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Numerical experiments on the problem of moving a flexible beam are discussed. An optimal control problem is formulated and transcribed into a form which can be solved using semi-infinite optimization techniques. All experiments were carried out on a SUN 3 microcomputer.
A novel, real-valued Genetic Algorithm (GA) was designed and implemented to minimize the reflectivity and/or transmissivity of an arbitrary number of homogeneous, lossy dielectric or magnetic layers of arbitrary thickness positioned at either the center of an infinitely long rectangular waveguide, or adjacent to the perfectly conducting backplate of a semi-infinite, shorted-out rectangular waveguide. Evolutionary processes extract the optimal physioelectric constants falling within specified constraints which minimize reflection and/or transmission over the frequency band of interest. This GA extracted the unphysical dielectric and magnetic constants of three layers of fictitious material placed adjacent to the conducting backplate of a shorted-out waveguide such that the reflectivity of the configuration was 55 dB or less over the entire X-band. Examples of the optimization of realistic multi-layer absorbers are also presented. Although typical Genetic Algorithms require populations of many thousands in order to function properly and obtain correct results, verified correct results were obtained for all test cases using this GA with a population of only four.
An innovative hyperbolic preconditioning technique is developed for the numerical solution of the Helmholtz equation which governs acoustic propagation in ducts. Two pseudo-time parameters are used to produce an explicit iterative finite difference scheme. This scheme eliminates the large matrix storage requirements normally associated with numerical solutions to the Helmholtz equation. The solution procedure is very fast when compared to other transient and steady methods. Optimization and an error analysis of the preconditioning factors are present. For validation, the method is applied to sound propagation in a 2D semi-infinite hard wall duct.
The acoustic fields and transmission losses produced in semi-infinite circular ducts with peripherally segmented liners are analyzed using a series expansion of hard-wall duct modes. The coefficients of the series are computed using Galerkin's method. Unlike finite element approaches, this analysis includes the effects of realistic sources and the number of peripheral strips need not be small. It is shown that peripherally segmented liners redistribute the acoustic energy in waves composed of only a single circumferential mode at the source into other waves which contain a multitude of circumferential modes in the lined section. The accuracy of eigenfunctions computed from the analysis was observed to increase as either the frequency or radial mode order increased. The transmission losses were found to be accurate at frequencies above the cut-on value of the first-order radial mode in a hard-wall duct. The results show that for plane wave sources, peripherally segmented liners may attenuate as much sound as an optimized uniform liner at the optimal point while giving more noise suppression at most other frequencies.