16.810 (16.682) 16.810 (16.682) Engineering Design and Rapid Prototyping Design Optimization - Structural Design Optimization Instructor(s) Prof. Olivier de Weck Dr. Il Yong Kim January 23, 2004 Course Concept 16.810 (16.682) 2 today Course Flow Diagram 16.810 (16.682) CAD/CAM/CAE Intro Overview Manufacturing Training Structural Test “Training” Design Optimization Hand sketching CAD design FEM analysis Produce Part 1 Test Produce Part 2 Optimization Problem statement Final Review Test Learning/Review Deliverables Design Sketch v1 Part v1 Experiment data v1 Design/Analysis output v2 Part v2 Experiment data v2 Drawing v1 Design Intro today Wednesday FEM/Solid Mechanics Analysis output v1 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 Minimize Subject to f g h (x) () d 0 x () 0 x 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: g(x), h(x) 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 ( ) () 0 () 0 f h d x x x Minimize Subject to g 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 Improve Design Computer Simulation START Converge ? Y N END ( ) Subj ( ) 0 () 0 f g h d x x x Change x Determine an initial design (x 0 ) termination criterion? Minimize ect to Evaluate f(x), g(x), h(x) Does your design meet a 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 ( ) j ( ) 0 () 0 f g h d x x x BC’s are given Loads are given minimize sub ect to 1. To make the structure strong Min. f(x) e.g. Minimize displacement at the tip g(x) d 0 2. Total mass d M C 16.810 (16.682) 13 Size Optimization Beams ( ) j ( ) 0 () 0 f g h d x x x minimize sub ect to 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 ( ) j ( ) 0 () 0 f g h d x x x Hermite, Bezier, B-spline, NURBS, etc. minimize sub ect to 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 ( ) j ( ) 0 () 0 f g h d x x x minimize sub ect to 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 = 4.16u10 5 Distributed loading Obj = 3.29u10 5 Minimize 3 * ii d z F *, )to Subject U( d x d: M , o 3 : 0 d U(x) d1 Obj = 2.88u10 5 Mass constraints: 35% Obj = 2.73u10 5 16.810 (16.682) 22 Topology Optimization DongJak Bridge in Seoul, Korea H L H 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) x: design variable f(x): displacement Optimum solution (x*) x 16.810 (16.682) 25 Optimization Methods Gradient-based methods Heuristic methods 16.810 (16.682) 26 Gradient-based Methods f(x) Start Move Gradient=0 Stop! You do no c ore optimization Check gradient Check gradient t know this fun tion bef No active constraints Optimum solution (x*) 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 Global optimum Local optimum Local 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 16.810 (16.682) Design domain First Order Method 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 Centroid Jitter on Focal Plane [RSS LOS] NASA Nexus Spacecraft Concept 60 T=5 sec 14.97 Pm 1 pixel Requirement: J =5 Pm z,2 OTA 40 20 Centroi d Y [ P m] 0 -20 Sunshield Instrument -40 Module 012 -60 -60 -40 -20 0 20 40 60 meters Centroid X [Pm] 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 le Approx. 480 passengers each Approx. 8,700 nm range each Takeoff BWB A3XX-50R 18% BWB A3XX-50R 19% Total Sea-Level 19% BWB A3XX-50R Operators Empty Fuel Burn per Seat 32% BWB A3XX-50R Boeing Blended Wing Body Concept Goal Shown to Same Sca Maximum Weight Static Thrust Weight : Find a design for a family of blended wing aircraft that will combine aerodynamics, structures, propulsion and controls such that a competitive system emerges - as measured by a set of operator metrics. 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 Baseline Design Performance Natural frequency analysis Design requirements 16.810 (16.682) 44 Baseline Design Performance and cost G 1 0.070 mm G 2 0.011 mm f 245 Hz m 0.224 lbs C 5.16 $ 16.810 (16.682) 45 Baseline Design 245 Hz 421 Hz f1=0 f2=0 f3=0 f4=0 f5=0 f6=0 f7=421 Hz f8=1284 Hz f9=1310 Hz f1=245 Hz f2=490 Hz f3=1656 Hz 16.810 (16.682) 46 Design Requirement for Each Team # Product name mass (m) Cost (c) Disp (G1) Disp (G2) Nat Freq (f) Qual ity F1 (lbs) F2 (lbs) F3 (lbs) Const Optim Acc 0 Base line 0.224 lbs 5.16 $ 0.070 mm 0.011 mm 245 Hz 3 50 50 100 c m GGf 1 Family economy 20% -30% 10% 10% -20% 2 50 50 100 c m GGf 2 Family deluxe 10% -10% -10% -10% 10% 4 50 50 100 m c GGf 3 Cross over 20% 0% -15% -15% 20% 4 50 75 75 m c GGf 4 City bike -20% -20% 0% 0% 0% 3 50 75 75 c m GGf 5 Racing -30% 50% 0% 0% 20% 5 100 100 50 m GG f c 6 Mountain 30% 30% -20% -20% 30% 4 50 100 50 GGf mc 7 BMX 0% 65% -15% -15% 40% 4 75 100 75 GGf mc 8 Acrobatic -30% 100% -10% -10% 50% 5 100 100 100 GGf mc 9 Motor bike 50% 10% -20% -20% 0% 3 50 75 100 GGf cm 16.810 (16.682) 47 Design Optimization Topology optimization Design domain Shape optimization 16.810 (16.682) 48 Design Freedom 1 bar G 2.50 mm G 2 bars G 0.80 mm Volume is the same. 17 bars G 0.63 mm 16.810 (16.682) 49 Design Freedom 1 bar 2 bars 2.50 mmG G 0.80 mm 17 bars More design freedom More complex (Better performance) (More difficult to optimize) G 0.63 mm 16.810 (16.682) 50 Cost versus Performance 17 bars 0 1 2 3 4 5 6 7 8 9 Cost [$] 1 bar 2 bars 0 0.5 1 1.5 2 2.5 3 Displacement [mm] 16.810 (16.682) 51 Plan for the rest of the course Class Survey Jan 24 (Saturday) 7 am – Jan 26 (Monday) 11am Company tour Jan 26 (Monday) : 1 pm – 4 pm Guest Lecture (Prof. Wilson, Bicycle Science) Jan 28 (Wednesday) : 2 pm – 3:30 pm Manufacturing Bicycle Frames (Version 2) Jan 28 (Wednesday) : 9 am – 4:30 pm Jan 29 (Thursday) : 9 am – 12 pm Testing Jan 29 (Thursday) : 10 am – 2 pm GA Games Jan 29 (Thursday) : 1 pm – 5 pm Guest Lecture, Student Presentation (5~10 min/team) Jan 30 (Friday) : 1 pm – 4 pm 16.810 (16.682) 52 References P. Y. Papalambros, Principles of optimal design, Cambridge University Press, 2000 O. de Weck and K. Willcox, Multidisciplinary System Design Optimization, MIT lecture note, 2003 M. O. Bendsoe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” comp. Meth. Appl. Mech. Engng, Vol. 71, pp. 197-224, 1988 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 Il Yong Kim and Byung Man Kwak, “Design space optimization using a numerical design continuation method,” International Journal for Numerical Methods in Engineering, Vol. 53, Issue 8, pp. 1979-2002, March 20, 2002. 16.810 (16.682) 53 Web-based topology optimization program Developed and maintained by Dmitri Tcherniak, Ole Sigmund, Thomas A. Poulsen and Thomas Buhl. Features: 1.2-D 2.Rectangular design domain 3.1000 design variables (1000 square elements) 4. Objective function: compliance (FuG) 5. Constraint: volume 16.810 (16.682) 54 Web-based topology optimization program Objective function -Compliance (FuG) Constraint -Volume Design variables - Density of each design cell 16.810 (16.682) 55 Web-based topology optimization program No numerical results are obtained. Optimum layout is obtained. 16.810 (16.682) 56 Web-based topology optimization program P 2P 3P Absolute magnitude of load does not affect optimum solution 16.810 (16.682) 57 Web-based topology optimization program http://www.topopt.dtu.dk 16.810 (16.682) 58