LaRC/SMC/ACMBNASA Langley Research CenterHampton, VA 23681; MS240757-864-2799; [email protected] NASA, Jaroslaw Sobieski, 2003Issues in OptimizationJaroslaw SobieskiNASA Langley Research CenterHampton VirginiaHow to know whetheroptimization is neededHow to recognize thatthe problem at hand needsoptimization.• General Rule of the Thumb: there must be at least two opposing trends as functions of a design variablexAnalysisxf1f2f1f2f1f2Power Line Cabletout cableslack cablehhLength(h)A(h)Volume(h)• Given:• Ice load• self-weight small• h/span smallALVmintoutslackWing Thin-Walled BoxLift•Top cover panelsare compressedbthickness t•Buckling stress= f(t/b)2bfewribsmanyribs Cover weightRib total weightWing box weightminMultistageRocketSaturn Vfueldrop whenburnednumberof segmentsfuel weightsegmentjunctionsweightrocket weight23min• More segments (stages) = lessweight to carry up = less fuel• More segments = more junctions =more weight to carry up• Typical optimum: 2 to 4.Under-wing NacellePlacementshock wavedragnacellewingunderside• Inlet ahead of wing max. depth =shock wave impinges on forwardslope = drag• Nacelle moved aft = landing gearmoves with it = larger tail (or longer body to rotate for take-off =more weightnacellefore aftdragweightRangemaxNational Taxationrevenue collectedincentive to worktax paid on $ earnedaveragetax rate0 %100 %max• More tax/last $ = less reason to strive to earn• More tax/$ = more $ collected per “unit of economic activity”National Taxationrevenue collectedincentive to worktax paid on $ earnedaveragetax rate0 %100 %max• 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 OptimizeP NewtonA cm2σ N/cm2A• Monotonic trend• No counter-trend• Nothing to optimizeσ allowableRodVarious types of design optimaDesign Definition: Sharp vs.ShallowXXXconstraints - 0 contours12constraints - 0 contours12ObjectiveConstraint- bad side of• Near-orthogonal intersectiondefines a design point• Tangential definition identifies a band of of designsbandpointdescentMultiobjective Optimizationdesign & manufacturing sophistication$Q = 1/(quality & performance & comfort)V&W R&Rboth both trade-off$Qpareto-frontierpareto-optimum12341234f1f2A Few Pareto-OptimizationTechniques• Reduce to a single objective: F = Σi wi fiwhere 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 fi independently; Get n optimal designs;Find a compromise design equidistant from all the above.• Pareto-optimization intrinsically depends on judgmentalpreferencesImparting Attributes byOptimizationF = Σi wi fi• Changing wi inmodifies 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 wi as control, optimization serves as a tool to “steer” the design toward a desired behavior or having pre-determined, desired attributes.Optimum: Global vs. LocalWhy the problem:constraintObjective contoursX1X2 LG•Nonconvexobjective orconstraints(wiggly contours)massSpring k N/cmddPP = p cos (ωt)resonancek•Disjoint design space• Local information, e.g., derivatives, does not distinguishlocal from global optima - the Grand Unsolved Problem in AnalysisWhat to do about itXFStartM1M2<M1Tunnel•“Tunneling” algorithmfinds a better minimumOpt.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 itXFStartM1M2<M1Tunnel•“Tunneling” algorithmfinds a better minimumOpt.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. shotgunMultiprocessorcomputerWhat to do about itXFStartM1M2<M1Tunnel•“Tunneling” algorithmfinds a better minimumOpt.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.Using Optimization to Impart Desired AttributesLarger scale example: EDOF = 11400; Des. Var. = 126; Constraints = 24048;Built-up, trapezoidal, slender transport aircraft wing70 ft span• 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.• Four attributes: • structural mass• 1st bending frequency• tip rotation• internal volumeCase : F = w1 (M/M0) + w2 (Rotat/Rotat0)NormalizedMass M/M0•Broadvariation:52 % to 180 %Rotationweight factorMassweight factorRotat = wingtip twist angleOptimization Crossing the Traditional Walls of SeparationOptimization AcrossConventional BarriersVehicle designFabricationdata• Focus on vehicle physicsand variables directlyrelated to it• E.g, range; wing aspect ratio• Focus on manufacturingprocess and its variables• E.g., cost; riveting head speedTwo Loosely Connected Optimizations•Seek design variablesto maximize performanceunder constraints of:PhysicsCostManufacturing difficulty • Seek process variablesto reduce the fabrication cost.The return on investment (ROI) is a unifying factorROI = f(Performance, Cost of Fabrication)Integrated OptimizationROI = f(Range, Cost of Fabrication)• Required: Sensitivity analysis on both sides∂Range/