16.810 16.810 Engineering Design and Rapid Prototyping Engineering Design and Rapid PrototypingLecture 6 Design Optimization -Structural Design Optimization - Instructor(s) Prof. Olivier de Weck January 11, 2005What 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 2Optimization Statement Minimize Subject to f gh (x) ( ) ≤ 0 x () = 0 x 16.810 3Design 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) Lift = f(design) What is “f” for each case? Drag = f(design) Mass = f(design) 16.810 416.8105f(x) : Objective function to be minimizedg(x) : Inequality constraintsh(x) : Equality constraintsx : Design variablesMinimize ( )() 0() 0fSubject to gh≤=xxxOptimization StatementOptimization Procedure Improve Design Computer Simulation START Converge ? Y N END ( ) Subj( ) 0 () 0 f g h ≤ = x x x Change x Determine an initial design (x0) termination criterion? Minimize ect to Evaluate f(x), g(x), h(x) Does your design meet a 16.810 6Structural Optimization Selecting the best “structural” design -Size Optimization - Shape Optimization - Topology Optimization 16.810 7Structural Optimization ( ) j ( ) 0 () 0 f g h ≤ = 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 tipg(x) ≤ 02. Total mass ≤ MC 16.810 8Size Optimization Beams ( ) ( ) 0 () 0 f g h ≤ = x x x minimize subject to Design variables (x) f(x) : compliance x : thickness of each beam g(x) : mass Number of design variables (ndv) ndv = 5 16.810 9Size Optimization -Shape are givenTopology - Optimize cross sections 16.810 10Shape Optimization B-spline( ) ( ) 0 () 0 f g h ≤ = x x x minimize subject to 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 11Shape Optimization Fillet problem Hook problem Arm problem 16.810 12Shape Optimization Multiobjective & Multidisciplinary Shape Optimization Objective function 1. Drag coefficient, 2. Amplitude of backscattered wave Analysis 1. Computational Fluid Dynamics Analysis2. 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 13Topology Optimization Cells ( ) ( ) 0 () 0 f g h ≤ = x x x minimize subject to Design variables (x) f(x) : compliance x : density of each cell g(x) : mass Number of design variables (ndv) ndv = 27 16.810 14Topology Optimization Short Cantilever problem Initial Optimized 16.810 15Topology Optimization Bridge problem Obj = 4.16× 105 Distributed loading Obj = 3.29× 105 Minimize ∫Γ i i d z F Γ , )to Subject ρ ( d x ≤ Ω M ,o∫Ω 0 ≤ρ (x) ≤ 1 Obj = 2.88× 105 Mass constraints: 35% Obj = 2.73× 105 16.810 16Topology Optimization DongJak Bridge in Seoul, Korea H L H 16.810 17Structural Optimization What determines the type of structural optimization? Type of the design variable (How to describe the design?) 16.810 18Optimum Solution – Graphical Representation f(x) x: design variable f(x): displacement Optimum solution (x*) x 16.810 19Optimization Methods Gradient-based methods Heuristic methods 16.810 20Gradient-based Methods f(x) Start Move Gradient=0 Stop! You do no c ore optimization Check gradient Check gradient t know this fun tion befNo active constraints Optimum solution (x*) x (Termination criterion: Gradient=0) 16.810 21Gradient-based Methods 16.810 22Global optimum vs. Local optimum f(x) Termination criterion: Gradient=0 Global optimum Local optimum Local optimum x No active constraints 16.810 23Heuristic Methods  Heuristics Often Incorporate Randomization  3 Most Common Heuristic Techniques  Genetic Algorithms  Simulated Annealing  Tabu Search 16.810 24Optimization Software - iSIGHT -DOT - Matlab (fmincon) 16.810 25Topology Optimization Software  ANSYS Static Topology Optimization Dynamic Topology Optimization Electromagnetic Topology Optimization Subproblem Approximation Method First Order Method Design domain 16.810 26Topology Optimization Software  MSC. Visual Nastran FEA Elements of lowest stress are removed gradually. Optimization results Optimization results illustration 16.810 27Design Freedom 1 bar δ= 2.50 mm δ 2 bars δ= 0.80 mm Volume is the same. 17 bars δ= 0.63 mm 16.810 28Design Freedom 1 bar 2 bars 2.50 mmδ= δ= 0.80 mm 17 bars More design freedom More complex (Better performance) (More difficult to optimize) δ= 0.63 mm 16.810 29Cost versus Performance 17 bars 0123456789 Cost [$]1 bar2 bars 0 0.5 1 1.5 2 2.5 3 Displacement [mm] 16.810 30References 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 31Web-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 (F×δ) 5. Constraint: volume16.810 33Web-based topology optimization program Objective function -Compliance (F×δ) Constraint -Volume Design variables - Density of each design cell 16.810