6.888/ESD.77J Multidisciplinary System Multidisciplinary Design Optimization of the Space Shuttle External Tank Mo-Han Hsieh Christopher Michael Lawson May 7th, 24
Outline Motivation Problem Statement Model Formulation Design of Experiment Single Objective Optimization Multi-objective Optimization Conclusions
Motivation The safety and profitability of the Space Transportation System (STS) has been increasingly scrutinized. Decision makers are interested in the future fiscal viability of manned spaceflight. Use the simplified External Tank of the STS as a learning tool to practice our MSDO knowledge. Feedback to the space industry. Source: NASA
Problem Statement Designing the External Tank of the STS. Multidisciplinary (e.g. structures, aero., finance, etc.) Single Objective: To maximize Return on Investment. ROI = (Rev (Cfix + TC)) / (Cfix + TC) where ROI: Return on Investment. Rev: Revenue from Customers. C fix : Fixed Cost. TC: Tank Cost. Multi-Objective: To maximize ROI while minimizing TC.
Design Variables Approximation: Cone, Cylinder, and Hemisphere. Design Variables: Height of Nose Cone: H Length of Cylindrical Body: L Radius of Hemisphere: R Thickness of Hemisphere: t hs Thickness of Cylinder: t cy H L t co t cy Thickness of Cone: t co R t hs
Design Constraints Total Volume: V total 3, (m 3 ). Equivalent Stress Cone: F co 4 (kn/cm 2 ). Cylinder: F cy 4 (kn/cm 2 ). Hemisphere: F hs 4 (kn/cm 2 ). Vibration of st Bending Mode: Vibration Factor.8. http://www.kettering.edu/ ~drussell/demos/membranesquare/ Square.html
Benchmarking Simulation Model Constraint on Total Volume: V total 3, m 3. Equivalent Stress: Cone: 4 kn/cm 2. Cylinder: 4 kn/cm 2. Hemisphere: 4 kn/cm 2. Revenue: $2, /mission/kg External Tank in Use Total Volume: 2,4 m 3 Pressure of Cryogenic Fuel: Cone: 2 kn/cm 2. Cylinder: 25 kn/cm 2. Hemisphere: 25 kn/cm 2. Avg. Revenue: $23, /mission/kg. H L R t co t cy t hs Tank Cost: $.8 million. Tank Weight: 3, kg. Avg. Payload: 8, kg. H = 52 cm. L = 2792 cm. R = 42 cm. tco = tcy = ths =.635 cm. Benchmark for postoptimization Ref. [, 2]
Model Fidelity Model from an Excel data file. Low fidelity: simplified approximation of the physics involved. Useful to understand the performance tradeoffs of the External Tank features. Leads to better results in later design.
N 2 Diagram of the System Design Variables Stress Surface and Volume Aero. Drag Penalty Weight and Cost of Material Vibration on st Bending Mode Cost of Seams Tank Cost Return on Investment. H, 2. L, 3. R, 4. t co, 5. t cy, 6. t hs, 3, 4, 5, 6, 2, 3, 3 4, 5, 6, 2, 3, 5 4, 5, 6. g2, 2.g3, 3. g4 (constrains). Surface 2. Volume 3. Seam Length 4. g (constraint) 3. Delta Payload. Tank Weight 2. Material Cost 2. g5 (constraint). Total Seam Cost. Total Tank Cost. Payload Launched 2. Return on Investment (ROI)
Design of Experiment (I) Orthogonal Array - OA(8, 7, 3, 2). 8 experiments and 3 levels. Design factors and their levels: Factor Level Level Level 2 H 64 78 93 L 477 49 55 R 45 48 4 t hs.36.78.2 t cy.62.66.7 t co.62.73.84-2 -8-4 4 8 2 - -2-3 -4-5 -6 Initial Guess! ROI =.4-7
Design of Experiment (II) Mean response of ROI =.38. Factor effect of ROI: Level H L R t hs t cy t co.7.29.7.42.5.24.3..... 2 -. -.29 -.8 -.42 -.5 -.25 H 64 L 477 R 45 t hs.36 t cy.62 t co.62 Initial Start Point: x
Single Objective Optimization Objective: maximize ROI Gradient Based Method (SQP) Initial Start Point (not all constraints are satisfied.) H 64. L 477. R 45. = ths.36 t cy.62 tco.62 ROI =.24-2 -8-4 4 8 2 - -2-3 -4-5 -6-2 -8-4 4 8 2 - -2-3 -4-5 -6 Optimal Design H 77.3 L 5279.4 R 388.7 = ths.342 t.5892 cy tco.5967 ROI =.26-7 -7-8 -8
Sensitivity Analysis (with respect to design variables) J* =.6772 4.433 8.5922.538 3.674.7947 H L R t hs t cy t co Graphical Representation of Normalized Sensitivity Graphical Representation of Normalized Sensitivity H L R ts tcy tco 2 3 4 5 6 7 8 9 Magnitude % change in objective per % change in design variable. The optimum is most sensitive to the radius (R). i.e. % increase of R will result in an 8.59% decrease of ROI.
Sensitivity Analysis (with respect to fixed parameters) Fixed Parameter Original Value Cost of Material (dollar/kg): c unit 6 Cost of Seam (dollar/m): c seam Fixed Cost of Launch (dollar/kg): k 2 Payload (Oxygen): p 5 Payload (Hydrogen): p 2 5 2 J p J c J c = J J J unit seam 2 k p p =.486.486..437.97 The optimum is most sensitive to p payload. i.e. % increase of p will result in an.44% increase of ROI.
Scaling of Design Variable Design Variable Hessian Scaling Scaled Design Variable New Hessian H 8.68E-8-4 4 H -2.5779 L -4.32E- -5 5 L -.3528 R.4E-5-2 2 R.246 ths 2.32E-5-2 2 t hs.2292 tcy -4.28E-4-2 2 t cy -5.68 tco 4.2E-7-3 3 t co.5882 Improvement of the Optimal Solution: from H 77.3 L 5279.4 R 388.7 = ts.342 t cy.5892 tco.5967-2 -8-4 4 8 2 - -2-3 -4-5 -6-7 to H 69.6 L 647.5 R 356.5 = ts.32 t cy.543 tco.552 ROI =.26 ROI =.25-8 -2-8 -4 4 8 2 - -2-3 -4-5 -6-7 -8 OK.
Genetic Algorithm Used GA toolbox in MATLAB. Alternative crossover and selection rules. An exhaustive search was made of the design space. More than 5 million runs! (25 hours 5 days.) ROI =.346 ( 68% improvement over the gradient search method.).4.346 Return on Investment (ROI).3.2...25 SQP GA
Multi-objective Optimization Objectives: ROI (J) & Total Tank Cost (J2) Determine the Pareto front by Adaptive Weighted Sum Method (de Weck & Kim) (AWSM): min J { ( ) ( ) ( ) } w = α J x + α J 2 x Impose additional inequality constraints J ( x) P J 2 x ( x) P y 2 δ Advantages of AWSM: (Ref. [3]). Produces evenly distributed solutions. 2. Finds Pareto optimal in non-convex regions (NCR). 3. Neglects non-pareto optimal solutions in NCR. δ 2
Pareto Optimal Solutions ROI Utopia: ROI =.346 TC = $632,83-2 -8-4 4 8 2-2 -8-4 4 8 2 - - -2-2 -3-3 -4-4 -5-5 -6-6 -7-7 -8 Utopia -8 TC Utopia: ROI =.638 TC = $544,32
65 Sensitivity of Pareto Front to Weighting Factors ROI =.28 Total Tank Cost 63 6 59 57 55 α =. -2-8 -4 4 8 2 - -2-3 -4. 53.5.5 2 2.5 Sensitivity 2.5 2.5 2-5 -6.5.5 α =. -7-8. Return on Investment (ROI)..2.3.4.5
Design Evolution STS External Tank in Use Post-Optimization Benchmarking: Initial Design: ROI =.24 (not all constraints are satisfied.) SQP: ROI =.26 ROI =.439 (violates volume constraint.) Scaling: ROI =.25 GA: ROI =.346-2 -8-4 4 8 2-2 -8-4 4 8 2-2 -8-4 4 8 2-2 -8-4 4 8 2 - - - - -2-2 -2-2 -3-3 -3-3 -4-4 -4-4 -5-5 -5-5 -6-6 -6-6 -7-7 -7-7 -8-8 -8-8
Summary and Future Development A better understanding of the MSDO techniques. (e.g. SQP, GA, Sensitivity Analysis, Scaling, etc.) A better understanding of the performance tradeoffs of the STS External Tank features. Future Development Increase the fidelity of the External Tank model. Integrate other systems of the STS into analysis. Incorporate decision making process into design.
References. Heppenheimer, T. A., 22, Development of the Space Shuttle, 972-98, Smithsonian Institution Press, Washington, D.C. 2. Jenkins, Dennis R., 2, Space Shuttle: The History of the National Space Transportation System: The First Missions, Specialty Pr Pub & Wholesalers. 3. de Weck, O.L. and Kim, I.Y., Adaptive Weighted Sum Method for Multiobjective Optimization, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Paper AIAA-24-68, Palm Springs, California, April 9-22, 24.
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