Issues in Optimization Jaroslaw Sobieski NASA Langley Research Center Hampton Virginia NASA Langley Research Center LaRC/SMC/ACMB Copyright NASA, Jaroslaw Sobieski, 2003 How to know whether optimization is needed How to recognize that the problem at hand needs optimization. General Rule of the Thumb: there must be at least two opposing trends as functions of a design variable Analysis x f1 f1 f2 f2 f1 f2 x Power Line Cable tout cable slack cable h Length(h) Given: A(h) Ice load Volume(h) self-weight small h/span small A L V min tout h slack Wing Thin-Walled Box Lift Top cover panels are compressed b thickness t Buckling stress = f(t/b) 2 b many Rib total weight min few Cover weight Wing box weight ribs ribs Multistage Rocket drop when burned number segment junctions weight rocket weight 2 3 min fu el fuel weight of segments More segments (stages) = less weight to carry up = less fuel More segments = more junctions = more weight to carry up Typical optimum: 2 to 4. Saturn V more weight fore nacelle aft Under-wing Nacelle Placement shock wave drag nacelle wing underside Inlet ahead of wing max. depth = shock wave impinges on forward slope = drag Nacelle moved aft = landing gear l drag weight Range max moves with it = larger tail (or onger body to rotate for take-off = National Taxation tax paid on $ earned revenue collected max incentive to work 0 % average 100 % tax rate More tax/last $ = less reason to strive to earn More tax/$ = more $ collected per unit of economic activity National Taxation revenue collected 0 % max tax paid on $ earned incentive to work average 100 % tax rate More tax/last $ = less reason to strive to earn More tax/$ = more $ collected per unit of economic activity What to do: If we are left of max = increase taxes If we are right of max = cut taxes Nothing to Optimize Rod P Newton A cm 2 Monotonic trend No counter-trend σ σ allowable Nothing to optimize N/cm 2 A Various types of design optima Design Definition: Sharp vs. constraints - 0 contours Shallow - bad side of 1 2 constraints - 0 contours 1 2 band point X X Constraint descent Objective Near-orthogonal intersection defines a design point Tangential definition identifies a band of of designs X Multiobjective Optimization trade- both Q = 1/(quality & f 1 off both performance & f 2 comfort) $ 1 4 $ 4 pareto-frontier 2 3 3 2 design & manufacturing sophistication 1 Q pareto-optimum V&W R&R A Few Pareto-Optimization Techniques Reduce to a single objective: F = Σ w i f ii where w s are judgmental weighting factors Optimize for f 1 ; Get f* 1; ; Set a floor f 1 >= f* i ; Optimize for f 2 ; get f 2 ; Keep floor f 1 , add floor f 2 ; Optimize for f 3 ; Repeat in this pattern to exhaust all f s; The order of f s matters and is judgmental Optimize for each f independently; Get n optimal designs; i Find a compromise design equidistant from all the above. Pareto-optimization intrinsically depends on judgmental preferences Imparting Attributes by Optimization Changing w i in F = Σ i w i f i modifies the design within broad range Example: Two objectives setting w 1 = 1; w 2 = 0 produces design whose F = f 1 setting w 1 = 0; w 2 = 1 produces design whose F = f 2 setting w 1 = 0.5; w 2 = 0.5 produces design whose F is in between. Using w as control, optimization serves as a tool i to steer the design toward a desired behavior or having pre-determined, desired attributes. Optimum: Global vs. Local X2 Why the problem: Objective contours Nonconvex objective or constraint constraints (wiggly contours) X1 L G resonance d Spring k N/cm Disjoint design mass space d P P = p cos (ωt) k Local information, e.g., derivatives, does not distinguish local from global optima - the Grand Unsolved Problem in Analysis Use a multiprocessor computer Start from many initial designs Execute multipath optimization Increase probability of locating global minimum Probability, no certainty Multiprocessor computing = analyze many in time of one = new situation = can do what could not be done before. What to do about it A shotgun approach: F Start M1 Opt. Tunnel M2<M1 X Tunneling algorithm finds a better minimum A shotgun approach: Use a multiprocessor computer Start from many initial designs Execute multipath optimization Increase probability of locating global minimum Probability, no certainty Multiprocessor computing = analyze many in time of one = new situation = can do what could not be done before. What to do about it F Start M1 Opt. Tunnel shotgun Multiprocessor computer M2<M1 X Tunneling algorithm finds a better minimum What to do about it F Start M1 Opt. Tunnel A shotgun approach: Use a multiprocessor computer Start from many initial designs Execute multipath M2<M1 optimization X Increase probability of locating global minimum Tunneling algorithm Probability, no certainty finds a better minimum Multiprocessor computing = analyze many in time of one = new situation = can do what could not be done before. Using Optimization to Impart Desired Attributes Larger scale example: EDOF = 11400; Des. Var. = 126; Constraints = 24048; Built-up, trapezoidal, slender transport aircraft wing Design variables: thicknesses of sheet metal, rod cross-sectional areas, inner volume (constant span and chord/depth ratio Constraints: equivalent stress and tip displacement Two loading cases: horizontal, 1 g flight with engine weight relief, and landing. n p a s f t 7 0 Four attributes: structural mass 1st bending frequency tip rotation internal volume Case : F = w 1 (M/M 0 ) + w 2 (Rotat/Rotat 0 ) Normalized Mass M/M 0 variation: 52 % to 180 % weight factor Mass weight factor Broad Rotation Rotat = wingtip twist angle Optimization Crossing the Traditional Walls of Separation Optimization Across Conventional Barriers data Vehicle design Fabrication Focus on vehicle physics Focus on manufacturing and variables directly related to it process and its variables E.g, range; E.g., cost; wing aspect ratio riveting head speed Two Loosely Connected Optimizations Seek design variables Seek process variables to maximize performance to reduce the fabrication cost. under constraints of: Physics Cost Manufacturing difficulty The return on investment (ROI) is a unifying factor ROI = f(Performance, Cost of Fabrication) Integrated Optimization Required: Sensitivity analysis on both sides ?Range/ ?(AspectRatio) ?Cost/ ?(Rivet head speed) ?(Rivet head speed)/ ?(AspectRatio) ROI = f(Range, Cost of Fabrication) ?ROI/ ?AspectRatio = ?ROI/ ?Cost ?Cost/ ?(Rivet h.s.) ?(Rivet h.s)/ ?(AspectRatio) + + ?(ROI)/?Range ?Range/?(AspectRatio) Integrated Optimization Design < --- > Fabrication Given the derivatives on both sides Design Fabrication Unified optimization may be constructed to seek vehicle design variable, e.g., AspectRatio, for AR maximum ROI incorporating AR effect on Range and on Opt. fabrication cost. ROI ROI Range Cost Range; Cost Optimization Applied to Complex Multidisciplinary Systems Multidisciplinary Optimization MDO Coupling Decomposition What to optimize for at the discipline level Approximations Sensitivity Wing drag and weight both influence the flight range R. R is the system objective direct i P P Displ a l i that affect drag Displ Wing - structure Wing - aerodynamics Loads acements a = sweep angle Structure influences R by y by weight ndirectly by st ffness that affect displacements Loads & acements must be consistent R = (k/Drag) LOG [( W o + W s + W f )/ (W o + W s )] Dilemma: What to optimize the structure for? Lightness? Displacements = 1/Stiffness? An optimal mix of the two? Trade-off between opposing objectives of lightness and stiffness Weight Displacement Weight Displacement ~ 1/Stiffness Thickness limited by stress Wing cover sheet thickness Lightness Stiffness What to optimize for? Answer: minimum of f = w1 Weight + w2 Displacement vary w1, w2 to generate a population of wings of diverse Weight/Displacement ratios Let system choose w1, w2. Approximations Why Approximations: Analyzer Analyzer Approximate Model Human judgment problems ice for large problems to reduce and control cost $$ cents a.k.a. Surrogate Models Optimizer Optimizer OK for sma Now-standard pract ll Design of Experiments(DOE) & Response Surfaces (RS) RS provides a domain guidance , rather than local guidance, to system optimizer DOE Placing design points in design space in a pattern Example: Star pattern (shown incomplete) RS X1 X2 F(X) F(X) = a + {b} {X} + {X} [c]X quadratic polynomial hundreds of variables Response Surface Approximation A Response Surface is an n- dimensional hypersurface relating n Design of Experiments (DOE) methods used to disperse data points in design space. More detail on RS in section on Approximations inputs to a single response (output). V a r i a b l e 1 V a r i a b l e 2 Respon s e BLISS 2000: MDO Massive Computational Problem Solved by RS (or alternative approximations) or di or di or di System optimization X1 X2 ) X1 X2 ) X1 X2 ) RS RS i in parallel I n s t a n t a n e o u s r e s p o n s e MC D A T A B A S E Optimization of subsystem scipline Analysis of subsystem scipline Optimization of subsystem scipline F(X F(X F(X Precompute off-l ne cloud Radical conceptual simplification at the price of a lot more computing. Concurrent processing exploited. Coupled System Sensitivity Consider a multidisciplinary Y A system with two subsystems A and B (e.g. Aero. & Struct.) system equations can be written in symboli A[( X ,Y ),Y ] = 0 A B A B[( X ,Y ),Y ] = 0 B A B rewrite these as follows Y A = Y A ( X ,Y ) A B Y B = Y B ( X ,Y A ) B A B A X B X B Y B Y A Y c form as these governing equations define as implicit functions. Implicit Function Theorem applies. Coupled System Sensitivity - Equations These equations can be represented in matrix notation as Y dY dX ? ? ? ?? ? ? ? ? = ? ? ? A A A I ? Y A X ? ? A ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Y B Y dY? ? ? ? ? ? ? ? ? ? ? ? B B I? 0 Y dX different A A same Y dY? ? ? ?? ? Right Hand Sides A A I ? 0 = ? ? ? ? ? ? matrix ? ? ? ? ? ? ? ? ? ? ? ? ? Y dX ? ? ? Y? ?X B B B Y dY dX ? ? ? ? ? ? ? ? ? ? ? ? B B B I ? B Y A Total derivatives can be computed if partial sensitivities computed in each subsystem are known Linear, algebraical equations with multiple RHS Example of System Derivative for Elastic Wing Example of partial and system sensitivities Angle of atta ck deg 10 id wing partial derivative 7.0 Based on rig Based on elastic wing system derivative 4.0 -40 -30 -20 -10 0 … chord sweep angle -deg In this example, the system coupling reverses the derivative sign X Flowchart of the System Optimization Process System Analysis α β γ System Sensitivity Analysis α β γ Start Sensitivity solution Approximate Analysis Optimizer X Y γ Y α β Y β Stop System Internal Couplings Co u p l i n g B r e a d t h Quantified All-in-One Decompose ( ( D e c o m p o s e ) ) ( D e c o m p o s e ) Strength: relatively large ? YO/ ?YI Breadth: {YO} and {YI} are long [? YO/ ?YI] large and full Coupling Strength A Few Recent Application Examples Multiprocessor Computers create a new situation for MDO Supersonic Business Jet Test Case Structures (ELAPS) ) ) ) Aerodynamics (lift, drag, trim supersonic wave drag by A - Wave Propulsion (look-up tables Performance (Breguet equation for Range Some stats: Xlocal: struct. 18 aero 3 propuls. 1 X shared: 9 Y coupl.: 9 Examples: Xsh - wing aspect ratio, Engine scale factor Xloc - wing cover thickness, throttle setting Y - aerodynamic loads, wing deformation. System of Modules (Black Boxes) for Supersonic Business Jet Test Case Struct. Perform. Aero Propulsion Data Dependence Graph RS - quadratic polynomials, adjusted for error control 0 1 1 10 Flight Range as the Objective Normalized Cycles 0.2 0.4 0.6 0.8 1.2 1.4 2 3 4 5 6 7 8 9 Series1 Series2 RS 1 10 1 0 Analysis Histogram of RS predictions and actual analysis for Range 12 Air Borne Laser System Design: another application of the similar scheme Beam Control System System Level Design Turret Assembly Boeing Large Optics CDR 25-27 April Four Axis gimbals Transfer optics 747F Aircraft -t Beam Transfer Assembly Boeing Sensor Suite Active Mirrors CDR 29 Feb - 3 Mar Illuminators Electronics BMC 4 I Boeing Software/Processors Chemical Oxygen Iodineen Laser (COIL) TRW 21-23 March 8-10 March 500000 A Candidate for Shuttle Replacement: Two-stage Orbital Transport Collaborated with GWU, 2 nd stage separates and continues and ASCAC Branches: System to destination Analysis and Vehicle Analysis LB x UB 900000 810000 720000 630000 450000 360000 270000 180000 90000 0 RS True Result sample: System Weight (lb) Fly-back Variance over MDO iterations. booster Initial design was infeasible NVH Model A Body-In-Prime (BIP) Model - Trimmed Body Structure without the powertrain and suspension subsystems MSC/NASTRAN Finite Element Model of 350,000+ edof; Normal Modes, Static Stress, & Design Sensitivity analysis using Solution Sequence 200; 29 design variables (sizing, spring stiffness); Computational Performance Fine grain parallelism of Crash Code was an important factor in reducing the optimization procedure total elapsed time: 291 hours cut to 24 hours for a single analysis using 12 processors. Response Surface Approximation for crash responses that enabled coarse grain parallel computing provided significant reduction in total elapsed time: 21 concurrent crash analysis using 12 processors each over 24 hours (252 processors total). For effective utilization of a multiprocessor computer, user has to become acquainted with the machine architecture. 255 days of elapsed computing time cut to 1 day Computer Power vs. Mental Power Quantity vs Quality Invention by Optimization? P A I b P {X} = {A, I, b}; Minimize weight; See b Zero Optimization transformed frame into truss A qualitative change Why: structural efficiency is ranked: Tension best Compression Bending worst If one did not know this, and would not know the concept of a truss, this transformation would look as invention of truss. Optimizing Minimum Drag/Constant Lift Airfoil for Transonic Regime New (he use a fil i Base If this was done before Wh e & w Drag minimized while holding constant lift by geometrically adding the base airfoils. Each base airfoil had some aerodynamic merit Result: a new type, flat-top Whitcomb airfoil . itcomb invented the flat-top airfoil nd tunnel), this would look like an invention. Continuous quantitative transformation vs. conceptual quantum jump Common feature in both previous examples: Variable(s) existed whose continuous change enabled transformation to qualitatively new design X no seed for 2nd wing OK Second wing may Counter-example: wither away Optimization may reduce but cannot grow what is not there, at least implicitly, in the initial design. Technology Progress: Sigmoidal Staircase piston/ vacuum tube/transistor /digital camera Performance jet film Time exhaustion rapi inception d advance optimizat Optimization assists in rapid advance phase ; ion Human creativity shifts gears to next step Augmenting number crunching power of computer with good practice rules members In compression constraints because the slender members are not defined until the end. Subtle point: it is difficult to keep the analysis valid when the imparted change calls for new constraints. Topology Optimization Modern version of what Michelangelo said 500 years ago: (paraphrased) to create a sculpture just remove the unnecessary material QuickTime and a Base material TIFF (Uncompressed) decompressor are needed to see this picture. This optimization cannot include buckling Topology optimization removes pixels from base material Topology Optimization - 2 Base material theoretical as built QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. members This optimization can not include buckling In compression constraints because the slender members do not emerge as such until the end. Subtle point: it is difficult to keep the analysis valid when the imparted change requires new constraints. Design by Rules Tension Bending Compression Structural efficiency ranking Structural weight String Truss Problem Solution Problem Solution Problem Solution narrow Problem obstacle Complications Solution 1 Solution 2 .things are getting too complicated Human eye-brain apparatus excels in handling geometrical complexities amplified by abundance of choices By some evidence, eye-brain apparatus may process 250 MB data in a fraction of a second. Optimization in Design Process feedback Need or Concept Design Design Proto- Production ld Oppor- tunity Preliminary Detailed type Qualitative Quantitative Firm footho research extension trend Optimization most useful where quantitative content is high Closure Optimization became an engineer s partner in design It excels at handling the quantitative side of design It s applications range from component to systems It s utility is dramatically increasing with the advent of massively concurrent computing Current trend: extend optimization to entire life cycle with emphasis on economics, include uncertainties. Engineer remains the principal creator, data interpreter, and design decision maker. LaRC/SMC/ACMB Copyright NASA, Jaroslaw Sobieski, 2003