1© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxAnas AlfarisDecomposition and CouplingLecture 4Multidisciplinary System Design Optimization (MSDO)2© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxToday’s Topics• Information Flow and Coupling • MDO frameworks– Single-Level (Distributed analysis)– Multi-Level (Distributed design)• Collaborative Optimization• Analytical Target Cascading• (Hierarchical Decomposition & Multi-Domain Formulation)3© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxStandard Optimization ProblemGivenmin ( )s.t. ( ) 0JxgxSolve the problem**That is, find s.t. J( ) ( ), dom( ) dom( )x x f x x J gOptimization EngineFunction Evaluatorx()Jx()gx0x*xmnznngJx::4© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxInformation FlowABA BABABA BABAABA BBABA BABDependent Tasks (Series) Independent Tasks (Parallel) Interdependent Tasks (Coupled)Image by MIT OpenCourseWare.5© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxInformation FlowCDABGHEFKLIJC DAB GHEFK L IJ••••••••••••Feed ForwardFeed Backward6© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxInformation FlowCDABGHEFKLIJC DAB GHEFK L IJSequentialParallelCoupled••••••••••••7© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxAdvantages of DecouplingComputation of g(x) can be very time consuming, want to divide the work and compute in parallel.For example, if121 2 1 21 1 2 2( , ), where ,and g( ) ( ( ), ( ))nnx x x x xx g x g xThen g1and g2can be computed in parallel. Graphically,OptimizerSS1 SS21x1g2g2xSS1SS2Optim1x2x1g2gRR8© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxCoupling The decoupled constraints assumption is not general. Subsystemscan be coupled and loops can arise. For example,OptimizerSS1 SS21x2x1u2u2w1wx: decision variablesw: SS outputs (constraint, cost)u: SS input (dependent)SS1SS2Optimvline: SS inputhline: SS output1w2w1u1x2u2x1w2wLoopComputation of w1and w2requires an iterative method.9© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxCoupling• An example where such a loop happens is as follows:121 1 1 2 2 12 2 2 1 1 2min ( , )( , ( , )) 0s.t. ( , ( , )) 0J x xw g x g x ww g x g x w1212where , , : , 1,2nni i i ix x g x u w i• w1and w2satisfy 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 w1and w2, which canbe time consuming.Want a way to return to the situation of decoupled constraints.RR10© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxSurrogate Variables (“Tearing”)Information loop can be broken by introducing surrogate variables.121 1 1 2 2 12 2 2 1 1 2min ( , )( , ( , )) 0s.t. ( , ( , )) 0J x xw g x g x ww g x g x w• u1and u2are 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.121112222 1 1 11 2 2 2min ( , )s.t.( , ) 0( , ) 0( , ) 0( , ) 0J x xg x ug x uu g x uu g x u11© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxNumerical Example1212221 1 2222 3 4331 1 2 2332 3 4 1mins.t. 0 0where ( 3) ( 4) 2 2JJwwJ x xJ x xw x x ww x x w2 2 2 21 2 3 41 1 1 2 3 42 2 1 2 3 4min ( 3) ( 4)s.t. ( , , , ) 0 ( , , , ) 0x x x xw g x x x xw g x x x xcoupled2 2 2 21 2 3 4331 1 2 5332 3 4 6331 2 5 6333 4 6 5min ( 3) ( 4)s.t. 2 0 2 0 2 0 2 0x x x xw x x xw x x xx x x xx x x xdecoupledSolution:1233(0,0,4,3,12 ,24 )xMATLAB® 5.3coupled: 356,423 FLOPS 4.844suncoupled: 281,379 FLOPS 0.453s12© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxSingle-level and Multi-Level FrameworksSingle-level(Distributed Analysis)-disciplinary models provide analysis-all optimization done at system levelnon-hierarchical decompositionMulti-level(Distributed Design)-provide disciplinary models with design tasks-optimization at subsystem and system levelshierarchical decomposition13© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxSingle-level (Distributed Analysis)• Disciplinary models provide analysis• Optimization is controlled by some overseeing code or databasee.g. ISight (Optimizer)SystemOptimizerShared dataLocal dataStructuresLocal dataAeroOptimizerdesign variablesconstraintsiSightxJ(x),g(x),h(x)subsystem analyses14© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxOptimizerobjectivedesign variablesconstraintsxJ(x)performance analysisaerodynamic analysisstructural analysisxg(x)h(x)xg(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 Single-level (Distributed Analysis)15© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox• Multi-level Optimization methods distribute decision making throughout the system• Subsystem level models are provided with design tasks• Optimization is performed at a subsystem level in addition to the system level • Provide some autonomy to design groups and reduces communication requirements.Multi-level (Distributed Design)16© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxMulti-level (Distributed Design)System level optimizerSS1optimizerSS2optimizerSSNoptimizerSS1analyzerSS2analyzerSSNanalyzer……command/resultcommand/resultcommand/resultSubsystemblack box (BB)17© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxCollaborative OptimizationCollaborative Optimization (CO)• disciplinary teams satisfy local constraints while trying to match target values specified by a system coordinator• preserves disciplinary-level design freedom.• CO is used typically to solve discipline-based decomposed system optimization problems.18© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxCollaborative OptimizationOPTIMIZERTARGET STATECoupledUncoupled19© Massachusetts Institute of Technology - Prof. de Weck and Prof. WillcoxCollaborative OptimizationTwo levels of optimization:• A system-level optimizer provides a set of targets. – These