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Design Optimization - Structural Design Optimization

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Design Optimization - Structural Design Optimization 16.810 (16.682) Engineering Design and Rapid Prototyping Design Optimization - Structural Design Optimization Instructor(s) Prof. Olivier de Weck Dr. Il Yong Kim [email protected] [email protected] January 23, 2004 Course Concept today 16.810 (16.682) 2 Course Flow Diagram Learning/Review Problem statement Deliverables Design Intro Hand sketching Design Sketch v1 CAD/CAM/CAE Intro CAD design Drawing v1 FEM/Solid Mechanics FEM analysis Analysis output v1 Overview Manufacturing Produce Part 1 Part v1 Training Structural Test Test Experiment data v1 “Training” today Design/Analysis Design Optimization Optimization output v2 Wednesday Produce Part 2 Part v2 Test Experiment data v2 Final Review 16.810 (16.682) 3 What Is Design Optimization? Selecting the “best” design within the available means 1. What is our criterion for “best” design? Objective function 2. What are the available means? Constraints (design requirements) 3. How do we describe different designs? Design Variables 16.810 (16.682) 4 Optimization Statement Minimizef (x ) Subject tog (x )≤ 0 h()x = 0 16.810 (16.682) 5 Constraints - Design requirements Inequality constraints Equality constraints 16.810 (16.682) 6 Objective Function - A criterion for best design (or goodness of a design) Objective function 16.810 (16.682) 7 Design Variables Parameters that are chosen to describe the design of a system Design variables are “controlled” by the designers The position of upper holes along the design freedom line 16.810 (16.682) 8 Design Variables For computational design optimization, Objective function and constraints must be expressed as a function of design variables (or design vector X) Objective function:f (x ) Constraints:gh (xx ), ( ) Cost = f(design) Displacement = f(design) What is “f” for each case? Natural frequency = f(design) Mass = f(design) 16.810 (16.682) 9 Optimization Statement Minimizef (x ) Subject to g()x ≤ 0 h()x = 0 f(x) : Objective function to be minimized g(x) : Inequality constraints h(x) : Equality constraints x : Design variables 16.810 (16.682) 10 Optimization Procedure Minimizef (x ) Determine an initial design (x ) Subject tog (x )≤ 0 START 0 h()x = 0 Improve Design Computer Simulation Evaluate f(x), g(x), h(x) Change x Converge ? Does your design meet a N termination criterion? Y END 16.810 (16.682) 11 Structural Optimization Selecting the best “structural” design -Size Optimization - Shape Optimization - Topology Optimization 16.810 (16.682) 12 Structural Optimization minimizef (x ) subject tog (x )≤ 0 h()x = 0 BC’s are given Loads are given 1. To make the structure strong Min. f(x) e.g. Minimize displacement at the tip 2. Total mass ≤ MC g(x) ≤ 0 16.810 (16.682) 13 Size Optimization Beams minimizef (x ) subject tog (x )≤ 0 h()x = 0 Design variables (x) f(x) : compliance x : thickness of each beam g(x) : mass Number of design variables (ndv) ndv = 5 16.810 (16.682) 14 Size Optimization -Shape are given Topology - Optimize cross sections 16.810 (16.682) 15 Shape Optimization B-spline minimizef (x ) subject tog (x )≤ 0 h()x = 0 Hermite, Bezier, B-spline, NURBS, etc. Design variables (x) f(x) : compliance x : control points of the B-spline g(x) : mass (position of each control point) Number of design variables (ndv) ndv = 8 16.810 (16.682) 16 Shape Optimization Fillet problem Hook problem Arm problem 16.810 (16.682) 17 Shape Optimization Multiobjective & Multidisciplinary Shape Optimization Objective function 1. Drag coefficient, 2. Amplitude of backscattered wave Analysis 1. Computational Fluid Dynamics Analysis 2. Computational Electromagnetic Wave Field Analysis Obtain Pareto Front Raino A.E. Makinen et al., “Multidisciplinary shape optimization in aerodynamics and electromagnetics using genetic algorithms,” International Journal for Numerical Methods in Fluids, Vol. 30, pp. 149-159, 1999 16.810 (16.682) 18 Topology Optimization Cells minimizef (x ) subject tog (x )≤ 0 h()x = 0 Design variables (x) f(x) : compliance x : density of each cell g(x) : mass Number of design variables (ndv) ndv = 27 16.810 (16.682) 19 Topology Optimization Short Cantilever problem Initial Optimized 16.810 (16.682) 20 Topology Optimization 16.810 (16.682) 21 Topology Optimization Bridge problem Obj = 4.16×105 Distributed loading Obj = 3.29×105 Minimize F i zidΓ, ∫ Γ Subject to ρ(x)dΩ ≤ M o , ∫ Ω 0 ≤ ρ(x) ≤1 Obj = 2.88×105 Mass constraints: 35% Obj = 2.73×105 16.810 (16.682) 22 Topology Optimization DongJak Bridge in Seoul, Korea H H L 16.810 (16.682) 23 Structural Optimization What determines the type of structural optimization? Type of the design variable (How to describe the design?) 16.810 (16.682) 24 Optimum Solution – Graphical Representation f(x): displacement f(x) x: design variable x Optimum solution (x*) 16.810 (16.682) 25 Optimization Methods Gradient-based methods Heuristic methods 16.810 (16.682) 26 Gradient-based Methods You do not know this function before optimization f(x) Start Check gradient Move Check gradient Gradient=0 Stop! x No active constraints Optimum solution (x*) (Termination criterion: Gradient=0) 16.810 (16.682) 27 Gradient-based Methods 16.810 (16.682) 28 Global optimum vs. Local optimum f(x) Termination criterion: Gradient=0 Local optimum Local optimum Global optimum x No active constraints 16.810 (16.682) 29 Heuristic Methods A Heuristic is simply a rule of thumb that hopefully will find a good answer. Why use a Heuristic? Heuristics are typically used to solve complex optimization problems that are difficult to solve to optimality. Heuristics are good at dealing with local optima without getting stuck in them while searching for the global optimum. Schulz, A.S., “Metaheuristics,” 15.057 Systems Optimization Course Notes, MIT, 1999. 16.810 (16.682) 30 Genetic Algorithm Principle by Charles Darwin - Natural Selection 16.810 (16.682) 31 Heuristic Methods Heuristics Often Incorporate Randomization 3 Most Common Heuristic Techniques Genetic Algorithms Simulated Annealing Tabu Search 16.810 (16.682) 32 Optimization Software - iSIGHT -DOT - Matlab (fmincon) 16.810 (16.682) 33 Topology Optimization Software ANSYS Static Topology Optimization Dynamic Topology Optimization Electromagnetic Topology Optimization Subproblem Approximation Method First Order Method Design domain 16.810 (16.682) 34 Topology Optimization Software MSC. Visual Nastran FEA Elements of lowest stress are removed gradually. Optimization results Optimization results illustration 16.810 (16.682) 35 MDO Multidisciplinary Design Optimization 16.810 (16.682) 36 Multidisciplinary Design Optimization NASA Nexus Spacecraft Concept Centroid Jitter on Focal Plane [RSS LOS] 60 40 1 pixel T=5 sec m] µ 20 OTA 0 14.97 µm Centroid Y [ -20 Sunshield Instrument -40 Module Requirement: Jz,2=5 µm 012 -60 -60 -40 -20 0 20 40 60 meters Centroid X [µm] Goal: Find a “balanced” system design, where the flexible structure, the optics and the control systems work together to achieve a desired pointing performance, given various constraints 16.810 (16.682) 37 Multidisciplinary Design Optimization Aircraft Comparison Shown to Same Scale Boeing Blended Wing Body Concept Approx. 480 passengers each Approx. 8,700 nm range each 19% Operators BWB Empty A3XX-50R Weight 18% Maximum BWB Takeoff Weight A3XX-50R Goal: Find a design for a family of blended wing aircraft 19% that will combine aerodynamics, structures, propulsion Total BWB Sea-Level and controls such that a competitive system emerges - as A3XX-50R measured by a set of operator metrics. Static Thrust 32% Fuel BWB Burn A3XX-50R per Seat ©Boeing 16.810 (16.682) 38 Multidisciplinary Design Optimization Ferrari 360 Spider Goal: High end vehicle shape optimization while improving car safety for fixed performance level and given geometric constraints Reference: G. Lombardi, A. Vicere, H. Paap, G. Manacorda, “Optimized Aerodynamic Design for High Performance Cars”, AIAA-98- 4789, MAO Conference, St. Louis, 1998 16.810 (16.682) 39 Multidisciplinary Design Optimization 16.810 (16.682) 40 Multidisciplinary Design Optimization 16.810 (16.682) 41 Multidisciplinary Design Optimization Do you want to learn more about MDO? Take this course! 16.888/ESD.77 Multidisciplinary System Design Optimization (MSDO) Prof. Olivier de Weck Prof. Karen Willcox 16.810 (16.682) 42 Genetic Algorithm Do you want to learn more about GA? Take part in this GA game experiment! 16.810 (16.682) 43 Baseline Design Performance Natural frequency analysis Design requirements 16.810 (16.682) 44 Baseline Design Performance and cost δ1 = 0.070 mm δ2 = 0.011 mm fHz= 245 mlbs= 0.224 C = 5.16 $ 16.810 (16.682) 45 Baseline Design 245 Hz 421 Hz f1=245 Hz f1=0 f2=0 f2=490 Hz f3=0 f3=1656 Hz f4=0 f5=0 f6=0 f7=421 Hz f8=1284 Hz f9=1310 Hz 16.810 (16.682) 46 Design Requirement for Each Team Nat Product mass Cost Disp Disp Qual F1 F2 F3 # Freq Const Optim Acc name (m) (c) (δ1) (δ2) ity (lbs) (lbs) (lbs) (f) Base 0.224 5.16 0.070 0.011 245 0 3 50 50 100 c m δ1, δ2,f line lbs $ mm mm Hz Family 1 20% -30% 10% 10% -20% 2 50 50 100 c m δ1, δ2,f economy Family 2 10% -10% -10% -10% 10% 4 50 50 100 m c δ1, δ2,f deluxe Cross 3 20% 0% -15% -15% 20% 4 50 75 75 m c δ1, δ2,f over 4 City bike -20% -20% 0% 0% 0% 3 50 75 75 c m δ1, δ2,f 5 Racing -30% 50% 0% 0% 20% 5 100 100 50 m δ1, δ2, f c 6 Mountain 30% 30% -20% -20% 30% 4 50 100 50 δ1, δ2,f m c 7 BMX 0% 65% -15% -15% 40% 4 75 100 75 δ1, δ2,f m c 8 Acrobatic -30% 100% -10% -10% 50% 5 100 100 100 δ1, δ2,f m c Motor 9 50% 10% -20% -20% 0% 3 50 75 100 δ1, δ2,f c m bike 16.810 (16.682) 47 Design Optimization Topology optimization Design domain Shape optimization 16.810 (16.682) 48 Design Freedom 1 bar δ = 2.50 mm δ 2 bars δ = 0.80 mm Volume is the same.
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