Tuning of idiophones using shape optimization on finite and boundary element 1D models

Abstract

We address the multi-modal tuning of idiophones and approach it as a shape optimization design problem. The Finite Element Method (FEM) and the Boundary Element Method (BEM), are adopted to discretize continuous models and lead to eigenvalue problems for the shape optimization of both the idiophone vibrating bars and the resonators. The methods presented here consist of two main components. The first component involves solving the eigenproblem of a discrete dynamical system that approximates a real idiophone bar (beam) or a resonator (acoustic tube), while the second component is the implementation of an appropriate design procedure to achieve the desired frequency ratios of the bar and resonator eigenfrequencies through optimization. For the shape optimization problem, both a gradient algorithm and a population–based algorithm are tested. We introduce the use of analytically computed sensitivity indices, which significantly accelerate the optimization when gradient–based algorithms are used in conjunction with the FEM. For illustrative purposes, the presented methods are applied to 1D formulation however, these can be expanded in a straightforward way to more involved/realistic 2D or 3D models.