Response surface methodology: advantages and challenges

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1 100 FORUM The forum series invites readers to discuss issues and suggest possible improvements that can help in developing new methods towards the advancing of current existing processes and applied methods for design optimization practice. Response surface methodology: advantages and challenges S. Nagendra General Electric Corporate R&D, Schenectady, New York, USA Response surface methods (RSM) are information based techniques and perform well when there is a rich tapestry of accurate information provided. They have gained immense popularity due to their ease of use as well as dexterity of being applied across the board of industrial applications. RSM offers techniques for mapping multidimensional patterns of responses to varying levels of control factors that are identified to govern physical processes. RSM is dependent on the use of regression analysis on data from experiments carried out at multiple levels and can be used to find approximate minima or maxima in response patterns provided such optima are within the design space of regressed points. It is recognized that RSM is a mathematical tool based on statistics (Box et al., 1978; Myers and Montgomery, 1995). Several commercial tools have currently come to the market (e.g. MINITAB, Design Expert, E-Chip, Statistica, SAS etc.) that are prevalently used. Using RSM makes it possible to shorten learning curves associated with new processes and product definitions. When process or product complexity escalates rapidly, the demands on skills of individuals increases. Reconfiguring skills and organization around intelligence inherent in technology works. Commercial products are designed in keeping with physical laws of the continuum. Owing to lack of appropriate descriptive mathematical structure to handle the mathematics of the physical continuum, information is not translated in a continuum sense into RSM methodology prevalent in product design today (e.g. unlike the finite element method that is based on a weak variational principle and in which the discretized domain is modeled using polynomials that have to satisfy norms associated with completeness and exactness, there is no basis requirement for the polynomials used for RSM). In this sense the RSM approach can be viewed as having no rigorous mathematical requirement to model the continuum (except of course at the

2 discrete set of points chosen) at present and needs further development to actually be an effective mathematical tool that can predict the behavior of the continuum. Predictive accuracy and information transfer RSM techniques, however, provide an efficient method of transfering information. The strength of the method lies in capturing accurate efficient smooth approximations for accurate data garnered from numerical or practical experiments at discrete data points in the design space (e.g. Venter et al., 1997). The selection of the set of equations used to represent the system behavior has a strong influence on the accuracy of the approximation. Chosen functions must be able to capture the non-linearity (if any) of the function as accurately as possible. The design of the set of experiments has an important influence on the accuracy and cost of computing the response surface. Furthermore, designed response surfaces are valid only in portions of the design space called ``region of interest''. The smooth nature of chosen polynomial-based approximations eliminate numerical noise and can efficiently transform the regressed information from a discrete set of points. The statistical assumption of an infinite sample size is reduced to a discrete set capturing a domain in the design space bounded by the set of chosen points. Furthermore, response surface approximations allow for derivative based optimization, where derivatives may not exist or are difficult to evaluate (Rowe et al., 1998; Sobieski and Haftka, 1997). This is beneficial in providing a progression of the optimization process itself, however, the quality of the derivative is dependent on the quality of the response surface approximation. Herein lies the need for diagnostic measures for judging the quality of the response surface approximation. The quality of the response surface approximation is dependent on the selection of discrete set of data points. The current approaches of optimality criteria in the design of experiments (Atkinson and Donev, 1996; Pukelsheim, 1992) like the A-, E-, T- optimality or the more popular D-optimality criteria based on minimizing the variance of information described by the data points could be used to improve the quality of the predicted response. Errors due to bias, modeling or truncation etc. are not accounted for. Augmentation of the data set using additional points that satisfy any of the above criteria provides a logical step in arriving at a high quality response surface. Model accuracy and predictive capabilities of the response surface have to be evaluated. However, once the response surface is created there is no criteria to diagnose the range of applicability of the response surface. Traditional methods of model checking relied principally on the variety of plots of the least squares residual. These were identified as omnibus tests for departures from fitted models. Engineering judgment would anticipate that when the response surface prediction deviates from exact analyses, it would be a possible start of the breakdown of the response surface approximation. Forum 101

3 102 Specific tests should be designed to answer precise formulation questions about model adequacy and predictive range. If two models are constructed from two independent discrete data sets of the same response, the discriminatory nature between individual models in predicting the same response has to be evaluated. This would provide a restriction to the design domain if used in optimum design. If the criteria are to have as many terms as possible in the regression equations for accurate prediction of the response, care must be taken to see that the regression is not over fitting, as this leads to poorer predictive capabilities of the model. For a diagnostic procedure to be useful it should not require appreciable computation compared to the creation of the actual response surface model. To avoid dangers of over interpretation, some calibration of diagnostics is essential. It is much simpler if the diagnostic has a mathematical basis such as a known null distribution of a test statistic. Quality of optimum Response surfaces can be considered as approximations that replace the objective and constraint surfaces with simple polynomial functions which are regressed to data at selected discrete points. After initial investment in computing data at selected points these approximations can be quite inexpensive as data estimators. The quality of optimum obtained from the approximate problem using response surfaces has to be determined. One suggested approach is to perform a complete exact analyses executed at that optimum and comparing the solution. There is a definite need to develop criteria to evaluate the quality of the optimum solution thus obtained. Response surface model issues RSM is still in its nascent stage as a design tool of broad acceptance. The wide range of applicability on designed experiments is fortunate but comes with a caveat: the lack of mathematical formalism and basis with which the physical continuum is tied to model (e.g. a simply supported plate can be modeled for buckling response if the variables are assumed continuous. However, the same plate cannot be modeled for accurate response when we consider discrete variables, i.e. modes and mode shapes which indicate the lack of mathematical connection between the physics of the continuum and the regressed polynomial approximation). The family of possible polynomials is enormous and, depending on geometry of the physical system of interest, an appropriate family is chosen for modeling the physical system (e.g. for a simply supported rectangular plate a sin series is chosen using variational principles to satisfy the boundary conditions, or for a circular plate various orders of Bessel functions are chosen, this is not the case with RSM approaches ± the polynomial approximation depends on the points and its accuracy). This restriction limits the range of applicability as well as the variety of problems that can be solved using RSM.

4 In a sense it prevents the method from maturing into problems of complexity like interacting structures of varying geometry as well as products with complicated load paths. Discrete variable designs cause a great deal of trouble for the RSM approach as the choice of smooth polynomials causes the approach to approximate the discrete design as a continuous one (e.g. composite plates with design variables being the discrete ply orientations as function of the plate aspect ratio). Design of stiffened or unstiffened plates or shells for frequency or stability depends on the mode shapes which are inherently discrete. The dimension of discreteness needs to be addressed as part of the response surface approach and this is a prevalent challenge to the capability of the procedure being universally accepted as a design tool. Engineering components go through a machining operation or welding operation, tend to have various deformations as part of the process (e.g. shrinkage distortion due to welding, or coolant induced shrinkage during milling). These are localized highly non-uniform behaviors which are difficult to model physically and lead to issues of dimensional tolerance issues. Response surface methods when used for component assembly or tolerancing issues see the problem of developing an assembled geometry or final assembly which depends on the sign as well as magnitude of the associated sensitivity derivative with the associated dimension to form the response surface. This is a discrete-continuous problem which has real product significance and the derivative influences the choice of polynomial. Response surface techniques have been found to suffer due to lack of diagnostic procedures in this case. Practical physical systems have non-potential loading situations (e.g. beam with a follower load (Bolotin, 1996)) or non-conservative forces (e.g. friction) cannot be adequately modeled using response surfaces due to the lack of mathematical richness to capture singular behavior (e.g. flutter in the beam with follower case (Bolotin, 1996)). Designs with multiple eigenvalues (common as the optimum is approached) cannot be modeled as the information (set of discrete data points) matrix tends to become singular and the response surface approximation cannot be constructed. The nature of the physical system sometimes cannot be adequately captured by the response surface approximation. The difference in the beam-column under compressive load, with the load being a follower force in one case and a dead load in the other (Bolotin, 1996), are two different situations from a physics perspective. The response is similar except in the region of flutter instability, the follower force creates a singular behavior remarkably different from the divergent behavior of the beam with a dead load. Stability, multiple load paths, bifurcations and mode jumping phenomena are very common non smooth phenomena that RSM finds difficulty in dealing with. Predictions from response surfaces and appropriate comparison with actual experiments and their correlation is needed to correctly improve the model and evaluate its validity in design. Development of techniques that will provide efficient response surface correction to correlated data is essential. Forum 103

5 104 Conclusion RSM definitely provides a numerical structure for dealing with possible discrepancies in the design space. It provides an avenue for efficiently smoothing out numerical noise. There are tremendous opportunities in increasing the efficacy of RSM methods by incorporating the rich family of polynomials and tying the same to governing differential equations of the system in a continuum sense rather than to a set of discrete points. The challenge for RSM to be a viable design tool is to model the continuum and capture the rich repertoire of physical phenomena accurately. There is, of course, no way to know how these challenges will play out in any particular case. In an environment shaped by intelligent technology, there are few, if any, hard wired solutions. Individuals will not discover what to do by deducing proper courses of action from first principles. Instead they will need to experiment, prototype and repeatedly test proactively. Being intelligent about intelligent technology does not mean knowing all the answers or doing all the analyses. It means being aware of when and where appropriate technology can be used effectively. References Atkinson, A.C. and Donev, A.N. (1996), Optimum Experimental Designs, Oxford Science Publications, Oxford. Bolotin, V.V. (1996), Stability Problems in Fracture Mechanics, John Wiley and Sons, New York, NY. Box, G.E.P., Hunter, W.G. and Hunter, J.S. (1978), Statistics For Experimenters, John Wiley and Sons, New York, NY. Myers, R.H., and Montgomery, D.C. (1995), Response Surface Methodology, John Wiley and Sons, New York, NY. Pukelsheim, F. (1992), Optimal Design of Experiments, John Wiley and Sons, New York, NY. Roux, W.J., Stander, N. and Haftka, R.T. (1998), ``Response surface approximations for structural optimization'', Intl. Journal of Numerical Methods in Engineering, Vol. 42, pp Sobieski, J.S. and Haftka, R.T. (1997), ``Multidisciplinary aerospace design optimization: survey of recent developments'', Structural, Vol. 14, pp Venter, G., Haftka, R.T. and Chiredast, M. (1997), ``Response surface approximations for fatigue life prediction'', AIAA , 38'th SDM, pt. 2., Kissimmee, FL, pp