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Part II: MDO Architechtures

System Design – New Paradigms. Part II: MDO Architechtures. Prof. P.M. Mujumdar, Prof. K. Sudhakar Dept. of Aerospace Engineering, IIT Bombay & Umakant Joysula, DRDL, Hyderabad. OUTLINE. Coupled System Engg. Design Optimization Problem Statement Analyzer & Evaluator

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Part II: MDO Architechtures

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

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

  3. S1 DISCIPLINE 1 Z Z S2 DISCIPLINE 2 COUPLEDSYSTEM • 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 MULTIDISCIPLINARY ANALYSIS(MDA) Colloquium on MDO, VSSC Thiruvananthapuram

  4. MDO ARCHITECTURE / FORMULATION • 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 Colloquium on MDO, VSSC Thiruvananthapuram

  5. OPTIMIZER Z f, g, h Interface Z S ANALYSIS 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 Nested ANalysis and Design (NAND) Colloquium on MDO, VSSC Thiruvananthapuram

  6. Z so S Evaluator r = A(Z, S) Iterator updateS r Analysis Converged S ANALYSIS AND EVALUATOR Closed Analysis Nested ANalysis and Design (NAND) Colloquium on MDO, VSSC Thiruvananthapuram

  7. Optimizer Z, S f, h, r, g Interface Z, S r Evaluator ALTERNATE STATEMENT • 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) Colloquium on MDO, VSSC Thiruvananthapuram

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

  9. SYSTEM AND DISCIPLINE LEVEL SYSTEM LEVEL (DISCIPLINE COORDINATOR) ZS ZS1 ZS2 Y DISCIPLINE ‘2’ ZL2 DISCIPLINE 1 ZL1 Z = ( ZLi)  (ZSi ) ZL : Local to discipline (Disciplinary Variables) ZS : Shared by more than one discipline (System Variables) Y : Coupling functions Colloquium on MDO, VSSC Thiruvananthapuram

  10. 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 Colloquium on MDO, VSSC Thiruvananthapuram

  11. CLASSIFICATION OF MDO ARCHITECTURES • 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) Colloquium on MDO, VSSC Thiruvananthapuram

  12. Iterator Analyzer1 f, g0 z1 System Coordinator y21, y31 Z g1 y12, y13 Analyzer 2 System Optimizer z2 y12, y32 f, G g2 y21, y23 Analyzer 3 z3 y13, y23 g3 y31, y32 NAND-NAND FORMULATION (MDF) Colloquium on MDO, VSSC Thiruvananthapuram

  13. NAND-NAND FORMULATION MATHEMATICAL STATEMENT Find Z which Minimize f (Z ) subject to g 0  0 (System Design Constraints) g1 0 ; g2 0 ; g3 0 (Disciplinary Design Constraints) Colloquium on MDO, VSSC Thiruvananthapuram

  14. System Coordinator Analyzer 1 f, g0 y21, y31 * * z1 Z, Y* g1 y12, y13 Analyzer 2 System Optimizer y12, y32 * * z2 g2 y21, y23 Analyzer 3 y13, y23 * * f, G z3 ICC g3 y31, y32 SAND-NAND FORMULATION (IDF) Zaug = { design variables Z, coupling variables Y*} ; y*13 -y13 = 0 Colloquium on MDO, VSSC Thiruvananthapuram

  15. SAND-NAND FORMULATION (IDF) • Augmented Design Variable Vector • Zaug = ( Z ,y*12 ,y*13, y*21, y*23, y*31, y*32) • Design Constraints (DC): • g0  0 ( system design constraints) • g1 0 ; g2 0 ; g3 0 (disciplinary design constraints) • Auxiliary Constraints: • ( Inter disciplinary Consistency Constraints) • y21 - y*21 = 0; y31 - y*31 = 0 • y12 - y*12 = 0; y32 - y*32 = 0 ( ICC) • y13 - y*13 = 0; y23 - y*23 = 0 • Min f (Zaug) ; subject to constraints ‘DC’ and ‘ICC’ Colloquium on MDO, VSSC Thiruvananthapuram

  16. System Coordinator f, g0 Evaluator 1 * y21, y31 * z1, s1 Z, S, Y* r1 g1 y12, y13 Evaluator 2 System Optimizer * * y12, y32 z2, s2 r2 g2 y21, y23 Evaluator3 * * y13, y23 f, G, R z3, s3 r3 g3 y31, y32 SAND-SAND FORMULATION Zaug = { design variables Z, coupling variables Y*, state variables S} Colloquium on MDO, VSSC Thiruvananthapuram

  17. SAND-SAND FORMULATION (AAO) • Augmented Design Variable Vector • Zaug = ( Z ,S, y*12 , y*13, y*21, y*23, y*31, y*32 ) • Design Constraints (DC): • g0  0 ( system design constraints) • g1 0 ; g2 0 ; g3 0 (disciplinary design constraints) • Auxiliary Constraints: • y21 - y*21 = 0; y31 - y*31 = 0 • y12 - y*12 = 0; y32 - y*32 = 0 ( ICC) • y13 - y*13 = 0; y23 - y*23 = 0 Colloquium on MDO, VSSC Thiruvananthapuram

  18. SAND-SAND FORMULATION • Auxiliary Constraints:(Disciplinary Analysis Constraints) • r1 = s1 – E1( z1, y*21 ,y*31) = 0 • r2 = s2 – E2( z2, y*12 ,y*32) = 0 (DAC) • r3 = s3 – E3( z3, y*13 ,y*23) = 0 • Optimization problem statement: • Find Zaug which • Minimize f (Zaug ) • Subject to • ‘DC’ , ‘ICC’ and ‘DAC’ as stated above. Colloquium on MDO, VSSC Thiruvananthapuram

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

  20. COMPARISON OF SINGLE LEVEL FORMULATIONS NAND - NAND SAND-NAND SAND-SAND Z Z, 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 1 2 3 4 5 Colloquium on MDO, VSSC Thiruvananthapuram

  21. 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 Colloquium on MDO, VSSC Thiruvananthapuram

  22. BI-LEVEL PROBLEM DECOMPOSITION DESIGN VECTOR SINGLE LEVEL BI-LEVEL (‘CO’) Z = ZL ZS ; System level Zaug = Z  Y*Zaug = ZS ZC ZS =  zSi , ZC =  zci zci = zcIi zcOi Discipline level X =  xi xi = xLi xsi xcIi  xcOi Colloquium on MDO, VSSC Thiruvananthapuram

  23. COLLABORATIVE OPTIMIZATION FORMULATION * * rn r1 Subspace optimizer N Min rn(xn) =xsn-zsn + xcIn-zcIn  + xcOn– zcOn s.t. gn(xn)  0 Subspace Optimizer 1 Min r1(x1) = xs1-zs1 + xcI1-zcI1 + xcO1 – zcO1 s.t. g1(x1)  0 System level Optimizer Min f(Z) s.t. rj (Z) = 0 ; j = 1, N z1 zn * xn x1 g1 , xcO1 gn , xcOn Analysis 1 Analysis N zSi shared variables ; zcIi & zcOi coupling variables xsi , xcIi & xcOi copies of system targets at discipline level Colloquium on MDO, VSSC Thiruvananthapuram

  24. COLLABORATIVE OPTIMIZATION • System level Optimization Problem • Find Z aug • which Minimize F (ZS) • s.t. r* (Zaug) = 0 • F : objective function • Zaug : 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 XL = XL** Colloquium on MDO, VSSC Thiruvananthapuram

  25. COLLABORATIVE OPTIMIZATION • Discipline / Subspace Optimization Problem • For a ‘n’ discipline problem, there will be ‘n’ sub-space • optimization problems. • Mathematical statement for an ith sub-space: • Find xi • Min ri (xi ) = xsi - zsi  + xcIi - zcIi +  ycOi - zcOi  • s.t gi (xi )  0 ; hi (xi ) = 0 • ri = r*i ; xi = x*i • The norm in the objective function ri (xi ) is generally, • calculated as L2 norm. Colloquium on MDO, VSSC Thiruvananthapuram

  26. A3 A1 A2 A3 A1 A2 Convergence System Level Coordination ? Approximation Model SS01 SS02 SS03 Process flow Information flow CONCURRENT SUB-SPACE OPTIMIZATION Colloquium on MDO, VSSC Thiruvananthapuram

  27. 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 Colloquium on MDO, VSSC Thiruvananthapuram

  28. 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. Colloquium on MDO, VSSC Thiruvananthapuram

  29. Opportunity for Concurrent Processing Discipline 1 Optimization and Optm. Sensitivity Analysis initialize X & Z System Analysis and Sensitivity Analysis Discipline 2j Optimization and Optm. Sensitivity Analysis System Optimization Update Variables Discipline k Optimization and Optm. Sensitivity Analysis X = X0 + DXOPT Z = Z0 + DZOPT Human Intervention BLISS CYCLE Bi-Level Integrated System Synthesis - BLISS X = X0 + DXOPT Z = Z0 +DZOPT Colloquium on MDO, VSSC Thiruvananthapuram

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

  31. Step – 3 Obtain sensitivity of X and F (optimal) wrt ZS 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(ZS) is obtained as a linear extrapolation based on the optimum design sensitivity obtained in each subsystem Colloquium on MDO, VSSC Thiruvananthapuram

  32. Thank You Visithttp://www.casde.iitb.ac.in/MDO/ 4th Meeting of SIG-MDO in March 2004 Colloquium on MDO, VSSC Thiruvananthapuram

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