Mest 16888 ES077 Multidisciplinary System Design Optimization(MSDO) Design Space Exploration Lecture 5 18 February 2004 Karen willcox C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Multidisciplinary System Multidisciplinary System Design Optimization (MSDO) Design Optimization (MSDO) Design Space Exploration Lecture 5 18 February 2004 Karen Willcox
Mest Today's Topics 16888 ESD.J7 Design of Experiments Overview Full Factorial Design · Parameter stud One at a time Latin Hypercubes Orthogonal Arrays Effects DoE Paper Airplane Experiment C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Today’s Topics Today’s Topics • Design of Experiments Overview • Full Factorial Design • Parameter Study • One at a Time • Latin Hypercubes • Orthogonal Arrays • Effects • DoE Paper Airplane Experiment
Mlsd Design of Experiments 16888 E77 A collection of statistical techniques providing a systematic way to sample the design space Useful when tackling a new problem for which you know very little about the design space Study the effects of multiple input variables on one or more output parameters Often used before setting up a formal optimization problem Identify key drivers among potential design variables Identify appropriate design variable ranges Identify achievable objective function values Often, DOE is used in the context of robust design. Today we will just talk about it for design space exploration C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Design of Experiments Design of Experiments • A collection of statistical techniques providing a systematic way to sample the design space • Useful when tackling a new problem for which you know very little about the design space. • Study the effects of multiple input variables on one or more output parameters • Often used before setting up a formal optimization problem – Identify key drivers among potential design variables – Identify appropriate design variable ranges – Identify achievable objective function values • Often, DOE is used in the context of robust design. Today we will just talk about it for design space exploration
Mlsd Design of Experiments 16888 E77 Design variables= factors Values of design variables levels noise factors variables over which we have no control e.g. manufacturing variation in blade thickness Control factors variables we can control e.g. nominal blade thickness Outputs =observations objective functions Factors +—“ Experiment Observation Levels Often an analysis code C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Design of Experiments Design of Experiments Design variables = factors Values of design variables = levels Noise factors = variables over which we have no control e.g. manufacturing variation in blade thickness Control factors = variables we can control e.g. nominal blade thickness Outputs = observations (= objective functions) Factors + Levels “Experiment” Observation (Often an analysis code)
Mest Matrix Experiments 16888 E77 Each row of the matrix corresponds to one experiment Each column of the matrix corresponds to one factor Each experiment corresponds to a different combination of factor levels and provides one observation Expt Ne Factora Factorb Observation A1 B1 n1 A1 B2 A2 B1 n A2 B2 n4 Here, we have two factors each of which can take two levels C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Matrix Experiments Matrix Experiments • Each row of the matrix corresponds to one experiment. • Each column of the matrix corresponds to one factor. • Each experiment corresponds to a different combination of factor levels and provides one observation. Expt No. Factor A Factor B Observation 1 A1 B1 K1 2 A1 B2 K 2 3 A2 B1 K 3 4 A2 B2 K 4 Here, we have two factors, each of which can take two levels
M Full-Factorial Experiment 16888 E77 Specify levels for each factor Evaluate outputs at every combination of values n factors complete but expensive In observations / levels E Factor A B Al B1 2 factors, 3 levels each A1 B2 Al B3 In=32 =9 expts B1 4 factors 3 levels each 6 B3 7 A3 B1 n= 34=81 expts A3 B2 A3 B3 o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Full -Factorial Experiment Factorial Experiment • Specify levels for each factor • Evaluate outputs at every combination of values n factors – complete but expensive! l levels l n observations Expt Factor No. A B 1 A1 B1 2 A1 B2 3 A1 B3 4 A2 B1 5 A2 B2 6 A2 B3 7 A3 B1 8 A3 B2 9 A3 B3 2 factors, 3 levels each: l n = 3 2 = 9 expts 4 factors, 3 levels each: l n = 3 4 = 81 expts
Mlesd Fractional Factorial Experiments 16888 E77 Due to the combinatorial explosion, We cannot usually perform a full factorial experiment So instead we consider just some of the possible combinations · Questions: How many experiments do need? Which combination of levels should i choose? Need to balance experimental cost with design space coverage C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Fractional Factorial Experiments Fractional Factorial Experiments • Due to the combinatorial explosion, we cannot usually perform a full factorial experiment • So instead we consider just some of the possible combinations • Questions: – How many experiments do I need? – Which combination of levels should I choose? • Need to balance experimental cost with design space coverage
Mlesd Fractional Factorial Design 16888 E77 Initially, it may be useful to look at a large number of factors superficially rather than a small number of factors in detail: 1112 1112:113:143… 21:122 213122::23:24…… VS 311325:133:1343 many levels n15n2 many factors 8 C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Fractional Factorial Design Fractional Factorial Design Initially, it may be useful to look at a large number of factors superficially rather than a small number of factors in detail: 1 11 12 2 21 22 1 2 , , , n nn f ll f ll f ll # # 1 11 12 13 14 2 21 22 23 24 3 31 32 33 34 ,,,, , ,, , , , ,, , , f llll f ll ll f ll ll ! ! ! vs. many levels many factors
Mlesd DoE Techniques Overview 16888 ES077 TECHNIQUE COMMENT EXPENSE ( levels, n=# factors Full factorial Evaluates all possible in -grows design designs exponentially with number of factors Orthogonal arrays Dont always seem to Moderate-depends work -interactions? on which array One at a time Order of factors?1+n(l-1)-cheap Latin hypercubes Not reproducible, 1-cheap poor coverage if divisions are large Parameter studyCaptures no 1+n(l-1)-cheap interactions o Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics DoE Techniques Overview Techniques Overview TECHNIQUE COMMENT EXPENSE (l=# levels, n=# factors) Full factorial design Evaluates all possible designs. ln - grows exponentially with number of factors Orthogonal arrays Don’t always seem to work - interactions? Moderate – depends on which array One at a time Order of factors? 1+n(l-1) - cheap Latin hypercubes Not reproducible, poor coverage if divisions are large. l - cheap Parameter study Captures no interactions. 1+n(l-1) - cheap
Mest Parameter Stud 16888 E77 Specify levels for each factor Change one factor at a time, all others at base level Consider each factor at every level Expt Factor No n factors A D 1+n(-1) B1 D1 evaluations B1 C1 D1 /levels A3 B1 C1 D1 B2 C1 D1 4 factors 3 levels each 12345678q Al B3 D1 Al B1 D1 1+n(1-1) Al B1 C3 A1 B1 D2 1+4(3-1)=9 expts B1 Baseline: Al. B1. C1. D1 o Massachusetts Institute of Technology -Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
10 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Parameter Study Parameter Study • Specify levels for each factor • Change one factor at a time, all others at base level • Consider each factor at every level Expt Factor No. AB C D C1 D1 D1 D1 D1 D1 D1 D1 D2 D3 C1 C1 C1 C1 C2 C3 C1 C1 4 A1 B2 5 A1 B3 6 A1 B1 7 A1 B1 8 A1 B1 1 A1 B1 2 A2 B1 3 A3 B1 9 A1 B1 n factors 1+ n ( l-1) evaluations l levels 4 factors, 3 levels each: 1+ n ( l-1) = 1+4(3-1) = 9 expts Baseline : A1, B1, C1, D1
