Response Surface Modelling

Topics: Factorial experiment, Design of experiments, Experimental design Pages: 8 (1475 words) Published: June 18, 2013
CHAPTER 3

Response surface methodology

3.1

Introduction

Response surface methodology (RSM) is a collection of mathematical and statistical techniques for empirical model building. By careful design of experiments, the objective is to optimize a response (output variable) which is influenced by several independent variables (input variables). An experiment is a series of tests, called runs, in which changes are made in the input variables in order to identify the reasons for changes in the output response. Originally, RSM was developed to model experimental responses (Box and Draper, 1987), and then migrated into the modelling of numerical experiments. The difference is in the type of error generated by the response. In physical experiments, inaccuracy can be due, for example, to measurement errors while, in computer experiments, numerical noise is a result of incomplete convergence of iterative processes, round-off errors or the discrete representation of continuous physical phenomena (Giunta et al., 1996; van Campen et al., 1990, Toropov et al., 1996). In RSM, the errors are assumed to be random.

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The application of RSM to design optimization is aimed at reducing the cost of expensive analysis methods (e.g. finite element method or CFD analysis) and their associated numerical noise. The problem can be approximated as described in Chapter 2 with smooth functions that improve the convergence of the optimization process because they reduce the effects of noise and they allow for the use of derivative-based algorithms. Venter et al. (1996) have discussed the advantages of using RSM for design optimization applications. For example, in the case of the optimization of the calcination of Roman cement described in Section 6.3, the engineer wants to find the levels of temperature (x1) and time (x2) that maximize the early age strength (y) of the cement. The early age strength is a function of the levels of temperature and time, as follows: y = f (x1, x2) + ε where ε represents the noise or error observed in the response y. represented by f(x1, x2) is called a response surface. The response can be represented graphically, either in the three-dimensional space or as contour plots that help visualize the shape of the response surface. Contours are curves of constant response drawn in the xi, xj plane keeping all other variables fixed. Each contour corresponds to a particular height of the response surface, as shown in Figure 3.1. (3.1) The surface

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300

200

900 100 1000 120 160 200

x1

x2

240

280

320

1100

Figure 3.1 Three-dimensional response surface and the corresponding contour plot for the early age strength of Roman cement where x1 is the calcination temperature (°C) and x2 is the residence time (mins). This chapter reviews the two basic concepts in RSM, first the choice of the approximate model and, second, the plan of experiments where the response has to be evaluated.

3.2

Approximate model function

Generally, the structure of the relationship between the response and the independent variables is unknown. The first step in RSM is to find a suitable approximation to the true relationship. The most common forms are low-order polynomials (first or second-order). In this thesis a new approach using genetic programming is suggested. The advantage is that the structure of the approximation is not assumed in advance, but is given as part of the solution, thus leading to a function structure of the best possible quality. In addition, the complexity of the function is not limited to a polynomial but can be generalised with the inclusion of any mathematical operator (e.g.

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trigonometric functions), depending on the engineering understanding of the problem. The regression coefficients included in the approximation model are called the tuning parameters and are estimated by...
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