1. Optimal Multilevel System Design under Uncertainty NSF Workshop on Reliable Engineering Computing Savannah, 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. Optimal System Design Design Target Problem

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. VEHICLE ELECTRONICSCLIMATE CONTROLBODYCHASSISPOWERTRAIN ENGINEDRIVELINETRANSMISSION CYLINDER BLOCK … Decomposition Example VALVETRAIN

6. 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 system subsystem 1subsystem 2 component 1component 2 component m subsystem n

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 r p are functions of –Children response variables r c1, r c2, …, r cn, (required) –Local design variables x p (optional) –Shared design variables y p (optional) In the following formulations: –Subscript index pairs denote level and element –Superscript indices denote computation “location”

9. Mathematical Formulation

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

11. Multilevel System Design under 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 system subsystem 1subsystem 2 component 1component 2 component m subsystem n

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) X XX consistency constraints in ATC formulation secure validity YY

18. Examples

19. Results Linear.MCS *  lin % E[Z 1 ] Var[Z 1 ] 1/2 3.6321 1.9386 3.4921 0.9327 4.00 107.85 E[Z 2 ] Var[Z 2 ] 1/2 200 44.721 205.04 45.101 -2.45 -0.84 E[Z 3 ] Var[Z 3 ] 1/2 -5.25 0.8385 -5.3114 0.8407 -1.15 -0.26 E[Z 4 ] Var[Z 4 ] 1/2 -1.0333 0.1166 -1.0404 0.1653 -0.68 29.46 E[Z 5 ] Var[Z 5 ] 1/2 -0.1428 0.00627 -0.1448 0.00630 -1.3 -0.47 * 1,000,000 samples

20. Moment Approximation Using Advanced Mean Value Method 1.Consider Z=g(X) 2.Discretize “  range” (from  4 (P f = 0.003%) to  4 (P f = 99.997%)) 3.Find MPP for P[g(X)>0]<  (-  i  for all i 4.Evaluate Z=g(X MPP ), i.e., generate CDF of Z 5.Derive PDF of Z by differentiating CDF numerically 6.Integrate PDF numerically to estimate moments

21. Results Linear.MAMMCS *  lin %  MAM % E[Z 1 ] Var[Z 1 ] 1/2 3.6321 1.9386 3.6029 0.9013 3.4921 0.9327 4.00 107.85 3.17 -3.36 E[Z 2 ] Var[Z 2 ] 1/2 200 44.721 203.37 45.203 205.04 45.101 -2.45 -0.84 -0.81 0.22 E[Z 3 ] Var[Z 3 ] 1/2 -5.25 0.8385 -5.3495 0.8423 -5.3114 0.8407 -1.15 -0.26 0.71 0.19 E[Z 4 ] Var[Z 4 ] 1/2 -1.0333 0.1166 -1.0380 0.1653 -1.0404 0.1653 -0.68 29.46 -0.23 0 E[Z 5 ] Var[Z 5 ] 1/2 -0.1428 0.00627 -0.1454 0.00631 -0.1448 0.00630 -1.3 -0.47 0.41 0.15 * 1,000,000 samples

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

23. Lower-level Problem Formulation

24. Results and Reliability Assessment ActiveP f, %MCS * Liner Wear RateNo< 0.130 Blow-byNo< 0.130 Oil ConsumptionYes0.130.16 0.03% less reliable than assumed * 1,000,000 samples

25. Statistical Properties of Power Loss MAM - PDF MCS – PDF (1,000,000 samples) LinearizationMAMMCS Lin.  MAM  E[pl]0.39500.39220.39320.45%-0.25% Var[pl] 1/2 0.04810.03090.031154.6%-0.64%

26. Upper-level Problem Formulation LinearizationMAMMCS* Lin.  MAM  E[fuel]0.5341 0.5342-0.01% Var[fuel] 1/2 0.007570.007600.00759-0.25%0.13% * 1,000,000 samples

27. Probability Distribution of BSFC MAMMCS 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) +  model +  metamodel +  data +  num +  unc. prop. Need to keep ALL errors relatively low

31. Q & A

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

33. 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. Global Least Squares : a : vector of constants ; Moving Least Squares : ; where :

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 Relative errors, %MaximumMean Standard Deviation Power loss8.620.370.77 Wear rate9.780.721.32 Blow-by3.980.370.63 Oil consumption41.81.743.68 BSFC0.010.0050.004

35. Accuracy and Efficiency of Monte Carlo Method 99% confidence  99.9% confidence  99.99% confidence 