Design Optimization - Structural Design Optimization

Transcription

1 (16.682) Engineering Design and Rapid Prototyping Design Optimization - Structural Design Optimization Instructor(s) Prof. Olivier de Weck deweck@mit.edu Dr. Il Yong Kim kiy@mit.edu January 23, 2004

2 Course Concept today (16.682) 2

3 Course Flow Diagram Learning/Review Design Intro CAD/CAM/CAE Intro FEM/Solid Mechanics Overview Manufacturing Training Structural Test Training Design Optimization Problem statement Hand sketching CAD design FEM analysis Produce Part 1 Test today Optimization Produce Part 2 Test Deliverables Design Sketch v1 Drawing v1 Analysis output v1 Part v1 Experiment data v1 Design/Analysis output v2 Part v2 Wednesday Experiment data v2 Final Review (16.682) 3

4 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? 3. How do we describe different designs? Constraints (design requirements) Design Variables (16.682) 4

5 Optimization Statement Minimize f ( x) Subject to g( x) 0 h( x) = (16.682) 5

6 Constraints - Design requirements Inequality constraints Equality constraints (16.682) 6

7 Objective Function - A criterion for best design (or goodness of a design) Objective function (16.682) 7

8 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 (16.682) 8

9 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) Natural frequency = f(design) Mass = f(design) What is f for each case? (16.682) 9

10 Optimization Statement Minimize f ( 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 (16.682) 10

11 Optimization Procedure Minimize f ( x) Subject to g( x) 0 h( x) = 0 START Determine an initial design (x 0 ) Improve Design Computer Simulation Evaluate f(x), g(x), h(x) Change x N Converge? Y END Does your design meet a termination criterion? (16.682) 11

12 Structural Optimization Selecting the best structural design -Size Optimization - Shape Optimization - Topology Optimization (16.682) 12

13 Structural Optimization minimize f ( x) subject to g( x) 0 h() x = 0 BC s are given Loads are given 1. To make the structure strong e.g. Minimize displacement at the tip 2. Total mass M C Min. f(x) g(x) (16.682) 13

14 Size Optimization minimize f ( x) subject to g( x) 0 h() x = 0 Beams Design variables (x) x : thickness of each beam f(x) : compliance g(x) : mass Number of design variables (ndv) ndv = (16.682) 14

15 Size Optimization - Shape are given Topology - Optimize cross sections (16.682) 15

16 Shape Optimization minimize f ( x) subject to g( x) 0 h() x = 0 B-spline Hermite, Bezier, B-spline, NURBS, etc. Design variables (x) x : control points of the B-spline (position of each control point) f(x) : compliance g(x) : mass Number of design variables (ndv) ndv = (16.682) 16

17 Shape Optimization Fillet problem Hook problem Arm problem (16.682) 17

18 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 , (16.682) 18

19 Topology Optimization minimize f ( x) subject to g( x) 0 h() x = 0 Cells Design variables (x) x : density of each cell f(x) : compliance g(x) : mass Number of design variables (ndv) ndv = (16.682) 19

20 Topology Optimization Short Cantilever problem Initial Optimized (16.682) 20

21 Topology Optimization (16.682) 21

22 Topology Optimization Bridge problem Distributed loading Obj = Minimize Subject to Γ Ω F i z dγ, ρ( x) dω M 0 ρ( x) 1 Mass constraints: 35% i o, Obj = Obj = Obj = (16.682) 22

23 Topology Optimization DongJak Bridge in Seoul, Korea H H L (16.682) 23

24 Structural Optimization What determines the type of structural optimization? Type of the design variable (How to describe the design?) (16.682) 24

25 Optimum Solution Graphical Representation f(x) f(x): displacement x: design variable Optimum solution (x*) x (16.682) 25

26 Optimization Methods Gradient-based methods Heuristic methods (16.682) 26

27 Gradient-based Methods f(x) You do not know this function before optimization Start Check gradient Move Check gradient Gradient=0 Stop! No active constraints Optimum solution (x*) (Termination criterion: Gradient=0) x (16.682) 27

28 Gradient-based Methods (16.682) 28

29 Global optimum vs. Local optimum f(x) Termination criterion: Gradient=0 Local optimum Local optimum Global optimum No active constraints x (16.682) 29

30 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, Systems Optimization Course Notes, MIT, (16.682) 30

31 Genetic Algorithm Principle by Charles Darwin - Natural Selection (16.682) 31

32 Heuristic Methods Heuristics Often Incorporate Randomization 3 Most Common Heuristic Techniques Genetic Algorithms Simulated Annealing Tabu Search (16.682) 32

33 Optimization Software - isight -DOT - Matlab (fmincon) (16.682) 33

34 Topology Optimization Software ANSYS Static Topology Optimization Dynamic Topology Optimization Electromagnetic Topology Optimization Subproblem Approximation Method First Order Method Design domain (16.682) 34

35 Topology Optimization Software MSC. Visual Nastran FEA Elements of lowest stress are removed gradually. Optimization results Optimization results illustration (16.682) 35

36 MDO Multidisciplinary Design Optimization (16.682) 36

37 Multidisciplinary Design Optimization Centroid Jitter on Focal Plane [RSS LOS] NASA Nexus Spacecraft Concept pixel Centroid Y [µm] T=5 sec OTA µm -20 Sunshield Instrument Module meters Requirement: Jz,2=5 µm 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 (16.682) 37

38 Multidisciplinary Design Optimization Boeing Blended Wing Body Concept Aircraft Comparison Shown to Same Scale Approx. 480 passengers each Approx. 8,700 nm range each Goal: 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. Operators Empty Weight Maximum Takeoff Weight Total Sea-Level Static Thrust Fuel Burn per Seat BWB A3XX-50R BWB A3XX-50R BWB A3XX-50R BWB A3XX-50R 19% 32% 19% 18% Boeing (16.682) 38

39 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 , MAO Conference, St. Louis, (16.682) 39

40 Multidisciplinary Design Optimization (16.682) 40

41 Multidisciplinary Design Optimization (16.682) 41

42 Multidisciplinary Design Optimization Do you want to learn more about MDO? Take this course! /ESD.77 Multidisciplinary System Design Optimization (MSDO) Prof. Olivier de Weck Prof. Karen Willcox (16.682) 42

43 Genetic Algorithm Do you want to learn more about GA? Take part in this GA game experiment! (16.682) 43

44 Baseline Design Performance Natural frequency analysis Design requirements (16.682) 44

45 Baseline Design Performance and cost δ1 = mm δ = mm 2 f = 245 Hz m C = lbs = 5.16 $ (16.682) 45

46 Baseline Design 245 Hz 421 Hz f1=245 Hz f2=490 Hz f3=1656 Hz f1=0 f2=0 f3=0 f4=0 f5=0 f6=0 f7=421 Hz f8=1284 Hz f9=1310 Hz (16.682) 46

47 Design Requirement for Each Team # Product name mass (m) Cost (c) Disp (δ1) Disp (δ2) Nat Freq (f) Qual ity F1 (lbs) F2 (lbs) F3 (lbs) Const Optim Acc 0 Base line lbs 5.16 $ mm mm 245 Hz c m δ1, δ2,f 1 Family economy 20% -30% 10% 10% -20% c m δ1, δ2,f 2 Family deluxe 10% -10% -10% -10% 10% m c δ1, δ2,f 3 Cross over 20% 0% -15% -15% 20% m c δ1, δ2,f 4 City bike -20% -20% 0% 0% 0% c m δ1, δ2,f 5 Racing -30% 50% 0% 0% 20% m δ1, δ2, f c 6 Mountain 30% 30% -20% -20% 30% δ1, δ2,f m c 7 BMX 0% 65% -15% -15% 40% δ1, δ2,f m c 8 Acrobatic -30% 100% -10% -10% 50% δ1, δ2,f m c 9 Motor bike 50% 10% -20% -20% 0% δ1, δ2,f c m (16.682) 47

48 Design Optimization Topology optimization Design domain Shape optimization (16.682) 48

49 Design Freedom 1 bar δ = 2.50 mm δ 2 bars δ = 0.80 mm Volume is the same. 17 bars δ = 0.63 mm (16.682) 49

50 Design Freedom 1 bar δ = 2.50 mm 2 bars δ = 0.80 mm 17 bars More design freedom (Better performance) More complex (More difficult to optimize) δ = 0.63 mm (16.682) 50

51 Cost versus Performance 17 bars Cost [$] bars 1 bar Displacement [mm] (16.682) 51

52 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 (16.682) 52

53 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 , 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 , 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 , March 20, (16.682) 53

54 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 design variables (1000 square elements) 4. Objective function: compliance (F δ) 5. Constraint: volume (16.682) 54

55 Web-based topology optimization program Objective function -Compliance (F δ) Constraint -Volume Design variables - Density of each design cell (16.682) 55

56 Web-based topology optimization program No numerical results are obtained. Optimum layout is obtained (16.682) 56

57 Web-based topology optimization program P 2P 3P Absolute magnitude of load does not affect optimum solution (16.682) 57

58 Web-based topology optimization program (16.682) 58