYNAMICS, STRUCTURAL DYNAMICS … …