Issues in Optimization Jaroslaw Sobieski NASA Langley Research Center Hampton Virginia NASA Langley Research Center Hampton, VA 23681; MS240 LaRC/SMC/ACMB Copyright NASA, Jaroslaw Sobieski, 2003How to know whether optimization is neededHow 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 xPower Line Cable tout cable slack cable h Length(h) A(h) Volume(h) A L V min • Given: • Ice load • self-weight small • h/span small tout h slackWing Thin-Walled Box Lift •Top cover panels are compressed b thickness t •Buckling stress = f(t/b)2 b fewmany Cover weight Rib total weight Wing box weight min ribs ribsMultistage Rocket fueldrop when burned number of segments fuel weight segment junctions weight rocket weight 2 3 min weight to carry up = less fuel more weight to carry up • More segments (stages) = less • More segments = more junctions = • Typical optimum: 2 to 4. Saturn VUnder-wing Nacelle Placement longer body to rotate for take-off = more weight fore nacelle aft shock wave drag nacelle wing underside shock wave impinges on forward slope = drag moves with it = larger tail (or drag weight Range max• Inlet ahead of wing max. depth = • Nacelle moved aft = landing gearNational 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 collectedmax tax paid on $ earned 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” • What to do: • If we are left of max = increase taxes • If we are right of max = cut taxesNothing to Optimize Rod P Newton A cm2 • Monotonic trend • No counter-trend σ σ allowable • Nothing to optimize N/cm2 AVarious types of design optimaDesign Definition: Sharp vs. constraints - 0 contours Shallow - bad side of 1 2 12 bandpoint constraints - 0 contours XX Objective Constraint descent • Near-orthogonal intersection defines a design point • Tangential definition identifies a band of of designs XMultiobjective Optimization trade-both Q = 1/(quality &f1 off both performance &f2 comfort)$ 1 4$ 4 pareto-frontier 2 3 3 2 design & manufacturing sophistication 1 Q pareto-optimum V&W R&RA Few Pareto-Optimization Techniques • Reduce to a single objective: F = Σi wi fi where w’s are judgmental weighting factors • Optimize for f1; Get f*1;; •Set a floor f1 >= f*i ; Optimize for f2; get f2 ; • Keep floor f1, add floor f2 ; Optimize for f3 ; • 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 preferencesImparting Attributes by Optimization • Changing wi in F = Σi wi fi modifies the design within broad range • Example: Two objectives • setting w1 = 1; w2 = 0 produces design whose F = f1 • setting w1 = 0; w2 = 1 produces design whose F = f2 • setting w1 = 0.5; w2 = 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 •Nonconvexobjective 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 locatingglobal 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” algorithmfinds a better minimumA “shotgun” approach:• Use a multiprocessor computer• Start from many initial designs• Execute multipath optimization• Increase probability of locatingglobal 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” algorithmfinds a better minimumWhat to do about it A “shotgun” approach: X F Start M1 M2<M1 Tunnel Opt. •“Tunneling” algorithmfinds a better minimum • 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.Using Optimization to Impart Desired AttributesLarger 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. npas ft70 • Four attributes: • structural mass • 1st bending frequency • tip rotation • internal volumeCase : F = w1 (M/M0) + w2 (Rotat/Rotat0) Normalized Mass M/M0 •Broad variation: 52 % to 180 % Rotation weight factor Mass weight factor Rotat = wingtip twist angleOptimization Crossing the Traditional Walls of SeparationOptimization 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 speedTwo 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