Function approximation methods are often applied independently of the type of optimization algorithms. Many optimization algorithms use approximation concepts to allow a simpler representation of the actual objective function. An excellent survey about approximation concepts has been provided by van Houten [van Houten 98]. Fig. 3.3 shows the considered approximation methods for this study. In general, we distinguish local, global and mid-range approximation. These methods are shortly explained in the following subsections.
Global Neural Networks DesignofExperiments Mid-Range Single-Point Multi-Points Local Taylorseries
Figure 3.3: Various approximation methods considered in this study.
3.3.1 Local Approximation Methods
The term local refers to this fact that only information about an objective function from the neighborhood of the current approximation is used in updating the approximation. There- fore, these approximations are valid only in a small subregion of the total search region. In their simplest form, linear approximations are used. Local approximation is based on a reference design and, mostly, a Taylor series approximation. First order approximation is using gradient approximation, second order requires the Hessian too.
3.3.2 Global Approximation Methods
Global methods are used to nd an approximation of the objective function for the entire design space or, at least, for a large region of it. This requires a relatively large number of points to t the approximate function to the data. Typical members of global approximation methods are neural networks [Papadrakakis 99] and design of experiments [Montgomery 01].
• NN Neural Networks: The NN method is more suitable for the applications in
which is no way to describe the problem with a function. A trained network presents a rapid mapping of given input into the desired output quantities, thereby enhancing the e ciency of the redesign process. The NN training comprises the following tasks. At rst select the proper training set, then nd the suitable network architecture and determine the appropriate values of characteristic parameters such as the learning rate and momentum term [Papadrakakis 99].
• DOE Design of Experiments: DOE method is a systematic approach for the
investigation of a system or process. A series of structured tests are designed in which planned changes are made to the input variables of a process or system. The eects of these changes on a prede ned output are then assessed [Montgomery 01]. The construction of response surface models and the design of experiments is an iterative process and consists of several steps. The rst step is to select a function approximation model (linear, quadratic etc.) for the objective function. Next, data points are selected to run numerical analysis. After the function models have been tted to the response data, the functions must be checked for their validity to represent the real behavior of the response functions. Decisions on how to proceed to the next step must be made by the designer and in uence the success of the optimization problem solution.
3.3.3 Mid-Range Approximation Methods
Methods that combine most of the strengths of local and global methods are known as mid-range approximation methods [Kessels 01]. They are classi ed as single-point or multi-
points. The corresponding approximate response functions are simple and often explicit in terms of the design variables. Mid-range function approximations of objective function and constraints are valid in a region larger than local methods but smaller than global methods. First a model function has to be selected. Next the set of design points to be used in the tting of the model are either newly generated or used from previous cycles. Finally the unknown function parameters are calculated. To restrict the region of validity, usually bounds to the design variables are imposed, so-called move limits [Etman 97,Keulen 97].
In local methods, all information from previous cycles is discarded. When data from earlier cycles is stored and used in later steps, more accurate approximations and larger search regions can be applied. Such methods are termed mid-range single point path methods, since a single new point is added in each cycle. The multi-points approximation method replaces an optimization problem by a sequence of approximate ones [van Houten 98]. In a multi- points path method, in each cycle one or more new design points are generated within the move limits to base the approximations on, but data from previous steps can also be used.
In this dissertation, a mid-range multi-points approximation approach is used. A linear approximation of the objective function within the move limit area is considered for the optimization process. The minimum of the approximated objective function is searched by MFD.
Chapter 4
Optimization Procedure
In this chapter, the optimization procedure constituting the numerical structural acoustic optimization programs is described. Sections 4.1, 4.2 and 4.3 provide some information on issues that are common to all of optimization algorithms employed. Sec. 4.1 presents the general framework of an optimization procedure. Then, Sec. 4.2 introduces the formulation of the optimization problem and some necessary initializations for the beginning of an opti- mization process. The convergence criteria is presented in Sec. 4.3. The last two sections of this chapter focus on aspects of the optimization procedure that are unique to each of opti- mization algorithms as well as the control parameter setting and the calculation of objective function.