Multidisciplinary System Multidisciplinary System Design Optimization (MSDO) Decomposition and Coupling Lecture 4 17 February 2004 Olivier de Weck 1 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Today’s Topics Last time discussed standard approach: Sequential modular analysis (Lecture 3). Modules are executed sequentially with or without feedback loops. ? MDO frameworks Other Approaches: – Distributed analysis – Distributed design ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 2 Fundamentally different approaches in MDO Distributed Analysis -disciplinary models provide analysis -all optimization done at system level non-hierarchical decomposition hierarchical decomposition Distributed Design -provide disciplinary models with design tasks CSSO -optimization at subsystem and system levels CO BLISS ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 3 Standard Optimization Problem Given * xx 0 ()J x x ()gx Optimization Engine Function Evaluator ∈ n x ! n J : ! → → ! n m g : ! ! Solve the problem (min Jx) (s.t. gx) ≥ 0 * * That is, find x s.t. J( x ) ≤ f x ? ∈ J ( x), dom( ) ∩ dom( )g ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 4 Distributed Analysis ? Disciplinary models provide analysis ? Optimization is controlled by some overseeing code or database e.g. GenIE database system (Stanford) ISight (Optimizer) iSight GenIE NPSol Shared data Local data Structures Local data Aero Optimizer design variables constraints x J(x),g(x),h(x) subsystem analyses ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 5 Distributed Analysis Optimizer objective design variables constraints x J(x) performance analysis aerodynamic analysis structural analysis x g(x) h(x) x g(x) h(x) ? During the optimization, the overseeing code keeps track of the values of the design variables and objective ? The values of the design variables are changed according to the optimization algorithm ? Disciplinary models are asked to evaluate constraints/objective ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 6 Distributed Design System level optimizer SS1 optimizer SS2 optimizer SSN optimizer SS1 analyzer SS2 analyzer SSN analyzer …… command/result command/result command/result Subsystem black box (BB) 7 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Advantages of Decoupling Computation of g(x) can be very time consuming, want to divide the work and compute in parallel. n2 For example, if x = (,x x 2 ), where x ∈! n1 , x ∈! 1 1 2 and g(x) = (g x g x )) ( ), ( 1 1 2 2 Then g 1 and g 2 can be computed in parallel. Graphically, Optimizer SS1 SS2 1 x 1 g 2 g x 2 g g 2 SS1 SS1 Optim 1 x 2 x 1 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 8 Coupled Situation d Situation The decoupled constraints assumption is not general. Subsystems can be coupled and loops can arise. For example, Optimizer SS1 SS2 1 x 2 x 1 u 2 u 2 w 1 w SS1 SS2 Optim 1 w 2 w 1 u 1 x 2 u 2 x 1 w 2 w Loop x: decision variables vline: SS input w: SS outputs (constraint, cost) hline: SS output u: SS input (dependent) Computation of w 1 and w 2 requires an iterative method. ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 9 Information Flow Loop (2) ? An example where such a loop happens is as follows: ( 1 , min Jxx 2 ) s.t. 1 = (, 2 ( 2 , 1 w g x g x w )) ≥ 0 1 1 (, ( 1 ,w 2 = g x g x w )) ≥ 0 2 2 1 2 n2 × i ,where x 1 ∈ ! n1 , x ∈ ! , g : x i " w i = 1, 2 2 i i ? w 1 and w 2 satisfy coupled relations at each optimization iteration. At each constraint evaluation, nonlinear equations must be solved (e.g. by Newton’s method) in order to obtain w 1 and w , which can 2 be time consuming. Want a way to return to the situation of decoupled constraints. ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 10 Surrogate Variables (“Tearing”) Information loop can be broken by introducing surrogate variables. ( 1 ,min Jxx 2 ) ( 1 ,min Jxx 2 ) s.t. s.t. 1 = (, 2 ( 2 , 1 w g x g x w )) ≥ 0 (,gxu) ≥ 0 1 1 1 1 1 (,gxu) ≥ 0 = (, ( 1 ,w g x g x w )) ≥ 0 2 2 2 1 2 2 2 2 (,u g x u ) = 0 2 ? 1 1 1 (,u g x u 1 ? 2 2 2 ) = 0 ? u 1 and u 2 are decision variables acting as the inputs to g1(SS1) and g2 (SS2). Introducing surrogate variables breaks information loop but increases the number of decision variables. ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 11 Numerical Example 1 + 2 + 2 min J J 2 decoupled min x x 2 + (x ? 3) 2 + (x ? 4) 2 1 3 4 s.t. w 1 ≥ 0 3 s.t. w x x 2 3 + 2x 5 ≥ 0= ? 1 1 w 2 ≥ 0 3 = 2 + 2 w 2 = x x 4 + 2x 6 ≥ 0? 3 where J x x 2 3 3 ? 5 ? 1 J 2 1 = (x 1 ? 3) 2 + (x ? 4) 2 x x 2 3 + 2x x 6 = 0 3 4 3 = 3 ? 3 2 x x 4 3 + 2x x 5 = 0 3 w x x 2 + 2w ? 6 ? 1 1 3 ? 3 w 2 = x x 4 + 2w 3 1 Solution: coupled x = (0, 0, 4, 3,12 , 24 1 3 ) 2 3 2 + 2 2 2 MATLAB 5.3 min x x 2 + (x ? 3) + (x ? 4) 1 3 4 s.t. w g x x x x ) ≥ 0 coupled: 356,423 FLOPS 4.844s = ( 1 , 2 , 3 , 1 1 4 = ( 1 , 2 , 3 , 4 ) ≥ 0 uncoupled: 281,379 FLOPS 0.453s w g x x x x 2 2 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 12 Distributed Design Methods Distributed Design Methods ? Disciplinary models are provided with design tasks ? Optimization is performed at a subsystem level in addition to the system level Concurrent Subspace Optimization (CSSO) ? divide the design problem into several discipline- related subspaces ? each subspace shares responsibility for satisfying constraints while trying to reduce a global objective Collaborative Optimization (CO) ? disciplinary teams satisfy local constraints while trying to match target values specified by a system coordinator ? preserves disciplinary-level design freedom ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 13 Collaborative Optimization OPTIMIZER TARGET STATE Coupled Uncoupled 14 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Collaborative Optimization Two levels of optimization: ? A system-level optimizer provides a set of targets. – These targets are chosen to optimize the system-level objective function ? A subsystem optimizer finds a design that minimizes the difference between current states and the targets. – Subject to local constraints ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 15 Collaborative Optimization min J sys { x 0 } wrt: x 0 = { target variables } sys s.t. J k = 0 ? subproblems J performance analysis k { x 0 } J 1 { x 0 } J k 2 2 local local min J 1 = target - { variables variables }{ variables variables } min J k = target x = { local variables } x = { local variables } s.t. {local constraints} s.t. {local constraints} analysis for subsystem 1 analysis for subsystem k { x } computed { x } computed results results ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 16 CO CO – Subsystem Level 2 local min J 1 = target { variables variables } x = { local variables } s.t. {local constraints} ? The subsystem optimizer modifies local variables to achieve the best design for which the set of local variables and computed results most nearly matches the system targets ? The local constraints must also be satisfied ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 17 CO CO – System Level min J sys wrt: x 0 = { target variables } s.t. J k = 0 ? subproblems k ? System-level optimizer changes target variables to improve objective and reduce differences J k – J k =0 are called compatibility constraints – compatibility constraints are driven to zero, but may be violated during the optimization – CO may therefore discover parts of the design space that cannot be reached by sequential optimization ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 18 CO Example: Aircraft Design Consider a simple aircraft design problem: maximize range for a given take-off weight by choosing wing area, aspect ratio, twist angle, L/D, and wing weight. aero struct perf modified from Kroo et al. AIAA 94-4325 wing area, S aspect ratio, AR twist angle, θ range, R L/D wing weight, W ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 19 x CO Example: Aircraft Design max R 0 T x 0 = [R 0 S 0 AR 0 θ 0 L/D 0 W 0 ] s.t. J 1 =0, J 2 =0, J 3 =0 x 0 J 1 x 0 J 2 min J 2 J 2 =(AR-AR 0 ) 2 + (θ-θ 0 ) 2 + 2 (S-S 0 ) 2 +(W-W 0 ) T x = [S AR] x θ, W struct analysis x 0 J 3 min J 3 J 3 =(R-R 0 ) 2 + (L/D-L/D 0 ) 2 + (W-W 0 ) T x = [L/D W] x R perf analysis min J 1 J 1 =(AR-AR 0 ) 2 + (θ-θ 0 ) 2 + 2 (L/D-L/D 0 ) 2 + (S-S 0 ) T x = [AR θ] L/D aero analysis ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 20 2 Collaborative Optimization min J sys { x 0 } wrt: x 0 = { target variables } s.t. J k = 0 ? subproblems J sysk performance analysis { x 0 } J 1 { x 0 } J k 2 2 local local min J 1 = target - + min J k = target - + { variables variables } { variables variables } 2 2 coupling local coupling local - - { variables variables } { } { y 1k } { variables varia les } x = { local variables } x = { local variables } s.t. {local constraints} y k1 s.t. {local constr in s}t computed analysis for subsystem 1 { x } computed { x } results results analysis for subsystem k ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 21 Collaborative Optimization y x 0 = system-level target variable values x = subsystem local variables ij = coupling functions ? y ij =outputs of subsystem j which are needed as inputs to subsystem i. ? Coupling equations must also be satisfied, so coupling variables are included in subsystem objective. ? Used to reduce the number of system-level parameters. ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 22 BLISS 2000 Schematic Q = { X ,Y * BBi } i sh BBi , w SOptim BB1 BB2 BB3 BB4 ^ 1BB Y ^ 2BB Y ^ 3BB Y ^ 4BB Y 1 Q 2 Q 3 Q 4 Q BB1 BB2 BB3 BB4 X X loc X loc X loc X loc 23 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Black Box (BB) A black box has the following properties: 1. BB has its own local variables (Xloc) and has the exclusive right to determine Xloc. Xloc is a subset of decision variables that can appear explicitly only in the associated BB. 2. BB must satisfies its constraints at each system level iteration. 3. BB operates independently of other BB’s. Neither its inputs nor its outputs are directly communicated between other BB’s. Also, BB assumes no knowledge (e.g. Xloc) of other BB’s. Instead, BB connection is done implicitly via the system optimizer, by the use of Y*. 4. Computation methods within a BB are not restricted by BLISS. (It can be simulation or just an intelligent guess.) ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 24 BLISS 2000 Formulation ? BLISS is a bi-level optimization algorithm. The subsystem optimization formulation is as follows: * Given: Q = { X , Y w } X sh : share decision variable sh , variables: U = { X loc , Y ^ } Y * : input to BB from other BB (surrogate var) min : fU ) = ∑ wY w: weight used in BB optimization ( ^ ii X loc : local decision variable i (s.t. gU) ≤ 0, for each BB Y ^ : output of BB (to system and/or other BB) (hU ) = 0, for each BB g(i) : BB inequali y constraints U U l ≤≤U u h(i) : BB equality constraints output: Y ^ U lower : lower bound on local variables keep: X loc U : upper bound on local variables upper ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 25 Insert Insert – Slides by Dr. Sobieski 26 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Wing drag and weight both influence the flight range R. R is the system objective direct i P P Displ a l i that affect drag Displ Wing - structure Wing - aerodynamics Loads acements a = sweep angle Structure influences R by y by weight ndirectly by st ffness that affect displacements Loads & acements must be consistent R = (k/Drag) LOG [( W o + W s + W f )/ (W o + W s )] Dilemma: What to optimize the structure for? Lightness? Displacements = 1/Stiffness? An optimal mix of the two? Courtesy of Jaroslaw Sobieski. Used with permission. Trade-off between opposing objectives of lightness and stiffness Weight Displacement Weight Displacement ~ 1/Stiffness Thickness limited by stress Wing cover sheet thickness Lightness Stiffness What to optimize for? Answer: minimum of f = w1 Weight + w2 Displacement vary w1, w2 to generate a population of wings of diverse Weight/Displacement ratios Let system choose w1, w2. Courtesy of Jaroslaw Sobieski. Used with permission. Approximations Why Approximations: Analyzer Analyzer Approximate Model Human judgment problems ice for large problems to reduce and control cost $$ cents a.k.a. Surrogate Models Optimizer Optimizer OK for small Now-standard pract Courtesy of Jaroslaw Sobieski. Used with permission. Design of Experiments(DOE) & Response Surfaces (RS) RS provides a domain guidance , rather than local guidance, to system optimizer DOE Placing design points in design space in a pattern Example: Star pattern (shown incomplete) RS X1 X2 F(X) F(X) = a + {b} {X} + {X} [c]X quadratic polynomial hundreds of variables Courtesy of Jaroslaw Sobieski. Used with permission. BLISS 2000: MDO Massive Computational Problem Solved by RS (or alternative approximations) or di or di or di System optimization X1 X2 ) X1 X2 ) X1 X2 ) RS RS ine in parallel I n s t a n t a n e o u s r e s p o n s e MC D A T A B A S E Optimization of subsystem scipline Analysis of subsystem scipline Optimization of subsystem scipline F(X F(X F(X Precompute off-l cloud Radical conceptual simplification at the price of a lot more computing. Concurrent processing exploited. Courtesy of Jaroslaw Sobieski. Used with permission. Coupled System Sensitivity Consider a multidisciplinary Y A system with two subsystems X A and B (e.g. Aero. & Struct.) system equations can be written in symboli [( X A A ,Y ),Y ] = 0 B A B [( X ,Y ),Y ] = 0 B A B rewrite these as follows Y A = Y A ( X ,Y ) A B Y B = Y B ( X ,Y A ) B A B A B X B Y B Y A Y c form as these governing equations define as implicit functions. Implicit Function Theorem applies. Courtesy of Jaroslaw Sobieski. Used with permission. Coupled System Sensitivity - Equations These equations can be represented in matrix notation as Y dY dX ? ? ? ?? ? ? ? ? = ? ? ? A A A I ? Y A X ? ? A ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Y B Y dY? ? ? ? ? ? ? ? ? ? ? ? B B I? 0 Y dX different A A same Y dY? ? ? ?? ? Right Hand Sides A A I ? 0 = ? ? ? ? ? ? matrix ? ? ? ? ? ? ? ? ? ? ? ? ? Y dX ? ? ? Y? ?X B B B Y dY dX ? ? ? ? ? ? ? ? ? ? ? ? B B B I ? B Y A Total derivatives can be computed if partial sensitivities computed in each subsystem are known Linear, algebraical equations with multiple RHS Courtesy of Jaroslaw Sobieski. Used with permission. X Flowchart of the System Optimization Process System Analysis α β γ System Sensitivity Analysis α β γ Start Sensitivity solution Approximate Analysis Optimizer X Y γ Y α β Y β Stop Courtesy of Jaroslaw Sobieski. Used with permission. System Internal Couplings Co u p l i n g B r e a d t h Quantified All-in-One Decompose ( ( D e c o m p o s e ) ) ( D e c o m p o s e ) Strength: relatively large ? YO/ ?YI Breadth: {YO} and {YI} are long [? YO/ ?YI] large and full Coupling Strength Courtesy of Jaroslaw Sobieski. Used with permission. Supersonic Business Jet Test Case Structures (ELAPS) ) ) ) Aerodynamics (lift, drag, trim supersonic wave drag by A - Wave Propulsion (look-up tables Performance (Breguet equation for Range Some stats: Xlocal: struct. 18 aero 3 propuls. 1 X shared: 9 Y coupl.: 9 Examples: Xsh - wing aspect ratio, Engine scale factor Xloc - wing cover thickness, throttle setting Y - aerodynamic loads, wing deformation. Courtesy of Jaroslaw Sobieski. Used with permission. System of Modules (Black Boxes) for Supersonic Business Jet Test Case Struct. Perform. Aero Propulsion Data Dependence Graph RS - quadratic polynomials, adjusted for error control Courtesy of Jaroslaw Sobieski. Used with permission. 0 1 1 10 Flight Range as the Objective Normalized Cycles 0.2 0.4 0.6 0.8 1.2 1.4 2 3 4 5 6 7 8 9 Series1 Series2 RS 1 10 1 0 Analysis Histogram of RS predictions and actual analysis for Range Courtesy of Jaroslaw Sobieski. Used with permission. References (I) Jaroslaw, Sobieszczanski-Sobieski et al. Bi-level Intergrated System Synthesis (BLISS) For Concurrent And Distributed Processing. AIAA 2002-5409. Updated Journal Article (handout): Jaroslaw Sobieski, Altus, Phillips, Sandusky, “Bi-level Integrated System Synthesis for Concurrent and Distributed Processing” AIAA Journal, Vol. 41, No.10, October 2003, pp. 1996-2003 I.P. Sobieski and I.M. Kroo. Collaborative Optimization Using Response Surface Estimation. AIAA Journal Vol. 38 No. 10. Oct 2000. R.D. Braun and I.M. Kroo. Development and Application of the Collaborative Optimization Architecture in a Multidisciplinary Design Environment. ICASE/NASA Langley Workshop on MDO, March 13-16, 1995 Erin J. Cramer et al. Problem Formulation for Multidisciplinary Optimization. SIAM Journal of Optimization. Vol. 4, No. 4 pp. 754-776, Nov 1994 Natalia M. Alexandrov (ed). Multidisciplinary Design Optimization – State of the Art. SIAM. 1994. ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 27 References (II) Kroo, I.: “MDO applications in preliminary design: status and directions,” AIAA Paper 97-1408, 1997. Kroo, I. and Manning, V.: “Collaborative optimization: status and directions,” AIAA Paper 2000-4721, 2000. Sobieski, I. and Kroo, I.: “Aircraft design using collaborative optimization,” AIAA Paper 96-0715, 1996. Balling, R. and Wilkinson, C.: “Execution of multidisciplinary design optimization approaches on common test problems,” AIAA Paper 96 4033, 1996. Giesing, J. and Barthelemy, J.: “A summary of industry MDO applications and needs”, AIAA White Paper, 1998. AIAA MDO Technical Committee: “Current state-of-the-art in multidisciplinary design optimization”, 1991. “Optimal Design in Multidisciplinary Systems,” AIAA Professional Development Short Course Notes, September 2002. ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 28