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

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 SCIENCE FOR 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 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

Key Points A GOOD THEORY CAN LEAD TO GREAT ALGORITHMS MATHEMATICS IS OFTEN THE ENABLING TECHNOLOGY 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  …

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 DATA GENERATED AT Mach # = 2.0 AND LONG FOREBODY FOREBODY RESTRICTED TO 1/2 LENGTH MATCH

Long and Short Forebody LONG FOREBODY SHORT FOREBODY

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

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

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 E i =.1 eV E i = 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) 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 =q = : 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) q 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 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,

Numerical Solutions WHAT ABOUT THE CONTROL PROBLEM?

In Real Life … NEED FEEDBACK CONTROL

Infinite Dimensional Theory u opt (t)= :  h (t,x,y)  OBSERVER y= C [h(t,x,y)] Sensor Information COMPUTATIONAL PROBLEM THE FUNCTIONAL GAIN

LQG Feedback Control

“ ABSTRACT ” MATHEMATICS MADE THE DIFFERENCE

NEXT APPLICATION LARGE SPACE STRUCTURES (THE ULTRA LARGE)

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

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

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

Inflatable Truss Structures Deploy and assemble into large structures

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

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

NEXT APPLICATION DESIGN OF JET ENGINES

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

Design of Injection Scram Jets  Objective: l Prioritization of Design / Control Variables  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

N 2 and H 2 O Contours  Wedge Angle: 15 deg  Shock Angle: 32 deg  Flow Solver l GASP™ l Marching – 2nd Order Upwind – 3rd Order l Converges 70 planes l 3 OM in Iters/plane l Grid Sizes: – Zone 1: 41 x 57 x 2 – Zone 2: 31 x 81 x 2

H 2 O Mass Fraction Sensitivity  Slip line shifts down  Sensitivity to q 3 =  j l 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 l PREVIOUS ENGINEERING DESIGN METHODOLOGY REQUIRED 8400 CPU HRS ~ 1 YEAR l USING A HYBRID SEM DEVELOPED AT VA TECH AS IMPLEMENTED BY AEROSOFT IN SENSE™ REDUCED THE DESIGN CYCLE TIME FROM CPU HRS ~ 1 YEAR TO 480 CPU HRS ~ 3 WEEKS NEW MATHEMATICS WAS THE ENABLING TECHNOLOGY

OTHER SENSE™ APPLICATIONS SENSITIVITIES FOR 3D SHAPE OPTIMIZATION WITH … COMPLEX GEOMETRIES

NEXT APPLICATION SYSTEM BIOLOGY/EPIDEMICS

Epidemic Models SusceptibleInfected Removed ASSUME A WELL MIXED UNIFORM POPULATION

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 EXTENSION OF CLASSICAL SIR Models (Kermak – McKendrick, 1927)

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

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

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

“ SEIJR ” PDE Equations DIFFUSION CONVECTION VERY COMPLEX SYSTEMS OF PDEs COMMON LINK BETWEEN ALL THE PROBLEMS

Common Link WIND TUNNEL EQUATIONS: q = ( M 0, q 1, q 2  ) q q qq q (x,y) in  (q) q q q  NANO-FILM EQUATIONS: q = ( , , , , d ) q

Common Link LARGE STRUCTURE EQUATIONS: q = q(x) q(x) JET ENGINE EQUATIONS: q = ( U , U j,  j  ) q q qq q (x,y) in  (q) q q q 

Remarks MATH COMBINED WITH COMPUTATIONAL SCIENCE WILL BE THE KEY TO FUTURE TECHNOLOGY BREAKTHROUGHS COMPUTATIONS 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 l PHYSICS, CHEMISTRY, BIOLOGY … l ENGINEERING, FLUID & STRUCTURAL DYNAMICS, NANO-SCIENCE …

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