麻省理工学院：《Multidisciplinary System》Lecture 4 17 February

Mes 16888 ESO.77 Multidisciplinary System Design Optimization( MSDO) Decomposition and Coupling Lecture 4 17 February 2004 Olivier de weck o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Multidisciplinary System Multidisciplinary System Design Optimization (MSDO) 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

M Today's Topics 6888 ESO.77 ast 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 o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Today’s Topics 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

M esd Fundamentally different approaches in MDO 16883 ESO.77 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 optimization at subsystem and system levels CO BLISS o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Fundamentally different approaches in MDO 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

Mlesd Standard Optimization Problem 50. 9 Given x x∈R J:R”→>R Optimization Engine g:R”→>R J(x) Solve the problem x g(x min(x) st,g(x)≥0 Function Evaluator That is, findx S t J()sf(x), vxe dom()ndom(g) o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Standard Optimization Problem Standard Optimization Problem Given * x x 0 J ( ) x x gx ( ) Optimization Engine Function Evaluator ∈ n x ! n J : ! → → ! n m g : ! ! Solve the problem min J x( ) s.t. g x( ) ≥ 0 * * That is, find x s.t. J( x ) ≤ f x ( ), ∀ ∈ x dom( J ) ∩ dom( ) g © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox 4

Mlesd Distributed Analysis 16888 ESO.77 Disciplinary models provide analysis Optimization is controlled by some overseeing code or database e.g. Genie database system(Stanford) ISight(Optimizer) iSight Gene optimizer design variables NPSoL Shared data X subsystem analyses Structures Aero Local data ocal data o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Distributed Analysis 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

Mes Distributed Analysis 16888 ESO.77 Optimizer objective design variables constraints X//9( g(x X h(x) h(x) aerodynamic performance structural analysis analysis analysis 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 o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Distributed Analysis 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

Mes Distributed Design 6888 ES077 System level optimizer command/result command/result command/result SS1 SS2 SSN optimizer optimize optimizer SS1 SS2 SSN analyzer analyzer analyzer Subsystem black boX (BB C Massachusetts Institute of Technology -Prof. de Weck and Prof WillcoX

Distributed Design 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

Mes Advantages of Decoupling 6888 ES077 Computation of g(x)can be very time consuming, want to divide the work and compute in parallel For example, if x=(x,-x), where, xER and g(x)=(g(),g,(x Then g, and g2 can be computed in parallel. Graphically, Optimizer Optim g SS1 SS1 SS2 SS1 o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Advantages of Decoupling Advantages of Decoupling Computation of g(x) can be very time consuming, want to divide the work and compute in parallel. n 2 For example, if x = ( , x x 2 ), where x ∈ ! n1 1 1 , x2 ∈ ! and g( x) = (g x g x ( ), ( )) 1 1 2 2 Then g1 and g2 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

M Coupled Situation 16888 ESO.77 The decoupled constraints assumption is not general Subsystems can be coupled and loops can arise. For example Optimizer Optim x11 w SSI SS2 p 001 X. decision variables vline: SS input w: SS outputs(constraint, cost u:SS input( dependent) hline: SS output Computation of w, and w, requires an iterative method o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Coupled Situation Coupled 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 u 1 u 2 w 1 w2 SS1 SS2 Optim w 1 w 2 u 1 1 x u 2 2 x w 1 w 2 Loop x: decision variables vline: SS input w: SS outputs (constraint, cost) hline: SS output u: SS input (dependent) Computation of w1 and w2 requires an iterative method. 9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox

Mlesd Information Flow Loop(2) 16888 ESO.77 An example where such a loop happens is as follows minD(x,x2) W=81(x,g2(x2,w1)20 6y2=8M1(x,m12)≥0 where x∈R,x2∈R,g1:x×;b,i=1,2 W,and w2 satisfy coupled relations at each optimization iteration At each constraint evaluation, nonlinear equations must be solved (e.g. by Newtons method)in order to obtain w and w2, which can be time consuming Want a way to return to the situation of decoupled constraints o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox

Information Flow Loop (2) Information Flow Loop (2) • An example where such a loop happens is as follows: ( 1 min Jx x, 2 ) s.t. 1 = (, 2 ( 2 , w g x g x w1)) ≥ 0 1 1 ( , ( 1 w , 2 = g x g x w )) ≥ 0 2 2 1 2 n 2 × 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 2, which can 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