M. Kokkolaras and P.Y Papalambros University of Michigan

1. Optimal Multilevel SystemDesign under UncertaintyNSF Workshop on Reliable Engineering ComputingSavannah, Georgia, 16 September 2004 M. Kokkolaras and P.Y Papalambros University of Michigan Z. Mourelatos Oakland University

2. Outline • Design by Decomposition • Hierarchical Multilevel Systems • Analytical Target Cascading • Deterministic Formulation • Nondeterministic Formulations • Propagation of Uncertainty • Practical Issues • Example

3. Design Target Problem Optimal System Design

4. Design by Decomposition • When dealing with large and complex engineering systems, an “all-at-once” formulation of the optimal design problem is often impossible to solve • Original problem is decomposed into a set of linked subproblems • Typically, the partitioning reflects the hierarchical structure of the organization (different design teams are assigned with different subproblems according to expertise)

5. Decomposition Example VEHICLE BODY ELECTRONICS POWERTRAIN CHASSIS CLIMATE CONTROL TRANSMISSION ENGINE DRIVELINE … VALVETRAIN CYLINDER BLOCK

6. system subsystem 1 subsystem 2 subsystem n component m component 1 component 2 Multilevel System Design • Multilevel hierarchy of single-level (sub)problems • Responses of higher-level elements are depend on responses of lower-level elements in the hierarchy

7. Challenges • Need to assign design targets for the subproblems to the design teams • Design teams may focus on own goals without taking into consideration interactions with other subproblems; this will compromise design consistency and optimality of the original problem

8. Analytical Target Cascading • Operates by formulating and solving deviation minimization problems to coordinate what higher-level elements “want” and what lower-level elements “can” • Parent responses rp are functions of • Children response variables rc1,rc2, …,rcn, (required) • Local design variables xp (optional) • Shared design variables yp(optional) • In the following formulations: • Subscript index pairs denote level and element • Superscript indices denote computation “location”

9. Mathematical Formulation

10. optimization inputs optimization outputs response and shared variable values cascaded down from the parent response and shared variable values passed up to the parent response and shared variable values passed up from the children response and shared variable values cascaded down to the children Information Exchange element optimization problem pij, where rij is provided by the analysis/simulation model

11. system subsystem 1 subsystem 2 subsystem n component m component 1 component 2 Multilevel System Designunder Uncertainty • Multilevel hierarchy of single-level (sub)problems • Outputs of lower-level problems are inputs to higher-level problems: need to obtain statistical properties of responses

12. Nondeterministic Formulations • For simplicity, and without loss of generalization, assume uncertainty in all design variables only • Introduce random variables (and functions of random variables) • Identify (assume) distributions • Use means as design variables assuming known variance

13. Stochastic Formulation

14. Constraints • “Hard” and “soft” inequalities • “Hard” and “soft” equalities • Typically, a target reliability of satisfying constraints is desired

15. Probabilistic Formulation

16. Propagation of Uncertainty State of the Art (?): Since functions are generally nonlinear, use first-order approximation (Taylor series expansion around the means of the random variables)

17. Validity of Linearization Y(X) mY mX X consistency constraints in ATC formulation secure validity

18. Examples

19. Results * 1,000,000 samples

20. Moment Approximation UsingAdvanced Mean Value Method • Consider Z=g(X) • Discretize “b-range” (from b = 4 (Pf = 0.003%) to b = -4 (Pf = 99.997%)) • Find MPP for P[g(X)>0]<F(-bi) for all i • Evaluate Z=g(XMPP), i.e., generate CDF of Z • Derive PDF of Z by differentiating CDF numerically • Integrate PDF numerically to estimate moments

21. Results * 1,000,000 samples

22. Example:Piston Ring/Liner Subassembly Brake-specific fuel consumption (BSFC) GT Power Power loss due to friction Oil consumption Blow-by Liner wear rate RingPak Ring and liner surface roughness Liner material properties

23. Lower-level Problem Formulation

24. Results and Reliability Assessment 0.03% less reliable than assumed * 1,000,000 samples

25. Statistical Properties of Power Loss MCS – PDF (1,000,000 samples) MAM - PDF

26. Upper-level Problem Formulation * 1,000,000 samples

27. Probability Distribution of BSFC MAM MCS with 1,000,000 samples

28. Practical Issues • Computational cost • Noise/accuracy in the model vs. magnitude of uncertainty in inputs • Convergence of multilevel approach

29. Concluding Remarks • Practical yet rational decision-making support • Value of optimization results is in trends not in numbers • Strategies should involve a mix of deterministic optimization and stochastic “refinement” • Need for accurate uncertainty quantification (and propagation)

30. Error Issues • y=f(x) + emodel + emetamodel + edata + enum + eunc. prop. • Need to keep ALL errors relatively low

31. Q & A

32. Optimum Symmetric Latin Hypercube (OSLH) Sampling OSLH Samples Partitioned Group #2 Partitioned Group #1

33. a : vector of constants ; Global Least Squares : ; Moving Least Squares : where : Cross-Validated Moving Least Squares (CVMLS) Method • Polynomial Regression using Moving Least Squares (MLS) Method • In MLS, sample points are weighted so that nearby samples have more influence on the prediction.

34. Metamodel Errors • Optimal symmetric Latin hypercube sampling (200 train points and 150 trial points for Ringpak, 45 train points and 40 trial points for GT-power) • Moving least squares approximations

35. Accuracy and Efficiency of Monte Carlo Method