Optimization for an Acoustic Problem

In this paper, an optimal shape design for an interfacial boundary between different media, through which sound propagates, is discussed. For example, designs for soundproof walls along high-speed train routes or highways, walls of concert halls, etc are included in the same category. For the above-mentioned problems, an algorithm to search for an optimal shape was proposed and tested numerically in the three-dimensional problems in [KS99], in which Fuzzy Optimization Method (FOM)[KS97] was used effectively. Originally, FOM was invented as a local minimizer search algorithm. In order to look for a global minimizer, Multi-start Fuzzy Optimization Method (MS-FOM), which is a hybrid algorithm with FOM and Genetic Algorithms (GAs), has been developed on the basis of FOM[KOPS98]. An application of MS-FOM to such optimization problems makes it possible not only to look for a global minimizer but also to clarify the structure of the manifold of the cost functional defined in the parameter space. This fact depends mainly upon the functions of MS-FOM, one of which is counting-up of all local minimizers. Here, the algorithm to search for an optimal shape by use of MSFOM is briefly stated and numerical results using it are presented. An observation of such results indicates the rather precise structure of the cost manifold, i.e., the distribution of local maximizers and minimizers in the parameter space. Such observation may be impossible by an application of other global minimizer searching algorithms, for example, Genetic Algorithms. Finally, physical meanings of a set of local maximizers will be discussed from the view point of resonance phenomena corresponding to the variations of eigenfrequencies of coupled media based on the shape change of the interfacial boundary. Through the discussion mentioned above, shape optimization for an acoustic problem arouses careful treatment to look for a global minimizer.