Shape optimization techniques for musical instrument design
Luı́s Henrique, José V. Antunes, João Soeiro de Carvalho · The Journal of the Acoustical Society of America · 2001
The design of musical instruments is still mostly based on empirical knowledge and costly experimentation. One interesting improvement is the shape optimization of resonating components, given a number of constraints (allowed parameter ranges, shape smoothness, etc.), so that vibrations occur at specified modal frequencies. Each admissible geometrical configuration generates an error between computed eigenfrequencies and the target set. Typically, error surfaces present many local minima, corresponding to suboptimal designs. This difficulty can be overcome using global optimization techniques, such as simulated annealing. However these methods are greedy, concerning the number of function evaluations required. Thus, computational times can be unacceptable if complex problems, such as bell optimization, are tackled. Those issues are addressed in this paper, and a method for improving optimization procedures is proposed. Instead of using the (local) parameters Pf(r) as searched variables, the system geometry is modeled in terms of truncated series of orthogonal space-functions Pj(r)=∑najnψjn(r), and optimization is performed on the (global) coefficients ajn. Fourier series and orthogonal polynomials are typical functions ψjn(r). This technique reduces considerably the number of search variables, and has a potential for significant computational savings in complex problems. It is illustrated by optimizing the shapes of both current and uncommon marimba bars.