Computational Methods for Design Motivating Applications and Introduction to Modeling John A. Burns Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia 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
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Computational Methods for Design Motivating Applications and Introduction to Modeling John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary.
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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)
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 )
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.