1. 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 C enter for A pplied M athematics 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

2. Who, What and Why ? WHO MIGHT BE INTERESTED ? l STUDENTS IN MATH, ENGINEERING and SCIENCES ? WHAT WILL I TALK ABOUT ? l 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 l A COLLECTION OF CURRENT REAL WORLD APPLICATIONS WHERE NEW MATHEMATICS HAD TO BE DEVELOPED IN ORDER TO SOLVE THESE PROBLEMS l AN INTRODUCTION TO NUMERICAL METHODS NEEDED FOR OPTIMAL DESIGN AND CONTROL OF PHYSICAL AND BIOLOGICAL SYSTEMS l INTRODUCE THE CONTINUOUS SENSITIVITY EQUATION METHODS

3. Who, What and Why ? WHY DO THIS ?  FOR THE STUDENT… l IT IS FUN (AT LEAST IT CAN BE FUN) l TO SEE WHY MATHEMATICS IS SO IMPORTANT … MATHEMATICS IS THE ENABLING SCIENCE FOR MOST OF THE GREAT BREAKTHROUGHS IN MODERN SCIENCE AND TECHNOLOGY  FOR ME … l IT IS FUN (AT LEAST IT CAN BE FUN) l I CAN TALK ABOUT THE RESEARCH PROJECTS AT ICAM l I CAN TRY TO EXPLAIN WHY … I HAVE THE BEST JOB IN THE WORLD

4. Joint Effort  Virginia Tech l J. Borggaard, J. Burns, E. Cliff, T. Herdman, T. Iliescu, D. Inman, B. King, E. Sachs l J. Singler, E. Vugrin  Texas Tech l D. Gilliam, V. Shubov  George Mason University l L. Zietsman OTHERS... D. Rubio (U. Buenos Aires) J. Myatt (AFRL) A. Godfrey (AeroSoft, Inc.) M. Eppard (Aerosoft, Inc.) K. Belvin (NASA) …. FUNDING FROM AFOSR DARPA NASA FBI

5. 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 Lecture From Nano-Technology to Large Space Structures or How Mathematical Research is Becoming the Enabling Science From the Ultra Small to the Ultra Large

6. 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

7. Thing to Remember A GOOD THEORY CAN LEAD TO GREAT ALGORITHMS MATHEMATICS IS OFTEN THE ENABLING SCIENCE BIG TECHNOLOGICAL ADVANCES HAVE COME BECAUSE WE HAVE GENERATED NEW MATHEMATICS  Differentiation of functions with respect to shapes  Integration of set-valued functions  Control of infinite dimensional systems  …

8. FIRST APPLICATION AERODYNAMIC DESIGN

9. Free-Jet Test Concept WIND TUNNEL

10. 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.

11. 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

12. Design of Optimal Forebody DATA GENERATED AT Mach # = 2.0 AND LONG FOREBODY FOREBODY RESTRICTED TO 1/2 LENGTH MATCH

13. Long and Short Forebody LONG FOREBODY SHORT FOREBODY

14. Design of Optimal Test Forebody DataOptimal DesignInitial Design Momentum in x-direction - m(x,y)

15. Design of Optimal Test Forebody DEVELOPED A NEW MATHEMATICAL METHOD “ CONTINUOUS SENSITIVITY EQUATION METHOD ” HOW WELL DID WE DO ??? HOW DID WE DO IT?

16. Design of Optimal Test Forebody OPTIMIZATION LOOPS (TRUST REGION METHOD) INITIALITR # 1ITR # 5ITR # 2ITR # 12 THE “ SENSITIVITY EQUATION METHOD ” WAS 100 TIMES FASTER THAN PREVIOUS “ STATE OF THE ART ” METHODS

17. NEXT APPLICATION SYSTEM BIOLOGY/EPIDEMICS

18. Epidemic Models SusceptibleInfected Removed ASSUME A WELL MIXED UNIFORM POPULATION

19. Epidemic Models  SIR Models (Kermak – McKendrick, 1927) l S usceptible – I nfected – R ecovered/Removed

20. Epidemic Models (SARS)  SEIJR: S usceptibles – E xposed - I nfected - Re moved Model of SARS Outbreak in Canada by Chowell, Fenimore, Castillo-Garsow & Castillo-Chavez (J. Theo. Bio.) MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT

21. Other Problems  Cancer l Cell Growth l Vascularization l Capillary Formulation –Reaction diffusion –Moving boundary problems  Heart Models l Nerve Membranes l Blood flows –FitzHugh-Nagumo –Navier-Stokes  Enzyme Kinetics l Biochemistry l Cell Growth –Michaelis-Menton –Extensions … J. D. Murray, Mathematical Biology: I and II, Springer, 2002 (2003). Reference FAR OUT PROBLEMS

22. TRANSIMS - EpiSIMS C. Barrett - Los Alamos R. Laubenbacher - VBI 10 years for transportation modelClearly a “fake” cloud …

23. Dynamic Pathogen & Migration MODELS? ID? SENSITIVITY? COMPUTATIONAL TOOLS? WHAT ARE THE (SOME) PROBLEMS?

24. “ SEIJR ” Model: Improved DIFFUSION CONVECTION HIGHLY COMPLEX

25. NEXT APPLICATION DESIGN OF PRINTERS

26. Design of Ink Jet Printers Tektronix Graphics, Printing & Imaging Division (FUNDING - NSF)

27. Design of Ink Jet Printers ADJUST THE ACTUATOR SENSOR CONTROL

28. NEXT APPLICATION NANO-TECHNOLOGY

29. Control of Thin Film Growth E i =.1 eV E i = 5.0 eV “VARIABLE ENERGY ION SOURCE” OR

30. 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) Cambridge Hydrodynamics, SC Solutions, U. Colorado, Oak Ridge National Lab Atomistic Model-Based Design of GMR Processes. Virginia (PI: H. Wadley)

31. Control of Thin Film Growth MD SIMULATION

32. Control of Thin Film Growth h(t,x,y ) q =q = : Sensitivity of h(t,x,y, , , , , d ) to  -   h(t,x,y, , , , , d )

33. ? Model: The Equations ? Phenomenological models (Ortiz, Repetteo, Si, Zangwill, … 1990s) q Molecular Dynamic Models (Alder, Wainwright, … 1950s) Position of N - atoms q q N  10 9 ORDINARY DIFF EQUATIONS

34. Models (Ortiz, Repetteo, Si) Raistrick, I. And Hawley, M., Scanning Tunneling and Atomic Force Microscope Studies of Thin Sputtered Films of YBa 2 Cu 3 O 7, Interfaces in High Tc Superconducting Systems, Shinde, S. L. and Rudman, D. A. (eds.), 1993, 28-70.

35. Control of Thin Film Growth Phenomenological models (Ortiz, Repetteo, Si, Zangwill, … 1990s) u Transition Function Predicts negative film growth Parameter identification impossible Not even necessary in YBCO films! Removes negative film growth Parameters can be tuned Include more deposition processes Generalized Transition Function (Stein, VA TECH)

36. Need “ Reasonable ” Model Negative Film Height ! Mean Film Height

37. Mean Height For YBCO Film NO TRANSITION FUNCTION

38. Parameterized Models General transition function provides flexibility However, need to include deposition energy +... MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT

39. NEXT APPLICATION JET ENGINES

40. Design of Injection Scram Jets q 1  U  q 2  U j q 3  j Design/Control Variables Slip LIne Air H2 H2 UU UJUJ jj H2 H2

41. Design of Injection Scram Jets  Objective: l DETERMINE BEST ANGLE  Free-stream & Design Variables l Free-stream: N 2 / O 2 mixture M = 3, T = 800 K l Injectant: H 2 M = 1.7, T = 291 K l Momentum ratio = 1.7 Slip LIne Air H2 H2 Virginia Tech Gene Cliff & AeroSoft, Inc. Andy Godfrey Mark Eppard q 1  U  q 2  U j q 3  j Design/Control Variables UU UJUJ jj SHAPE

42. NEXT APPLICATION LARGE SPACE STRUCTURES

43. Control of Large Space Structures NIA Active Shape And Vibration Control Skilled R&D Workforce Inflatable/Rigidizable And Assembled Structures VT- ICAM Modeling VT- ICAM NASA LaRC FUNDING FROM DARPA and NASA

44. 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 shade sunlight AVOID THESE PROBLEMS IN FUTURE SPACE STRUCTURES NEW APPLICATIONS REQUIRE STRUCTURES > 100 m 2

45. Inflatable Assembled Structures  UV Curing  Thermosets  Thermoplastics  Elastic Memory  Stem  Aluminum Temperature, ºC Psi, Pa Inflatable/Rigidizable And Assembled Structures

46. Inflatable Truss Structures Deploy and assemble into large structures

47. New Mathematical Theory SENSOR (MFC TM ) Flexible Actuators INFINITE DIMENSIONAL OPTIMAL CONTROL THEORY IMPLIES VERY PRACTICAL INFORMATION

48. New Mathematical Models Including Thermal Effects Changes Everything ADD THERMAL EQUATIONS MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT

49. 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 l PHYSICS, CHEMISTRY, BIOLOGY … l FLUID DYNAMICS, STRUCTURAL DYNAMICS … l…l…