Shape optimization of acoustic scattering bodies E.A. Divo a, * , A.J. Kassab b , M.S. Ingber c a Department of Engineering Technology, University of Central Florida, 4000 Central Florida Boulevard, Orlando, FL 32816-2450, USA b Department of Mechanical, Materials, and Aerospace Engineering, University of Central Florida, 4000 Central Florida Boulevard, Orlando, FL 32816-2450, USA c Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, USA Received 3 April 2002; revised 26 June 2002; accepted 27 June 2002 Abstract Shape optimization of acoustic scattering bodies is carried out using genetic algorithms (GA) coupled to a boundary element method for exterior acoustics. The BEM formulation relies on a modified Burton-Miller algorithm to resolve exterior acoustics and to address the uniqueness issue of the representation problem associated with the Helmholtz integral equation at the eigenvalues of the associated interior problem. The particular problem of interest considers an incident wave approaching an axisymmetric shaped body. The objective is to arrive at a geometric configuration that minimizes the acoustic intensity captured by a receiver located at a distance from the scattering body. In particular, the acoustic intensity is required to be minimum as measured proportional to the integral of the product of the potential and its complex conjugate over a volume of space which models the receiver. This is opposed to the more traditional measure of the potential at a single point in space. q 2003 Published by Elsevier Ltd. Keywords: Scattering body; Burton-Miller algorithm; Genetic algorithms 1. Introduction Shape optimization of acoustic scattering bodies is achieved using genetic algorithms (GA) as a minimization tool coupled to a boundary element method solver for the exterior acoustics problem. The BEM formulation relies on a modified Burton-Miller algorithm to resolve exterior acoustics and to address the uniqueness issue of the representation problem associated with the Helmholtz integral equation (HIE) at the eigenvalues of the associated interior problem. The particular problem of interest considers an incident wave approaching an axisymmetric shaped body. The objective is to arrive at a geometric configuration which minimizes the acoustic intensity captured by a receiver located at a distance from the scattering body. In particular, the acoustic intensity is required to be minimum as measured proportional to the integral of the product of the potential and its complex conjugate over a volume of space which models the receiver. This is opposed to the more traditional measure of the potential at a single point in space. Minimizing the acoustic intensity in a volume of space in the far field has wider application in practice than minimizing point potentials. Nevertheless, as for the case of minimizing point potentials, there are many local minima associated with any shape optimization rendering gradient-based methods of optimization ineffective. An overview of non-gradient based optimization methods in multidisciplinary design is provided by Hajela [1]. Hajela discussed simulated annealing, genetic algorithms, TABU search, and rule-based expert systems for problems involving discontinuous design variables, constraints, large parameter spaces, and local optimum. Several investigators [2–5] have considered structural shape optimization but, in all these studies, the objective was to optimize the mechanical properties of the structure, not the acoustical properties. Several studies have also been performed to optimize the acoustic properties structures. However, these studies have considered either structural modifications or active control to perform the optimization as opposed to shape modification. For example, Cunefare et al. [6] minimized the interior acoustic pressure for shells 0955-7997/03/$ - see front matter q 2003 Published by Elsevier Ltd. doi:10.1016/S0955-7997(03)00022-5 Engineering Analysis with Boundary Elements 27 (2003) 695–703 www.elsevier.com/locate/enganabound * Corresponding author. Tel.: þ 1-407-823-5778. E-mail address: edivo@mail.ucf.edu (E.A. Divo).