Introduction to System Partitioning and Coordination

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1 Introduction to System Partitioning and Coordination James T. Allison ME 555 April 4, 2005 University of Michigan Department of Mechanical Engineering

2 Your Subsystem Optimization Problems are Complete Now What? Methods for coordinating subsystems: Sequential Optimization MDF (a.k.a. All-in-One) IDF AAO ATC

3 Can Subsystem Design Problems be Solved Independently? Not if interaction is present. An interaction is present when the response of one element is not only dependent on its own state, but also on the state of other members in the system. Result: The whole is different than the sum of the parts. Case I: Members collaborate to synergistically improve system performance, even at the expense of individual member objectives. (Teamwork) Case II: Members considering only their own performance may experience interactions that collude to degrade system performance.

4 Interaction, Coupling, and Coupling Strength Coupling Interaction Coupling indicates the existence of interaction between system elements (subspaces). Coupling Strength Interaction Strength If a subspace response is highly dependent (sensitive) on the state of another subspace, the coupling between the subspace is strong. If a subspace response is weakly dependent (insensitive) on the state of another subspace, the coupling between the subspace is weak. The coupling strength of an entire system depends on the number and strength of all couplings present.

5 Illustration of Interaction/Coupling Existence: DSM SS1 SS2 SS3 SS4

6 Physical Example: 2DOF Mass Spring System Governing Equations of Motion: [ m1 0 0 m 2 ] { ẍ1 ẍ 2 } + [ k1 + k 2 k 2 k 2 k 2 ] { x1 x 2 } = { 0 0 } Coupled in stiffness: Motion of m 1 (i.e. x 1 (t)) is dependent of the value of k 2 ( x 1 (t) depends on the state of x 2 ), and visa-versa. m 1 k 1 x 1 k 2 m 2 x 2

7 Sequential Optimization Sequential Approach: Optimization Commonly used process in industry. Deals with, but does not exploit, interactions. Design freedom is progressively reduced. Lack of parallelization and iterative optimization increases design cycle time. May not find a system optimal design, even after iteration. SS1 Optimizer SS2 Optimizer SS3 Optimizer SS1 Analysis SS2 Analysis SS3 Analysis Fix x l1, x s1 Fix x l2, x s2 Final Design

8 Sequential Optimization Example f(x 1, x 2 ) = a 1 (x 1 a 2 ) 2 + a 3 (x 2 a 4 ) 2 + a 5 x 1 x 2 Case I: No Interaction a = (1, 1, 1, 1, 0) T 6 5 (x 1-1) 2 +(x 2-1) 2 4 x x 1

9 Sequential Optimization Example f(x 1, x 2 ) = a 1 (x 1 a 2 ) 2 + a 3 (x 2 a 4 ) 2 + a 5 x 1 x 2 Case II: Interaction Present a = (1, 1, 1, 1, 1) T 6 5 (x 1-1) 2 +(x 2-1) 2 -x 1 x 2 4 x 2 3 OFAT solution 2 optimum x 1

10 What are Some Alternatives to Sequential Optimization? MDF Multidisciplinary Feasible approach a complete analysis is performed at every optimization iteration. Also known as the All-in-One approach. IDF Individual Disciplinary Feasible approach system analysis is performed simultaneously with system optimization. AAO All-at-Once approach system analysis, optimization, and determination of state variables performed simultaneously. Useful for systems involving differential equations. ATC Analytical Target Cascading each system element has its own optimizer. Parent design problems coordinate the design of child design problems. Useful for systems with a hierarchical structure.

11 x l Local design variables: Design variables that are each inputs to only one subspace. x s Shared design variables: Design variables that are each inputs to more than one subspace. y Coupling variables: Quantities passed between subspaces that are not original design variables, but rather artifacts of decomposition. MDF/IDF/AAO Terminology { s 1 x= x, x,, x x x x 1 = 1, s1 SS1 N f1, g1, h1 y 21 y 12 y j1 y 1j } System Analysis x x x 2 = 2, s2 SS2 f2, g2, h ( 1,, N ) = { 1,, N} = { 1,, N} f f f g g g h h h x = x, x N N sn SSN y Nj y jn fn, gn, hn

12 MDF Formulation and Architecture min x={x l,xs} f(x) subject to g(x) = {g 1, g 2,...} 0 h(x) = {h 1, h 2,...} = 0 { s, 1, 2} x= x x x System Analyzer ( 1, 2) = { g1, g2} = h, h } f f f g h { 1 2 Optimizer x 1, xs f, g, h x 2, xs f2, g2, h2 x f, g, h Subspace Analyzer Subspace Analyzer Analyzer s y 12 s s SS1 SS2 y 21

13 Example Problem Single Element Aeroelasticity Aeroelasticity: Requires both aerodynamic and structural analysis Application: Air-flow sensor design Implementation: Will be used to demonstrate MDF, IDF, and ATC.

14 Air-flow Sensor Analysis Structural Analysis: M = kθ = 1 2 F l cos θ Given a design (l, w) and a drag force F, solve for the corresponding deflection θ. Aerodynamic Analysis: F = CA f v 2 = Clw cos θv 2 Given a design (l, w) and a deflection θ, find the drag force F. System Analysis: Given a design (l, w), find the equilibrium values F and θ. v F 1/2 cos k cos

15 Air-flow Sensor Design Sensor Calibration: Find the sensor design that achieves a deflection θ for a given air speed v that is as close as possible to a target deflection value θ T, while ensuring the drag force remains below an upper limit, and the sensor surface area remains a constant. min l,w (θ θ T ) 2 subject to F F max 0 lw A = 0 Design Parameters: k, A, F max, C, v

16 Solution by Monotonicity For cases where meeting the deflection target requires a drag force greater than F max, the target cannot be met, and the inequality constraint will be active. This, along with the equality constraint lw = A, facilitates analytical solution for l and w. F max = Clw cos θv 2 kθ 1 2 F maxl cos θ = 0 l = ( ) θ = cos 1 Fmax CAv 2 ( ) 2k cos 1 Fmax CAv 2 CAv 2 F 2 max Now that F, θ and l are known, we can solve for w using w = A/l. This solution technique provides a check for other approaches.

17 Partitioning by Discipline What are the shared design variable(s)? What are the local design variable(s)? What are the coupling variable(s)? How might this partitioned analysis be executed? (i.e., find F and θ for a given l and w. l l,w Structural Analysis Aerodynamic Analysis F

18 Matlab Demonstration Fixed Point Iteration Analysis

19 Analysis Discussion Would increasing k increase or decrease the number of required iterations? Would increasing w increase or decrease the number of required iterations?

20 MDF Design Solution MDF: Complete analysis for every optimization iteration. Monotonicity Solution: x = [l, w ] = [0.3651, ] Matlab Demonstration MDF

21 MDF Discussion Advantages: Intuitive procedure/no specialized knowledge required Easy to incorporate existing models Disadvantages: Proven that FPI will not converge in many cases If multiple analysis solutions exist, MDF may not choose the best one Cannot be parallelized Decision making is centralized

22 IDF Formulation and Architecture min x={x l,xs},y f(x, y) subject to g(x, y) = {g 1, g 2,...} 0 h(x, y) = {h 1, h 2,...} = 0 f 1, g 1, h 1, y i1 Optimizer f 2, g 2, h 2, y i2 h aux (x, y) = y(x, y) y = 0 x 1, x s1, y 1j x 2, x s2, y 2j Analyzer Analyzer s s

23 IDF Formulation of Air Flow Sensor Problem min l,w,θ,f (θ θ T ) 2 subject to F F max 0 lw A = 0 θ θ(l, F ) = 0 F F (l, w, θ) = 0 Structural Analysis System Optimizer l, F l, w, F Aero Analysis

24 IDF Design Solution IDF: Analysis and Design performed simultaneously Monotonicity Solution: x = [l, w ] = [0.3651, ] Matlab Demonstration IDF

25 IDF Discussion Advantages: Exploits the power of optimization algorithms to drive analysis (another example is regression) optimizer controls both x and y All subspaces executed in parallel Analysis and Design performed simultaneously reduces computation time In many cases will converge when MDF will not, and may find better solutions than MDF Faster convergence for tightly coupled systems Disadvantages: Increased dimension of the optimization problem over MDF Increased centralization of decision making Must meet hard equality constraints

26 AAO Formulation and Architecture min x={x l,xs},y,s f(x, y) subject to g(x, y, s) = {g 1, g 2,...} 0 h(x, y, s) = {h 1, h 2,...} = 0. { } y(x, y, s) y h aux (x, y, s) = w(x, y, s) Optimizer f 1, g 1, h 1, y i1, w 1 f 2, g 2, h 2, y i2, w 2 x 1, x s1, y 1j, s 1 x 2, x s2, y 2j, s 2 = 0. Evaluator Evaluator

27 Advantages: AAO Discussion (No Demo) Exploits the power of optimization algorithms to drive analysis and DEQ solution optimizer controls x, y and s Convenient for systems of PDE s All subspaces executed in parallel Highly efficient process Excellent convergence properties Faster convergence for tightly coupled systems Disadvantages: Increased dimension of the optimization problem over IDF Highly centralized decision making increased communication requirements Must meet hard equality constraints Difficult to integrate simulations

28 February 23, :42 Proceedings of DETC ASME Design Engineering Technical Conferences Long Beach, California, USA, September 24-28, 2005 DRAFT DETC More details on MDF, IDF, and AAO are available in this upcoming DETC publication, which is now available on the course website. ON THE IMPACT OF COUPLING STRENGTH ON COMPLEX SYSTEM OPTIMIZATION FOR SINGLE-LEVEL FORMULATIONS James Allison, Michael Kokkolaras, and Panos Papalambros Optimal Design Laboratory Department of Mechanical Engineering University of Michigan, Ann Arbor, Michigan ABSTRACT Design of modern engineering products requires complexity management. Several methodologies for complex system optimization have been developed in response. Single-level strategies centralize decision-making authority, while multi-level strategies distribute the decision-making process. This article studies the impact of coupling strength on single-level Multidisciplinary Design Optimization formulations, particularly the Multidisciplinary Feasible (MDF) and Individual Disciplinary Feasible (IDF) formulations. The Fixed Point Iteration solution strategy is used to motivate the analysis. A new example problem with variable coupling strength is introduced, involving the design of a turbine blade and a fully analytic mathematical model. The example facilitates a clear illustration of MDF and IDF and provides an insightful comparison between these two formulations. Specifically, it is shown that MDF is sensitive to variations in coupling strength, while IDF is not. KEYWORDS Complex System Design, Multidisciplinary Design Optimization, Coupling Strength, Multidisciplinary Feasible, Individual Disciplinary Feasible, Fixed Point Iteration 1 INTRODUCTION This article endeavors to illustrate the implementation of single-level formulations for complex system optimization via an analytical example. Specifically, the effect that subsystem interdependence (coupling) has on the performance of these formulations is explored. Background in complex system optimization is provided, and the mathematical model for the illustrative example is developed and presented. This work aims at improving general understanding of techniques for complex system optimization, and it is part of a greater effort to review single and multi-level MDO formulations [1, 2]. Design of products classified as complex systems poses substantive challenges to both analysis and optimization, necessitating specialized solution techniques. A complex system is defined as an assembly of interacting members that is difficult to understand as a whole. An interaction between members exists if some aspect of one member affects how the system responds to changes in another member. A system is difficult to understand if an individual cannot understand the details of all members and all interactions between members. A system may qualify as complex due to its large scale (large number of members or inputs), or due to strong interactions. These interactions complicate optimization, but provide opportunity to exploit synergy between system members. Analysis of complex systems as an undivided whole can be inefficient, if not intractable. An alternative is to partition the system into smaller subsystems (or subspaces). Wagner [3] identified four categories of system partitioning methods: by object, by aspect, sequential, or matrix. The aspect (discipline) partitioning paradigm is used in this article, and the term subspace is used to refer to system members. System partitioning is also characterized by the structure of its communication pathways. A non-hierarchic system has no restrictions on these pathways 1 Copyright c 2005 by ASME

29 Introduction to Analytical Target Cascading ATC: Target setting methodology intended for product development. Based on a hierarchical problem structure. Typically partitioned by physical object (subsystems/components). Early product development tool. Can be used for either target setting or for design optimization. Originally intended for problems with unidirectional information flow, but can be used where feedback coupling is present. Multilevel optimization methodology each subspace has its own optimizer (rather than a single centralized optimizer).

30 Analysis Structure Amenable to ATC System-level design variables Subsystem Responses: Subsystem performance Geometry System Analysis: Vehicle System Integration Packaging Control Subsystem-level design variables Subsystem Analysis: Powertrain integration and simulation Component Responses: Engine map Geometry Component-level design variables Component Analysis: Engine Simulation......

31 Using ATC for Product Development 1. Specify top-level targets. 2. Propagate targets through the system. (a) Develop low-fidelity analysis models. (b) Partition the system. (c) Formulate the target cascading problem. (d) Solve the target cascading problem with a coordination strategy. 3. Perform subspace detail design to meet targets. 4. Verify system consistency and system level target matching.

32 How is the Target Cascading Problem Formulated and Solved? 1. Choose a partitioning structure 2. Identify the input/output relations 3. Identify shared values 4. Formulate the optimization problem for each system element (a) Minimize local objective function (b) Create penalties for shared values that do not match (c) Subject to local design constraints 5. Choose a strategy for executing element optimization problems (explained later) 6. Run the process until system inconsistency meets convergence criteria

33 ATC Formulation Air Flow Sensor Problem What values are shared between subspaces? l Structural Analysis F l,w Aerodynamic Analysis

34 ATC Formulation Air Flow Sensor Problem What values are shared between subspaces? l, θ, and F l Structural Analysis F l,w Aerodynamic Analysis

35 ATC Formulation Air Flow Sensor Problem Structural Problem (arbitrarily chosen as master problem) min l,f (f 1 f ) 2 + (l 1 l 2 ) 2 + (F 1 F 2 ) 2 + (θ 1 θ 2 ) 2 subject to F F max 0 Aerodynamic Problem (arbitrarily chosen as subproblem) min l,w,θ (l 1 l 2 ) 2 + (F 1 F 2 ) 2 + (θ 1 θ 2 ) 2 subject to lw A = 0 where f = (θ θ T ) 2, and f is the known optimal value of f.

36 ATC Implementation 1. Begin with a guess for subproblem responses to solve the master problem. 2. Use the master problem responses to solve the subproblem. 3. Iterate until system consistency stops changing.

37 ATC Implementation 1. Begin with a guess for subproblem responses to solve the master problem. 2. Use the master problem responses to solve the subproblem. 3. Iterate until system consistency stops changing. Note: Other coordination strategies are possible when more than two levels exist. Penalty function weights may be required to improve ATC convergence. In theory, if all targets are attainable, ATC will converge to an exactly consistent system. If targets are unattainable, some type of weighting update method is required.

38 Matlab Demonstration ATC

39 Additional ATC Example Problem Illustrative example with slightly increased complexity to demonstrate additional aspects of ATC. Statically Indeterminate Structural Analysis (SISA) of an anchoring system. Can be expanded to an arbitrarily large number of subspaces.

40 L Ø d nb beam n b rod n b -1 Ø d r(nb-1) l r(nb-1) Ø d 3 beam 3 Ø d r2 rod 2 l r2 Ø d 2 beam 2 Ø d r1 Ø d 1 rod 1 l r1 beam 1 F 1

41 Anchor Design Problem min x={x l,xs} f(x) = δ 1 (x) subject to g 1i (x) = σ bi (x) σ allow 0 i = 1... n b g 2j (x) = σ aj (x) σ allow 0 j = 1... n r g 3 (x) = n b i=1 m bi + nr i=1 m rj m allow 0 g 4i (x) = F t i F t allow 0 i = 1... n b x = {d 1 d 2 d 3 d r1 d r2 } T

42 Analytical Target Cascading Example Analysis structure manipulated into a sequential form with only one feedback. Analysis becomes a root-finding problem: ˆF1 (F 3 ) F 1 = 0 Present two-level partitioning scheme. F 3 3 a d r2 f 2 b F 2 e r1 g ˆ 1 F 1 c

43 F 1 Subspace 1 d 1, d r1 F Subspace 2 Subspace 3 d 2 d 3, d r2 F 3 F 3 subspace 3 subspace 2 subspace 1 F 3 3 a d r2 f 2 b F 2 e r1 g ˆ F 1 1 c

44 min x 11 =[d 1,d r1,f 2,F 3,δ 2,m 2,m 3 ] T (δ 1(11) 1 0)2 + (F 1(11) 1 F 1(11) 0 )2 + (F 2(22) 1 F 2(22) 2 )2 SS1-Top +(δ 1 2(23) δ2 2(23) )2 + (m 1 2(22) m2 2(22) )2 +(m 1 3(23) m2 3(23) )2 + (F 1 3(21) F 2 3(22) )2 + (F 1 3(21) F 2 3(23) )2 subject to g 11 (x 11 ) = σ b1 (x 11 ) σ allow 0 g 21 (x 11 ) = σ a1 (x 11 ) σ allow 0 g 3 (x 11 ) = m 1 (x 11 ) + m 2 + m 3 m allow 0 g 41 (x 11 ) = F t 1 (x 11 ) F t allow 0 min x 22 =[d 2,F 3,δ 2 ] T (F 2(22) 1 F 2(22) 2 )2 + (m 1 2(22) m2 2(22) )2 SS2-Bottom +(F 1 3(21) F 2 3(22) )2 + (δ 1 2(23) δ2 2(23) )2 subject to g 12 (x 22 ) = σ b2 (x 22 ) σ allow 0 g 42 (x 22 ) = F t 2 (x 22 ) F t allow 0 min x 23 =[d 3,d r2,f 3 ] T (δ 2(23) 1 δ2 2(23) )2 + (m 1 3(23) m2 3(23) )2 + (F 3(21) 1 F 3(23) 2 )2 SS3-Bottom subject to g 13 (x 23 ) = σ b3 (x 23 ) σ allow 0 g 22 (x 23 ) = σ a2 (x 23 ) σ allow 0 g 43 (x 23 ) = F t 3 (x 23 ) F t allow 0

Lecture Notes: Introduction to IDF and ATC

Lecture Notes: Introduction to IDF and ATC James T. Allison April 5, 2006 This lecture is an introductory tutorial on the mechanics of implementing two methods for optimal system design: the individual