Jan. 28, 2004MDO Algorithms - 1 Colloquium on MDO, VSSC Thiruvananthapuram Part II: MDO Architechtures Prof. P.M. Mujumdar, Prof. K. Sudhakar Dept. of Aerospace Engineering, IIT Bombay & Umakant Joysula, DRDL, Hyderabad System Design – New Paradigms

Jan. 28, 2004MDO Algorithms - 2 Colloquium on MDO, VSSC Thiruvananthapuram Coupled System Engg. Design Optimization Problem Statement Analyzer & Evaluator Classification of MDO Architectures Single level Architectures / formulations Bi-level Architectures / formulations OUTLINE

Jan. 28, 2004MDO Algorithms - 3 Colloquium on MDO, VSSC Thiruvananthapuram COUPLED SYSTEM Comprises of several modules or components or disciplines Output of one module affects another module and vice- versa Analysis of one discipline requires information from analysis of another discipline DISCIPLINE 1 S1 Z DISCIPLINE 2 S2 Z M ULTI D ISCIPLINARY A NALYSIS (MDA)

Jan. 28, 2004MDO Algorithms - 4 Colloquium on MDO, VSSC Thiruvananthapuram Stating the design problem as a Formal Engineering Optimization problem Integration of Optimization and Analysis of Coupled Systems - MDAO MDAO can be accomplished in several ways leading to different MDO architectures MDO ARCHITECTURE / FORMULATION

Jan. 28, 2004MDO Algorithms - 5 Colloquium on MDO, VSSC Thiruvananthapuram ENGINEERING DESIGN PROBLEM Min f (Z, S(Z) ) subject to h(Z,S(Z)) = 0; g(Z,S(Z))  0; S(Z) is a solution of A (Z, S(Z)) = 0; A(Z,S) = 0; Non-linear, Iterative, Fully Converged Coupled Multi- Disciplinary Analysis (MDA) – Time Intensive OPTIMIZER Interface Z ANALYSIS Z S f, g, h Nested ANalysis and Design (NAND)

Jan. 28, 2004MDO Algorithms - 6 Colloquium on MDO, VSSC Thiruvananthapuram ANALYSIS AND EVALUATOR Analysis S Converged Z S r Evaluator r = A(Z, S) soso Iterator update S Closed Analysis Nested ANalysis and Design (NAND)

Jan. 28, 2004MDO Algorithms - 7 Colloquium on MDO, VSSC Thiruvananthapuram ALTERNATE STATEMENT Evaluator r Z, S Interface f, h, r, g Z, S Optimizer Optimizer searches for solution Evaluator light on time Converged analysis not sought when far away from optimum? Analysis Open Analysis feasible only at optimum Design & Constraint vectors are augmented Simultaneous ANalysis & Design (SAND)

Jan. 28, 2004MDO Algorithms - 8 Colloquium on MDO, VSSC Thiruvananthapuram Analysis v/s Evaluators * Solving pushed to optimization level Conventional approach: INTERFACE Solve OPTIMIZER 2. Calculates 3. Calculates Evaluator: Does not solve Evaluates residues for given Computationally inexpensive OPTIMIZER INTERFACE EVALUATOR A different approach * : Analysis: Conservation laws of system Nonlinear, iterative Multidisciplinary Time intensive 1. Generates 2. Calculates

Jan. 28, 2004MDO Algorithms - 9 Colloquium on MDO, VSSC Thiruvananthapuram SYSTEM AND DISCIPLINE LEVEL SYSTEM LEVEL (DISCIPLINE COORDINATOR) Z S DISCIPLINE 1 Z L1 DISCIPLINE ‘2’ Z L2 Z = (  Z Li )  (Z Si ) Z L : Local to discipline (Disciplinary Variables) Z S : Shared by more than one discipline (System Variables) Y : Coupling functions Z S1 Y Z S2

Jan. 28, 2004MDO Algorithms - 10 Colloquium on MDO, VSSC Thiruvananthapuram CLASSIFICATION OF MDO ARCHITECTURES Based on the fact whether the optimization is carried out at Single level Bi-level * One optimizer * System Optimizer - controls all - System variables design variables * Disciplinary Optimizer - Disciplinary variables

Jan. 28, 2004MDO Algorithms - 11 Colloquium on MDO, VSSC Thiruvananthapuram Based on manner in which the Inter-Disciplinary Feasibility and Multi-Disciplinary Analysis (MDA) is carried out. Disciplinary Consistent solution implies ‘NAND’ at discipline level. Otherwise ‘SAND’ Interdisciplinary Consistent Solution implies ‘NAND’ at system Level. Otherwise ‘SAND’ Basic Single Level Formulations *NAND-NAND * SAND-NAND * SAND-SAND (MDF) (IDF) (AAO) CLASSIFICATION OF MDO ARCHITECTURES

Jan. 28, 2004MDO Algorithms - 12 Colloquium on MDO, VSSC Thiruvananthapuram NAND-NAND FORMULATION (MDF) System Optimizer z1z1 z2z2 z3z3 f, G Analyzer 1 g1g1 y 12, y 13 Analyzer 3 Analyzer 2 y 21, y 31 y 12, y 32 y 13, y 23 g2g2 y 21, y 23 y 31, y 32 g3g3 f, g 0 Z SystemCoordinatorSystemCoordinator Iterator

Jan. 28, 2004MDO Algorithms - 13 Colloquium on MDO, VSSC Thiruvananthapuram MATHEMATICAL STATEMENT Find Z which Minimize f (Z ) subject to g 0  0 (System Design Constraints) g 1  0 ; g 2  0 ; g 3  0 (Disciplinary Design Constraints) NAND-NAND FORMULATION

Jan. 28, 2004MDO Algorithms - 14 Colloquium on MDO, VSSC Thiruvananthapuram SAND-NAND FORMULATION (IDF) Zaug = { design variables Z, coupling variables Y*} ; y* 13 -y 13 = 0 System Optimizer Analyzer 1 f, g 0 z1z1 g1g1 y 12, y 13 z2z2 g2g2 y 21, y 23 Analyzer 3 Analyzer 2 z3z3 y 31, y 32 g3g3 Z, Y * f, G System Coordinator y 13, y 23 * * y 21, y 31 * * y 12, y 32 * * ICC

Jan. 28, 2004MDO Algorithms - 15 Colloquium on MDO, VSSC Thiruvananthapuram SAND-NAND FORMULATION (IDF) Augmented Design Variable Vector Z aug = ( Z, y* 12, y* 13, y* 21, y* 23, y* 31, y* 32 ) Design Constraints (DC): g 0  0 ( system design constraints) g 1  0 ; g 2  0 ; g 3  0 (disciplinary design constraints) Auxiliary Constraints: ( Inter disciplinary Consistency Constraints) y 21 - y* 21 = 0; y 31 - y* 31 = 0 y 12 - y* 12 = 0; y 32 - y* 32 = 0 ( ICC) y 13 - y* 13 = 0; y 23 - y* 23 = 0 Min f (Z aug ) ; subject to constraints ‘DC’ and ‘ICC’

Jan. 28, 2004MDO Algorithms - 16 Colloquium on MDO, VSSC Thiruvananthapuram SAND-SAND FORMULATION Zaug = { design variables Z, coupling variables Y*, state variables S} System Optimizer Evaluator 1 f, g 0 z 1, s 1 r1r1 g1g1 y 12, y 13 z 2, s 2 g2g2 r2r2 y 21, y 23 Evaluator 3 Evaluator 2 z 3, s 3 r3r3 y 31, y 32 g3g3 Z, S, Y * f, G, R System Coordinator y 13, y 23 y 21, y 31 * y 12, y 32 * * * **

Jan. 28, 2004MDO Algorithms - 17 Colloquium on MDO, VSSC Thiruvananthapuram SAND-SAND FORMULATION (AAO) Augmented Design Variable Vector Z aug = ( Z, S, y* 12, y* 13, y* 21, y* 23, y* 31, y* 32 ) Design Constraints (DC): g 0  0 ( system design constraints) g 1  0 ; g 2  0 ; g 3  0 (disciplinary design constraints) Auxiliary Constraints: y 21 - y* 21 = 0; y 31 - y* 31 = 0 y 12 - y* 12 = 0; y 32 - y* 32 = 0 ( ICC) y 13 - y* 13 = 0; y 23 - y* 23 = 0

Jan. 28, 2004MDO Algorithms - 18 Colloquium on MDO, VSSC Thiruvananthapuram SAND-SAND FORMULATION Auxiliary Constraints:(Disciplinary Analysis Constraints) r 1 = s 1 – E 1 ( z 1, y* 21,y* 31 ) = 0 r 2 = s 2 – E 2 ( z 2, y* 12,y* 32 ) = 0 (DAC) r 3 = s 3 – E 3 ( z 3, y* 13,y* 23 ) = 0 Optimization problem statement: Find Z aug which Minimize f (Z aug ) Subject to ‘DC’, ‘ICC’ and ‘DAC’ as stated above.

Jan. 28, 2004MDO Algorithms - 19 Colloquium on MDO, VSSC Thiruvananthapuram Single Level MDO Architectures Analysis 1 Iterations till convergence Analysis 2 Iterations till convergence Multi-Disciplinary Analysis (MDA) Interface Optimizer Analysis 1 Iterations till convergence Analysis 2 Iterations till convergence Disciplinary Analysis Interface Optimizer Evaluator 1 No iterations Evaluator 2 No iterations Disciplinary Evaluation Interface Optimizer Individual Discipline Feasible (IDF) All At Once (AAO) 1. Minimum load on optimizer 2. Complete interdisciplinary consistency is assured at each optimization call 3. Each MDA i Computationally expensive ii Sequential 1. Complete interdisciplinary consistency is assured only at successful termination of optimization 2. Intermediate between MDF and AAO 3. Analysis in parallel 1. Optimizer load increases tremendously 2. No useful results are generated till the end of optimization 3. Parallel evaluation 4. Evaluation cost relatively trivial Iterative; coupled Multi-Disciplinary Feasible (MDF) Uncoupled Non-iterative; Uncoupled

Jan. 28, 2004MDO Algorithms - 20 Colloquium on MDO, VSSC Thiruvananthapuram COMPARISON OF SINGLE LEVEL FORMULATIONS NAND - NAND SAND-NAND SAND-SAND ZZ, y* Z, S, y* Analyzer/ Evaluator/ Evaluator/ Analyzer Analyzer Evaluator Inter- Discipline Consistent Disciplinary Consistent Solution at Consistent Solution Optimality Solution MDF IDF All-at-Once Extreme In-Between Extreme

Jan. 28, 2004MDO Algorithms - 21 Colloquium on MDO, VSSC Thiruvananthapuram BI-LEVEL FORMULATIONS Industry design environment Distributed approach Disciplines retain control over their respective design variables Coordination through Project Office Bi-level formulations attempt to incorporate such features in the Mathematical definition of the Problem statement

Jan. 28, 2004MDO Algorithms - 22 Colloquium on MDO, VSSC Thiruvananthapuram SINGLE LEVEL BI-LEVEL (‘CO’) Z = Z L  Z S ; System level Z aug = Z  Y* Z aug = Z S  Z C Z S =  z Si, Z C =  z ci z ci = z cIi  z cOi Discipline level X =  x i x i = x Li  x si  x cIi  x cOi BI-LEVEL PROBLEM DECOMPOSITION DESIGN VECTOR

Jan. 28, 2004MDO Algorithms - 23 Colloquium on MDO, VSSC Thiruvananthapuram znzn x1x1 g 1, x cO1 System level Optimizer Min f(Z) s.t. r j (Z) = 0 ; j = 1, N * Analysis 1Analysis N z1z1 r1r1 * rnrn * xnxn Subspace Optimizer 1 Min r 1 (x 1 ) =  x s1 -z s1  +  x cI1 -z cI1  +  x cO1 – z cO1  s.t. g 1 (x 1 )  0 Subspace optimizer N Min r n (x n ) =  x sn -z sn  +  x cIn -z cIn  +  x cOn – z cOn  s.t. g n (x n )  0 COLLABORATIVE OPTIMIZATION FORMULATION z Si shared variables ; z cIi & z cOi coupling variables x si, x cIi & x cOi copies of system targets at discipline level g n, x cOn

Jan. 28, 2004MDO Algorithms - 24 Colloquium on MDO, VSSC Thiruvananthapuram COLLABORATIVE OPTIMIZATION System level Optimization Problem Find Z aug which Minimize F (Z S ) s.t. r* (Z aug ) = 0 F : objective function Z aug : design variable vector(targets issued to sub-spaces) r* : non-linear constraint vector, whose elements are discrepancy functions returned from solution of the sub –space optimization problems The system-level solution is defined as, F = F** and Z = Z** and X L = X L **

Jan. 28, 2004MDO Algorithms - 25 Colloquium on MDO, VSSC Thiruvananthapuram Discipline / Subspace Optimization Problem For a ‘n’ discipline problem, there will be ‘n’ sub-space optimization problems. Mathematical statement for an i th sub-space: Find x i Min r i (x i ) =  x si - z si  +  x cIi - z cIi  +  y cOi - z cOi  s.t g i (x i )  0 ; h i (x i ) = 0 r i = r* i ; x i = x* i The norm in the objective function r i (x i ) is generally, calculated as L 2 norm. COLLABORATIVE OPTIMIZATION

Jan. 28, 2004MDO Algorithms - 26 Colloquium on MDO, VSSC Thiruvananthapuram ? System Level Coordination Approximation Model Process flowInformation flow Convergence SS01 SS02 SS03 A1A1 A2A2 A3A3 A1A1 A2A2 A3A3 CONCURRENT SUB-SPACE OPTIMIZATION

Jan. 28, 2004MDO Algorithms - 27 Colloquium on MDO, VSSC Thiruvananthapuram CONCURRENT SUB-SPACE OPTIMIZATION Step 1 – System Analysis at initial system design vector, local sensitivities Step 2 – Total System Sensitivities using GSE Step 3 – Concurrent Subspace Optimizations  Each Subspace solves the system level optimization problem (same objective and constraints)  Subspace design vector is a subset of the system design vector local to the subspace. Non-local variables kept fixed  Non-local states approximated linearly using sensitivities. Local states obtained from disciplinary analysis  Each subspace return different optima

Jan. 28, 2004MDO Algorithms - 28 Colloquium on MDO, VSSC Thiruvananthapuram CONCURRENT SUB-SPACE OPTIMIZATION Step 4 – Design database updated during subspace optimizations Step 5 – System level co-ordination for compromise/trade-off  Database used to create second order response surfaces for objective and constraints  System optimization based on these approximations with all design variables used to direct system convergence  The approximate system optimum generated by the co- ordination process is used as the next design iterate in Step 1.

Jan. 28, 2004MDO Algorithms - 29 Colloquium on MDO, VSSC Thiruvananthapuram System Analysis and Sensitivity Analysis Update Variables Discipline 1 Optimization and Optm. Sensitivity Analysis initialize X & Z X = X 0 +  X OPT Z = Z 0 +  Z OPT Opportunity for Concurrent Processing Discipline 2 j Optimization and Optm. Sensitivity Analysis Discipline k Optimization and Optm. Sensitivity Analysis System Optimization Human Intervention BLISS CYCLE Bi-Level Integrated System Synthesis - BLISS X = X 0 +  X OPT Z = Z 0 +  Z OPT

Jan. 28, 2004MDO Algorithms - 30 Colloquium on MDO, VSSC Thiruvananthapuram Bi-Level Integrated System Synthesis - BLISS Step – 1 System Analysis + Sensitivity (GSE) Step – 2 Subsystem objective F s ={df/dX} T  X s Subsystem optimization Linear approximation for the coupling variables for evaluating constraints Shared variables (system var.) & Y * held constant during subsystem optimization

Jan. 28, 2004MDO Algorithms - 31 Colloquium on MDO, VSSC Thiruvananthapuram Step – 3 Obtain sensitivity of X and F (optimal) wrt Z S and Y* These sensitivities link the system and subsystem level optimizations (Optimal Design Sensitivities) At system level use shared variables to further improve system objective Step – 4 System level optimization problem Bi-Level Integrated System Synthesis - BLISS F(Z S ) is obtained as a linear extrapolation based on the optimum design sensitivity obtained in each subsystem

Jan. 28, 2004MDO Algorithms - 32 Colloquium on MDO, VSSC Thiruvananthapuram Thank You Visit 4 th Meeting of SIG-MDO in March 2004