Design-of-Experiments Methods and their Application to Robust. Multicriteria Optimization Problems. Design-of-experiments (DOE) methods are utilized to ...
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LAUTENSCHLAGER, U.; ESCHENAUER, H.A.; MISTREE,F.
Design-of-Experiments Methods and their Application to Robust Multicriteria Optimization Problems Design-of-experiments (DOE) methods are utilized to explore the design space and t o build response surface models in order to facilitate the effective solution of multiobjective optimization problems. Response surface models provide an eficient means to rapadly model the trade-off between conflicting goals. For robust design applications these approximation models are used efficiently and effectively t o calculate variances due to noise factors using Taylor series expansions. This combined-army approach is compared to Taguchi’s original crossed-array approach. The shape optimization of a flywheel with two conjlacting design goals is used to illustrate the approach.
1. Design-of-Experiments and Response Surface Methodology The formal techniques which support the design and analysis of experiments are called DOE techniques. DOE is a statistical approach for solving problems from engineering to social science [l, 21. Statistical experimentation allows to estimate factor effects and to develop response surface approximations from the generated data which are frequently used to replace expensive function evaluations in structural optimization [3-51. An experimental design describes a sequence of experiments, which are mathematically defined by a matrix X . The rows of X denote experimental runs and the columns the specified factor settings for each run. A large variety of experimental designs exists, e.g., Full and Fractional Factorial Designs, Orthogonal Arrays (OA), or Central Composite Designs (CCD) [l, 21. In a full factorial design, all possible combinations m k of factor levels are investigated, where k factors are selected at rn levels. It is possible to estimate all main and all interaction effects. The number of experiments increases exponentially with the number of factors. To reduce the number of experiments to a more practical level, fractional factorial designs of the form mk-p are introduced, where the generator p describes the fraction of the full design. It is assumed that certain higher order interactions are negligible. OAs are the smallest fractional factorial design to study main factor effects. It is assumed that factor interactions do not exist, therefore main effects can be confounded with two-factor interactions. CCDs are probably the most widely used experimental designs to fit response surface models. A CCD generally consists of a two-level full or fractional factorial design portion, where the two levels are coded by -1 and +1, augmented by two star points on the axis of each factor at a distance of -a and +a from the center, and one (or more) center points. Response surface models are created to replace the original structural analysis with an efficient, usually second-order polynomial approximation of the form k
k
i=l
i=l
k-1
k
i=l j=i+l
These models include main effects, two-factor interactions and quadratic terms and the unknown coeflicients b are estimated by means of regression analysis based on a least-square method, according to
b = [X’X] X’y. 2. Robust Design Realization
The notion of Robust Design was originally developed by G. TAGUCHI, a Japanese quality consultant. Robust design methods have been widely used in industry to implement parameter design and tolerance design, while seeking solutions that minimize performance variations ( m i n u’) and achieving specified performance targets( min ( P - PO)). The fundamental principle for achieving robustness is to improve the quality of a product/process by minimizing the eflects of the causes (noise factors) of variation without eliminating the causes. Robustness is the product’s ability to function according to the specifications besides environmental changes and noise. Instead of
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eliminating or controlling noise, the nonlinear relationship between a quality (performance) characteristic and the control and noise factors are exploited, to find control factor settings that minimize the effects of noise factors. To implement the robust design approach, the effects of control and noise factors on mean ( p y ) and variance (02) of a response must be made available. Response surface models for each response are created to model performance and performance variation. Two approaches for creating these models are used in this work: (i) the so-called Dual Response Surface Approach ( Combined-Array, [S]), where a single experimental design that includes control and noise factors is used, and (ii) Taguchi’s Crossed-Array Approach [7], where the inner control factor array is crossed with an outer noise factor array for each experimental run. For the dual response surface approach, the design is augmented to include noise factors z and a formal regression model 6 = f(x,z) is postulated. Eq. (1) is extended to include noise factors, k
k
k-1
k
m
i=l
i=l
i=l
j=i+l
j=1
k
m
i = l j=1
where no quadratic terms are required for noise factors. The exploration of the interactions xizj forms the core of robust design. This model allows to analytically derive individual mean (pclp) and variance (ui)models (expected values, E) by means of first-order Taylor series expansions according to
where uz,represents the standard deviation of a noise factor. It is assumed for this approach that all noise factors vary approximately normally. In the crossed-array approach, mean and variance for each response are calculated for the outer array experiments of each run of the inner array using standard statistics. Response surface models are then derived, where the responses S = f(x) only depend on x. 3. Formulation of the Robust Multicriteria Optimization Problem
The application of Multicriteria (multiobjective , vector) Optimization techniques is primarily due to the fact that systems design does not only require a minimization of costs, but also other objectives like weight, reliability, accuracy, etc. A multicriteria optimization problem can be solved through defining a substitute problem and by that reducing it to a scalar optimization problem. Such substitute problems are based on the objective weighting, the constraint-oriented transformation (trade-off method), or modeling different priority levels for the objectives. The objectives which are mostly competitive and nonlinear do not lead to one solution point for the optimum but rather to a ”functional-efficient” (PARETO-optimal) solution set, i.e., the decision maker selects the most efficient compromise solution out of such a set. This optimization strategy is the basic part of the optimization model and the general nonlinear multicriteria optimization problem is defined as follows [8]:
) ) M i n ” ( f ( x I) h(x) = O,g(x)20) 1
XEW
(5)
where 8” is the set of real numbers (n-dimensional), f the vector of the k objective functions, 2 the vector of the n design variables, h the vector of the q equality constraints, and g the vector of the p inequality constraints. A scalar optimization problem exists, if only one objective is considered. If a robust design formulation is considered, the minimization of the objective function leads to a value that should be achieved with a certain probability. In the formulation of the objective function, n standard deviations are added to the mean to express this situation. A deterministic objective function within Eq. (6) is now formulated as the new stochastic objective function
+
E ( x , 2 ) = ~ f , ( x ) noji(x, 2 ) .
(6)
Similarly, a stochastic formulation has to be considered for the constraints. Otherwise, design points with mean values within the feasible design space may violate a constraint and are actually infeasible with a certain probability. The new stochastic constraint is formulated as
The number n is selected based on the desired reliability, a value of n = 3 or higher is typical. The larger n , the smaller is the remaining, feasible design space.
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Figure 1: Concept for Robust MulticTiteTia Optimization For the solution of optimization problems in structural design, the Three-Columns-Model [9] has been applied successfully in many cases. It consists of the three columns l’structuml analysis model”, ”optimization model”, and ”optimization algorithms” which are integrated within the optimization procedure SAPOP (Structural Analysis Program and Optimization Procedure). The robust design approach is implemented into the optimization loop (Fig. 1). Experiments are performed for model screening, i.e., using low-order experiments to identify important factors, and for creating approximation models. Mean and variance models are developed for robust design applications, which then enter the optimization loop.
4. Robust Multicriteria Optimization of a Flywheel
The classical design objective for a flywheel is to maximize the capacity of the stored kinetic energy while satisfying constraints on the mass and stress limits. This problem has been studied extensively in the design literature. For the current model, the role of the shaft is neglected for simplicity. The stress state, the mass and the kinetic energy are analyzed. The mathematical model for calculating the stress distribution in the disk with nonuniform profile is adopted from [lo]. Shear stresses are neglected, therefore, the radial and tangential stresses, urr and r o e , respectively, are the principal stresses and assumed to be uniform across the thickness h ( r ) . Failure is expected, if the reference stress
determined by the shear strain energy hypothesis according to V O N MISES,exceeds the admissible value stress bodm. The disk mass rn and the kinetic energy T a r e calculated by solving the integrals (9)
numerically, where the required points are derived from a BEZIER-curve definition with four control points, Pi, that lead to a total of eight design variables ti. Details on the structural and optimization model, on creating response surfaces and optimal flywheel shapes are found in [ll]. The optimization results are presented in Fig. 2. The three curves represent functional-efficient curves for the conflicting goals of maximizing kinetic energy (see optimal flywheel shape) while minimizing the reference stress. The lowest curve (0 - g) represents the deterministic solution. The two upper curves (3 - u) that represent the solutions for the two robust design approaches are very close together, which indicates nearly equal quality of the models. However, for a similar stress value, the achievable kinetic energy is lower than in the deterministic case. Furthermore, the crossed-array approach requires about five times more experiments than the combined-array approach. This supports the efficiency and effectiveness of the proposed dual response surface approach.
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c
Figure 2: PARETO-Optimal Robust Design Solutions 5. Closure The work in this paper represents an approach for the early stages of designing complex structures when multiple objectives require to efficiently and effectively model the trade-off between these objectives. Response surface models are used to approximate the structural behavior as accurate as possible and provide an efficient means for a designer to rapidly come u p with various design solutions. Robust design aspects are addressed in the current model to include quality considerations. Two robust design approaches are presented that are contained within a concept for robust multicriteria optimization. The results indicate the necessity for considering robustness during structural optimization and favor the dual response surface approach because of its eficiency.
Acknowledgements This research is funded by the German Academic Exchange Service (DAAD) with a "DAAD Post Graduate Fellowship supported by the Third Special University Program".
6. References 1 BOX,G.,W.HUNTER, HUNTER, J.: Statistics for Experimenters; Wiley, Inc., New York 1978. 2 MONTGOMERY, D.: Design and Analysis of Experiments; Third ed. John Wiley and Sons, New York 1991. 3 BARTHELEMY, J.F.M.,HAFTKA, R.T.: Approximation concepts for optimum structural design - a review; Structural Optimization, vol. 5 (1993),pp. 129-144. 4 ROUX,W.J., STANDER, N., HAFTKA, R.T.: Response Surface Approximations for Structural Optimization; Sixth AIAA/ USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA (1996),pp. 565-578. 5 SCHOOFS, A.J.G., KLINK, M.B.M.,VANCAMPEN, D.H.: Approximation of structural optimization problems by means of designed numerical experiments; Structural Optimization, Vol. 4 (1992), pp. 206-212. 6 MYERS,R.H., MONTGOMERY, D.: Response Surface Methodology, Process and Product Optimization Using Designed Experiments; John Wiley and Sons, New York (1995). 7 PHADKE,M.S.:Quality Engineering using Robust Design. Prentice Hall,Englewood Cliffs,New Jersey (1989). 8 ESCHENAUER, H.A.,KOSKI,J . , OSYCZKA, A,: Multicriteria Design Optimization - Procedures and Applications; Springer-Verlag, Berlin, Heidelberg, New York (1990). 9 ESCHENAUER, H.A., OLHOFF,N., SCHNELL, W.: Applied Structural Mechanics; Springer-Verlag, Berlin, Heidelberg, New York (1997). 10 SANDGREN, E., RAGSDELL, K.M.: Optimal Flywheel Design with a general Thickness Form Representation; Journal of Mechanism, Transmissions, and Automation in Design, Vol. 105 (1983), pp. 425-433. 11 LAUTENSCHLAGER, U . , ESCHENAUER, H.A., MISTREE, F.: Flywheel Design: A Multiobjective DOEBased Exploration Method; ASME Design Automation Conference, Sacramento, CA, September 14-17 (1997),DETC97/DAC3961.
Address: DIPL.-ING. UWE LAUTENSCHLAGER (M.Sc.), PROF. DR.-ING. HANSA. ESCHENAUER, FOMAAS, University of Siegen, 57068 Siegen, Germany. PROF.DR. FARROKH MISTREE,The G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA.
