Presentation Agenda Computational Multidisciplinary What is Design Optimization? MDO: Multidisciplinary Design Optimization Design Optimization MOO: Multiobjective Optimization Optimization Methods Applications IL YONG KIM, PhD Dept of Mechanical and Materials Engineering Queen’s University Summary November 1, 2005 2 What is Design Optimization? Selecting the “best” design within the available means What is 1. What is our criterion for “best” design? Objective function Design Optimization? 2. What are the available means? Constraints (design requirements) 3. How do we describe different designs? Design Variables 4 Optimization Statement Optimization Procedure Minimize J (x) Subject to g(x) ≤ 0 h(x) ≤ 0 Minimize J (x) START Subject to g(x) ≤ 0 h(x) ≤ 0 Improve Design J(x) : Objective function to be minimized g(x) : Inequality constraints h(x) : Equality constraints x : Design variables Determine an initial design (x0) Computer Simulation Evaluate J(x), g(x), h(x) Converge ? Does your design meet a termination criterion? Change x N Y END 5 6 1 Examples Topology Optimization by DSO MDO: L H Multidisciplinary Design Optimization H 7 MDO Definition Definition of “Discipline” What is MDO ? Optimal design of complex engineering systems that requires analysis that accounts for interactions amongst the disciplines A discipline can often be defined by a set of equations that govern the underlying physical processes of interest. “How to decide what to change, and to what extent to change it, when everything influences everything else.” Ref: AIAA MDO website http://endo.sandia.gov/AIAA_MDOTC/main.html 9 Engineering Design Disciplines Aircraft: Spacecraft: Automobiles: Aerodynamics Propulsion Structures Controls Avionics/Software Manufacturing others Astrodynamics Thermodynamics Communications Payload & Sensor Structures Optics Guidance & Control Engines Body/chassis Aerodynamics Electronics Hydraulics Industrial design others 10 Traditional Pairings • Aerodynamics + Structures = Aeroelasticity • Optics + Controls = Adaptive Optics • Thermodynamics + Structures = Thermostructures • Acoustics + Structures = Acoustic Structures •Finance + Manufacturing = Lean Production • Scheduling + Manufacturing = Just-in-Time Progress often occurs within disciplines and at the intersection of traditional disciplines • But there can be more than two disciplines interacting • Some can be non-technical, e.g. cost estimation Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 11 12 2 Multidisciplinary Aspects of Design MDO Framework Emphasis is on the multidisciplinary nature of the complex engineering systems design process. Structures Control Aerodynamics Emphasis in recent years has been on advances that can be achieved due to the interaction of two or more disciplines. Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 13 Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 14 Optimization Statement Minimize J (x) MOO: Subject to g(x) ≤ 0 h(x) ≤ 0 Multiobjective Optimization J(x) : Objective function to be minimized g(x) : Inequality constraints h(x) : Equality constraints x : Design variables 16 Multiobjective Optimization Problem Formal Definition Multiple Objectives The objective can be a vector J of z system responses or characteristics we are trying to maximize or minimize When multiple objectives (criteria) are present min J(x) s.t. g(x) ≤ 0 h(x) ≤ 0 Often the objective is a where J =  J1 ( x ) " J z ( x )   J1   cost [$]  scalar function, but for  J   - range [km]  real systems often we  2   attempt multi-objective  J 3   weight [kg]  J= =  optimization:  J i   - data rate [bps] x 6 J(x) #   # Objectives often      J z   - ROI [%]  conflict with each other! T x = [ x1 " xi " xn ] T g =  g1 ( x ) " g m1 ( x )  T h =  h1 (x ) " hm2 ( x )  T 17 Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 18 3 Mapping Single objective n-dimension T Pareto Frontier 1-dimension [ x1 x2 " xn ] J2: x2 Pareto frontier Weight x 6 J (x) J 1 3 x1 Multiobjective n-dimension J2 [ x1 x2 " xn ]T m-dimension 4 2 x2 5 x 6 J(x) J1 x1 J1: Manufacturing cost 19 20 Pareto Frontier Pareto Frontier In a two-dimensional trade space (i.e. two decision criteria), the Pareto Optimal set represents the boundary of the most design efficient solutions. TPF System Trade Space Pareto-Optimal Front 2200 Dominated Solutions Non-Dom inated S olutions 2000 “Take from Peter to pay Paul” $1M/Im age 1800 $2M /Im age 1600 $0.5M /Im age SSI 1400 1200 1000 SCI 800 0 500 $0.25M/Im age 1000 1500 2000 2500 3000 3500 4000 Performance (total # of images) Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 21 22 MDO and MOO single objective MDO and MOO single discipline cantilever beam m F l $ δ Minimize displacement s.t. mass and loading constraint single discipline multiple obj. Lifecycle Cost ($M) Pareto Optimal means ….. Zang, Thomas and Green, Lawrence, “Multidisciplinary Design Optimization Techniques: Implications and Opportunities for Fluid Dynamics Research,” 30th AIAA Fluid Dynamics Conference Norfolk, VA June 28 July 1, 1999 23 vo α multiple disciplines support bracket F Minimize stamping costs (mfg) subject to loading and geometry constraint D multiple disciplines commercial aircraft airfoil V fuel (x,y) Maximize CL/CD and maximize Minimize cost and maximize cruise wing fuel volume for specified α, vo speed s.t. fixed range and payload Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 24 4 Computational Computational Optimization Methods Optimization Methods (1) Gradient-based Methods (1) Gradient-based Methods (2) Heuristic Methods (2) Heuristic Methods Optimum Solution – Graphical Representation Gradient-based Methods You do not know this function before optimization You do not know this function before optimization J(x) J(x) Start Check gradient Start Move Check gradient Gradient=0 No active constraints Optimum solution (x*) x No active constraints Stop! Optimum solution (x*) x (Termination criterion: Gradient=0) 27 Gradient-based Methods 28 Global vs. Local Optimum Two steps are repeated until a local optimum is found. J(x) (1) Sensitivity Analysis : Which direction to go? Local Optimum (2) Line Search Local Optimum : How much to go? (to the direction that was determined by sensitivity analysis) Global Optimum No active constraints 29 x 30 5 Sequential Quadratic Programming Gradient-based Methods • Create a quadratic approximation to the Lagrangian • Solve the quadratic problem to find the search direction, S • Perform the 1-D search • Update the approximation to the Lagrangian Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary System Design Optimization,” Lecture notes, MIT, 2004 31 32 Heuristic Methods Computational A Heuristic is simply a rule of thumb that hopefully Optimization Methods 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 (1) Gradient-based Methods without getting stuck in them while searching for the global optimum. (2) Heuristic Methods Schulz, A.S., “Metaheuristics,” 15.057 Systems Optimization Course Notes, MIT, 1999. 34 Genetic Algorithms Heuristic Methods Principle by Charles Darwin - Natural Selection Most Common Heuristic Techniques • Genetic Algorithms • Simulated Annealing • Tabu Search • Particle Swarm Method 35 36 6 Genetic Algorithms: Procedures Genetic Algorithms Gradient search - Treats one design at one time Genetic algorithms - Treats a set of designs at one time Gradient search Genetic algorithms 20 40 60 80 37 How To Choose an Algorithm? 38 Algorithm Selection Matrix Linear J and g and h • Linear/nonlinear • Type of design variables (real/integer, continuous/discrete) Nonlinear J or g or h Continuous, real Simplex SQP (constraine x (all) Barrier Methods d) Newton (unconstrained) Discrete Branch-and-Bou GA nd x (at least one) SA, Tabu Search PSO • Equality/inequality constraints • Discontinuous feasible spaces • Initial solution feasible/infeasible • Simulation code runtime 39 40 Range of Objectives Part vs. System Do you really need to obtain the global optimum? Feasible Improved Local Optimum Global Optimum 41 Single-discipline Multidiscipline Single-objective Multiobjective Easy to optimize Difficult to optimize Parts System 42 7 Valveless Micropump MIT, AIST* Applications Net flow Glass (Pyrex) Fluid Si MDO is a useful tool in the design of Inlet choke virtually all complex, multidisciplinary Outlet choke (heating) PZT systems… * AIST (Japan): National Institute of Advanced Science and Technology 44 Structure of Valveless Micropump FE modeled region 45 Result – Viscosity Change 46 Result – Rectification efficiency right choke (outlet) Rectification efficiency t=0.5 ms ε= t=1.5 ms t=2.5 ms u m+ − u m− u m+ + u m− u m+ = Mean velocity at inlet t=3.5 ms u m− = Mean velocity at outlet Case I : u m+ = u m− (ε = 0) Prototype by Dr. Mastumura at AIST Case Unit: g/mm-sec 47 II : u m− = 0 (ε = 1) Rectification efficiency t=0 ms 0.25 0.20 0.15 0.10 0.05 0.00 0 2 4 6 8 10 Time (ms) 48 8 Semiconductor equipment: X-ray mask Optimization Formulation [Optimization I] maximize Q(b1 , b2 , b3 ) subject to ε 0 − ε (b1 , b2 , b3 ) ≤ 0 Tmax (b1 , b2 , b3 ) − T0 ≤ 0 bilower ≤ bi ≤ biupper , i = 1,",3 [Optimization II] maximize ε (b1 , b2 , b3 ) subject to Q0 − Q (b1 , b2 , b3 ) ≤ 0 Tmax (b1 , b2 , b3 ) − T0 ≤ 0 Reduce distortion due to change in the gravity direction bilower ≤ bi ≤ biupper , i = 1,",3 Determine dynamic thermal distortions and pattern blur. 49 50 Universal Joint Precision Machine Design Universal Joints transmit rotation between shafts whos e axis are coplanar, but not coinciding Assembly comprised of 3 components: [1] – Flange yoke, Weld yoke, Cross Trunion – Conduct optimization on individual parts & assembly At non-zero joint angles, the output shaft will experienc e both acceleration & deceleration every revolution, lea ding to dynamic instability Multi-Disciplinary Design Optimization considers: – – – – – – Reduce structural and thermal distortions. 51 [2] Application of practical dynamic loading conditions Reduction in overall mass (weight savings) Increase in maximum joint angle (performance gain) Decrease of manufacturing cost (economics) Monitor Von-Mises stress and Strain Energy Density Modify up to 10 design variables per part Employ ‘Adaptive Weighted Sum’ to construct a Pareto Surface representing optimal designs 52 [1] http://www.nrg.com.au/~hemp/bigjoint/bigjoint.htm&h=225&w=300&sz=12&tbnid=IfuOOO7gop0J:&tbnh=83&tbnw=111&start=10&prev=/images%3Fq%3Duniversal%2Bjoint%26hl%3Den%26lr%3D%26client%3Dfirefoxa%26rls%3Dorg.mozilla:en-US:official_s%26sa%3DG [2] Parker, Sybil P. “Encyclopedia of Engineering.” McGraw-Hill Book Company, New York, United States of America, 1983 pp 1151-1153. Cementless hip prosthesis Motivation: – – Un-cemented implants should be used for younger patient s since younger patients are more active, and more bone s tock is preserved for revision surgery Un-cemented implants with a longer life reduce the numbe r times the patient has to undergo revision surgery Design Objectives: – Minimize wear of the bearing and acetabulum surfaces – Minimize relative motion at the implant-bone interface • • – Micro-movements at implant-bone interface inhibit bone in-grow th into the implant surface which is necessary for long term impl ant fixation Minimize cost Femoral Osteotomy Implant shape/size Bearing shape/size Acetabular component shape/size Chromosome length change Loading conditions: – – Wear debris can cause an inflammatory reaction eventually lead ing to bone degradation at the implant-bone interface Design Variables – – – – Variable Chromosome Length GA Normal Walking Stair climbing Design representation with gradual refinement Contact Analysis: – – – Bearing-acetabulum Taper-bearing Implant-bone 53 54 9 Design for Flexibility Design for Flexibility Bridge Customers such as the military would like a simply-designed bridge that can be used for various applications - Various span lengths Short creek Large river - Various loading conditions Support tanks, trucks, cars, etc. Flexibility, Changeability, Extensibility, Reconfigurability, Modularity, etc. 55 Optimization for Manufacturability 56 Design Under Uncertainty J(x) f ( x1 ) < f ( x2 ) But, ∆f ( x1 ) > ∆f ( x2 ) for the same ∆x Determine x2 using design optimization ∆J1 ∆J 2 x1 ∆x With William Nadir (Master’s student) x2 x Design variable ∆x 57 58 Summary Reference Il Yong Kim, “MECH465: Computer Aided Design,” Lecture notes, min J(x) s.t. g(x) ≤ 0 h(x) ≤ 0 Queen’s University, 2005 Panos Y. Papalambros and Douglass J. Wilde, Principles of Optimal Design, 2nd edition, Cambridge University Press Olivier de Weck and Karen Willcox, “MSDO: Multidisciplinary MDO is a design tool that help create advanced and System Design Optimization,” Lecture notes, MIT, 2004 complex engineering systems that are competitive AIAA MDO website http://endo.sandia.gov/AIAA_MDOTC/main.html not only in terms of performance, but also in terms of Jasbir S. Arora, Introduction to Optimum Design, 2nd edition, manufacturability, serviceability, and overall life-cycle McGraw-Hill cost effectiveness. 59 60 10