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# chapt3 - Chapter 3 Response Surface Approximations Most...

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28 Chapter 3 Response Surface Approximations Most optimization algorithms that are in use for solving analytical engineering optimization problems are sequential in nature. That is, the objective function and constraints are evaluated at one point at a time, and the values at that point, as well as previous design point contribute to a decision on where in design space to move to for the next evaluation. When the objective functions and (or) the constraints are evaluated by experiments rather When the objective functions and (or) the constraints are evaluated by experiments rather than by analytical evaluations, there is usually an incentive to perform the experiments in batches rather than singly. One reason for batching experiments is that most require set-up time, advance planning and reservations of experimental facilities or technicians. Another reason is that experimental errors make it difficult to interpret the results of a single experiment. When a batch of experiments is performed, errors in one or two experiments tend to stand out. Duplicating experiments for identical nominal conditions permits us to estimate the magnitude of experimental scatter due to errors and variability in the properties of the tested designs. Finally, some of the experimental scatter can be averaged out by performing a large number of experiments. Because of these advantages of running experiments in batches, experimental optimization has followed a different route than analytical optimization. The standard approach is to use an optimization strategy that is based on the results of a batch of experiments. On the basis of the experiments, we construct approximations to objective functions and/or constraints and perform optimization on the basis of these approximations. In most cases, the optimum obtained is then tested, and if satisfactory results are obtained the design procedure is terminated. In some cases, the optimum is used as the central design point for a new batch of experiments, and the process is repeated once or twice. This process is sometimes called sequential approximate optimization. However, because of the cost and time associated with conducting experiments, it is rare that the process is iterated to convergence. When analytical calculations were mostly based on closed form solutions or numerical models that required minimal modeling and computations, the difference between analysis and experiments was very clear. However, today numerical evaluations of objective functions and constraints often share many of the properties of experimental evaluations. First, numerical models such as finite element structural models require substantial investment of time to set up and debug. Furthermore, the evaluation of such models may require large computational resources, so that the cost of numerical simulation may be comparable to the cost of experiments. Second, with analytical simulations based on complex numerical models, many sources of noise are often found in the results of numerical simulations. These includes round-off errors as well as errors due to incomplete convergence of iterative processes. Additionally,

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chapt3 - Chapter 3 Response Surface Approximations Most...

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