Plan for the Session Quiz on Constructing Orthogonal Arrays 10 minutes Complete some advanced topics on OAs Lecture on Computer Aided Robust Design Recitation on HW 5 16 881 Robust System Design Session 11 MIT How to Estimate Error variance in an L18 Consider Phadke pg 89 How would the two unassigned columns contribute to error variance Remember L18 21x37 Has 1 1 2 1 7 3 1 16 DOF But 18 rows Therefore 2 DOF can be used to estimate the sum square due to error 16 881 Robust System Design Session 11 MIT Breakdown of Sum Squares GTSS SS due to mean SS due to factor A 16 881 Total SS SS due to factor B Robust System Design Session 11 etc SS due to error MIT Column Merging Can turn 2 two level factors into a 4 level factor Can turn 2 three level factors into a six level factor Need to strike out interaction column account for the right number of DOF Example on an L8 16 881 Robust System Design Session 11 MIT Column Merging in an L8 Eliminate the column which is confounded with interactions Create a new four level column Exp no 1 2 3 4 5 6 7 8 16 881 A 1 1 1 1 2 2 2 2 B 1 1 2 2 1 1 2 2 Control Factors C D E 1 1 1 1 2 2 2 1 1 2 2 2 2 1 2 2 2 1 1 1 2 1 2 1 Robust System Design Session 11 F 1 2 2 1 1 2 2 1 G 1 2 2 1 2 1 1 2 MIT Computer Aided Robust Design 16 881 Robust System Design Session 11 MIT Engineering Simulations Many engineering systems can be modeled accurately by computer simulations Finite Element Analysis Digital and analog circuit simulations Computational Fluid Dynamics Do you use simulations in design analysis How accurate reliable are your simulations 16 881 Robust System Design Session 11 MIT Simulation Design Optimization Formal mathematical form minimize subject to y f x h x 0 g x 0 minimize weight subject to height 23 max stress 0 8Y f x x Simulation h x g x 16 881 Robust System Design Session 11 MIT Robust Design Optimization Vector of design variables x Control factors discrete vs continuous Objective function f x S N ratio noise must be induced Constraints h x g x Not commonly employed Sliding levels may be used to handle equality constraints in some cases 16 881 Robust System Design Session 11 MIT Noise Distributions Normal Arises when many independent random variables are summed Uniform Arises when other distributions are truncated Lognormal Lognormal Distribution Arises when normally distributed variables are multiplied or transformed p x x 16 881 Robust System Design Session 11 MIT Covariance COV x y E x E x y E y n m Size error Pin 2 Correlation of Noise Factors COV x y xi x y j y Size error Pin 1 i 1 j 1 Correlation coefficient What does k 1 imply What does negative k imply What does k 0 imply 16 881 Robust System Design Session 11 k COV x y VAR x VAR y MIT Question About Noise Does the distribution of noise affect the S N ratio of the simulation If so under what conditions Does correlation of noise factors affect S N ratios If so in what way raise lower 16 881 Robust System Design Session 11 MIT Simulating Variation in Noise Factors Taylor series expansion Linearize the system response Apply closed form solutions Monte Carlo Generate random numbers Use as input to the simulation Orthogonal array based simulation Create an ordered set of test conditions Use as input to the simulation 16 881 Robust System Design Session 11 MIT Taylor Series Expansion Approximate system response f f x y f xo yo x x xo f x xo y Apply rules of probability To get y yo h o t y yo VAR aX aVAR X VAR X Y VAR X VAR Y iff x y independent y 2 y X i i 1 x 2 16 881 n Robust System Design Session 11 MIT Local Linearity of the System Response Surface wrt Noise Holds for q1 Machining most CMMs n q qi n f t n j t j n j 1 n t ij f t 1 f n Fails for t1 Dimensions of form Dual head valve grinding t2 n1 E n n2 16 881 Robust System Design Session 11 15 MIT Key Limitations of Taylor Series Expansion System response must be approximately linear w r t noise factors Linear over what range What if it isn t quite linear Noise factors must be statistically independent How common is correlation of noise What happens when you neglect correlation 16 881 Robust System Design Session 11 MIT Monte Carlo Simulation 2 01 p x1 x1 ytrial f x1 x2 1 59 p x2 2y trials 1 2 y y trial trials 1 i 1 x2 16 881 Robust System Design Session 11 MIT Monte Carlo Simulation Pros and Cons No assumptions about system response f x You may simulate correlation among noises How can this be accomplished Accuracy not a function of the number of noises 95 confidence interval 1 96 trials It s easy too It takes a large number of trials to get very accurate answers 16 881 Robust System Design Session 11 MIT Othogonal Array Based Simulation Define noise factors and levels Two level factors Level 1 i i Level 2 i i Three level factors Level 1 i 3 i 2 Level 2 i Level 3 i 3 i 2 Choose an appropriate othogonal array Use as the outer array to induce noise 16 881 Robust System Design Session 11 MIT Setting Levels for Lognormal Distributions p x x 7 p x log 0 p x log 7 3 x x 16 881 Robust System Design Session 11 MIT Using Sliding Levels to Simulate Correlation Try it for RFP Mean is defined as RFM Tolerance is 2 Fill out rows 1 and 19 of the noise array 16 881 Robust System Design Session 11 MIT Run the Noise Array At the baseline control factor settings Run the simulation at all of the noise factor settings Find the system response for each row of the array Perform ANOVA on the data Percent of total SS is percent contribution to variance in system response 16 881 Robust System Design Session 11 MIT Othogonal Array Pros and Cons Can handle some degree of non linearity Can accommodate correlation Provides a direct evaluation of noise factor contributions Usually requires orders of magnitude fewer function evaluations than OA simulation 16 881 Robust System Design Session 11 MIT Optimization Choose control factors and levels Set up an inner array of control factors For each row induce noise from the outer array Compute mean variance and S N Select control factors based on the additive model Run a confirmation experiment 16 881 Robust System Design Session 11 MIT Next Steps Homework 8 due on Lecture 13 Next session Read Phadke …