Shape optimization of an acoustic horn is performed with the goal to minimize the portion of the wave that is reflected. The analysis of the acoustical properties of the horn is performed using a finite element method for the Helmholtz equation.
The optimization is performed employing a BFGS Quasi-Newton algorithm, where the gradients are provided by solving the associated adjoint equations. To avoid local solutions to the optimization problem corresponding to irregular shapes of the horn, a filtering technique is used that applies smoothing to the design updates and the gradient. This smoothing technique can be combined with Tikhonov regularization. However, experiments indicate that regularization is redundant for the optimization problems we consider here. However, the use of smoothing is crucial to obtain sensible solutions. The smoothing technique we use is equivalent to choosing a representation of the gradient of the objective function in an inner product involving second derivatives along the design boundary. Optimization is performed for a number of single frequencies as well as for a band of frequencies. For single frequency optimization, the method shows particularly fast convergence with indications of super-linear convergence close to optimum. For optimization on a range of frequencies, a design was achieved providing a low and even reflection throughout the entire frequency band of interest.
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