THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

On robust estimation of the location parameter

Preface. About the Authors. Foreword. Prologue. Part I: Fundamentals. 1. Sampling Plans. 1.1 The 'Curse of Dimensionality' and How to Avoid It. 1.2 Physical versus Computational Experiments. 1.3 Designing Preliminary Experiments (Screening). 1.3.1 Estimating the Distribution of Elementary Effects. 1.4 Designing a Sampling Plan. 1.4.1 Stratification. 1.4.2 Latin Squares and Random Latin Hypercubes. 1.4.3 Space-filling Latin Hypercubes. 1.4.4 Space-filling Subsets. 1.5 A Note on Harmonic Responses. 1.6 Some Pointers for Further Reading. References. 2. Constructing a Surrogate. 2.1 The Modelling Process. 2.1.1 Stage One: Preparing the Data and Choosing a Modelling Approach. 2.1.2 Stage Two: Parameter Estimation and Training. 2.1.3 Stage Three: Model Testing. 2.2 Polynomial Models. 2.2.1 Example One: Aerofoil Drag. 2.2.2 Example Two: a Multimodal Testcase. 2.2.3 What About the k -variable Case? 2.3 Radial Basis Function Models. 2.3.1 Fitting Noise-Free Data. 2.3.2 Radial Basis Function Models of Noisy Data. 2.4 Kriging. 2.4.1 Building the Kriging Model. 2.4.2 Kriging Prediction. 2.5 Support Vector Regression. 2.5.1 The Support Vector Predictor. 2.5.2 The Kernel Trick. 2.5.3 Finding the Support Vectors. 2.5.4 Finding . 2.5.5 Choosing C and epsilon. 2.5.6 Computing epsilon : v -SVR 71. 2.6 The Big(ger) Picture. References. 3. Exploring and Exploiting a Surrogate. 3.1 Searching the Surrogate. 3.2 Infill Criteria. 3.2.1 Prediction Based Exploitation. 3.2.2 Error Based Exploration. 3.2.3 Balanced Exploitation and Exploration. 3.2.4 Conditional Likelihood Approaches. 3.2.5 Other Methods. 3.3 Managing a Surrogate Based Optimization Process. 3.3.1 Which Surrogate for What Use? 3.3.2 How Many Sample Plan and Infill Points? 3.3.3 Convergence Criteria. 3.3.4 Search of the Vibration Isolator Geometry Feasibility Using Kriging Goal Seeking. References. Part II: Advanced Concepts. 4. Visualization. 4.1 Matrices of Contour Plots. 4.2 Nested Dimensions. Reference. 5. Constraints. 5.1 Satisfaction of Constraints by Construction. 5.2 Penalty Functions. 5.3 Example Constrained Problem. 5.3.1 Using a Kriging Model of the Constraint Function. 5.3.2 Using a Kriging Model of the Objective Function. 5.4 Expected Improvement Based Approaches. 5.4.1 Expected Improvement With Simple Penalty Function. 5.4.2 Constrained Expected Improvement. 5.5 Missing Data. 5.5.1 Imputing Data for Infeasible Designs. 5.6 Design of a Helical Compression Spring Using Constrained Expected Improvement. 5.7 Summary. References. 6. Infill Criteria With Noisy Data. 6.1 Regressing Kriging. 6.2 Searching the Regression Model. 6.2.1 Re-Interpolation. 6.2.2 Re-Interpolation With Conditional Likelihood Approaches. 6.3 A Note on Matrix Ill-Conditioning. 6.4 Summary. References. 7. Exploiting Gradient Information. 7.1 Obtaining Gradients. 7.1.1 Finite Differencing. 7.1.2 Complex Step Approximation. 7.1.3 Adjoint Methods and Algorithmic Differentiation. 7.2 Gradient-enhanced Modelling. 7.3 Hessian-enhanced Modelling. 7.4 Summary. References. 8. Multi-fidelity Analysis. 8.1 Co-Kriging. 8.2 One-variable Demonstration. 8.3 Choosing X c and X e . 8.4 Summary. References. 9. Multiple Design Objectives. 9.1 Pareto Optimization. 9.2 Multi-objective Expected Improvement. 9.3 Design of the Nowacki Cantilever Beam Using Multi-objective, Constrained Expected Improvement. 9.4 Design of a Helical Compression Spring Using Multi-objective, Constrained Expected Improvement. 9.5 Summary. References. Appendix: Example Problems. A.1 One-Variable Test Function. A.2 Branin Test Function. A.3 Aerofoil Design. A.4 The Nowacki Beam. A.5 Multi-objective, Constrained Optimal Design of a Helical Compression Spring. A.6 Novel Passive Vibration Isolator Feasibility. References. Index.

Engineering Design via Surrogate Modelling: A Practical Guide

Surrogate-based Analysis and Optimization

The use of statistical techniques to build approximations of expensive computer analysis codes pervades much of today’s engineering design. These statistical approximations, or metamodels, are used to replace the actual expensive computer analyses, facilitating multidisciplinary, multiobjective optimization and concept exploration. In this paper, we review several of these techniques, including design of experiments, response surface methodology, Taguchi methods, neural networks, inductive learning and kriging. We survey their existing application in engineering design, and then address the dangers of applying traditional statistical techniques to approximate deterministic computer analysis codes. We conclude with recommendations for the appropriate use of statistical approximation techniques in given situations, and how common pitfalls can be avoided.

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Metamodels for Computer-Based Engineering Design: Survey and Recommendations

A procedure for generating and using a polynomial approximation to wing bending material weight of a High Speed Civil Transport (HSCT) is presented. Response surface methodology is used to fit a quadratic polynomial to data gathered from a series of structural optimizations. Several techniques are employed in order to minimize the number of required structural optimizations and to maintain accuracy. First, another weight function based on statistical data is used to identify a suitable model function for the response surface. In a similar manner, geometric and loading parameters that are likely to appear in the response surface model are also identified. Next, simple analysis techniques are used to find regions of the design space where reasonable HCST designs could occur. The use of intervening variables along with analysis of variance reduce the number of polynomial terms in the response surface model function. Structural optimization is then performed by the program GENESIS on a 28-node Intel Paragon. Finally, optimizations of the HSCT are completed both with and without the response surface.

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Variable-Complexity Response Surface Approximations for Wing Structural Weight in HSCT Design

Flux-split algorithms for flows with non-equilibrium chemistry and vibrational relaxation

A new approach for combining conceptual and preliminary design techniques for wing optimization is presented for the high-speed civil transport (HSCT). A wing-shape parametrization procedure is developed which allows the linking of planform and airfoil design variables. Variable-complexity design strategies are used to combine conceptual and preliminary-design approaches, both to preserve interdisciplinary design influences and to reduce computational expense. In the study, conceptual-design-level algebraic equations are used to estimate aircraft weight, supersonic wave drag, friction drag, and drag due to lift. The drag due to lift and wave drag are also evaluated using more detailed, preliminary-design-level techniques. The methodology is applied to the minimization of the gross weight of an HSCT that flies at Mach 3 with a range of 6500 mi.

Variable-complexity aerodynamic optimization of a high-speed civil transport wing

A method has been developed to efficiently implement supersonic aerodynamic predictions from Euler solutions into a highly constrained, muItidisciplinary design optimization of a High-Speed Civil Transport. The method alleviates the large computational burden associated with performing computational fluid dynamics analyses through the use of variable-complexity modeling techniques, response surface (RS) methodologies, and coarse-grained parallel computing. Using information gained from lower-fidelity aerodynamic models, reduced-term RS models representing a correction to the linear theory RS model predictions are constructed using Euler solutions. Studies into 5-, 10-, 15-, and 20-variable design problems show that accurate results can be obtained with the reduced-term models at a fraction of the cost of creating the full-term quadratic RS models. Specifically, a savings of 255 CPU hours out of 392 CPU hours required to create the full-term RS model is obtained for the 20-variable problem on a single 75MHz IP21 processor of a Silicon Graphics, Inc. Power Challenge.

Response Surface Models Combining Linear and Euler Aerodynamics for Supersonic Transport Design

Convergence difficulties were encountered in our recent efforts towards a combined aerodynamic-structural optimization of the High Speed Civil Transport (HSCT). The underlying causes of the convergence problems were traced to numerical noise in the calculation of aerodynamic drag components for obstacles to convergence. The first technique employed a sequential approximation optimization method which used large initial move limits on the design variables. This helped dislodge the optimizer out of the local minima in the design space created by the noisy drag data. The second method utilized the aircraft. Two techniques were developed to circumvent the response surface methods to construct smooth approximations to the noisy data. The response surfaces were formed by analyzing several individual HSCT configuration and then fitting polynomial functions to selected objective function data. A simplified example design problem was used to demonstrate the response surface technique and to investigate various other issues relating to the construction of the response surfaces.

Bernard Grossman

Papers

Variable-Complexity Response Surface Approximations for Wing Structural Weight in HSCT Design

Flux-split algorithms for flows with non-equilibrium chemistry and vibrational relaxation

Variable-complexity aerodynamic optimization of a high-speed civil transport wing

Response Surface Models Combining Linear and Euler Aerodynamics for Supersonic Transport Design

Noisy Aerodynamic Response and Smooth Approximations in HSCT Design