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Gaussian Quadrature for Computer Aided Robust Design

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Gaussian Quadrature for Computer Aided Robust Design Page 1 of 88 Gaussian Quadrature for Computer Aided Robust Design by Geoffrey Scott Reber S.B. Aeronautics and Astronautics Massachusetts Institute of Technology, 2000 Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Masters of Science in Aeronautics and Astronautics at the Massachusetts Institute of Technology OF TECHN3LOGY JUL 0 1 2004 June, 2004 LIBRARIES © 2004 Massachusetts Institute of Technology. All rights reserved t AERO Signature of Author: Department o auticsd Astronautics -Myay 7, 2004 Certified by: aniel Frey Professor of Mechan' al Engineering Thesis Supervisor Approved by: Edward Greitzer H.N. Slater Professor of Aeronautics and Astronautics Chair, Committee on Graduate Students Page 2 of 88 Gaussian Quadrature for Computer Aided Robust Design by Geoffrey Scott Reber Submitted to the Department of Aeronautics and Astronautics on May 7, 2004 in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics ABSTRACT Computer aided design has allowed many design decisions to be made before hardware is built through "virtual" prototypes: computer simulations of an engineering design. To produce robust systems noise factors must also be considered (robust design), and should they should be considered as early as possible to reduce the impact of late design changes. Robust design on the computer requires a method to analyze the effects of uncertainty. Unfortunately, the most commonly used computer uncertainty analysis technique (Monte Carlo Simulation) requires thousands more simulation runs than needed if noises are ignored. For complex simulations such as Computational Fluid Dynamics, such a drastic increase in the time required to evaluate an engineering design may be probative early in the design process. Several uncertainty analysis techniques have been developed to decrease the number of simulation runs required, but none have supplanted Monte Carlo. Gaussian Quadrature (GQ) is presented here as a new option with significant benefits for many types of engineering problems. Advantages of GQ include: as few as 2*(number of noise factors) simulation runs required to estimate performance mean and variance, errors dependent only on the ability to approximate performance using polynomials for each noise factor, and the ability to estimate gradients without further simulation runs for use in computer aided optimization of mean or variance. The mathematically basis for GQ is discussed along with case studies demonstrating its utility. Thesis Supervisor: Professor Daniel Frey Title: Gaussian Quadrature for Computer Aided Robust Design Page 3 of 88 Biographical Note I would first like to thank Professor Frey for not only introducing me to robust design, but for being a wonderful advisor. I wish all the best for the entire Robust Design Group, whose excellent work is just beginning. I could not have completed this thesis without the support of my family and friends. Aryatna, thank you for you eternal love and encouragement. Noah, as the newest member of "clan Reber", you inspired me to give it my all and get it done. And to everyone back home in California: you were the light at the end of the tunnel. Geoff Reber MIT, May 2004 Page 4 of 88 Table of Contents S INTRODUCTION.......................................................................................................................................8 1.1 WHAT Is ROBUST DESIGN AND WHY IS IT IMPORTANT?.........................................................................8 1.2 A BRIEF BACKGROUND ON ROBUST DESIGN......................................................................................... 9 1.3 THE ELEMENTS OF ROBUST DESIGN .................................................................................................... 12 1.4 STATEM ENT O F PURPO SE..................................................................................................................... 13 2 EXISTING METHODS FOR UNCERTAINTY ANALYSIS................................................................14 2.1 MONTE CARLO SIMULATION ............................................................................................................. 15 2.2 QUASI-MONTE CARLO SIMULATION .................................................................................................... 16 2.2. 1 Latin Hypercube Samp ling............................................................................................................. 17 2.2.2 Hammersley Sequence Sampling................................................................................................. 18 2.3 DESIGN OF EXPERIMENTS (ORTHOGONAL ARRAYS) ........................................................................... 20 3 GAUSSIAN QUADRATURE .................................................................................................................... 23 3.1 BACKGROUND: WHAT IS GAUSSIAN QUADRATURE? ........................... ............................................ .23 3.2 CUBATURE FOR COMPUTATION OF EXPECTED PERFORMANCE ........................................................... 27 3.3 GAUSSIAN QUADRATURE FOR COMPUTATION OF EXPECTED PERFORMANCE ......................................... 28 3.3.1 Non-StandardMultivariate Normal or Uniform Noise Factors................................................... 31 3.3.2 Why Compute the Nominal Performance and the Deviationfrom Nominal?................................ 32 3.3.3 A Note on Polynomial Approximations........................................................................................ 33 3.4 GAUSSIAN QUADRATURE FOR COMPUTATION OF PERFORMANCE VARIANCE ....................................... 33 3.5 REQUIREMENTS FOR USE OF GAUSSIAN QUADRATURE ....................................................................... 36 3.6 IMPLEMENTATION FOR ROBUST DESIGN ............................................................................................ 36 4 COMPARISON OF GAUSSIAN QUADRATURE TO EXISTING METHODS................................39 4.1 B ASIC F UN CTION S .............................................................................................................................. 39 4.2 BUCKLING OF A TUBULAR COLUMN IN COMPRESSION .......................................................................... 42 4.2.1 Problem Descrip tion...................................................................................................................... 42 4.2.2 Uncertainty Analysis Using Gaussian Quadrature....................................................................... 43 4.2.3 Tubular Column Uncertainty Analysis Results............................................................................. 44 Page 5 of 88 4.3 SLIDER-CRANK M ECHANISM ............................................................................................................. 45 4.3.1 Problem Description...................................................................................................................... 45 4.3.2 Slider-Crank Mechanism Uncertainty Analysis Results............................................................... 46 4.4 CONTINUOUSLY STIRRED TANK REACTOR ........................................................................................ 47 4.4.1 Problem Description...................................................................................................................... 47 4.4.2 CSTR UncertaintyAnalysis Results............................................................................................. 50 4.5 OP-AMP CASE STUDY ......................................................................................................................... 52 4.5.1 Problem Description...................................................................................................................... 52 4.5.2 Op-Amp UncertaintyAnalysis Results......................................................................................... 54 5 GAUSSIAN QUADRATURE FOR COMPUTER AIDED OPTIMIZATION..................................... 56 5.1 GRADIENT BASED OPTIMIZATION ...................................................................................................... 56 5.2 GRADIENT CALCULATION USING GAUSSIAN Q UADRATURE .................................................................. 59 5.3 SOME RESULTS...................................................................................................................................62 6 CO NCLUSION.......................................................................................................................................... 64 7 REFERENCES.......................................................................................................................................... 66 8 A PPENDIX................................................................................................................................................ 67 8.1 M ATLAB CODE ................................................................................................................................... 67 8.1.1 Hammersley Sequence Sampling Locations................................................................................ 67 8.1.2 QuadratureLocations & Coefficientsfor Normal Variables ........................................................ 68 8.1.3 QuadratureLocations & Coefficientsfor Uniform Variables...................................................... 69 8.1.4 Gaussian Quadraturefor Normal Variables...............................................................................
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