THE DISCRETE ADJOINT APPROACH TO AERODYNAMIC SHAPE OPTIMIZATION

Transcription

1 THE DISCRETE ADJOINT APPROACH TO AERODYNAMIC SHAPE OPTIMIZATION a dissertatin submitted t the department f aernautics and astrnautics and the cmmittee n graduate studies f stanfrd university in partial fulfillment f the requirements fr the degree f dctr f philsphy Siva Kumaran Nadarajah January 003

2 c Cpyright by Siva Kumaran Nadarajah 003 All Rights Reserved ii

3 I certify that I have read this dissertatin and that, in my pinin, it is fully adequate in scpe and quality as a dissertatin fr the degree f Dctr f Philsphy. Antny Jamesn (Principal Adviser) I certify that I have read this dissertatin and that, in my pinin, it is fully adequate in scpe and quality as a dissertatin fr the degree f Dctr f Philsphy. Juan Alns I certify that I have read this dissertatin and that, in my pinin, it is fully adequate in scpe and quality as a dissertatin fr the degree f Dctr f Philsphy. Rbert MacCrmack Apprved fr the University Cmmittee n Graduate Studies: iii

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5 T My Parents, T.C. Nadarajah and Sushila v

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7 Abstract A viscus discrete adjint apprach t autmatic aerdynamic shape ptimizatin is develped, and the merits f the viscus discrete and cntinuus adjint appraches are discussed. The viscus discrete and cntinuus adjint gradients fr inverse design and drag minimizatin cst functins are cmpared with finite-difference and cmplex-step gradients. The ptimizatin f airfils in tw-dimensinal flw fr inverse design and drag minimizatin is illustrated. Bth the discrete and cntinuus adjint methds are used t frmulate tw new design prblems. First, the timedependent ptimal design prblem is established, and bth the time accurate discrete and cntinuus adjint equatins are derived. An applicatin t the reductin f the time-averaged drag cefficient while maintaining time-averaged lift and thickness distributin f a pitching airfil in transnic flw is demnstrated. Secnd, the remte inverse design prblem is frmulated. The ptimizatin f a three-dimensinal bicnvex wing in supersnic flw verifies the feasibility t reduce the near field pressure peak. Cupled drag minimizatin and remte inverse design cases prduce wings with a lwer drag and a reduced near field peak pressure signature. vii

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9 Acknwledgments I wuld like t express gratitude t my mentr, Prfessr Antny Jamesn, whse expertise, understanding, and patience added cnsiderably t my graduate experience. His vast knwledge and enthusiasm have inspired me greatly, and I thank him fr his time and supprt. A very special thanks t Dr. Juan Alns. He prvided me with directin, technical supprt and became a mentr and friend. I wuld like t thank the ther members f my cmmittee, Dr. Ilan Kr, Dr. Rbert MacCrmack, and Dr. Michael Saunders fr the assistance they prvided at all levels f the research prject. This research wuld nt have been pssible withut the financial assistance f the Airfrce Office f Scientific Research (AFOSR Grant N: F ), Defense Advanced Research Prject Agency (DARPA Grant N: MDA ), Advanced Simulatin and Cmputing: Department f Energy (ASC Grant N: B3449), and Mark Yng. I have enjyed the encuragement and warm cmpany f my clleagues in the Aerspace Cmputing Labratry. A special thanks t Sangh Kim fr the fruitful cnversatins abut ptimizatin and t Matthew McMullen fr the many discussins n research. Matt s humr was a wnderful distractin frm wrk. Life in the San Francisc Bay Area wuld have been meaningless withut the lve and friendship f the Sai Bay Area Yung Adults. I cherish the hurs f serving the cmmunity with such kind and lving friends. I am grateful t my Guru Sri Sathya Sai Baba fr inspiring me t believe in high ideals and t live up t what I believe in. My dream is t make my life Yur message. I am eternally grateful t my parents, T.C. Nadarajah and Sushila Nadarajah, ix

10 and my family fr their uncnditinal lve. Thanks fr encuraging me t be an independent thinker and having cnfidence in my abilities t g after new things that inspired me. Abve all, I thank Lisa Steffen, my wife and best friend, fr her unbunded lve and cnfidence in me. Lisa gives me smething t lk frward t each and every day. x

11 Cntents Abstract Acknwledgments vii ix Intrductin. The Discrete and Cntinuus Adjint Appraches Optimum Shape Design fr Unsteady Flws The Remte Inverse Design Prblem The Euler and Navier-Stkes Equatins 7. Mathematical Mdel Cnservatin f Mass Cnservatin f Mmentum Cnservatin f Energy Cnservative Frm f the Field Equatins Bundary Cnditins Cmpressible Reynlds Averaged Navier-Stkes Equatins Baldwin-Lmax Turbulence Mdel Numerical Discretizatin Finite-Vlume Technique Artificial Dissipatin Discrete Bundary Cnditin Time Stepping Scheme Cnvergence Acceleratin xi

12 3 Numerical Optimizatin Algrithms Calculus f Variatins Linearized Supersnic Flw Optimizatin Algrithms Steepest Descent Smthed Steepest Descent The Discrete and Cntinuus Adjint Appraches Frmulatin f the Optimal Design Prblem Derivatin f the Cntinuus Adjint Terms Numerical Discretizatin Cntinuus Adjint Bundary Cnditins Derivatin f the Discrete Adjint Terms Cntributins frm the Cnvective Flux Cntributins frm the Viscus Flux Cntributins frm the Artificial Dissipatin Flux Discrete Adjint Bundary Cnditins Inverse Design Drag Minimizatin Time Integratin and Cnvergence Acceleratin Grid Perturbatin Design Variables Mesh Pints Hicks-Henne Functins Finite-Difference and Cmplex-Step Gradients Outline f the Design Prcess Results Inviscid: Inverse Design Inviscid: Drag Minimizatin Viscus: Inverse Design Viscus: Drag Minimizatin xii

13 5 Optimum Shape Design fr Unsteady Flws Gverning Equatins Numerical Discretizatin Discretizatin f the Time Derivative Term Reduced Frequency General Frmulatin Time Accurate Cntinuus Adjint Equatins Time Accurate Discrete Adjint Equatins Design Prcess Full Unsteady Design (Unsteady-Flw Unsteady-Adjint) Partial Unsteady Design (Unsteady-Flw Steady-Adjint) Time-Averaged-Flw Steady-Adjint Design Multipint Design Results Cde and Grid Validatin RAE 8: Time-Averaged Drag Minimizatin with Fixed Time- Averaged Lift Cefficient VR-7: Time-Averaged Drag Minimizatin with Fixed Time- Averaged Lift Cefficient Cmparisn f Varius Design Appraches The Remte Inverse Design Prblem The Remte Inverse Design Prblem Frmulatin f the Cntinuus Adjint Equatins fr the Remte Inverse Prblem Frmulatin f the Discrete Adjint Equatin fr the Remte Inverse Prblem Implementatin f Remte Inverse Design Results Ni-Bump Bicnvex Wing: Verificatin Study xiii

14 6.3.3 Bicnvex Wing: Near Field Pressure Reductin, Withut Cnstraints Bicnvex Wing: Near Field Pressure Reductin, With Cnstraints Highly Swept Blunt LE Wing: Near Field Pressure and Drag Reductin Cnclusins 7 7. Discrete Versus Cntinuus Adjint Appraches Optimum Shape Design fr Unsteady Flws Remte Inverse Design Prblem Future Wrk A Viscus Cntinuus Adjint Equatins 3 A. Transfrmatin t Primitive Variables A. Cntributins frm the Mmentum Equatins A.3 Cntributins frm the Energy Equatin A.4 The Viscus Adjint Field Operatr A.5 Viscus Adjint Bundary Cnditins A.5. Inverse Design A.5. Drag Minimizatin Bibligraphy 45 xiv

15 List f Tables 3. Cmputatinal Cst f Gradient-Based Algrithms as a Functin f the Number f Design Variables fr the Brachistchrne Prblem L nrm f the Difference Between Adjint and Finite-Difference Gradients L nrm f the Difference Between Adjint and Finite-Difference Gradients L nrm f the Difference Between Adjint and Cmplex-Step Gradients4 5. Cmparisn f Cmputatinal Cst (Multigrid Cycles) Between Fur Design Appraches Euler Lens-Mesh Descriptins Initial and Final Time-Averaged Drag Cefficient fr Varius Reduced Frequencies using the Full Unsteady Design Apprach Cmparisn Between the Multipint and Full Unsteady Optimizatin Initial and Final Time-Averaged Drag Cefficient fr Varius Design Appraches Time-Averaged Drag Cefficient fr Varius Reduced Frequencies fr the Full Unsteady and Multipint Design Ttal Wing Drag Cefficient fr Varius Design Cases xv

16 List f Figures. Schematic f the Prpagatin f the Aircraft Pressure Signature Finite-Vlume Mesh fr Cell (i, j) Discretizatin f the Cnvective Fluxes Auxiliary Cntrl Vlume fr the Discretizatin f the Viscus Flux Cmplex-Step Versus Finite-Difference Gradient Errrs fr Inverse Design Case; ɛ = g g ref g ref Design Prcedure Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l Adjint Versus Finite Difference Gradients fr Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l. Carse Grid - 96 x 6, M = 0.74, C l = Adjint Versus Finite Difference Gradients fr Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l. Medium Grid - 9 x 3, M = 0.74, C l = Adjint Versus Finite Difference Gradients fr Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l. Fine Grid - 56 x 64, M = 0.74, C l = Adjint Versus Finite Difference Gradients fr Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l. Dissipative Cefficients Nt Frzen. Medium Grid - 9 x 3, M = 0.74, C l = xvi

17 4.9 Cntinuus Adjint Gradients fr Varying Flw Slver Cnvergence fr the Inviscid Inverse Design Case Discrete Adjint Gradients fr Varying Flw Slver Cnvergence fr the Inviscid Inverse Design Case Cntinuus Adjint Gradients fr Varying Adjint Slver Cnvergence fr the Inviscid Inverse Design Case Discrete Adjint Gradients fr Varying Adjint Slver Cnvergence fr the Inviscid Inverse Design Case Cmparisn f Cstate Values Between the Cntinuus and Discrete Adjint Methd fr the Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l. Medium Grid - 9 x 3, M = 0.74, C l = Cnvergence Histry fr the Cntinuus and Discrete Adjint fr the Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l. M = 0.74, C l = Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l Adjint Versus Finite Difference Gradients fr Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l. Carse Grid - 96 x 6, M = 0.78, C l = Adjint Versus Finite Difference Gradients fr Inviscid Inverse Design f Krn t NACA 64A40 at Fixed C l. Medium Grid - 9 x 3, M = 0.78, C l = Inviscid Drag Minimizatin f NACA 64A40 at Fixed C l Inviscid Drag Minimizatin f NACA 64A40 at Fixed C l Adjint Versus Finite Difference Gradients fr Inviscid Drag Minimizatin f NACA 64A40 at Fixed C l Adjint Versus Finite Difference Gradients fr Inviscid Drag Minimizatin f NACA 64A40 at Fixed C l Adjint Versus Finite Difference Gradients fr Inviscid Drag Minimizatin f NACA 64A40 at Fixed C l Adjint Versus Finite Difference Gradients fr Inviscid Drag Minimizatin f NACA 64A40 at Fixed C l. Dissipative Cefficients Nt Frzen. 8 xvii

18 4.4 Cntinuus Adjint Gradients fr Varying Flw Slver Cnvergence fr the Inviscid Drag Minimizatin Case Discrete Adjint Gradients fr Varying Flw Slver Cnvergence fr the Inviscid Drag Minimizatin Case Cntinuus Adjint Gradients fr Varying Adjint Slver Cnvergence fr the Inviscid Drag Minimizatin Case Discrete Adjint Gradients fr Varying Adjint Slver Cnvergence fr the Inviscid Drag Minimizatin Case Cmparisn f Cnvergence f the Objective Functin Between the Cntinuus and Discrete Adjint Methd fr Inviscid Drag Minimizatin Cmparisn f Cstate Values Between the Cntinuus and Discrete Adjint Methd fr Inviscid Drag Minimizatin f NACA 64A40 at Fixed C l. Medium Grid - 9 x 3, M = 0.75, C l = Cnvergence Histry fr the Cntinuus and Discrete Adjint fr Inviscid Drag Minimizatin f NACA 64A40 at Fixed C l. M = 0.75, C l = Inverse Design f NACA 00 t Onera M6 at Fixed C l Inverse Design f NACA 00 t Onera M6 at Fixed C l Inverse Design f RAE t NACA64A40 at Fixed C l Adjint Gradient Errrs fr Varying Flw Slver Cnvergence fr the Inverse Design Case; ɛ = g g ref g ref. Fine Grid - 5 x 64, M = 0.75, C l = Adjint Gradient Errrs fr Varying Adjint Slver Cnvergence fr the Inverse Design Case; ɛ = g g ref g ref. Fine Grid - 5 x 64, M = 0.75, C l = Adjint Versus Cmplex-Step Gradients fr Inverse Design f RAE t NACA64A40 at Fixed C l. Carse Grid x 64, M = 0.75, C l = Adjint Versus Cmplex-Step Gradients fr Inverse Design f RAE t NACA64A40 at Fixed C l. Medium Grid - 5 x 64, M = 0.75, C l = Adjint Versus Cmplex-Step Gradients fr Inverse Design f RAE t NACA64A40 at Fixed C l. Fine Grid - 04 x 64, M = 0.75, C l = xviii

19 4.39 Drag Minimizatin f RAE Airfil using the Cntinuus Adjint Frmulatin. Grid - 5 x 64, M = 0.75, Fixed C l = 0.65, α = degrees Drag Minimizatin f RAE Airfil using the Cntinuus Adjint Frmulatin. Grid - 5 x 64, M = 0.75, Fixed C l = 0.65, α = degrees Drag Minimizatin f RAE Airfil using the Discrete Adjint Frmulatin. Grid - 5 x 64, M = 0.75, Fixed C l = 0.65, α = degrees Drag Minimizatin f RAE Airfil using the Discrete Adjint Frmulatin. Grid - 5 x 64, M = 0.75, Fixed C l = 0.65, α = degrees Adjint Versus Cmplex-Step Gradients fr Pressure Drag Minimizatin at Fixed C l. Fine Grid - 5 x 64, M = 0.75, C l = Adjint Versus Cmplex-Step Gradients fr Ttal Drag Minimizatin at Fixed C l. Fine Grid - 5 x 64, M = 0.75, C l = Lens-Mesh 9x3: NACA 64A Clse-up View: Lens-Mesh 9x3: NACA 64A Cmparisn f Lift Cefficient versus Angle f Attack fr Varius Lenz- Mesh Grids and Experimental Results n a NACA 64A00 CT6 Case Cmparisn f Drag Cefficient versus Angle f Attack fr Varius Lenz-Mesh Grids n a NACA 64A00 CT6 Case Cmparisn f Lift Cefficient versus Angle f Attack fr Varius O- Mesh Grids and Experimental Results n a NACA 64A00 CT6 Case Cmparisn f Drag Cefficient versus Angle f Attack fr Varius O-Mesh Grids n a NACA 64A00 CT6 Case Cmparisn f Lift Cefficient versus Angle f Attack fr Varius O- Mesh, Lenz-Mesh Grids and Experimental Results n a NACA 64A00 CT6 Case Cmparisn f Drag Cefficient versus Angle f Attack fr Varius O- Mesh, Lenz-Mesh Grids and Experimental Results n a NACA 64A00 CT6 Case xix

20 5.9 Cmparisn f Lift Cefficient versus Angle f Attack fr Varius O- Mesh, Lenz-Mesh Grids and Experimental Results n a NACA 64A00 CT6 Case Cmparisn f Drag Cefficient versus Angle f Attack fr Varius O- Mesh, Lenz-Mesh Grids and Experimental Results n a NACA 64A00 CT6 Case Cnvergence Histry f the Steady, Unsteady Cntinuus, and Unsteady Discrete Adjint Equatins. 93x33 Lens-Mesh. RAE 8 Airfil, M = 0.78, ω r = 0.0, α = Initial and Final Gemetry fr a RAE 8 Airfil at M = 0.78, ω r = 0.0, ᾱ = Cnvergence f the Maximum and Time-Averaged Drag Cefficients fr the RAE 8 a M = 0.78, ω r = 0.0, ᾱ = Initial and Final Lift Cefficient Versus Angle f Attack fr a RAE 8 Airfil at M = 0.78, ω r = 0.0, ᾱ = Initial and Final Drag Cefficient Versus Angle f Attack fr a RAE 8 Airfil at M = 0.78, ω r = 0.0, ᾱ = Initial and Final Pressure Cefficients at Varius Phases fr a RAE 8 Airfil at M = 0.78, ω r = 0.0, ᾱ = Pressure Cntur Plt fr RAE 8 Airfil at Phase = Pressure Cntur Plt fr RAE 8 Airfil at Phase = Pressure Cntur Plt fr RAE 8 Airfil at Phase = Pressure Cntur Plt fr RAE 8 Airfil at Phase = Lift Cefficient Versus Angle f Attack fr Varius Reduce Frequencies fr the RAE 8 Airfil at M = 0.78, ᾱ = 0, Fixed C l = Cnvergence f the Time-Averaged Drag Cefficient fr Varius Reduced Frequencies fr the RAE 8 Airfil at M = 0.78, ᾱ = 0, Fixed C l = Initial and Final Pressure Cefficients at Varius Phases fr a RAE 8 Airfil at M = 0.76, ω r = 0.0, ᾱ = xx

21 5.4 Initial and Final Pressure Cefficients at Varius Phases fr a RAE 8 Airfil at M = 0.80, ω r = 0.0, ᾱ = Initial and Final Gemetry fr a VR-7 Airfil at M = 0.75, ω r = 0.0, ᾱ = Cnvergence f the Maximum and Time-Averaged Drag Cefficients fr the VR-7 a M = 0.75, ω r = 0.0, ᾱ = Initial and Final Lift Cefficient Versus Angle f Attack fr a VR-7 Airfil at M = 0.75, ω r = 0.0, ᾱ = Initial and Final Drag Cefficient Versus Angle f Attack fr a VR-7 Airfil at M = 0.75, ω r = 0.0, ᾱ = Initial and Final Pressure Cefficients at Varius Phases fr a VR-7 Airfil at M = 0.75, ω r = 0.0, ᾱ = A Cmparisn f the Lift Cefficient Versus Angle f Attack fr Varius Reduced Frequencies and the Multipint Apprach fr the RAE 8 Airfil at M = 0.78, ᾱ = 0,Fixed C l = A Cmparisn f Final Airfil Gemetries Between the Initial Airfil, Airfils Designed with Varius Reduced Frequencies, and Airfil Designed using the Multipint Apprach. M = 0.78, ᾱ = 0, Fixed C l = A Cmparisn f the Cnvergence f the Time-Averaged Drag Cefficient fr Varius Reduced Frequencies and the Multipint Apprach fr the RAE 8 Airfil at M = 0.78, ᾱ = 0, Fixed C l = Cmparisn f Time-Averaged-Flw Steady-Adjint, Partial Unsteady, Full Cntinuus and Discrete Unsteady Gradients. RAE 8 Airfil, M = 0.78, α = 0, Fixed C l = Near Field and Far-Field Dmains Lcatin f Near Field Pressure and Adjint Remte Sensitivity Surce Terms Final Slutin Pressure Cnturs fr Ni-Bump Gemetry Initial and Final Lwer Ni-bump Wall Gemetry xxi

22 6.5 Initial Lwer Surface Pressure Distributins Final Lwer Surface Pressure Distributins Initial Upper Surface Pressure Distributins Final Upper Surface Pressure Distributins Verificatin Study: Target, Initial, and Final Near Field Pressure Distributin. Mach =.5, α = 0 deg Snic Bm Minimizatin: Target, Initial, and Final Near Field Pressure Distributin after 50 Design Cycles. Mach =.5, α =.75 deg., N Lift Cefficient and Thickness Rati Cnstraints Snic Bm Minimizatin: Initial and Final Airfil Shape After 50 Design Cycles. Mach =.5, α =.75 deg, Fixed Lift Cefficient = 0., Fixed Thickness Rati Snic Bm Minimizatin: Target, Initial, and Final Near Field Pressure Distributin after 50 Design Cycles. Mach =.5, α =.75 deg., Fixed Lift Cefficient = 0., Fixed Thickness Rati Snic Bm and Drag Minimizatin: Initial and Final Airfil Shape After 50 Design Cycles. Mach =.5, α = 0.89 deg, Fixed Lift Cefficient = 0.05, Fixed Thickness Rati Snic Bm and Drag Minimizatin: Target, Initial, and Final Near Field Pressure Distributin after 50 Design Cycles. Mach =.5, α = 0.89 deg., Fixed Lift Cefficient = 0.05, Fixed Thickness Rati Snic Bm and Drag Minimizatin: Pressure Cnturs f Final Design f the Upper Surface f the Wing. Mach =.5, α = 0.89 deg, Fixed Lift Cefficient = 0.05, Fixed Thickness Rati Snic Bm and Drag Minimizatin: Pressure Cnturs f Final Design f the Upper Surface f the Wing. Mach =.5, α = 0.89 deg, Fixed Lift Cefficient = 0.05, Fixed Thickness Rati Snic Bm and Drag Minimizatin: Surface Pressure Cefficient. Mach =.5, α = 0.89 deg, Fixed Lift Cefficient = 0.05, Fixed Thickness Rati xxii

23 Chapter Intrductin Engineers cntinually strive t imprve their designs, bth t increase their peratinal effectiveness and their market appeal. While sme qualities such as aesthetics are hard t measure, the factrs cntributing t peratinal perfrmance and cst are generally amenable t quantitative analysis. In the absence f an ptimizatin apprach, crucial decisins during a design prcess that culd ultimately affect the efficiency f a system are ften left t the judgment f experienced persnnel, researchers, and engineers. Since these decisins ultimately determine whether a cmpany fails r prspers, the intrductin f quantitative ptimizatin methds can be crucial t imprving its cmpetitive strength. In the design f a cmplex engineering system, relatively small design changes can smetimes lead t significant benefits. Fr example, small changes in wing sectin shapes can lead t large reductins in shck strength in transnic flw. Changes f this type are unlikely t be discvered by trial and errr methds, and it is in this situatin that ptimizatin methds can play a particularly imprtant rle. Beightler, in Fundatins f Optimizatin, [6] lists three imprtant steps n hw t ptimize a system: first, understand the system and the varius variables that influence it; secnd, decide n a measure f effectiveness that depends n the system variables that have a great influence n the efficiency f the system; third, chse thse values f system variables that prduce the ptimum system. In the first step, the knwledge f the inner wrkings f the system prvides the

24 CHAPTER. INTRODUCTION engineer with the fundamentals f hw varius variables interact with each ther t influence the system. In an aircraft design grup that cnsists f a wide variety f subgrups such as cmputatinal fluid dynamics, stability and cntrl, aerelastic and flutter analysis, engineers wrking in a particular grup shuld ideally have a general verview f the research and develpment cnducted in ther grups. Knwledge f the varius design parameters frm the numerus sub-grups and its influence n the perfrmance f the aircraft wuld help them t frmulate a better design prblem. The secnd step requires the designer t define a measure f the system effectiveness. In aircraft design, there are many figures f merit that measure the verall perfrmance f the aircraft. The cruise lift-t-drag rati, L, is particularly imprtant, D since it prvides an indicatin f the efficiency f the aircraft. An aircraft with high L D either prduces a large lift lad r lw drag. High lift capability allws an aircraft in cruise t carry a larger paylad. Lw drag translates t lw fuel cnsumptin and ultimately maximizes the aircraft range. The chice fr a figure f merit depends n the missin f the aircraft. In the last step, the designer prceeds t apply an ptimizatin algrithm t prduce an ptimum result that satisfies the cnstraints f the prblem. There are varius ptimizatin algrithms that ne can chse frm. The chice is highly dependent n the type f prblem: linear r nnlinear, number f design variables, number f figures f merit, and whether the prblem is uncnstrained r cnstrained. A desire t increase L withut cnstraints can lead t an increase in the aspect rati and thus D prduce an aircraft with a very large span fr a given wing area, which wuld in turn increase the wing weight. Therefre cnstraints can be as imprtant as the figure f merit in ptimizatin. In the last hundred years, aircraft designers have emplyed varius methds t arrive at their final designs. The current design prcess in a typical aircraft design cmpany fllws three steps. First, the cnceptual design grup prpses new cnfiguratins in anticipatin f new market requirements. The grup generally is cmpsed f twenty t thirty peple and wrks n an nging basis t create new cncepts and cnfiguratins. Radical new designs are predminantly presented t the market at this stage. Hwever, mst current designs are based n past designs that have been

25 3 successful in the market. Secnd, the preliminary design stage requires designers t meet aerelastic, flutter, stability and cntrl, and ther perfrmance criteria using basic empirical methds and lw-fidelity simulatin cdes. Apprximately three hundred peple require tw t three years and cnsume an estimated $ millin. This estimate is based n the design f a medium cmmercial jet transprt. A cmplete external cnfiguratin and majr lad and stresses are determined at the end f this stage. The aircraft perfrmance is als frzen at this stage and rders frm airline cmpanies fllw thereafter. Third, the detailed design stage invlves the many different sub-grups manufacturing and testing the varius cmpnents f the aircraft. Infrmatin is transfered between the grups t satisfy the verall system requirements. Apprximately three t five thusand persnnel wrk n arriving at the design gals guaranteed t airline cmpanies during the preliminary design stage. This stage nrmally requires arund $5 billin and spans between three t fur years. During the preliminary and detailed design stages, large amunts f wind tunnel data are cllected t imprve the existing design until a satisfactry design is btained within the scheduled time. Such a design prcess des nt allw fr vast numbers f design iteratins r variables t be cnsidered. With the intrductin f cmputatinal methds, researchers in the last thirty years have used these pwerful tls t prvide a greater understanding f the prblem and their ability t prvide analysis fr a greater number f designs. The ultimate aim f bth the traditinal and cmputatinal appraches was t imprve the design within a predetermined schedule but it was still left t the design grup t make the imprtant decisins that wuld ultimately ptimize their design. This raises the need t intrduce ptimizatin thery int aircraft design. Histrically, ptimizatin thery has its rts in the Renaissance perid. During this era the slutin t the Brachistchrne prblem attracted many great philsphers and thinkers. Galile guessed that the circular arc wuld prvide the shrtest time fr an bject t travel frm an elevated pint A t the grund at pint B. Then, in 694, Jhann Bernulli prved mathematically that the ptimum shape was in fact a cyclid.

26 4 CHAPTER. INTRODUCTION In the past, aircraft design techniques have depended n analytical slutins t arrive at ptimum shapes. Examples include the ptimum lift distributin f mnplane wings, and the Sears-Haack slutin fr the minimum wave drag f a bdy f revlutin in supersnic flw. In 945 Lighthill [50] first emplyed the methd f cnfrmal mapping t design tw-dimensinal airfils t achieve a desired target pressure distributin. These methds were restricted t incmpressible flw, but later McFadden [56] extended the methd t the cmpressible flw regime. Bauer et al. [5] and Garabedian et al. [] established an alternate methd based n cmplex characteristics t slve the ptential equatins in the hdgraph plane. This methd successfully prduced shck-free transnic flws. Cnstrained ptimizatin fr aerdynamic design was initially explred by Hicks et al. [7]. They used the finite-difference methd t evaluate sensitivities. Since then ptimizatin techniques fr the design f aerspace vehicles have generally used gradient-based methds. Thrugh the mathematical thery fr cntrl systems gverned by partial differential equatins established by Lins et al. [5], Pirnneau et al. [68] created a framewrk fr the frmulatin f elliptic design prblems. The apprach significantly lwers the cmputatinal cst and is clearly an imprvement ver classical finite-difference methds. Using cntrl thery the gradient is calculated indirectly by slving an adjint equatin. Althugh there is the additinal verhead f slving the adjint equatin, nce it has been slved the cst f btaining the derivatives f the cst functin with respect t each design variable is negligible. Cnsequently, the ttal cst t btain these gradients is independent f the number f design variables and amunts t the cst f ne flw slutin and ne adjint slutin. The adjint prblem is a linear partial differential equatin f lwer cmplexity than the flw slver. Jamesn was the first t apply cntrl thery fr transnic design prblems [33, 34, 35]. Subsequently, Jamesn et al. [37, 4, 4] pineered the shape ptimizatin methd fr Euler and Navier-Stkes prblems. Autmatic aerdynamic design f aircraft cnfiguratins has yielded ptimized slutins f wing and wing-bdy cnfiguratins by Reuther et al. [69, 7] and Burgreen et al. [0]. The injectin f ptimizatin thery int the design prcess and innvatins in cmputer technlgy have allwed researchers t attempt mre cmplex prblems.

27 5 This launched a snw-ball effect, where bigger and faster cmputers have allwed engineers t tackle mre sphisticated systems, which then ultimately required even faster cmputers. Future ptimizatin techniques will allw designers t reinvent the design prcess. The traditinal apprach discussed earlier will be replaced with a multidisciplinary apprach, where the varius disciplines f the design prcess will be cupled t allw changes in the cnfiguratin during the design prcess, a feature nt allwed in the current apprach. Lw-fidelity flw mdels cupled with ptimizatin techniques that may be able t identify glbal minimum such as genetic algrithms will be used during the cnceptual and preliminary design stages. Higher fidelity and gradient-based methds with multidisciplinary capabilities will be used during the detailed design stage. This will allw the designer t determine accurately the ptimal values f parameters such as the wing thickness that affect bth the aerdynamics and structural perfrmance f the aircraft. Designers will be able t examine a larger number f design cycles and this will allw them t tackle mre multidisciplinary prblems and perhaps arrive at radically new designs. Optimizatin will enhance the designers ability t prduce a better system instead f reducing their rle in the design prcess. Mrever, the designers prductivity will increase with the means t explre new appraches and designs. Often the sensitivity f the figure f merit with respect t the system variables is as imprtant as the ptimum result itself. Knwledge f the sensitivities prvides the designer with new insights n hw varius system variables affect the perfrmance f the system. This allws the designer t understand the system better and devise better prblem statements t tackle the issues at hand. The ptential benefit f using ptimizatin thery has nly been realized recently with the advent f faster methds t btain the gradient. The cntrl thery apprach t shape ptimizatin has revlutinized the cncept f utilizing cmputatinal fluid dynamics as a design tl. The ability t btain gradients cheaply has allwed researchers t attempt new prblems in aircraft design. This thesis cntributes t the develpment f the discrete and cntinuus adjint appraches, examining three distinctively separate prblems f current interest. Cntinuus Versus Discrete Adjint. The mtivatin fr the develpment

28 6 CHAPTER. INTRODUCTION f the discrete viscus adjint equatins is t study the trade-ffs between the cmplexity f frmulating the discrete adjint equatin and the accuracy f the resulting estimate f the gradient when cmpared t the cntinuus adjint apprach. The gal f this research is t evaluate bth appraches fr inverse design and drag minimizatin prblems. The gradients frm each apprach are cmpared t gradients acquired using the finite-difference and cmplex-step methds. Optimum Shape Design fr Unsteady Flws. Helicpter rtrs and turbmachinery blades perate in unsteady flw. The frward flight speed f a helicpter is restricted by the retreating blade stall limit and the advancing blade cmpressibility limit. In turbmachinery the flutter and stall bundaries limit the peratinal efficiency. It is apparent that the develpment f ptimum design methds fr unsteady flws is fundamental t imprving the perfrmance f a variety f aerspace systems. The unsteady adjint equatins are develped in this wrk. They are applied t the design f airfils underging a pitching scillatin t reduce the time-averaged drag cefficient while maintaining the time-averaged lift cefficient. Remte Inverse Design. A majr barrier t the develpment f supersnic business jets is the snic bm. A new apprach t tailr the aircraft shape t minimize the snic bm signature is develped in this research. The applicatin f the methd is aimed at mdificatins f the near field pressure signature f tw-dimensinal airfils and wings in three-dimensinal flw. A cupled remte inverse design and drag minimizatin bjective functin is als used t tailr the near field signature and cntrl the wave drag f three-dimensinal wings. The fllwing sub-sectins develp the mtivatin in greater depth fr each f the research gals. A descriptin f the prblem statement and the challenges faced in each area f study are discussed. In the chapters t fllw a detailed study f the frmulatin f the equatins and the results f the test cases are described and illustrated. The cnclusins discuss the ptential rle f these methds in the future design f cmplex systems.

29 .. THE DISCRETE AND CONTINUOUS ADJOINT APPROACHES 7. The Discrete and Cntinuus Adjint Appraches There are tw appraches t develp the adjint equatins: cntinuus r discrete. In the cntinuus adjint apprach the cntrl thery is applied t the differential equatins gverning the flw. The variatin f the cst functin and field equatins with respect t the flw field variables and design variables are cmbined thrugh the use f Lagrange multipliers, als called cstate r adjint variables. Cllecting the terms assciated with the variatin f the flw field variables prduces the adjint equatin and its bundary cnditin. The terms assciated with the variatin f the design variables prduce the gradient. The field equatins and the adjint equatin with its bundary cnditins must finally be discretized t btain numerical slutins. In the discrete adjint apprach, cntrl thery is applied directly t the set f discrete field equatins. The discrete adjint equatin is derived by cllecting tgether all the terms multiplied by the variatin δw i,j f the discrete flw variables. If the discrete adjint equatin is slved exactly, then the resulting slutin fr the Lagrange multipliers prduces an exact gradient f the discrete cst functin, and the derivatives shuld be exactly cnsistent with gradients btained by the cmplex-step methd. The discrete and cntinuus appraches have been pursued by a number f researchers using a wide variety f schemes and methds. Shubin and Frank [77] presented a cmparisn between the cntinuus and discrete adjint fr quasi-nedimensinal flw. A variatin f the discrete field equatins prves t be cmplex fr higher rder schemes. Due t this limitatin f the discrete adjint apprach, early implementatins f the discretizatin f the adjint equatin were cnsistent nly with a first rder accurate slutin f the flw equatin. Burgreen et al. [0] carried ut a secnd rder implementatin f the discrete adjint methd fr three-dimensinal shape ptimizatin f wings fr structured grids. Ellit and Peraire [7] used the discrete adjint methd n unstructured meshes fr the inverse design f multi-element airfils and wing-bdy cnfiguratins in transnic flw t prduce specified pressure distributins. Andersn and Venkatakrishnan [3] cmputed ptimum shapes fr inviscid and viscus flw n unstructured grids using

30 8 CHAPTER. INTRODUCTION bth the cntinuus and discrete adjint methds. Ill et al. [9] used the cntinuus adjint apprach t investigate shape ptimizatin n ne-and tw-dimensinal flws. Ta saan et al. [78] used a ne-sht apprach with the cntinuus adjint frmulatin. Kim et al. [47] cnducted an extensive gradient accuracy study f the Euler and Navier-Stkes equatins, cncluding that gradients frm the cntinuus adjint methd were in clse agreement with thse cmputed by the finite-difference methd. A cmparisn f bth the inviscid and viscus cntinuus and discrete adjint appraches was cnducted by Nadarajah and Jamesn [59, 60]. A subject f n-ging research is the trade-ff between the cmplexity f the adjint discretizatin, the accuracy f the resulting estimate f the gradient, and its impact n the cmputatinal cst t apprach an ptimum slutin. An advantage f the cntinuus adjint apprach is that it prvides researchers with an analytical frm f the equatins which can be studied t understand the nature f the equatin and its bundary cnditins. Stability analysis and analytical slutins f ne-dimensinal mdel prblems can be develped t understand the characteristics and behavir f the equatin. The scheme used t discretize and march the flw field equatins t a steady-state slutin can als be emplyed t slve the discretized cntinuus adjint equatins. This simplifies the develpment f cdes t implement the methd. A disadvantage f the discrete adjint methd is the cmplexity f applying cntrl thery t the discrete field equatins. The cmplexity f the discretizatin depends n the sphisticatin f the flw slver. Reuther [7] stated that fr methds that use GMRES (Generalized Minimum Residual) t slve very large linear algebra prblems, the task f develping the discrete adjint equatin is as easy as transpsing the flux Jacbian matrix. Fr explicit schemes, such as thse used in Jamesn s cdes, the develpment f the discrete adjint equatin prves t be a tedius task as shwn in Chapter 4. The cmplete discretizatin f all the terms in the flw slver requires an extensive amunt f algebraic manipulatin. The additin f the viscus flux further increases the cmplexity f deriving its adjint cunterpart. The cntinuus adjint methd is much simpler t implement fr explicit schemes. Anther imprtant issue f interest is the relative accuracy f the gradients derived by the tw methds. The cntinuus adjint apprach prvides the inexact

31 .. THE DISCRETE AND CONTINUOUS ADJOINT APPROACHES 9 gradient t the exact cst functin. On the ther hand, the discrete adjint apprach prvides the exact gradient t the inexact cst functin. Here, the exact cst functin is defined as the cntinuus frm f the cst functin, and the inexact cst functin as the value cmputed frm the discrete field equatins and bundary cnditins. The cntinuus gradient is calculated frm the discretized cntinuus adjint equatin, derived frm the cntinuus field equatins and cst functin. Therefre, the cntinuus gradient is nt necessarily exactly cnsistent with the cst functin which is evaluated numerically. The advantage f the discrete adjint methd is that the resulting discrete gradient is exactly cnsistent with the discrete cst functin. If the discrete gradient is driven t zer, then a lcal ptimum f the discrete cst is attained. Hwever, in the case f the cntinuus adjint methd, even if its gradient is driven t zer, the discrete bjective functin may nt have cnverged t the exact discrete minimum. If line searches are used in the ptimizatin algrithm, there may be a cnflict, where the discrete minimum in the search directin is incnsistent with the discretized cntinuus gradient. The discrete adjint apprach des nt suffer frm this incnsistency. In the limit as the mesh size is reduced, the cntinuus and discrete adjint methds shuld bth yield the exact gradient f the cntinuus cst functin. These questins prvide the mtivatin fr a cmparisn between the cntinuus and discrete adjint appraches and methds using gradients btained by finitedifferences r cmplex-steps. The specific bjectives f this wrk are: first, t review the frmulatin and develpment f the viscus adjint equatins fr bth the cntinuus and discrete apprach; secnd, t investigate the differences in the implementatin f bundary cnditins fr each methd fr varius cst functins; third, t cmpare the gradients f the tw methds t cmplex-step gradients fr inverse pressure design and drag minimizatin. Test cases are calculated fr varius prblems and grid sizes.

32 0 CHAPTER. INTRODUCTION. Optimum Shape Design fr Unsteady Flws The majrity f wrk in aerdynamic shape ptimizatin has been fcused n the design f aerspace vehicles in a steady flw envirnment. Investigatrs have applied these advanced design algrithms, particularly the adjint methd, t numerus prblems, ranging frm the design f tw-dimensinal airfils t full aircraft cnfiguratins t decrease drag, increase range, and reduce snic bm as shwn in Chapters 4 and 6. These prblems have been tackled using many different numerical schemes n bth structured and unstructured grids. Unlike fixed wing aircraft, helicpter rtrs and turbmachinery blades perate in unsteady flw and are cnstantly subjected t unsteady lads. Therefre, ptimal cntrl techniques fr unsteady flws are needed t imprve the perfrmance f helicpter rtrs and turbmachinery, and t alleviate the unsteady effects that cntribute t flutter, buffeting, pr gust and acustic respnse, and dynamic stall. Helicpter Rtrs. The flight envelpe f a helicpter rtr is set by the cmpressibility effects experienced by the advancing rtr blade and the retreating blade dynamic stall. As the helicpter frward flight speed is increased, the freestream velcity bserved in the reference frame f the advancing blade is that f the sum f the helicpter frward flight speed and the speed f the advancing blade. At high cruise speeds, the freestream Mach number bserved by the advancing blade reaches levels where lcal supersnic znes n the surface f the rtr blade are present. These regins usually terminate with a shck wave which causes a sudden increase in wave drag. During the retreating phase, the blade incidence appraches the stall angle, causing separatin t ccur n the upper surface f the blade, which leads t a lss f lift. At the 38th Cierva Memrial Lecture, Wilby [8] indicated that during the retreating blade stall it is the dramatic change in pitching mment mre than the lss f lift that impses a greater cnstraint n the design f the rtr blades. The change in pitching mment causes a large scillatry lad n the blade pitch cntrl mechanism which shrtens its fatigue life and thus increases the perating cst f the helicpter. Over the years, researchers in the field f rtrcraft aerdynamics have

33 .. OPTIMUM SHAPE DESIGN FOR UNSTEADY FLOWS develped ingenius methds t slve these prblems. Their effrts have develped blade prfiles that have a high maximum lift cefficient which allws the retreating blade t avid stall incidence, lw wave drag during the advancing phase, and lw pitching mment t reduce blade twist and cntrl lads. The intrductin f the swept tip t reduce the wave drag and reflex camber twards the rear prtin f the upper surface t increase negative lading thus reducing the pitching mment are just sme f the technlgies intrduced ver the last twenty years. Unfrtunately these imprvements nly cnsider the perfrmance f the blade prfile fr a selected number f flight cnditins. Turbmachinery. The flw thrugh a turbmachine is highly three-dimensinal and unsteady due t variatin f the hub-t-tip distance alng the blade rw, bladet-blade interference, and interactins between the rtrs and statrs. The nnlinear effects within the turbmachine generally have an unfavrable effect n the flutter characteristics f the blades. Kielb [45] recmmended five imprtant questins that must be cnsidered by a designer befre venturing int aerelastic analysis f turbmachineries: first, tw-dimensinal r three-dimensinal flw analysis; secnd, linear r nnlinear unsteady analysis; third, inviscid r viscus slutins; furth, single r multi-rw setups; and lastly, the type f aer/structural mdel. He prpses that, fr flutter and frced respnse prblems the nn-linear unsteady apprach is necessary, since linear analysis methds have resulted in significant errrs. Apart frm aermechanical issues, turbmachines generate secndary flws, which reduce the efficiency f the system. Vrtex shedding and wake-rtr interactins cntribute t these lsses. These challenges substantiates the need fr unsteady design ptimizatin techniques. A multipint design apprach is ne pssible technique fr the ptimizatin f blade prfiles in an unsteady flw envirnment. This apprach nly requires a small extensin f a steady flw design cde in rder t redesign a blade r airfil prfile fr multiple flw cnditins. A typical multipint design methd requires the fllwing three steps: first, steady flw slutins are cmputed fr a number f cases by varying freestream cnditins; secnd, the gradients fr each case are cmputed using either a classical finite-difference methd r using an adjint apprach; third, the gradients are weighted and the blade prfile is redesigned t satisfy the design bjective. Since

34 CHAPTER. INTRODUCTION the steady flw equatins are used t design the blades, uncertainties surrunding the perfrmance f these blades in an unsteady flw envirnment still prevail. Recently, the design f blade prfiles using unsteady techniques have been attempted. Diverse methds have been emplyed in the design f rtrcraft and turbmachinery blades. The fllwing are a selected number f papers n this tpic. Ghayur and Baysal [3] slved the unsteady transnic small disturbance equatin and its cntinuus adjint equatin t perfrm an inverse design at Mach 0.6. Aerdynamic shape ptimizatin f rtr airfils in an unsteady viscus flw was attempted by Yee et al. [8] using a respnse surface methdlgy. Here the authrs used an upwind-biased-factrized implicit numerical scheme t slve the RANS equatins with a Baldwin-Lmax turbulence mdel. A respnse surface methdlgy was then emplyed t ptimize the rtr blade. The bjective functin was a sum f the L/D at three different azimuth angles and was later redefined t include unsteady aerdynamic effects. Flrea and Hall [0] mdeled a cascade f turbmachinery blades using the steady and time-linearized Euler equatins. Gradients fr aerelastic and aeracustic bjective functins were then cmputed using the discrete adjint apprach. Bth the flw and adjint equatins were slved using a finite-vlume Lax-Wendrff scheme. The gradients were then used t imprve the aerelastic stability and acustic respnse f the airfil. In this wrk, a framewrk is develped t perfrm sensitivity analysis fr unsteady transnic flw, and t use this infrmatin t mdify the shape f the prfile in a favrable manner. Optimal cntrl f time dependent trajectries is generally cmplicated by the need t slve the adjint equatin in reverse time frm a final bundary cnditin. This requires infrmatin frm the trajectry slutin, which in turn depends n the cntrl derived frm the adjint slutin. The methd presented in this dissertatin is restricted t peridic unsteady transnic flws. The time accurate adjint equatins are based n Jamesn s cell-centered multigrid-driven fully-implicit scheme with upwind-biased blended first and third rder artificial dissipatin fluxes [36]. In this dissertatin bth the time accurate cntinuus and discrete adjint equatins are derived, and then used in the redesign f the RAE 8 and VR-7 rtr airfils underging a pitching scillatin t achieve lwer time-averaged drag, while

35 .3. THE REMOTE INVERSE DESIGN PROBLEM 3 keeping the time-averaged lift cnstant [6]. The fully unsteady design technique is cmpared t multipint and steady adjint appraches t gauge its effectiveness..3 The Remte Inverse Design Prblem A 00 Natinal Research Cuncil study [67] cncluded that,... the snic bm is the majr barrier t the develpment f the supersnic business jet and a majr, but nt the nly, barrier t the develpment f supersnic transprts with verland capability. The cmmittee als determined that there is a ptential market fr at least 00 supersnic business jets ver a ten year perid. The 8-5 passenger jets will prbably fly at apprximately Mach.8 with a range f nautical miles. Currently, the nly cmmercial supersnic aircraft in peratin is the Cncrde, built jintly by France and Britain. Twenty aircraft have been built since 973 and currently nly twelve are in peratin. The aircraft was built t cruise at Mach.0 with a ttal range f 4090 nautical miles at 60,000 feet. Designs f supersnic transprts f the future will benefit frm multidisciplinary ptimizatin techniques that were nt available during the design and cnstructin f the Cncrde. Apart frm lw snic bm capability, the new breed f supersnic transprts must als pssess superir perfrmance characteristics cmpared t its predecessrs t cmpete in the mdern cmmercial jet industry. These include imprvements in structures, aerdynamics, and prpulsin. In particular, the experience f NASA s HSR prgram suggests that it shuld be pssible t imprve the lift-t-drag rati frm the value f 7.5 attained by the Cncrde t arund 9 []. Befre the snic bm reductin prblem can be attempted, it is imprtant t have the capability t calculate the snic bm r grund pressure signature accurately. Fr typical cruise altitudes required fr aircraft efficiency, the distance frm the surce f the acustic disturbance t the grund is typically greater than 50,000 ft. A reasnably accurate prpagatin f the pressure signature can nly be btained with small cmputatinal mesh spacings that wuld render the analysis f the prblem intractable fr even the largest parallel cmputers. An apprach that has been